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Discrimination and job search in imperfect labour markets Watts, Martin John 1976

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DISCRIMINATION AND JOB SEARCH IN IMPERFECT LABOUR MARKETS by MARTIN JOHN WATTS B.A. (Econ.), University of Essex, 1970 M.A. (Econ.), University of Manchester, 1971 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of Economics We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1976 © Martin John Watts, 1976.. In presenting this thesis i n p a r t i a l fulfilment of the requirements for an advanced degree at the "University of B r i t i s h Columbia, I agree that the Library s h a l l make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. I t i s understood that p u b l i -cation, i n part or i n whole, or the copying of this thesis f o r f i n a n c i a l gain s h a l l not be allowed without my written permission. Martin John Watts Department of Economics The University of B r i t i s h Columbia Vancouver V6T1W5 i i ABSTRACT In this study the behaviour of firms and workers i s modelled simultaneously i n a labour market characterised by one imperfection, namely the absence of complete information on the part of job seekers about job offers and wage rates over firms. The necessary and s u f f i c i e n t conditions for wage dispersion i n equilibrium i n this model are examined. In the absence of perfect information,unemployed individuals search randomly for job offers. Since th e i r behaviour i s based on general information acquired through search, workers' turnover and acceptance behaviour i s stochastic. Firms make wage offer and vacancy creation decisions, based on their current l e v e l of employment,to maximise p r o f i t s over an i n f i n i t e horizon. Firms enjoy intertemporal monopsony power. Equilibrium i n th i s market i s stochastic. I t i s characterised by wage and employment dispersion and the simultaneous existence of f r i c t i o n a l unemployment and vacancies. The l e v e l of vacancy creation i s not consistent with s t a t i c p r o f i t maximisation. Such vacancies are defined as speculative. They do not represent desired net hires. Firms are speculating on the basis of individuals' stochastic turnover and acceptance behaviour. The aggregate measure of vacancy creation represents job offers at different wage rates. Excess demand, as conventionally defined, does not e x i s t . In this study the concept of a vacancy and the construction of an index of market pressure are examined under different market structures. Comparative s t a t i c predictions are generated i n this model, Model I . i i i The main contribution of this study i s i n the formulation of models of imperfect markets i n which there are two types of worker who d i f f e r systematically i n their labour market behaviour. Type one individuals have a higher pro b a b i l i t y of q u i t t i n g , i f employed, i n response to a par t i c u l a r wage offer than type two individuals. To a firm,type two individuals are more valuable because, i n response to a stream of wage offers over time, the mean returns associated with h i r i n g a type two ind i v i d u a l exceed the returns associated with h i r i n g a type one i n d i v i d u a l . Two labour-market structures are considered. In Model II,firms are unable to discriminate e x p l i c i t l y i n wage offer and h i r i n g between individuals. The important result i s that, although firms behave competitively, i n t e r - f i r m wage d i f f e r e n t i a l s are observed. Firms make decisions based on their current levels and compositions of employment. Type one indi v i d u a l s , although less valuable, earn a higher mean wage offer than type two ind i v i d u a l s . By contrast, i n Model I I I firms can discriminate e x p l i c i t l y between individuals i n h i r i n g and wage of f e r . Now firms make offers d i r e c t l y related.to the individual's value to the firm and i n t r a -firm wage d i f f e r e n t i a l s are observed. Comparative s t a t i c predictions are derived with respect to both models. Within this framework, the arguments of economists, as to whether low turnover workers earn a higher or lower wage than high turnover workers, are evaluated. Furthermore, the impact of f a i r employment laws on th i s labour market are examined. i v Excess demand, as c o n v e n t i o n a l l y d e f i n e d , does not e x i s t i n these models. Vacancies do not represent d e s i r e d net h i r e s and so a measure of market p r e s s u r e based on aggregate vacancies and f r i c t i o n a l unem-ployment i s i n c o r r e c t . T h i s important r e s u l t from the models demonstrates the inadequacy of u s i n g vacancy and unemployment data to draw i n f e r e n c e s about excess supply or demand i n a labour market. A crude measure o f market p r e s s u r e based on ex post h i r e s and quits i s developed and i t s inadequacies o u t l i n e d . TABLE OF CONTENTS Chapter Page 1. THE PROBLEM AND.THE LITERATURE SURVEY . . . . . . . . 1 I. Introduction 1 I I . An I l l u s t r a t i v e Model of the Labour Market . . . 3 I I I . An Outline of the Models Developed i n the Thesis 10 IV. Review of the Literature on Imperfect Markets. . 15 V. The Literature on Discrimination . 37 VI. The Thesis 51 2. THE CONCEPTUAL FRAMEWORK 53 I. Individual Turnover and Acceptance Behaviour 53 I I . Firms' Wage and Vacancy Creation Decisions and the Algorithm 72 I I I . The Characteristics of Stochastic Equilibrium. . 87 3. THE DISTRIBUTION OF PERCEPTIONS, THE DATA, THE SIMULATION PROCEDURE AND THE METHOD OF COMPARATIVE STATICS , 105 I. Summary 105 I I . The Gamma Distri b u t i o n . . 105 I I I . The Data 110 IV. The Simulation Procedure 115 V. Problems A r i s i n g from the Simulation Procedure . 123 VI. Comparative S t a t i c Results .. 126 v i 4. RESULTS FROM MODEL I 127 I . I n t r o d u c t i o n 127 I I . Comparative S t a t i c P r e d i c t i o n s . . . . . 132 I I I . Vacancy and Unemployment S t a t i s t i c s . 168 5. RESULTS FROM MODELS I I AND I I I . . . 173 I . I n t r o d u c t i o n . . . . . 173 I I . R e s u l t s of Model I I 176 I I I . Results of Model I I I 206 IV. A Comparison o f Models I I and I I I 245 V. C o n c l u s i o n 256 6. THE CONCEPT OF VACANCY CREATION AND EXCESS DEMAND IN AN IMPERFECT MARKET . 258 I . I n t r o d u c t i o n 258 I I . The Vacancy Concept 259 I I I . The Concept and Measure of Excess Demand 262 I V . C o n c l u s i o n 267 v i i LIST OF TABLES Table Page 1. The I n i t i a l Exogeneous Parameter Values i n Model I . » 110 2. The I n i t i a l Exogeneous Parameter Values i n Models I I and I I I . . 112 3. The Basic Solution to Model I 131 4. Results and Comparative S t a t i c Predictions for Misperceptions of the Reservation Wage 135 5. Results and Comparative S t a t i c Predictions f o r Different Levels of Unemployment Compensation . . . . 144 6. Results and Comparative S t a t i c Predictions for Different Horizons over which Returns from Search Accrue 149 7. Results and Comparative S t a t i c Predictions for Different Period Lengths 153 8. Results and Comparative S t a t i c Predictions for Different Values of the Perception Parameter, . . . 161 9. A Comparison of P a r t i a l and F u l l Equilibrium . . . . 167 10. The Basic Solution to Model I I 177 11. The Solution to Model I I Corresponding to Perception Parameter Values [1.05,0.95] 181 12. Summary S t a t i s t i c s for Unequal Values of the Perception Parameters 185 13. . Comparative S t a t i c Results for Model I I 200 14. The Basic Solution to Model I I I 210 15. The Solution to Model I I I with Perception :< •,-Parameters, [1.05,0.95] . . . V . . . . . . . . . . ]' 220 16. Summary S t a t i s t i c s for Unequal Values of the Perception Parameters i n Model I I . . . . . . . . . . 222 17. Comparative Static Results for Model I I I 241 v i i i Table Page 18. Comparison of S t o c h a s t i c E q u i l i b r i a i n Models I I and I I I corresponding to the B a s i c Parameter Values 247 19. Comparison of S t o c h a s t i c E q u i l i b r i a i n Models I I and I I I corresponding to Parameter Values [1.05,0.95] 248 20. Comparison o f Summary S t a t i s t i c s of Models I I and I I I 250 i x LIST OF FIGURES F i g u r e Page 1. The F i r m ' s D e c i s i o n Problem i n the I l l u s t r a t i v e model . . 6 2. The Labour Market S t r u c t u r e s Modelled 12 3. S t o c h a s t i c E q u i l i b r i u m i n Reder's Model 27 4. The F i r m ' s D e c i s i o n Problem i n A r c h i b a l d ' s Model . . 32 5. The Gamma D i s t r i b u t i o n 107 6. Gamma D i s t r i b u t i o n s w i t h D i f f e r e n t Means 108 7. Gamma D i s t r i b u t i o n s w i t h D i f f e r e n t V a r i a n c i e s . . . . 109 8. The M a t r i x of Employment States 175 X ACKNOWLEDGEMENTS I would l i k e to extend my g r a t i t u d e to C u r t i s Eaton who s t i m u l a t e d my i n t e r e s t i n t h i s t o p i c and helped c o n s i d e r a b l y i n i t s f o r m u l a t i o n . Thanks are due to C h r i s A r c h i b a l d f o r h i s comments on the text and h i s encouragement. To my w i f e , L i n d a , I express my a p p r e c i a t i o n f o r her p a t i e n c e and forbearance. F i n a l l y , I am indebted to Ann MacLeod f o r her a l a c r i t y i n t r a n s l a t i n g s c r a w l to the f i n a l , typed p r o d u c t . 1 CHAPTER 1 THE PROBLEM AND THE LITERATURE SURVEY I. Introduction Economists argue that the existence of wage dispersion and f r i c t i o n a l unemployment as equilibrium phenomena i n labour markets i s certainly related to, and perhaps caused by, imperfect information on the part of market participants."*" An i n d i v i d u a l seeking a job i n a pa r t i c u l a r industry lacks s p e c i f i c information about job offers and wage rates. Consequently, he samples offers by searching and contacting different firms. According to his perception of the wage d i s t r i b u t i o n i n the industry, the i n d i v i d u a l computes an estimate of his reservation wage. The 2 reservation wage i s that wage offer at which the expected marginal returns from search are zero. Thus, i f offered that wage, the indi v i d u a l would be in d i f f e r e n t between search and accepting the offer. On the basis of this reservation wage,he evaluates each offer and decides whether to accept the part i c u l a r offer or indulge i n further search. I t i s sometimes alleged that r a t i o n a l decision making on the part of firms i n response to th i s search behaviour i s 3 s u f f i c i e n t to generate wage dispersion and f r i c t i o n a l unemployment. Unfortunately,the micro-theoretic foundations of this behaviour have not been spelt out. The l i t e r a t u r e f a i l s to model simultaneously the r a t i o n a l behaviour of both sets of participants i n this imperfect market. This l i t e r a t u r e can be divided into four d i s t i n c t areas : 1. S t i g l e r [1961] and Gronau [1971] model the optimal search rules for individuals who sample from a known di s t r i b u t i o n of product prices and wage off e r s , respectively. Dispersion i s asserted to exist due to imperfect information of searchers ,but there i s no 4 formal analysis of firms' behaviour i n either model. 2. Mortenson [1970] and Salop [1973b] analyse optimal firm decision making for prescribed non-stochastic turnover and acceptance behaviour by individuals. The micro-foundations of workers' behaviour are not developed and i t i s not clear that wage dispersion and consequently job search are equilibrium phenomena. 3. Reder [1969] develops a model of an imperfect market i n which the nature of equilibrium i s described but again r a t i o n a l behaviour by market participants i s not specified. 4. Archibald [1954] develops a s t a t i c model of an imperfect market. He attempts to define the concept of a vacancy, but i n d i v i d u a l search behaviour i s not formally modelled. Rothschild [1973] argues that a satisfactory model of the adjustment to equilibrium of a market must have three components, namely : 3 1. A description of how market participants behave i n disequilibrium; 2. A set of rules as to how the market operates i n response to participants' behaviour; and 3. A theorem demonstrating convergence to equilibrium. 5 A cursory glance at this l i t e r a t u r e indicates that these c r i t e r i a have not been s a t i s f i e d . In my thesis I wish to model simultaneously and analyse the behaviour of firms and workers i n a labour market characterised by imperfect information. The main contribution i s i n the extension of this framework to incorporate two types of worker who d i f f e r systematically i n their labour market behaviour. The implications for firm behaviour of this heterogeneous workforce are examined. In order.to indicate the essential elements of my model of an imperfect labour market and motivate the close examination of the l i t e r a t u r e , I now develop a simple i l l u s t r a t i v e model. The labour market i s assumed to consist of a number of technically homogeneous firms who enjoy different levels of employment. Firms h i r e from a labour force homogeneous i n i t s capacity to do the job. Each firm makes a wage and vacancy decision to maximise p r o f i t s over a one period horizon. This assumption s i m p l i f i e s the analysis but s t i l l allows the examination of the essential elements of an imperfect market. These decisions w i l l be related to the l e v e l of employment the firm currently faces. Wage dispersion and dispersion i n the l e v e l of employment over firms are assumed to e x i s t , i n i t i a l l y . The labour I I . An I l l u s t r a t i v e Model of the Labour Market 4 market environment i s characterised by the d i s t r i b u t i o n of firms over different levels of employment and th e i r corresponding wage and vacancy decisions. An unemployed in d i v i d u a l has imperfect information about where the most remunerative job offers may be found. Consequently,he searches randomly. Other unemployed individuals are also searching for job offers. Thus an ind i v i d u a l searcher i s not assured of a job offer,even i f he samples a firm which has created vacancies. Through his knowledge of the labour market environment, which i s generated through search, the in d i v i d u a l evaluates the probability of an offer at each firm. He then computes the mean and variance of th i s wage offer d i s t r i b u t i o n . For any wage offer received,the i n d i v i d u a l computes the returns from search. I f he accepts t h i s offer,he assumes he w i l l enjoy this wage rate for the rest of his working horizon. Otherwise, remaining unemployed, he searches further and assumes he w i l l receive an offer equal to the mean wage offer i n the industry after one search. He accepts t h i s offer and assumes that he w i l l enjoy this wage rate for the rest of his working horizon. By comparing these income streams, he estimates the returns from search. His perceived reservation wage i s that current offer which makes h i s returns from search zero. I f the i n d i v i d u a l receives an offer less than his perceived reservation wage, the returns from search are positive and he searches further. Otherwise,he accepts the job offer. 5 I f the i n d i v i d u a l has complete information about the d i s t r i b u t i o n of wage off e r s over firms and the p r o b a b i l i t y of o f f e r d i s t r i b u t i o n over firms, then he can estimate the true mean wage o f f e r and thus the true reservation wage. On the basis of t h i s reservation wage, the i n d i v i d u a l makes acceptance decisions. The i n d i v i d u a l ' s perception of t h i s labour market environment i s incomplete, however, since i t i s generated through non-exhaustive job search. Consequently, the i n d i v i d u a l ' s perceived reservation wage i s subject to a p r o b a b i l i t y d i s t r i b u t i o n . From the properties of random sampling, the p r o b a b i l i t y d i s t r i b u t i o n of the perceived reservation wage w i l l be re l a t e d to the true wage o f f e r d i s t r i b u t i o n . For s i m p l i c i t y , the mean and variance of the reservation wage d i s t r i b u t i o n are assumed to be functions of the true reservation wage and the true variance of o f f e r s r e s p e c t i v e l y . The perceived reservation wage f o r an i n d i v i d u a l can be regarded (from the point of view of the employer) as a random drawing from t h i s density function. I f an i n d i v i d u a l samples a p a r t i c u l a r f i r m and receives an o f f e r , then the p r o b a b i l i t y of acceptance i s given by the p r o b a b i l i t y that the i n d i v i d u a l ' s perceived reservation wage i s less than or equal to the wage o f f e r . This equals the cumulative p r o b a b i l i t y d i s t r i b u t i o n of the perceived reservation wage evaluated at the wage o f f e r . Thus acceptance behaviour by unemployed i n d i v i d u a l s i s stochastic. The expected flow supply of unemployed i n d i v i d u a l s (searchers) to a p a r t i c u l a r f i r m which o f f e r s a wage w i s S^(w), where 1 > 0. dw 6 Likewise, a currently employed in d i v i d u a l has imperfect information about other job offers and wage rates. Thus his perception of the reservation wage i s also subject to a probability d i s t r i b u t i o n whose parameters are functions of the true reservation wage and the true variance of offers. His quit decision i s therefore stochastic. The probability that an i n d i v i d u a l quits i n response to a wage offer i s given by the probability that the perceived reservation wage exceeds the wage offer. I f the firm currently employs e individuals and offers a wage, w, then the expected number of individuals who do not dS 2 dS 2 quit i s S2(e,w), where ~j^7~ > 0 and > 0. The t o t a l expected flow supply of individuals to a firm which currently faces employment, e, and offers a wage, w, i s given by the horizontal sum of S^(w) and S2(e,w) and i s denoted by S(e,w). Labour i s the only variable factor of production. The firm faces an unchanging downward sloping marginal revenue product function, MR(e). This specification.of the marginal revenue product function cuts the l i n k between price and the aggregate output of the industry but, i n order to i s o l a t e the elements of an imperfect labour market, i t i s necessary to abstract from the impact of changes i n the product price on firm behaviour. e l ^ e2 e* e3' e Figure 1 - The Firm's Decision Problem i n the I l l u s t r a t i v e Model 7 The marginal cost of labour schedule facing the firm i s denoted by MS(e,e). This function i s obtained from the derivative of the wage b i l l , wS(e,w), with respect to t o t a l employment, e. To maximise one period profits,the firm chooses the l e v e l of employment at which the marginal revenue product equals marginal cost. I t hopes to employ e* individuals at wage, w*. Due to stochastic search and acceptance behaviour,more than individuals, the mean flow supply of unemployed workers, may contact the firm and be prepared to accept the wage of f e r , w*. At wage rate, w*, the firm would choose to employ e^ i n d i v i d u a l s , i f they were forthcoming. At employment l e v e l , e^, the marginal revenue product equals the wage, w*, and so i f the firm can employ e^ individuals at wage, w*, then p r o f i t s are maximised. Correspondingly, the firm would not choose to employ more than e^ workers. The firm anticipates that e - e^ individuals w i l l quit. Consequently, the firm w i l l create e^ - e^ vacancies but i t expects that only e* - e^ vacancies w i l l be f i l l e d . Thus the firm speculates i n i t s vacancy creation. The firm's decision problem i s shown i n Figure 1. This model closely follows Archibald's model [1954]. I t i s evident that, i f wage dispersion exists i n i t i a l l y and firms face different levels of employment, wage dispersion w i l l persist as w i l l dispersion i n firms' employment l e v e l s . Individuals w i l l indulge i n job search and there w i l l be f r i c t i o n a l unemployment. This suggests that industry equilibrium, however defined, may be characterised by wage dispersion and employment dispersion over firms. 8 In general, the maximising decisions of firms w i l l not regenerate the i n i t i a l labour market environment. I f the environment changes, then workers' search and quit behaviour w i l l change and so the environment w i l l once again change as firms make a new set of optimal decisions. This indicates the fundamental simultaneity of the problem. Searchers' decisions to accept job o f f e r s and employed i n d i v i d u a l s ' decisions to quit are dependent upon the d i s t r i b u t i o n of wage rates and vacancies over firms i n the industry. But, firms' decisions with respect to wage rates and vacancy decisions are dependent upon the search and quit behaviour of the labour force. To say anything meaningful about wage dispersion, vacancies and f r i c t i o n a l unemployment, i t i s necessary to resolve these simultaneous r e l a t i o n s h i p s and examine the nature of equilibrium i n th i s model. This i l l u s t r a t i v e model then demonstrates the nature of the problem I am looking at. This model has severe t h e o r e t i c a l short-comings, however. Model I, the b a s i c model, which i s developed formally i n Chapter 2, remedies these d i f f i c u l t i e s to some extent but predictably makes the problem much more complex. The p r i n c i p a l d e f i c i e n c i e s of thi s i l l u s t r a t i v e model are examined below. Firm behaviour i s unconvincing. In taking the expected flow supply of labour to the fi r m } t h e s t o c h a s t i c elements of job search have been purged and firms are not maximising expected p r o f i t s . I f the flow supply of labour i s known with c e r t a i n t y , then the firm creates e* - e.. vacancies which are instantaneously f i l l e d . 9 I t i s evident that an i n d i v i d u a l h i r e d l a s t p e r i o d has some value to the f i r m t h i s p e r i o d , b e c a u s e the f i r m ' s c u r r e n t wage and vacancy d e c i s i o n s depend on the number of employees i n h e r i t e d from l a s t p e r i o d . T h e r e f o r e , i n h i r i n g an i n d i v i d u a l , the f i r m a n t i c i p a t e s a flow of returns over t ime. Since the m a r g i n a l c o s t f u n c t i o n f a c i n g the f i r m , M S ( e , e ) , i s upward s l o p i n g , t h e f i r m e x h i b i t s s t a t i c monopsony power. I t may be demonstrated t h a t , under p l a u s i b l e assumptions, the marginal cost f u n c t i o n s h i f t s to the r i g h t i n response to an i n c r e a s e i n e, the current l e v e l of employment.^ Thus, the f i r m e x h i b i t s dynamic monopsony power. Then, i t i s a p p r o p r i a t e to model f i r m behaviour i n a m u l t i - p e r i o d framework. In a d d i t i o n , i n s p e c i f y i n g a one p e r i o d h o r i z o n , the model a b s t r a c t s from the dynamic i n t e r -dependence of firms r e s u l t i n g from the impact of each f i r m ' s d e c i s i o n s on the s t a t e of the labour market, notably such parameters as the mean and v a r i a n c e of the wage o f f e r d i s t r i b u t i o n and the l e v e l of unemploy-ment. These parameters i n f l u e n c e the nature of i n d i v i d u a l s ' job s e a r c h . In s h o r t , a s a t i s f a c t o r y model of an imperfect labour market r e q u i r e s both formal s p e c i f i c a t i o n of i n d i v i d u a l behaviour, that i s search r u l e s for i n d i v i d u a l s faced w i t h a d i s t r i b u t i o n of wage o f f e r s , and, i n a d d i t i o n , choice t h e o r e t i c foundations of f i r m s ' d e c i s i o n s , namely wage and vacancy d e c i s i o n s which maximise expected p r o f i t s over an i n f i n i t e h o r i z o n i n response to the s t o c h a s t i c flow supply of l a b o u r . These elements a r e f u r t h e r emphasised i n the b r i e f o u t l i n e of Model I which f o l l o w s . 10 I I I . An O u t l i n e of the Models Developed  i n the Thesis A . Model I In the b a s i c model, Model I , which i s f u l l y analysed i n Chapter 2, workers are assumed to maximise expected income discounted over t h e i r job h o r i z o n . Due to imperfect i n f o r m a t i o n , however, they do not know where they w i l l r e c e i v e the most remunerative job o f f e r s . Consequently, they are forced to search randomly. Since i n d i v i d u a l s do not n e c e s s a r i l y search the same firms or r e c e i v e the same job o f f e r s , they have d i f f e r e n t labour market experiences. They make turnover and acceptance d e c i s i o n s i n the l i g h t of t h e i r perceptions of the true labour market environment. Thus, t h e i r labour market behaviour i s s t o c h a s t i c . An employed i n d i v i d u a l , who i s o f f e r e d a wage r a t e which i s l e s s than h i s p e r c e p t i o n of the r e s e r v a t i o n wage, q u i t s h i s job and searches f o r a b e t t e r p o s i t i o n . Firms make wage and vacancy d e c i s i o n s to maximise the present value of expected p r o f i t s discounted over an i n f i n i t e h o r i z o n . T h i s equals the net worth of the f i r m . They are faced by s t o c h a s t i c t u r n -over by t h e i r employees and s t o c h a s t i c acceptance by job seekers. The f i r m makes at most as many o f f e r s as vacancies i t creates and i t must employ any i n d i v i d u a l a c c e p t i n g an o f f e r . Through random s e a r c h , however, i n s u f f i c i e n t searchers may contact the f i r m . Vacancy c r e a t i o n i s n e c e s s a r i l y s p e c u l a t i v e because the flow of searchers i s s t o c h a s t i c and because not a l l o f f e r s may be accepted. These wage and vacancy d e c i s i o n s depend on the l e v e l of employment the f i r m c u r r e n t l y faces 11 and also how i t perceives the labour market environment. The l e v e l of employment attained i s s t o c h a s t i c . This model i s t r u l y simultaneous i n that each set of market p a r t i c i p a n t s responds to h i s perception of the environment and how the other set of p a r t i c i p a n t s behaves. Equ i l i b r i u m i n t h i s model i s st o c h a s t i c and i s characterised by a steady state d i s t r i b u t i o n of firms over d i f f e r e n t l e v e l s of employ-ment. I f the stochastic process occurs f o r a long period of time,then the expected proportion of time any firm faces a given l e v e l of employ-ment i s given by the corresponding element of the d i s t r i b u t i o n . Or, given a large number of firms,the proportion of firms i n a p a r t i c u l a r employment state i s given by the element of the d i s t r i b u t i o n . Given the s t o c h a s t i c nature of the process,the actual proportion of firms having a p a r t i c u l a r employment l e v e l exhibits v a r i a t i o n about the corresponding element. The equilibrium i s characterised by speculative vacancy creation and, i n addition, i f t h i s steady state d i s t r i b u t i o n i s non-degenerate, wage dispersion and f r i c t i o n a l unemployment. Two important questions emerge from the modelling of t h i s imperfect labour market which I hope to answer i n my t h e s i s . 1. Under what labour market structures i s wage dispersion an equilibrium phenomenon? 12 2. What i s the i n t e r p r e t a t i o n , of vacancy c r e a t i o n i n such models? In p a r t i c u l a r , i s i t p o s s i b l e to c o n s t r u c t an index of labour market t i g h t n e s s or excess demand u s i n g labour market parameters, such as aggregate vacancies and unemployment, such that when the market i s i n s t o c h a s t i c e q u i l i b r i u m the index has a zero value? I n s h o r t , i s the concept of excess demand v a l i d i n an imperfect market? B. Models of D i s c r i m i n a t i o n U s i n g the same general framework as Model I , two models of d i s c r i m i n a t i o n are formulated. In these models, the labour force c o n s i s t s of two types of i n d i v i d u a l s with the same s k i l l s , b u t d i f f e r i n g i n t h e i r turnover and acceptance behaviour,due to s y s t e m a t i c a l l y d i f f e r i n g perceptions of r e s e r v a t i o n wage. C r u c i a l to the f i r m i s whether i t can i d e n t i f y the nature of each i n d i v i d u a l & s labour market behaviour through some overt c h a r a c t e r i s t i c s ( e . g . young and o l d workers d i f f e r i n g i n t h e i r rates of time preference or horizons over which they b e l i e v e r e t u r n s from search accrue) and, furthermore, whether i t i s l e g a l l y allowed to d i s c r i m i n a t e i n wages p a i d and i n h i r i n g . There are four d i s t i n c t labour market s t r u c t u r e s c o n s i d e r e d . The a p p r o p r i a t e model i s i n d i c a t e d by the elements of a 2 x 2 m a t r i x . RC RC I I I I I I I I I F i g u r e 2 The Labour Market S t r u c t u r e s Modelled 13 In state RC, each firm can recognise the characteristics of an ind i v i d u a l searcher's labour market behaviour p r i o r to h i r i n g , i n addition to knowing the characteristics of each employee's behaviour and the general composition of the stock of unemployed workers. In state RC, no fir m i s able to recognise di f f e r e n t individuals p r i o r to h i r i n g but, at any point i n time, i t knows the general composition of i t s work force and the stock of unemployed workers. In state DI, each firm i s l e g a l l y allowed to discriminate e x p l i c i t l y between the different types of worker through i t s wage offer and vacancy creation decision. I t makes an ex ante wage and vacancy decision for each type of worker. In state DI, e x p l i c i t wage and h i r i n g discrimination are i l l e g a l . Each firm makes offers randomly to those searchers who sample i t and pays a l l individuals the same wage. In Model I I firms do not discriminate i n wage offer or h i r i n g between different individuals either because the practice i s i l l e g a l (row 2 of the matrix) or because they are unable to recognise the characteristics of each individual's behaviour (column 2 of the matrix). Each of these states i s a s u f f i c i e n t condition for no i n t r a - f i r m discrimination, but neither i s a necessary condition. Each firm knows the general composition of i t s workforce and the stock of unemployed individuals. In Model I I I each fi r m i s able to recognise the characteristics of i n d i v i d u a l searchers and each member of i t s workforce (column 1 14 of the m a t r i x ) . Furthermore, i t i s allowed to d i s c r i m i n a t e (row 1 of the m a t r i x ) . Then, i n response to the p e r c e i v e d labour market environment and the number of and composition of i t s workforce, i t makes an ex ante wage and vacancy c r e a t i o n d e c i s i o n for each type of worker. Through the examination of these two models, I wish to answer the f o l l o w i n g questions : 1. I f each f i r m recognises the d i f f e r e n t types of i n d i v i d u a l worker and i s allowed to d i s c r i m i n a t e i n wage o f f e r and employment (Model I I I ) , w i l l f i r m s ' workforces be segregated? 2. Assuming t h a t , ex a n t e , type one i n d i v i d u a l s have a h i g h e r p r o b a b i l i t y of q u i t t i n g than type two i n d i v i d u a l s i n response to a given wage o f f e r , w i l l type one i n d i v i d u a l s earn lower wages as argued by Sanborn (1964), or w i l l firms o f f e r h i g h -turnover i n d i v i d u a l s h i g h e r wages? 3. What i s the impact of ' f a i r employment' laws on expected i n d u s t r y p r o f i t s and on expected l i f e t i m e incomes of the d i f f e r e n t types of i n d i v i d u a l s ? 4. I f i n t r a J f i r m d i s c r i m i n a t i o n i s i l l e g a l , w i l l wage d i f f e r e n t i a l s p e r s i s t over firms w i t h the same l e v e l but d i f f e r e n t composition of employment? In other words, w i l l a n t i - d i s c r i m i n a t i o n laws be e f f e c t i v e i n a b o l i s h i n g wage and employment d i s c r i m i n a t i o n ? 15 IV. Review of the L i t e r a t u r e oh Imperfect Markets A . A Summary As already noted the l i t e r a t u r e on imperfect markets can be s u b - d i v i d e d i n t o four a r e a s , namely : 1. Models of opt imal search behaviour by i n d i v i d u a l s i n markets c h a r a c t e r i s e d by imperfect i n f o r m a t i o n ; 2. Models of i n t e r t e m p o r a l f i r m behaviour i n response to p r e s c r i b e d n o n - s t o c h a s t i c turnover and acceptance behaviour by i n d i v i d u a l s ; 3. Models of a labour market which i s viewed as a s t o c h a s t i c p r o c e s s ; and 4. S t a t i c monopsony models which examine the concept of a vacancy. B. Optimal Search In S t i g l e r ' s fundamental a r t i c l e [1961], he argues that ignorance and heterogeneity of goods or s e r v i c e s generate p r i c e d i s p e r s i o n . He develops a theory of how consumers ought to behave, given that there i s a v a r i e t y of unknown p r i c e s for a homogeneous good. T h i s a n a l y s i s i s a p p l i c a b l e to i n d i v i d u a l s s e a r c h i n g for the h i g h e s t wage i n an imperfect labour market where jobs are ranked s o l e l y on the wage o f f e r e d . I f the consumer knows the d i s t r i b u t i o n of p r i c e s , but not the p a r t i c u l a r p r i c e at each s t o r e , then he searches randomly. Each search i s a drawing from the d i s t r i b u t i o n of p r i c e s . S t i g l e r argues that the consumer d e c i d e s , ex ante, on the optimal number of searches 16 and then picks the minimum price. The expected minimum price from a sample of n observations each with cumulative probability d i s t r i b u t i o n F(p) i s CO p = / (l-F(p) n" 1pnF'(p)dp. . . . . (1) 0 The expected gain from a further search i s 00 ' g n = P n - P n + 1 = /' (l-F(p)) nF(p)dp . . . . (2) which i s a decreasing function of n. I f the cost of one search i s c^then n i s the optimal number of searches where Sn ± C - gn+l • • • • • ( 3 ) The marginal returns from the next search are equalled or exceeded by the cost of another search. S t i g l e r has s i m p l i f i e d the problem by assuming that one unit of the good i s purchased irrespective of the price charged. In this framework, i t i s not clear how returns from search are computed when quantity demanded varies with price. Several authors, notably McCall [1965], [1960], Nelson [1970] and Rothschild [1973] argue that the decision rule that S t i g l e r devised 9 i s sub-optimal and that the optimal rule of search i s sequential. I f s i s the minimum price of those sampled,.then the expected returns from one further search are 17 (s-p)dF(p) = [(s-p)F(p)]^ + fS0 F(p)dp = g(p)« . . . . (4) Individuals w i l l continue to search, i f the lowest price observed exceeds R, where R i s the solution to g(R) = c. In contrast to S t i g l e r ' s ex ante decision rul e , the sequential rule does not require the c o l l e c t i o n of offers. This r e s u l t i s p a r t i c u l a r l y useful when modelling search i n a labour market, where the technical homogeneity of workers makes offer c o l l e c t i o n implausible. I t i s important to note that R i s independent of the number of searches already undertaken, i f the marginal cost of search i s constant. Thus, an i n d i v i d u a l may search many stores u n t i l a price less than R i s observed, although his decision r u l e i s based on the computation of the returns from one further search. A s i m i l a r one search decision rule can be developed for job search i n a labour market. This decision r u l e , however, i s generally not the optimum rule of search. Gastwirth [1971] examines the robustness of these optimal decision rules when the searcher i n c o r r e c t l y estimates the d i s t r i b u t i o n of prices. He demonstrates that even a modest sp e c i f i c a t i o n error dramatically increased the expected number of searches. He proposes a mixed rule which i s both robust and preserves the reservation price property. 18 The r u l e i s that the i n d i v i d u a l keeps s e a r c h i n g u n t i l e i t h e r n til observations have been taken or the lowest o b s e r v a t i o n a f t e r the k o b s e r v a t i o n i s l e s s than some p r e s c r i b e d R . R o t h s c h i l d [1974] derives the optimal search r u l e s for an i n d i v i d u a l who l e a r n s about an unknown d i s t r i b u t i o n of p r i c e s through s e a r c h . He concludes that economists can c o n s i d e r without great l o s s that search r u l e s from unknown d i s t r i b u t i o n s do have the same q u a l i t a t i v e p r o p e r t i e s as the optimal r u l e s from known d i s t r i b u t i o n s . " ' " ^ T e l s e r [1973] adopts a d i f f e r e n t approach. Using Monte C a r l o techniques,he c a l c u l a t e s the gains a r i s i n g from an optimal search r u l e as compared w i t h a n a i v e search r u l e of p i c k i n g the f i r s t p r i c e sampled, f o r d i f f e r e n t known d i s t r i b u t i o n s of p r i c e s . H i s r e s u l t s show that the gains are an i n c r e a s i n g f u n c t i o n of the r a t i o of the range of each d i s t r i b u t i o n to i t s minimum p r i c e and a d e c l i n i n g f u n c t i o n of the cost of one s e a r c h . He concludes, however, that the gains from the optimal r u l e over the naive r u l e are s m a l l . In the r e s t of h i s a r t i c l e , S t i g l e r makes some l o o s e a s s e r t i o n s about the i n t e n s i t y of search and the determinants of the magnitude of p r i c e d i s p e r s i o n . In the absence of any formal s p e c i f i c a t i o n of f i r m behaviour which generates d i s p e r s i o n , these remarks l a c k c o n v i c t i o n . I t i s a p a r t i a l e q u i l i b r i u m model i n that consumers' behaviour i n response to p r i c e v a r i a t i o n i s s p e c i f i e d , b u t how such d i s p e r s i o n i s generated i s not e x p l a i n e d . 19 S t i g l e r models an in d i v i d u a l store's behaviour when i t believes that consumers make n searches from a rectangular d i s t r i b u t i o n of store prices on the unit i n t e r v a l [0,1]. The expected number of customers buying from a store charging price, p, i s D(p) = k ( l - p ) n _ 1 . . . . (5) where k i s a po s i t i v e constant. Given that other store prices are fixed,then D(p) represents the store's demand function. Denoting p r o f i t s by II, the firm maximises n = D(p)p - C[D(p)] . . . . (6) where C[D(p)] i s the store's cost function. I f each store has the same cost function,then each w i l l choose the same price and the uniform d i s t r i b u t i o n which was supposed to generate the demand function w i l l shrink to a point. The preceding discussion closely follows Rothschild's review [1973]. This s t a t i c framework i s inappropriate to model the dispersion. Stores have no intertemporal interdependance. Stores have no intertemporal monopoly power because no customers are attached to them over time. Thus, they are i d e n t i c a l i n a l l respects and no price dispersion i s generated. In a l a t e r a r t i c l e [1962], S t i g l e r discusses the amount of wage dispersion observed i n different professions and the determination of the returns from search. He points out that to compute the returns from search requires that the firm's perceived wage offer has a correlation of one over time. He argues that, even with r a t i o n a l search and i n the absence of exogeneous s h i f t s i n supply and demand, 20 wage dispersion w i l l persist i n the labour market. He cit e s changes i n jobs due to changes i n workers' tastes and a b i l i t i e s and employees' i d e n t i t i e s as i n s u f f i c i e n t to generate s u f f i c i e n t search to eliminate dispersion. These phenomena, he states, set a minimum to the amount of wage dispersion. These changes surely can be regarded as exogeneous to the labour market. I f the labour market i s continually i n flux,so that individuals actually leave the labour force and new firms appear,then dispersion i s l i k e l y to pe r s i s t . S t i g l e r argues that costs of search prevent s u f f i c i e n t search and thus dispersion p e r s i s t s . Costs of search alone, however, do not generate dispersion. There are two s u f f i c i e n t but not necessary conditions for zero dispersion. 1. Firms are technically homogeneous and have no intertemporal monopoly power,as S t i g l e r unwittingly demonstrated [1961]; and 2. Individuals are homogeneous i n their search behaviour i n the sense that, ex post, they respond i n the same way to any par t i c u l a r price or wage of f e r , that i s , they either accept or reject the price with a probability of one. I f a l l individuals adopt the same sequential decision rule,then a l l individuals have the same reservation or c r i t i c a l price and there w i l l be no dispersion. I f firms exhibit dynamic monopsony power and individuals adopt a non-exhaustive ex  ante decision rule for search, such as the one proposed by S t i g l e r , then dispersion p e r s i s t s , as shown below. The 21 magnitude of dispersion seems to be p o s i t i v e l y related, to the cost of search. These conditions suggest that i n some cases,individuals' expectations about the nature of the price or wage d i s t r i b u t i o n are s e l f r e a l i s i n g . In my model of an imperfect labour market, individuals' stochastic turnover and acceptance behaviour i s motivated by their b e l i e f that the wage d i s t r i b u t i o n exhibits non-zero variance and thus non-zero dispersion which j u s t i f i e s search. This stochastic behaviour i n turn reinforces dispersion as firms experience different levels of employment. I f individuals believe that there i s no d i s -persion and each has the same perception of the reservation wage, then a l l firms w i l l offer that reservation wage and there w i l l be no d i s -persion and no search. Thus, non-exhaustive search, per se, i s not s u f f i c i e n t to generate dispersion. Gronau [1971] develops a model of intertemporal sequential search i n which the t o t a l time devoted to work and search i s constant. Thus, further search leads to a shorter working horizon. The ind i v i d u a l has a perception of the wage d i s t r i b u t i o n over the horizon. For each period over the fixed horizon,the i n d i v i d u a l computes a reservation wage such that the discounted income stream associated with t h i s wage i s equal to the discounted income stream associated with the best alternative. This expression takes the form of a recurrence relationship. The optimal current reservation wage depends on the sequence of optimal reservation wages over the remainder of the horizon. I f unemployed 22 p r i o r to the l a s t p e r i o d , ah i n d i v i d u a l accepts any wage o f f e r unless i t i s exceeded by h i s a l t e r n a t i v e e a r n i n g s , namely unemployment compensation. I n d i v i d u a l s assume that the temporal c o r r e l a t i o n of a f i r m ' s wage o f f e r i s u n i t y . S o l v i n g the recurrence r e l a t i o n s , Gronau demonstrates t h a t , given a time i n v a r i a n t wage o f f e r d i s -t r i b u t i o n , the r e s e r v a t i o n or acceptance wage i s a d e c l i n i n g f u n c t i o n of t ime. This argument i s m i s l e a d i n g i n that the time devoted to search i s normally a s m a l l f r a c t i o n of that devoted to work. Con-sequently, the acceptance wage d e c l i n e s very l i t t l e over the search p e r i o d and i s constants i f the h o r i z o n of job tenure i s assumed independent of the d u r a t i o n of search (see C h a p t e r ^ I B ) . A more p l a u s i b l e e x p l a n a t i o n f o r the d e c l i n i n g acceptance wage i s that the i n d i v i d u a l faces imperfect c a p i t a l markets. L i k e S t i g l e r [1961], Gronau takes wage d i s p e r s i o n as given and f a i l s to p r o v i d e any formal s p e c i f i c a t i o n of f i r m behaviour to generate d i s p e r s i o n . Salop [1973a] a l s o discusses the i n d i v i d u a l ' s systematic job search i n an i n t e r t e m p o r a l framework. The i n d i v i d u a l has knowledge of each f i r m ' s wage o f f e r d i s t r i b u t i o n and the p r o b a b i l i t y of an o f f e r at that f i r m at each p o i n t i n t ime. He has to choose the optimal search order and sequence of acceptance wages over time to maximise the present value of expected wealth over an i n f i n i t e h o r i z o n . Re-sampling any p a r t i c u l a r f i r m i s not p e r m i t t e d . He demonstrates that an i n d i v i d u a l w i l l sample a h i g h wage-low p r o b a b i l i t y o f o f f e r f i r m before a low wage-high p r o b a b i l i t y of o f f e r f i r m , i f mean wage r a t e s are i d e n t i c a l . 23 Again, r a t i o n a l firm behaviour which generates dispersion i s not specified. I f searchers are homogeneous i n their knowledge of wage of f e r s , then a l l w i l l sample the same firm f i r s t and a l l have the same acceptance wage. Dispersion w i l l disappear according to the second s u f f i c i e n t condition discussed above. C. Firms' Intertemporal Behaviour Mortenson [1970] develops a non-stochastic model of firm behaviour i n a labour market characterised by imperfect information. Information i s incomplete because i t has to be acquired sequentially through search, which i s time consuming and costly, and because old information becomes rapidly obsolete. Workers' preferences over firms are ranked by the wage offered, so there i s no psychic income associated with employment at a p a r t i c u l a r firm. Individuals have perceptions about the mean wage offer which they update through the i r own experience. This i s modelled by specifying that each i n d i v i d u a l has a perception of the mean wage which i s subject to a pr o b a b i l i t y d i s t r i b u t i o n . Each in d i v i d u a l appears homogeneous, ex ante, i n his perception of the mean wage and his labour market experiences are captured by specifying stochastic misperceptions of the mean wage. Search behaviour i s not formally specified,but firms can expect the number of unemployed searchers who sample them to be i n proportion to their r e l a t i v e size i n the labour market. Unemployed searchers have a higher probability of contact, that i s receiving an o f f e r , than employed searchers. This r e f l e c t s the lower opportunity costs of search when unemployed. The probability of contact i s assumed constant, 24 but i s c l e a r l y a f u n c t i o n of the d i s t r i b u t i o n of vacancies and l e v e l of unemployment. Both employed and unemployed searchers assume that they are o f f e r e d a job w i t h c e r t a i n t y at t h e i r n o t i o n of the mean market wage a f t e r t h e i r mean l e n g t h of s e a r c h , namely the i n v e r s e of the p r o b a b i l i t y of making a contact . T h e i r q u i t , acceptance and i n t e r - f i r m turnover d e c i s i o n s are a l l made through comparing these c e r t a i n income streams. Summing the i n d i v i d u a l components and t a k i n g the mathematical e x p e c t a t i o n , Mortenson derives the flow supply of labour to the f i r m . The p r o p o r t i o n a l r a t e of change of the f i r m ' s labour force i s a f u n c t i o n of i t s own r e l a t i v e wage and the l e v e l of unemployment,but i s independent of the s i z e of i t s labour f o r c e . Assuming the form of the wage d i s t r i b u t i o n and t a k i n g the l e v e l of unemployment as constant, he obtains the o p t i m a l time path of wages f o r the f i r m by s o l v i n g the i n t e r t e m p o r a l p r o f i t maximising problem. While r e c o g n i s i n g the e x p l i c i t dynamic interdependence of firms i n the h i r i n g of l a b o u r , Mortenson f i n e s s e s the problem by making these p a r t i a l e q u i l i b r i u m assumptions. Adopting a n o n - s t o c h a s -t i c labour supply schedule, he has suppressed any ex post heterogeneity of workers' labour market behaviour generated by s t o c h a s t i c misperceptions of the mean wage. Furthermore, by assuming a f i x e d p r o b a b i l i t y of c o n t a c t , he has avoided the s t o c h a s t i c elements of job search i n a market c h a r a c t e r i s e d by changing d i s t r i b u t i o n s of wages and vacancies over time. A l l p o s i t i o n s .> c r e a t e d are i n s t a n t a n e o u s l y f i l l e d so v a c a n c i e s , as c o n v e n t i o n a l l y d e f i n e d , do not e x i s t , and n e i t h e r does excess demand. 25 S i n c e , ex p o s t , labour of the same employment or unemployment status i s homogeneous i n i t s labour market behaviour, then, from the second s u f f i c i e n t c o n d i t i o n noted above, wage d i s p e r s i o n r e q u i r e s the heterogeneity o f f i r m s . In t h i s p a r t i a l e q u i l i b r i u m framework, t h i s r e q u i r e s that firms face e i t h e r d i f f e r e n t t e c h n i c a l c o n d i t i o n s or d i f f e r e n t i n i t i a l l e v e l s of employment. I f the l a t t e r i s t r u e , then, i n the l i m i t , wage d i s p e r s i o n disappears as firms move towards the saddle p o i n t s o l u t i o n to t h e i r i n t e r t e m p o r a l maximization problem. While Mortenson's model i s i l l u m i n a t i n g i n that i t demonstrates t h a t , given imperfect i n f o r m a t i o n , firms e x h i b i t i n t e r t e m p o r a l monopsony power, i t i s p a r t i a l and does not r e f l e c t the i n h e r e n t s t o c h a s t i c i t y of an imperfect labour market. Salop [1973b] develops an i n t e r t e m p o r a l model of f i r m ' s behaviour when faced w i t h a known q u i t r a t e which i s s o l e l y a f u n c t i o n of i t s wage o f f e r . Formal search behaviour of workers i s again not s p e c i f i e d . L i k e Mortenson's model,the l e v e l s of vacancies and unemployment, which a f f e c t the q u i t r a t e and wage behaviour of other firms i n the i n d u s t r y , are constant . A l s o p o s i t i o n s created are not v a c a n c i e s , s i n c e they are i n s t a n t a n e o u s l y f i l l e d i n the absence of s t o c h a s t i c behaviour. Given t h i s d e t e r m i n i s t i c behaviour on the p a r t of a l l p a r t i c i p a n t s , d i s p e r s i o n r e q u i r e s that firms face d i f f e r e n t l e v e l s of employment i n i t i a l l y and again such d i s p e r s i o n disappears as firms move towards t h e i r saddle p o i n t s o l u t i o n . 26 The c r u c i a l shortcoming of these models then i s that, despite imperfect information and thus job search by individuals, the labour supply to the in d i v i d u a l firm i s deterministic. Vacancy creation and excess demand do not e x i s t . For u n f i l l e d positions to exist,job creation must be speculative, that i s , the supply of labour to the firm must be subject to stochastic va r i a t i o n . D. The Labour Market Viewed as a Stochastic Process Reder [1968] develops a stochastic model of the labour market characterised by imperfect information. Workers are faced with a choice of heterogeneous jobs offering the same wage rate. They quit the i r jobs i n order to search for a preferable one,but, through i n -complete information,,they are not instantly re-employed. Since there i s no systematic preference for p a r t i c u l a r jobs, each firm faces es s e n t i a l l y the same labour supply schedule. Firms create vacancies i n anticipation of workers' turnover and acceptance behaviour. Aggregating over firms and individuals yields an effec t i v e demand schedule for labour, D, and an effec t i v e supply schedule, S. The speculative demand for labour i s denoted by D'. I t measures the number of vacancies that must be created on average at a given wage, i n order to generate employment given by the corresponding point on the effective demand schedule. Analogously, the S' schedule indicates the number of workers who must search for a job at a given wage rate, i n order that the number represented by the S schedule be employed. In t h i s s t a t i c framework, firms r e h i r e t h e i r labour force at the b e g i n n i n g of each p e r i o d and, i g n o r i n g previous labour market experience, they o f f e r the same wage. In s t o c h a s t i c e q u i l i b r i u m ON i n d i v i d u a l s are employed at wage r a t e , cW. Aggregate unemployment i s given by N ^ - N , s a n d aggregate vacancies by ^ - N . S t o c h a s t i c e q u i l i b r i u m r e q u i r e s that on average the number of i n d i v i d u a l s q u i t t i n g t h e i r jobs equals the number of new vacancies f i l l e d . A p r i o r i , the average length of time a vacancy i s open i s not r e l a t e d i n any p a r t i c u l a r way to the average p e r i o d of unemployment. Thus s t o c h a s t i c e q u i l i b r i u m i s c o n s i s t e n t w i t h the simultaneous e x i s t e n c e of unemployment and v a c a n c i e s , the r e l a t i o n s h i p between them b e i n g i n d e t e r m i n a t e . The model i s i l l u s t r a t e d i n F i g u r e 3. F i g u r e 3. S t o c h a s t i c E q u i l i b r i u m i n Reder's Model 28 Reder provides considerable insight into the nature of equilibrium i n an imperfect market,but the s t a t i c framework i s inappropriate. He f a i l s to formulate e x p l i c i t l y the behaviour of market participants or discuss how the market converges to equilibrium. Search and thus f r i c t i o n a l unemployment i n this model are generated by non-systematic job preferences rather than wage dispersion. Holjtj and David [1966] and Holt [1970] also discuss the nature of stochastic equilibrium i n the labour market without indicating the choice theoretic foundations of market participants' behaviour. Bergman [1973] develops a simulation model of an imperfect labour market. To test hypotheses about the impact of d i f f e r i n g behaviour by male and female workers, she incorporates ideas from Holt and David [1960], namely, that an unemployed person becomes less fussy about the characteristics of a job offer as the period of unemployment increases and, likewise, firms become less fussy about the characteristics of job candidates and more w i l l i n g to increase the wage off e r , the longer a vacant position remains u n f i l l e d . The labour market i s a daily process. Each employer has a fixed number of job s l o t s . Each day a random selection of employed workers quit and become unemployed. A number of candidates, dependent on the number of unemployed workers, parade randomly past each vacant s l o t . The probability that an employer w i l l offer a candidate the vacant position i s a function of the employer's length of search. Likewise, the probability that an unemployed i n d i v i d u a l accepts a position i s a function of his length of search. Thus, some slots remain open t i l l the next day and some individuals remain unemployed. Parameter values are chosen f o r the labour force s i z e , the number of s l o t s , the turnover and acceptance functions of workers and the p r o b a b i l i t y of o f f e r f u n c t i o n of employers. The system i s then simulated and each month a survey taken to measure unemployment and vacancy rates and the d u r a t i o n of unemployment. Bergman defines the s l o t r a t e as the r a t i o of the t o t a l p o s i t i o n a v a i l a b l e ( i . e . , both f i l l e d and u n f i l l e d ) to the s i z e of the labour f o r c e . I f the s l o t r a t e i s l e s s than 100 p e r c e n t , then the l e v e l of unemployment i s an i n c o r r e c t measure of the l e v e l of f r i c t i o n a l unemployment, s i n c e not a l l i n d i v i d u a l s can be employed. The vacancy r a t e i s the c o r r e c t measure of f r i c t i o n a l unemployment. I f the s l o t r a t e i s equal to 100 percent or more, then, she argues, a l l i n d i v i d u a l s i n the labour force c o u l d be employed and t o t a l unemployment i s the a p p r o p r i a t e measure of f r i c t i o n a l unemployment. She s t a t e s that the amount of unemployment a t t r i b u t a b l e to search i s low,even when there i s a h i g h r a t e of turnover. I n t h i s labour market, a s e p a r a t i o n (when the s l o t r a t e i s f ixed) makes one i n d i v i d u a l unemployed,but immediately opens up an opportunity f o r one of the c u r r e n t l y unemployed, s i n c e separat ions occur before the f i r m attempts to f i l l vacant s l o t s . She considers an i n t e g r a t e d labour market with groups of i n d i v i d u a l s e x h i b i t i n g d i f f e r e n t r a t e s of turnover and no e x p l i c i t d i s c r i m i n a t i o n i n h i r i n g by f i r m s . She argues that the r a t i o of unemployment rates f o r the d i f f e r e n t types of people i s given by the r a t i o of the products of the s e p a r a t i o n 13 rates and the employment rates of each type. In some occupations, 30 women have a higher turnover rate than men but, i f women compete equally with men for vacant positions, these different rates of turnover are i n s u f f i c i e n t to generate the observed difference i n unemployment rates. She concludes that labour markets are to a large degree segmented, so that i t i s the higher overcrowding or lower s l o t rates i n women's labour markets which generate their s i g n i f i c a n t l y higher rates of unemployment. Bergman points out that, i n this world of stochastic behaviour, firms would speculate i n s l o t creation i n order to secure the i r desired employment. Thus^ Holt's measure of labour market tightness [Holt, 1970] namely, vacancies/unemployment i s inappropriate and (desired-actual employment)/unemployment i s a better measure of tightness, since the speculative elements of vacancy creation are purged. There are, however, considerable sampling problems i n obtaining such a measure. In arguing that an imperfect labour market behaves as a stochastic process and that conventional measures of excess demand are inappropriate due to speculative behaviour, Bergman has made an important contribution. She f a i l s , however, to provide e x p l i c i t micro foundations of firm and worker behaviour and so her conclusions must be subject to doubt. In the absence of wage and vacancy decisions by firms, i n response to t h e i r own experience and current labour market conditions, i t i s not evident that individuals choose to remain un-employed, unless somehow wage dispersion exists or individuals exhibit s p e c i f i c preferences for different firms. Likewise, unless workers 31 appear to be heterogeneous i n thei r capacity to perform the job, firms do not choose to leave vacant positions u n f i l l e d . A model i n which firms, aware of differences i n labour market behaviour by recognisable groups of ind i v i d u a l s , are able to discriminate, generates different conclusions, perhaps. In Model I I I , which i s analysed i n Chapter 2 IC3, firms can recognise systematic differences i n individuals' labour market behaviour and discriminate. A l l these models are characterised by the absence of any choice theoretic foundations for in d i v i d u a l market participants' behaviour. As Rothschild [1973] points out, models of disequilibrium behaviour do not make sense,unless they meet certain standards of coherence and consistency. He argues that e x i s t i n g models of adjustment to equilibrium s a c r i f i c e a discussion of rules that market participants adopt out of equilibrium, for a description of how the market operates 14 i n response to in d i v i d u a l behaviour and a convergence theorem. E. The Vacancy Concept Archibald [1954] develops a simple monopsony model i n which the firm maximises short run p r o f i t s . He attributes the r i s i n g labour supply schedule faced by the firm, either to the preference of workers for p a r t i c u l a r jobs, or to the large size of the firm. Short run p r o f i t maximisation leads the firm to hire labour up to the point at which marginal revenue product equals marginal costs. The firm hires L workers at wage, w, and exploits labour through not paying i t i t s marginal revenue product, namely w*. The firm's decision problem i s shown i n Figure 4. 32 w MC 0 L L x L F i g u r e 4 The F i r m ' s D e c i s i o n Problem i n A r c h i b a l d ' s model A r c h i b a l d argues that the f i r m would create v a c a n c i e s , L ^ - L , i f i t c o u l d r e c r u i t c o s t l e s s l y at wage r a t e , w. He was t r y i n g to demonstrate that e q u i l i b r i u m i n the l a b o u r market at the r u l i n g wage was c o n s i s t e n t w i t h an excess of vacancies over unemployment. Thus, Hansen's [1953] use of excess demand as a measure of i n f l a t i o n a r y p r e s s u r e at f u l l employment was i n c o r r e c t . Burgess [1969] argues that the concept of v a c a n c i e s i s i n c o r r e c t . The f i r m i s u n w i l l i n g to i n c u r costs of r e c r u i t i n g , n e i t h e r a d v e r t i s i n g expenditure to improve s e a r c h e r s ' i n f o r m a t i o n , n o r i n c r e a s i n g the wage o f f e r . Given that the f i r m i s at i t s p r o f i t maximising p o s i t i o n , ( w , L ) , i t has some knowledge of i t s labour supply schedule. Myers [1968] argues that measuring vacancies as L ^ - L i s to i n f e r that the f i r m p r e f e r s a more e l a s t i c supply of l a b o u r . 33 Devine and Marcus [1967] demonstrate that, i n this s t a t i c framework, i f the firm can choose both i t s wage offer and l e v e l of r e c r u i t i n g costs, then Archibald's result s t i l l holds. Job vacancies are consistent with monopsony equilibrium, both with respect to wage offer and recruitment costs. Now S i s interpreted as the schedule r e l a t i n g the wage offer to the number of individuals prepared to work, given that at each l e v e l of employment the firm chooses i t s cost minimising combination of wage offer and recruitment costs (advertising). MC represents the o v e r a l l marginal costs, namely marginal wage costs and marginal recruitment costs. Vacancies, L^-L, w i l l be created and the firm i s unwilling to offer a higher wage or spend more on recruitment. Burgess again claims that such vacancies are i l l u s i o n a r y , since firms w i l l not even incur costs of an interview. In the absence of prohibitive interview costs, this assertion i s untrue. The desired employment l e v e l i s that l e v e l bf employment at which the marginal cost of h i r i n g another worker, namely the p r o f i t maximising wage, w, plus the interview costs, equals the marginal revenue product. In this model, reported vacancies equal desired employment minus the p r o f i t maximising l e v e l of employment. I t i s questionable whether this concept of vacancies i s legitimate i n this framework. The supply schedule i n the Archibald and Devine and Marcus models shows the number of workers who would show up at the firm and are hired,given the wage offer and the l e v e l of r e c r u i t i n g costs. The firm makes i t s optimal wage offer and the 34 number of individuals represented by the corresponding point on the supply schedule are hired. Both authors are now asking the question:-i f some variation i n hires i s possible at t h i s wage rate, what l e v e l of vacancies would the firm create? I t i s wrong to consider stochastic variation i n hires at the optimal wage rate alone. I f there i s im-perfect information i n the market, then the whole schedule would be subject to stochastic variation and the resu l t i n g optimal wage decision may w e l l be dif f e r e n t . These models highlight a severe p r a c t i c a l problem i n defining a vacancy. I t i s agreed that employers should be actively r e c r u i t i n g to f i l l a position i n order that i t should c l a s s i f y as a vacancy. In the Devine and Marcus model, firms are spending a fixed sum on r e c r u i t -ment to improve workers' information,but the volume of expenditure does not indicate the number of positions for which i t i s actively r e c r u i t i n g . The marginal cost of advertising another vacancy i s zero. Attempts to formalise the vacancy concept i n this s t a t i c framework are unsatisfactory and misleading. Since there i s no intertemporal interdependence i n a one period model, firms are forced to rehire t h e i r labour forces each period. These so-called vacancies represent the magnitude of the misperceptions of the firm about the labour supply forthcoming at the p a r t i c u l a r wage. I t creates positions based on i t s perception of the labour supply schedule and L are f i l l e d . I f i t knew about the nature of the labour supply schedule with certainty, then i t would create L positions which would be instantaneously f i l l e d . 35 Hold and David [1966] define a vacancy as a job with requirements specified for which the firm i s actively r e c r u i t i n g . Vacancies equal the number of men who would be hired today, but not necessarily to s t a r t , i f they had the same s k i l l and wage requirements as those the company recently hired to f i l l corresponding positions. There i s the small problem of technical change generating job vacancies representing new s k i l l requirements. The unemployed in d i v i d u a l i s one who : 1. Is actively searching; and 2. I f asked i s prepared to work at his best previous job. The l a t t e r condition ensures that the concept of unemployment relates to job search alone rather than a l l instances of i n a b i l i t y to f i n d work at reasonable wages. Provision must be made that the man . s t i l l q u a l i f i e s for holding the old job, i f offered. G i l p a t r i c k [1966] points out that t h i s d e f i n i t i o n implies that workers displaced by technological change do not qualify as unemployed,since the old job w i l l not be offered to them. I f the 'best previous job' should s t i l l e x i s t , then those s t r u c t u r a l l y unemployed are not included i n the Holt and David d e f i n i t i o n , and Gil p a t r i c k i s correct. Hold and David require that the i n d i v i d u a l be searching for employment at reasonable wages. This appears to exclude those individuals who quit t h e i r jobs believing that the general l e v e l of wages i s higher than the i r current wage. At best, the i r d e f i n i t i o n 36 covers those searchers with reasonable wage demands who are unemployed through inadequate demand or st r u c t u r a l changes or because of imperfect information as to the existence of suitable vacancies. An ind i v i d u a l who actively searches, can be f r i c t i o n a l l y unemployed through sampling firms with no vacancies, or through misperceiving the wage dis t r i b u t i o n , or both. Holt and David's d e f i n i t i o n , then, seems to exclude searchers with unreasonable wage requirements. At the p r a c t i c a l l e v e l , then, Holt and David's measures of un-employment and vacancies require extensive sampling of firms to obtain information about wage offers and the number of positions they wish to f i l l at that wage offer. A homogeneous measure of t o t a l vacancies requires that the vacancies aggregated over firms be homogeneous both i n the wage offered and the s k i l l s required. Thus, i f any wage d i s -persion exists i n a labour market, homogeneous i n the s k i l l s of i t s labour force, i t i s necessary to construct some measure of a market wage. The unemployed are asked the hypothetical question as to whether they would actively search given this market wage. Likewise, firms are asked how many vacancies they would create at this market wage rate. For p r a c t i c a l purposes, individuals without work for seven days are c l a s s i f i e d as unemployed. A compatible d e f i n i t i o n of a vacancy i s required. While unemployment i s a d i s t i n c t and recognisable state for the ind i v i d u a l (again ignoring the active search d e f i n i t i o n ) , the existence of a vacancy i s less w e l l defined. An employer, not actively recruiting,may create a position for a w e l l q u a l i f i e d applicant. 37 Alternative measures are open to an employer with an i n s u f f i c i e n t labour force, namely : 1. He can insti g a t e overtime; 2. He can subcontract the work elsewhere;, and 3. He can purchase additional machinery, that i s c a p i t a l deepening, and make his labour force more productive. F. Conclusion As indicated throughout the section, the l i t e r a t u r e on im-perfect markets has f a i l e d to model i n d i v i d u a l behaviour based on micro-theoretic foundations, i n response to the perceived environment and the behaviour of other market participants. Furthermore, how the market operates i n response to such behaviour i s not specified. The shortcomings w i l l be further examined and emphasised i n the next chapter i n which the basic theoretical framework i s discussed. V. The Literature on Discrimination A. A Summary The l i t e r a t u r e on models of discrimination can be sub-divided into three areas : 1. Models examining the impact of f a i r employment laws on wage rates and employment; 2. Models of discrimination based on employer prejudice; and 3. Models of discrimination based on imperfect information. 38 B. The Impact of Fair Employment Laws In his model of firm behaviour, Salop [1973] presents two explanations of the impact of different turnover rates by individuals on thei r respective wage offers. He suggests that workers with a low quit rate command high wages because thei r fixed costs of employment may be amortised over a longer period of time. On the other hand, individuals with a low marginal quit rate w i l l be discriminated against i n wage offer because they have fewer alternative job opportunities. The p a r t i a l equilibrium framework he adopts i s inappropriate to model discriminating firm behaviour whence his arguments cannot be evaluated. These explanations are, however, the ad hoc arguments presented by economists to explain wage d i f f e r e n t i a l s generated by d i f f e r e n t i a l rates of turnover by members of the labour force. I t i s alleged, for example, by Women's Bureau [1969], that women factory operatives have a higher rate of turnover than t h e i r male counterparts. Sanborn [1964] attempts to analyze the factors causing the observed wage d i f f e r e n t i a l s between men and women. He argues, l i k e Salop, that,since men have a longer period of job tenure, the i r fixed costs of employment are amortised over a longer period of time. Consequently, i n the absence of discrimination, through prejudice or ignorance, men should earn more than women, i f firms behave r a t i o n a l l y and discriminate. Unfortunately, Sanborn does not present a formal model to support his conclusions. 39 Demsetz [1965] argues that some laws designed to a i d m i n o r i t i e s who s u f f e r from d i s c r i m i n a t i o n have the opposite e f f e c t . The economic welfare of these i n d i v i d u a l s i s reduced. He c o n s i d e r s a world i n which employers have a systematic preference f o r h i r i n g from a p a r t i c u l a r group of i n d i v i d u a l s . A n o n - p r e f e r r e d job a p p l i c a n t earns a lower wage than i s r e c e i v e d by h i s p r e f e r r e d , but e q u a l l y p r o d u c t i v e , f e l l o w worker. The employer r e c e i v e s wealth compensation f o r employing a n o n - p r e f e r r e d i n d i v i d u a l . I f a minimum wage law i s imposed, then the employer cannot h i r e a n o n - p r e f e r r e d worker at l e s s than the minimum wage. He i s p r o h i b i t e d from r e c e i v i n g wealth compensation from a n o n -p r e f e r r e d i n d i v i d u a l . Consequently, he w i l l choose between a p p l i c a n t s on the b a s i s of t h e i r p e r s o n a l c h a r a c t e r i s t i c s . N o n - p r e f e r r e d workers w i l l be f o r c e d to seek l e s s d e s i r a b l e p o s i t i o n s i n occupations not covered by the law, or w i l l become unemployed. Those n o n - p r e f e r r e d i n d i v i d u a l s who earn above minimum wages, but l e s s than t h e i r f e l l o w workers, do not b e n e f i t from the minimum wage laws. Firms continue to r e c e i v e wealth compensation f o r employing them. Demsetz then considers the f a i r employment laws enacted i n some s t a t e s i n the U . S . A . Since upholding the law i s expensive and dependent on s u b j e c t i v e e v a l u a t i o n of a f i r m ' s behaviour, he argues that the p e n a l t i e s to v i o l a t o r s w i l l be s m a l l . He suggests that the i m p o s i t i o n of such laws w i l l have l i t t l e impact on the employment p r a c t i c e s of f i r m s . He supports t h i s c o n c l u s i o n with evidence t h a t , over t ime, the r a t i o s of the unemployment rates of b l a c k and white 40 workers i n states i n which f a i r employment have been enacted do not s i g n i f i c a n t l y d i f f e r from the ratios of unemployment rates i n other states. In Model III, firms actively discriminate i n both wage offer and h i r i n g against type one individuals who are less valuable to them. In Model I I , firms are forced to make ex ante wage and vacancy creation decisions and make job offers randomly to searchers. Thus, the ' f a i r employment' laws do result i n non-discriminatory h i r i n g practices. Both types of individuals earn higher wages and incomes i n Model I I . The question arises as to whether the non-preferred type one individuals enjoy an increase i n thei r share of labour income under f a i r employment laws. C. Employer Discrimination based on Prejudice Most theoretical models of labour market discrimination are developed i n a s t a t i c u t i l i t y maximising framework (for example, Becker [1956]; Arrow [1972]; and Freeman [1973]). Black employees are discriminated against i n the wage offered because of the d i s u t i l i t y experienced by whites employing them or working with them. This section w i l l focus on models of employer discrimination. Becker [1956] argues that, i f an i n d i v i d u a l has a taste for discrimination, then he must be prepared to pay for i t . He states that discrimination arises through a lack of o b j e c t i v i t y i n the market place. He distinguishes between prejudice and ignorance. I f , for example, employers believe erroneously that negroes have a lower productivity than whites, then the i r behaviour would be d i s -41 criminatory but caused by ignorance. The spread of knowledge would eliminate this discrimination. I f employers discriminate, despite t h e i r knowledge that blacks are equally productive, then they are prejudiced. This preference i s independent of knowledge about the productivity of the workforce. By this d e f i n i t i o n , a more appropriate description of Models I I and I I I might be "Models of Inter and Intra-Firm Exploitation". Individuals are exploited.or discriminated against not through prejudice or ignorance on the part of firms, but because of the objective evidence that they d i f f e r systematically i n t h e i r market behaviour and can be recognised. Becker quantifies the tastes and preferences of discriminators by defining a discrimination c o e f f i c i e n t , d. A firm facing a money wage, I T , f or a factor of production behaves as i f the wage was Tf(l+d). He considers a labour force consisting of two groups, negroes, and whites, who are perfect substitutes i n production and receive wages T T ^ and ir^ respectively. The i 1 " * 1 employer has a fixed c o e f f i c i e n t of discrimination, d^, against individuals i n group N. Then, i f the market wage, T T ^ , i s less than ir^Cl+d^ , the i * " * 1 employer w i l l only h i r e group W workers. I f ir^. exceeds ir^Cl+d^), then only group N th workers are hired by the i employer. I f ir^ equals ir^(l+d^) , the til i employer i s i n d i f f e r e n t as to the composition of his labour force. The market discrimination c o e f f i c i e n t d s a t i s f i e s (7) 42 I f the d i s c r i m i n a t i o n c o e f f i c i e n t s vary over employers, then most firms w i l l be segregated and marginal f i r m s , whose d i s c r i m i n a t i o n c o e f f i c i e n t equals the market d i s c r i m i n a t i o n c o e f f i c i e n t , w i l l have mixed labour f o r c e s . The market d i s c r i m i n a t i o n c o e f f i c i e n t w i l l be r e l a t e d to the d i s t r i b u t i o n of d i s c r i m i n a t i o n c o e f f i c i e n t s and the s u p p l i e s of each f a c t o r . The demand curve f a c i n g the d i s c r i m i n a t e d group i s downward s l o p i n g as a d d i t i o n a l workers seek employment w i t h i n c r e a s i n g l y d i s c r i m i n a t i n g employers. I f the supply of group N workers i n c r e a s e s , then IT w i l l fal\. because d i s c r i m i n a t i n g firms have to be compensated f o r employing group N workers. A segregated white f i r m , f i r m i , say, w i l l now d i s c o v e r that the r e a l cost of h i r i n g group N workers, namely T T N ( l + d ^ ) , i s l e s s than the wage p a i d to a white worker. Consequently, the f i r m w i l l f i r e a l l i t s white workers and employ s o l e l y b l a c k workers. T h i s ad hoc approach i s an u n s a t i s f a c t o r y way to model any economic phenomena. An economic model of d i s c r i m i n a t i o n should attempt to e x p l a i n the existence of d i s c r i m i n a t i o n through the f o r m u l a t i o n of i n d i v i d u a l and f i r m behaviour. Becker has imposed the existence of wage d i s c r i m i n a t i o n through the s p e c i f i c a t i o n of the c L ' s which are not j u s t i f i e d by any r a t i o n a l behaviour on the part of labour market p a r t i c i p a n t s . H i s model i s p a r t i a l i n that the behaviour of firms and workers i s not f o r m a l l y s p e c i f i e d . In a d d i t i o n , the p r e d i c t i o n s of t h i s model are not compatible w i t h casual o b s e r v a t i o n . The labour force i s not segregated by f i r m and firms do not f i r e one workforce, say group i f workers : a n d ' h i r e a new group N workforce i n response to a change i n f a c t o r p r i c e s . " ^ 43 Arrow [1972] formally models discrimination i n the labour market,, but he too adopts ah ad hoc approach. In adopting this s p e c i f i c a t i o n , however, he demonstrates the i n t e r n a l contradictions of the model. He argues that the i n d i v i d u a l employer's discrimination c o e f f i c i e n t depends on the number of black and white employees, rather than being fixed and independent of the number of black and white employees. He assumes that firms face the same production functions, produce one commodity and hir e black and white labour who are perfect substitutes i n production and i n e l a s t i c a l l y supplied. The analysis i s short run and labour i s the only variable factor of production. I f W and N are the amounts of white and black labour hired by the firm, then output i s given by f(W + N). P r o f i t s of the firm are n = f (W + N) - WTTW - W N . . . . (8) W N where and are the wages of white and black workers respectively. The representative firm wishes to maximise a u t i l i t y function whose arguments are p r o f i t s and the degree of association with black and white workers. U(jl;, W, N) denotes the u t i l i t y function and % > 0 UW > 0 U N < 0 . . . . . (9) Since firms are i d e n t i c a l , each w i l l choose the same number of W and N i n equilibrium. Solving the f i r s t order conditions for a maximum yields 44 V f ' " V + U W = 0 • • • • <10> V f ' - V + U N = 0 • • • ' < U > and so f 1 = Ww + ^ = WN + . . . . (12) where ^ = " v — ( 1 3 ) U N djj = - f j - . . . . . (14) •it} d^ and d^ are B e c k e r ' s d i s c r i m i n a t i o n c o e f f i c i e n t s . P r o d u c t i o n i s e f f i c i e n t because firms h i r e an equal number of workers. I f the assumption of i d e n t i c a l p r o d u c t i o n functions i s r e l a x e d , then a c o n d i t i o n f o r e f f i c i e n c y i s t h a t cL^  and d^ are the same for a l l firms who employ a mixed labour force i n e q u i l i b r i u m . A s u f f i c i e n t c o n d i t i o n i s that d^ and dL^  are constant and independent of the arguments of the u t i l i t y f u n c t i o n . T h i s i s e q u i v a l e n t to s p e c i f y i n g a l i n e a r u t i l i t y f u n c t i o n , w h i c h i s p r e c i s e l y the model developed by Becker. D i s c r i m i n a t i o n by white employers thus leads to a pecuniary r e -d i s t r i b u t i o n from b l a c k workers to the white community of workers and d i s c r i m i n a t i n g employers. White workers earn an a d d i t i o n a l - d ^ per man, while d i s c r i m i n a t i n g employers face a change i n p r o f i t s of N*d^ + W*d^,where N * and W* denote the number of b l a c k and white workers employed by the f i r m i n e q u i l i b r i u m . The e x p r e s s i o n f o r the 45 change i n p r o f i t s has indeterminate sign. Equilibrium i n this model then i s characterised by mixed labour forces. In a response to a change i n factor prices,firms change the r e l a t i v e composition of their workforce. I t i s assumed that, i n the long run, firms have access to perfect c a p i t a l markets. For a given labour force, the firm borrows optimally at the long run rate of interest to purchase c a p i t a l . I f the production function displays constant returns to c a p i t a l and labour, then the ca p i t a l labour r a t i o i s a unique function of the rate of inter e s t . Let the production function, f-(W + N), represent output after the optimal acquisition of c a p i t a l . Then, this derived function displays constant returns to labour. I f firms have different u t i l i t y function, then implausible conclusions emerge. I t i s no longer necessary that firms employ both types of labour. Equilibrium conditions become f ' < ww + *w i f w = 0 f ' = ww + ^ i f w > °* • • • ( 1 5 a ) f• < WN + djj i f N = 0 f* = Wjj+djj i f N>0, . ._. (15b) Since equilibrium implies f u l l employment, these equalities hold for at least one firm. Hence, a l l whites are employed i n firms i n which d*^  i s the algebraic minimum and blacks are employed i n firms i n which i f i s a minimum. I f some firms do not discriminate against blacks, then d^ j = 0. Therefore from (15b),for a l l firms who employ blacks, d^ = 0. Then, there i s a black and white wage d i f f e r e n t i a l , i f < 0. 46 But, i t i s reasonable to postulate that any preference a firm has for h i r i n g whites arises as a resu l t of the presence of d i s l i k e d blacks. Furthermore, i t can be argued that, i f a firm does not discriminate against black workers, i t does not pay whites any more. Then, N = 0 or cL = 0 implies ci^ = 0. Since cL = 0 or N = 0 for a l l firms, d^ = 0 for a l l firms. Thus, f = Ww = WN . . . . (16) and there i s no discrimination. A more concise but si m i l a r argument for no discrimination i n the long run i s that, given constant returns to labour, the non-discriminating but more pr o f i t a b l e firms drive out discriminating firms, i f c a p i t a l markets are perfect. Freeman [1973] argues that, i f owners of firms who discriminate are prepared to accept a lower return, then discrimination p e r s i s t s . This i s c l a s s i f i e d as a c a p i t a l market imperfection. In a s i m i l a r framework, i t i s possible to model firm behaviour given white employees' discrimination against fellow black workers. White workers demand a wage related to the degree of association with black workers. I f whites and blacks are substitutes i n production and employees do not discriminate, then equilibrium i s characterised by segregation and no wage discrimination. Employers minimise the cost of h i r i n g white workers by segregating the labour force. In equilibrium they are i n d i f f e r e n t between h i r i n g a black or a white •labour force. Thus, there are no wage d i f f e r e n t i a l s . I f black and 47 white workers are complementary i n production and white employers or workers discriminate, then wage d i f f e r e n t i a l s p e r s i s t . The existence of wage d i f f e r e n t i a l s provides a return to non-discriminating behaviour. The higher p r o f i t s or incomes earned by non-discriminating employers or employees constitute an incentive to increase their respective supplies over time. In p a r t i c u l a r , the supply of black employers and employees w i l l increase. I f the supply of non-discriminating employers and employees increases,then wage d i f f e r e n t i a l s w i l l disappear i n the long run. Freeman argues that the problem of. explaining economic discrimination i n the long run translates into the problem of .explaining l i m i t a t i o n s on the supply of non-discriminating (black) employers or limit a t i o n s on th e i r market behaviour. He argues that there are three ways i n which governmental discrimination and p o l i c i e s can influence the r e l a t i v e position of blacks i n the labour market, namely : 1. By affecting black human c a p i t a l formation through discrimination i n public education; 2. By discrimination for and against blacks i n public employment, and, less d i r e c t l y , i n employment by public contractors, and 3. By l e g a l regulations of discriminatory practices i n the market and applications of laws pertaining to employment and income."^ 48 D. Discrimination based on Imperfect Information In the models formulated by Becker and Arrow and Freeman, i t i s assumed that black and white workers are perfect substitutes i n pro-duction and that firms have perfect information about prices, marginal products and other relevant variables. Such information i s not actually possessed by employers producing i n an environment characterised by uncertainty and costs of search.. I f i t i s believed that the product-i v i t y of blacks and whites d i f f e r , but not i n a systematic fashion, then an assessment of an individual's productivity may be expensive. McCall argues that firms w i l l attempt to u t i l i z e r e l a t i v e l y costless information devices such as age, sex and race i n the h i r i n g decision. This screening by employers leads to discrimination i n h i r i n g . In Models I I and I I I , by contrast, firms know that individuals d i f f e r i n their labour market behaviour i n a systematic fashion. I n t r a -firm wage and h i r i n g discrimination i s practised,if the firm recognises different individuals p r i o r to h i r i n g and i f they are allowed to discriminate. In McCall's model,the firm wishes to minimise the cost of h i r i n g one successful employee, that i s an employee with a marginal product i n excess of some prescribed m*. Using his notation, i f the expected costs of finding and evaluating a white and a black worker are c^ and C 2 , respectively, and i f p^, p 2 respectively denote the expected probability of h i r i n g a suitable candidate, then the firm chooses i to minimise _ i i = 1 , 2 . The expected number of candidates of type i a firm ^ i must interview i s 1/p.. and each interview costs c_.. The firm revises 49 i t s estimates of and i n Bayesian f a s h i o n by i n c o r p o r a t i n g the i n f o r m a t i o n gained by employing d i f f e r e n t i n d i v i d u a l s . At any p o i n t i n t i m e , t h e n , t h e f i r m makes a [ 0 , 1 ] • d e c i s i o n . M c C a l l argues t h a t , as the labour market t i g h t e n s , the average p r o d u c t i v i t y o f the white unemployed p o o l w i l l f a l l , s i n c e the p o o l w i l l c o n t a i n an i n c r e a s i n g number of i n d i v i d u a l s who have f a i l e d to be employed i n other firms or have been f i f e d . At some p o i n t , experiments w i t h non-white employees w i l l b e g i n as p r i o r assessments are r e v i s e d . I f the experience w i t h non-white employees i s s u c c e s s f u l , then the use o f c o l o u r as a s c r e e n i n g device may be d i s c o n t i n u e d . T h i s model, though i n t e r e s t i n g , i s p a r t i a l i n that the d e t e r m i n -ants of c^ and are not formulated. The r e s p e c t i v e costs of e v a l u a t i n g i n d i v i d u a l s are random v a r i a b l e s and are determined by the i n t e r a c t i o n of f i r m and i n d i v i d u a l search behaviour. These costs w i l l depend on the r e l a t i v e s u p p l i e s of unemployed s e a r c h e r s . In the absence o f the f u l l s p e c i f i c a t i o n of the l a b o u r market, i t i s not c l e a r how t h i s d i s c r i m i n a t i o n i n h i r i n g i n f l u e n c e s the wages of white and b l a c k workers. M c C a l l models s e p a r a t e l y the search behaviour of i n d i v i d u a l s . Again u s i n g h i s n o t a t i o n ^ c denotes the cost per p e r i o d of s e a r c h , x i s a random v a r i a b l e denoting the wage o f f e r and <f>(x) i s the p r o b -a b i l i t y density f u n c t i o n of o f f e r s . Then, the r e s e r v a t i o n or acceptance wage e, f o r an unemployed i n d i v i d u a l s , s a t i s f i e s the equation oo c = / (x-e)<Kx)dx. = H(e) . . . . (17) e where H ' ( e ) < 0 . 50. L e t e^ denote the r e t u r n s from remaining unemployed and set C Q = HCeg). I f the cost of s e a r c h , c , exceeds e^, then not s e a r c h i n g at a l l i s the best s t r a t e g y . The value of e s a t i s f y i n g (17) f o r c > C Q i s l e s s than e^. Consequently, the i n d i v i d u a l i s prepared to accept an o f f e r l e s s than the r e t u r n s from not s e a r c h i n g . Thus, the d e c i s i o n r u l e f o r the i n d i v i d u a l i s to drop out of the labour f o r c e , i f c > C Q , o t h e r w i s e , remain f r i c t i o n a l l y unemployed, u n t i l an o f f e r , x > e * , i s r e c e i v e d where c = H(e*) . . . . (18) 18 M c C a l l a s s e r t s that costs of search are h i g h e r f o r b l a c k s and the wage o f f e r d i s t r i b u t i o n f o r b l a c k s i s i n f e r i o r to that of whites because of d i s c r i m i n a t i o n . He s t a t e s that these f a c t o r s 19 account f o r the d i s p r o p o r t i o n a t e number of non-white dropouts. In the l o n g r u n , i f there are no systematic d i f f e r e n c e s i n the p r o d u c t i v i t y of white and b l a c k workers,then screening w i l l cease and wage d i f f e r e n t i a l s w i l l not p e r s i s t . C l e a r l y , i f whites and b l a c k s do d i f f e r i n t h e i r p r o d u c t i v i t y , then a long run model of t h i s k i n d resembles a model of f i r m behaviour i n which there are two f a c t o r s of p r o d u c t i o n . In the absence of simultaneous m o d e l l i n g of f i r m and i n d i v i d u a l search b e h a v i o u r , i t i s i m p o s s i b l e to d i s c u s s market e q u i l i b r i u m . I t i s evident though that the wage d i f f e r e n t i a l observed i s r e l a t e d to the amount that firms on average b e l i e v e that p r o d u c t i v i t i e s d i f f e r between b l a c k and white workers. 51. E. Conclusion In conclusion, most models of discrimination are based on a u t i l i t y theory approach. This approach i s unsatisfactory because the existence of any economic phenomenon can be j u s t i f i e d by the spe c i f i c a t i o n of a u t i l i t y function. I t does provide a framework, however, i n which simple models can be formulated and predictions compared to observation. Arrow underlines the inconsistencies of the models. In p a r t i c u l a r , given perfect c a p i t a l markets and unlimited supplies of non-discriminating individuals, wage d i f f e r e n t i a l s disappear i n the long run. These aire models of discrimination through prejudice. McCall 1s model i s a model of discrimination generated by ignorance. In the long run, i n the absence of exogeneous changes i n the labour market, wage d i f f e r e n t i a l s w i l l r e f l e c t true differences i n productivity. VI. The Thesis In my thesis, I wish to answer the questions posed i n I I I , by specifying and solving the three models of an imperfect labour market. A simple model i s developed and the l i t e r a t u r e i s reviewed i n this chapter and the conceptual framework adopted i s outlined i n Chapter 2 1. The complexity of the model precludes analytic solution, so i n order 20 to perform the desired comparative s t a t i c experiments, I solve the model numerically using the algorithm described i n Chapter 2 IIB. The results from Model I are presented i n Chapter 4. 52. An i n t e r e s t i n g t h e o r e t i c a l extension i s the development of models of d i s c r i m i n a t i o n i n which there are two types of l a b o u r , with the same p e r c e i v e d s k i l l s but d i f f e r i n g i n t h e i r turnover and acceptance behaviour through d i f f e r i n g p e r c e p t i o n s of market parameters, a t t i t u d e s to r i s k , r a t e of time preference or length of h o r i z o n . C r u c i a l to the f i r m i s whether i t can i d e n t i f y each type of labour through some overt c h a r a c t e r i s t i c s ( e . g . young and o l d workers d i f f e r i n g i n t h e i r l e n g t h of working h o r i z o n ) , and, f u r t h e r , whether i t can d i s c r i m i n a t e i n wage o f f e r or i n vacancy c r e a t i o n . Models II and I I I are examined i n Chapter 2 and Appendix I and r e s u l t s are presented i n Chapter 5. Comparative s t a t i c p r e d i c t i o n s are derived from these two models. 53 CHAPTER 2 THE CONCEPTUAL FRAMEWORK, I. Individual Turnover and Acceptance Behaviour A. A Summary This simultaneous model of an imperfect labour market was formulated by Curtis Eaton and myself. I t provides the conceptual framework for the analysis of other structures examined i n my thesis. Modelled simultaneously i s the search behaviour of workers and the profitJmaximising behaviour of firms i n a labour market character-ised by one imperfection, the absence of complete information. This simultaneity complicates the development and analysis of the model. Individuals, faced with imperfect information, are forced to search. Their search behaviour i s determined by the perceived wage and vacancy creation behaviour of firms. Wage and vacancy decisions of firms are determined by th e i r notions of individuals' search behaviour. Firms make wage and vacancy decisions to maximise expected p r o f i t s discounted over an i n f i n i t e horizon,while workers' search behaviour i s motivated by the maximisation of expected discounted income over the job horizon. The labour force i s constant and homogeneous i n i t s capacity to do the job. This general assumption ensures that any wage dispersion, characterising equilibrium, i s generated by imperfect information on the part of market participants, rather than i n d i v i d u a l differences i n productivity. 54 Each i n d i v i d u a l i n the labour force has general i n f o r m a t i o n about the labour market environment, such as the mean and v a r i a n c e of the p r e v a i l i n g wage d i s t r i b u t i o n , but must generate s p e c i f i c i n f o r m a t i o n or job o f f e r s through random s e a r c h . He must be unemployed to s e a r c h . When unemployed, each i n d i v i d u a l r e c e i v e s unemployment compensation. No searcher accepts an o f f e r l e s s than the l e v e l of unemployment compensation which i s a c e r t a i n r e t u r n . L i k e w i s e , each employed i n d i v i d u a l q u i t s and searches f o r a more remunerative job o f f e r , i f o f f e r e d a wage rate l e s s than the l e v e l of unemployment compensation. In order to secure some employees, then, a l l f irms o f f e r a wage at l e a s t as h i g h as the l e v e l of unemployment compensation. Thus, there 1 are e x p l i c i t costs of s e a r c h . Each i n d i v i d u a l , searcher samples one f i r m per d e c i s i o n p e r i o d . On the b a s i s of h i s general labour market i n f o r m a t i o n , he computes an estimate of h i s r e s e r v a t i o n wage. The r e s e r v a t i o n wage i s that wage, c u r r e n t l y o f f e r e d , which makes the expected net r e t u r n s from f u r t h e r search z e r o . I f he r e c e i v e d t h i s o f f e r from the f i r m most r e c e n t l y sampled, the searcher would be i n d i f f e r e n t between c o n t i n u i n g to search and a c c e p t i n g the o f f e r . I f a wage l e s s than the r e s e r v a t i o n wage was o f f e r e d , he would continue s e a r c h i n g . Otherwise, the unemployed 2 searcher accepts the job o f f e r . L i k e w i s e , each i n d i v i d u a l , who i s c u r r e n t l y employed, evaluates the expected returns and costs a s s o c i a t e d with q u i t t i n g and s e a r c h i n g f o r a new j o b , next p e r i o d . He computes h i s r e s e r v a t i o n wage. I f the 55 wage o f f e r from h i s f i r m for next p e r i o d i s l e s s than h i s r e s e r v a t i o n wage, then he q u i t s and searches for another j o b . Otherwise, he remains employed. I t w i l l be demonstrated i n the next s e c t i o n t h a t , i f each i n d i v i d u a l i n the labour f o r c e , i r r e s p e c t i v e of employment s t a t u s , has the same p e r c e p t i o n of the d i s t r i b u t i o n of wage o f f e r s , a t t i t u d e s to r i s k , r a t e of discount and time h o r i z o n over which he b e l i e v e s wage d i f f e r e n t i a l s p e r s i s t , then each i n d i v i d u a l has the same r e s e r v a t i o n wage. To c h a r a c t e r i s e incomplete s p e c i f i c i n f o r m a t i o n i n the labour market, each i n d i v i d u a l ' s p e r c e p t i o n of the r e s e r v a t i o n wage i s assumed to be subject to a p r o b a b i l i t y d i s t r i b u t i o n whose parameters are functions of the true r e s e r v a t i o n wage and true v a r i a n c e o f wage o f f e r s i n the i n d u s t r y . A l l i n d i v i d u a l s , i r r e s p e c t i v e of labour market experience, have the same p r o b a b i l i t y d i s t r i b u t i o n of the r e s e r v a t i o n wage. The p e r c e i v e d r e s e r v a t i o n wage f o r an i n d i v i d u a l can be regarded as a random drawing from t h i s d i s t r i b u t i o n . Turnover and acceptance d e c i s i o n s are made by comparing the wage o f f e r w i t h the p e r c e i v e d r e s e r v a t i o n wage. Consequently, turnover and acceptance d e c i s i o n s are s t o c h a s t i c from the f i r m ' s p o i n t of view. C e t e r i s p a r i b u s , each i n d i v i d u a l has a h i g h e r p r o b a b i l i t y , ex ante, of a c c e p t i n g a job o f f e r , i f unemployed, or not q u i t t i n g a p o s i t i o n , i f employed, the h i g h e r the wage o f f e r . The i n d u s t r y has a f i x e d number of t e c h n i c a l l y i d e n t i c a l firms who h i r e from a f i x e d labour f o r c e . Each f i r m faces the same exogeneous product demand. I t can s e l l any l e v e l of output at the p r e v a i l i n g i n d u s t r y p r i c e . T h i s assumption i s g e n e r a l , i n the sense that i t cuts 56 the l i n k between i n d u s t r y output and p r i c e , but i t al lows me to i s o l a t e and focus on the features of an imperfect labour market, r a t h e r than be forced to d i s e n t a n g l e the i n f l u e n c e s of the product market. The f i r m ' s marginal revenue product and the wage o f f e r are both expressed as r a t e s per p e r i o d of time equal to the f i r m ' s d e c i s i o n p e r i o d which i s c o i n c i d e n t and equal to the i n d i v i d u a l ' s d e c i s i o n p e r i o d . At the beginning of each p e r i o d , t h e f i r m makes i t s wage and vacancy d e c i s i o n s f o r the succeeding p e r i o d . At the end o f the p e r i o d , a l l q u i t s and new h i r e s occur. In view of the s t o c h a s t i c flow of searchers and s t o c h a s t i c turnover and acceptance generated through misperception of the r e s e r v a t i o n wage, vacancy c r e a t i o n i s n e c e s s a r i l y s p e c u l a t i v e . The f i r m makes at most as many o f f e r s as vacancies i t has c r e a t e d , i f s u f f i c i e n t searchers show up. I t must employ a l l those i n d i v i d u a l s who accept an o f f e r . Through i n s u f f i c i e n t searchers sampling the f i r m or searchers r e f u s i n g job o f f e r s , the f i r m may f a i l to f i l l a l l the p o s i t i o n s i t has c r e a t e d . . For any p a r t i c u l a r l e v e l of employment, the f i r m ' s vacancy d e c i s i o n r e f l e c t s the d e s i r e d number of net h i r e s , the number of i n d i v i d -uals expected to q u i t and the expected p r o p o r t i o n of those searchers o f f e r e d a p o s i t i o n who accept. C e t e r i s p a r i b u s , a f i r m ' s optimal d e c i s i o n depends on i t s c u r r e n t l e v e l of employment. For any wage and vacancy d e c i s i o n , the l e v e l of employment a t t a i n e d i s s t o c h a s t i c . C o n s i d e r i n g l e v e l s of employment faces by firms as s t a t e s , then the e v o l u t i o n of the labour market i s a n o n - s t a t i o n a r y Markov p r o c e s s . On r e a c h i n g s t o c h a s t i c e q u i l i b r i u m , the labour market behaves as a s t a t i o n a r y Markov p r o c e s s . E q u i l i b r i u m i s c h a r a c t e r i s e d by wage d i s p e r s i o n , vacancies and f r i c t i o n a l unemployment. 57 I t i s now appropriate to analyse formally i n d i v i d u a l and f i r m behaviour. The simultaneity of the model complicates t h i s . Each set of market p a r t i c i p a n t s makes decisions i n response to t h e i r perceptions of the other set of p a r t i c i p a n t s ' behaviour and the general labour market environment. I t appears to be easier to model i n d i v i d u a l behaviour f i r s t rather than fi r m behaviour. B. I n d i v i d u a l Turnover and Acceptance Behaviour i n Model I In t h i s s e c t i o n , i n d i v i d u a l s ' turnover and acceptance behaviour i n response to a 'labour market environment', x, i s modelled. This environment may be characterised by the d i s t r i b u t i o n of firms over employment states, 3v(n = 0,1,...,n),and a corresponding set of decisions, 6 (n = 0,1,...,n), where 6 = (w , v ),(n = 0,1 n ) . ' n n n n Then $ n denotes the actual proportion of firms currently employing n i n d i v i d u a l s . Each of these firms makes a wage o f f e r , w , and creates v vacancies. fL i s the maximum possible l e v e l of employment n that the fi r m chooses. Then, x = (3,6) = ( 3 0 > 3 ,...,B ,... , B f l , 6 0 , 6 1 , . . . , 6 n , . . . 6 f t ) . . . . . (1) There are N firms and L i n d i v i d u a l s i n the labour force. I f U i s the number of unemployed workers, then, _ _ n . U = L - E = L - N E ' n.g. . . . . (2) n n n=0 > where E i s the l e v e l of employment. 58 Individuals have general information about the labour market environment, such as the mean and variance of the wage offer d i s -t r i b u t i o n , which they generate through t h e i r perceptions of the number of searchers, the number of firms and the pr o b a b i l i t y d i s -t r i b u t i o n of wage rates and job vacancies over firms. To generate s p e c i f i c information or job offers, however, individuals search randomly. F i r s t , i t i s necessary to compute the true mean and variance of the wage offer d i s t r i b u t i o n facing the searcher. There are U searchers each making one contact during the period. Then, each searcher contacts a p a r t i c u l a r firm with p r o b a b i l i t y , 1/N, during the period. To exclude the searcher's previous firm of employment or the firm he searched previously, greatly complicates the analysis without substantially changing the conclusions, given that N i s large. The firm creates v n vacancies and at the end of the period the firm makes offers to at most v of the searchers who have contacted i t . n I f less than v searchers have contacted the firm, a l l the searchers n receive offers. Then, the probability that a searcher, having contacted a p a r t i c u l a r firm with n employees, w i l l obtain an offer i s Y„ = V (V) 1/Nj (1-1/N) U" 1 _ ; i min fc£=-, 1). . . . . (3) j=0 2 3 59 The expression i n the f i r s t set of parentheses indicates the probability that j other searchers contact the firm. The second expression denotes the probability that the searcher receives an off e r , given that j other individuals sample the firm. I f j+1, the t o t a l number of searchers sampling the firm, exceeds v , the number of vacancies created, then the p a r t i c u l a r i n d i v i d u a l obtains an v offer with probability „ Qtherwise }he obtains an offer with certainty. The probability of contacting a firm with n employees i n one search i s simply B n - Hence, the probability of receiving an offer from a firm with n employees i n one search i s 3 n Y n ' I f a firm with n employees offers a wage, w n > then the pr o b a b i l i t y of obtaining a wage offer wn, namely <jT \, i s 3 n Y n - The searcher w i l l get no offer from a firm with n employees with p r o b a b i l i t y , 8 (1-y )• No offer i s denoted as a zero offer. Then the probability of a zero offer i s _ n <f> = 1 - E 8 Y . . . . . (4) „ n n n=0 Let EW* denote the mean offer obtained i n one search. As noted above, an offer of less than the l e v e l of unemployment compensation, w, i s unacceptable. Then, n EW* = Z W <i>w + wd> . . . . . (5) n=0 n 60 I f VW denotes the v a r i a n c e of o f f e r s , t h e n , n 0 - 2 -VW* = Z (w -EW*) 4> + (w-EW*) <|> . . . . . (6) n=0 n Let W-j. denote the wage o f f e r the employed i n d i v i d u a l w i l l r e c e i v e from h i s f i r m next p e r i o d . Then, ,the expected gross r e t u r n s , R, from the d e c i s i o n to q u i t and commence s e a r c h i n g next p e r i o d are H . R = S ( . ^ ~ ) 1 (Etf^Wj) . . . . (7) i = l where D i s the i n d i v i d u a l ' s own r a t e of d i s c o u n t and i s the h o r i z o n over which the i n d i v i d u a l expects to enjoy the wage d i f f e r e n t i a l . The i n d i v i d u a l assumes that the d i s t r i b u t i o n of o f f e r s and, i n p a r t i c u l a r , h i s o f f e r , W .^, w i l l p e r s i s t over the h o r i z o n H ^ . How much i n f o r m a t i o n about future wage rates contained- i n a c u r r e n t wage o f f e r can be measured by the temporal c o r r e l a t i o n of wage o f f e r s , defined i n H I E . The costs of search f o r one p e r i o d are given by the d i f f e r e n c e between h i s wage o f f e r and the l e v e l o f unemployment compensation, namely W^- w. H i s d e c i s i o n to q u i t i s based on the comparison of expected r e t u r n s and costs from one s e a r c h . The i n d i v i d u a l w i l l q u i t i f R > (W I -w). . . . . (8) The s e a r c h e r ' s d e c i s i o n to accept an o f f e r , or r e j e c t i t and continue s e a r c h i n g , i s analogous. During the c u r r e n t p e r i o d the i n d i v i d u a l samples a f i r m and r e c e i v e s a wage o f f e r , W y , which may be 61 zero. Then, the i n d i v i d u a l w i l l choose to remain unemployed and search further, i f the inequality i n (8) i s s a t i s f i e d . There exists some -value of W , say W*, such that (8) becomes an equality. W* i s the true reservation wage. The reservation wage i s that wage offer which makes an in d i v i d u a l i n d i f f e r e n t between accepting the offer and either indulging in:further search., i f currently unemployed, 3 or q u i t t i n g , i f currently employed. The relationship represents the decision rule of an ind i v i d u a l who has a perfect perception of the prob a b i l i t y d i s t r i b u t i o n of offers facing him and i s attempting to maximise the present value of expected future income. This decision rule has the same general form as the one period sequential decision rule developed i n Chapter 1 IVB for search i n product markets. The rule i s based on the. choice between acceptance of the current o f f e r , and a further search, and acceptance of the corresponding offer. An individual's actual turnover and acceptance behaviour, however, i s based on the comparison of the reservation wage determined by this rule and his current wage offer. Since the horizon over which the indi v i d u a l expects to enjoy the wage d i f f e r e n t i a l , which determines the returns from a further search, i s assumed constant, the reservation wage i s constant and independent of the individual's actual duration of search. Such, a search rule i s not optimal, however, because the mean duration 4 of search associated with, the reservation wage i s not unity. Thus, the individual's actual labour market behaviour i s not consistent with the decision r u l e . 62 T e l s e r [1973] i n d i c a t e s , however, that the gains from search i n product markets a s s o c i a t e d w i t h such a one p e r i o d s e q u e n t i a l r u l e over the gains to a naive search r u l e are s m a l l . ^ Then, i t i s p l a u s i b l e that the gains a s s o c i a t e d w i t h the optimal search r u l e over t h i s one p e r i o d r u l e , d e r i v e d i n the t e x t , would be s m a l l . Consequently, i n the i n t e r e s t s of computational economy, the one p e r i o d search r u l e i s used i n the s i m u l a t i o n procedure. I f a l l i n d i v i d u a l s were i d e n t i c a l w i t h r e s p e c t to t h e i r r a t e of d i s c o u n t , time h o r i z o n , behaviour towards r i s k and each p e r c e i v e d the true d i s t r i b u t i o n of o f f e r s i n the labour market, then any f i r m o f f e r i n g a wage r a t e l e s s than the true r e s e r v a t i o n wage, W*, would l o s e a l l employees and would be unable to a t t r a c t any new employees. T h i s i m p l i e s that no f i r m o f f e r s a wage l e s s than W* and, l i k e w i s e , no f i r m o f f e r s a wage h i g h e r than W*. Thus, there i s no d i s p e r s i o n and no s e a r c h . Indeed, i f i n d i v i d u a l s are i d e n t i c a l l y motivated but do not p e r c e i v e the true d i s t r i b u t i o n of o f f e r s , then e q u i l i b r i u m i s s t i l l c h a r a c t e r i s e d by no d i s p e r s i o n and no s e a r c h . I f each i n d i v i d u a l computes the r e s e r v a t i o n wage as W*, not equal to W*, then a l l firms w i l l o f f e r W*. Wage d i s p e r s i o n , then, i s not a property of models i n which i n d i v i d u a l s are i d e n t i c a l l y motivated and p e r c e i v e the same, but not n e c e s s a r i l y t r u e , d i s t r i b u t i o n of wage o f f e r s , even though there are s u b s t a n t i a l costs of s e a r c h . R o t h s c h i l d [1973], i n d i s c u s s i n g 63 S t i g l e r ' s c o n t r i b u t i o n , makes the same p o i n t . Costs of s e a r c h , per  s e , are i n s u f f i c i e n t to generate p r i c e or wage d i s p e r s i o n . Some heterogeneity of market p a r t i c i p a n t s i s r e q u i r e d . In the absence of exhaustive search each p e r i o d and due to d i f f e r e n t labour market e x p e r i e n c e s , an i n d i v i d u a l ' s p e r c e p t i o n s of the labour market environment may not c o i n c i d e w i t h the r e a l i t y of the labour market. Consequently, he may i n c o r r e c t l y compute the true r e s e r v a t i o n wage. Let wc represent h i s p e r c e p t i o n of the r e s e r v a t i o n wage. Then, wc i s the minimum wage o f f e r which would induce him to remain at h i s c u r r e n t job or accept a p o s i t i o n , i f unemployed. The phenomena of misperceptions of the labour market environment are modelled most e a s i l y by assuming t h a t each i n d i v i d u a l ' s p e r c e p t i o n of h i s r e s e r v a t i o n wage i s drawn from some p r o b a b i l i t y d e n s i t y f u n c t i o n , p(wc), defined f o r non-negative wc. The s p e c i f i c a t i o n of t h i s density f u n c t i o n allows the computation of the p r o b a b i l i t y of q u i t t i n g , given a wage, o f f e r , w. oo t(w) = / p(wc)dwc w t '(w) < 0 t"(w) > 0 where t(w) i s the p r o b a b i l i t y of t u r n o v e r . (9) 64 I f the i n d i v i d u a l receives an o f f e r , w, then, he w i l l q u i t , i f he perceives the reservation wage, wc, as exceeding w. This i s given by the cumulative density function i n (9). Analogously, the p r o b a b i l i t y , a(w), that a firm's wage o f f e r , w, i s accepted by a searcher i s given by The s p e c i f i c a t i o n of misperceptions, formulated i n (9) and (10) i s p l a u s i b l e because i n d i v i d u a l s ' knowledge of the labour market i s generated through random search and o f f e r c o l l e c t i o n . Thus, t h e i r perception of labour market parameters w i l l show stochastic v a r i a t i o n . Their perceptions, based on incomplete s p e c i f i c information, are rel a t e d to the true labour market environment. Then, the mean of the density function should be r e l a t e d to the true reservation wage and the variance of the density function should be r e l a t e d to the variance of the wage of f e r d i s t r i b u t i o n . Let M, VP represent the mean and variance, r e s p e c t i v e l y , of the density function of i n d i v i d u a l perceptions of the reservation wage, then these r e l a t i o n s may be written a(w) = p(wc)dwc = l-t(w) a'(w) > 0 a"(w) < 0. . . . . (10) 6 M = c,W* ( I D VP = c2VW* (12) 65 where W* i s the true r e s e r v a t i o n wage, W * i s the t r u e v a r i a n c e of o f f e r s and c^, are constants. I f c^ = 1, then the i n d i v i d u a l has an unbiased estimate of the true r e s e r v a t i o n wage. I f c^ < 1, then h i s estimate of the r e s e r v a t i o n wage i s b i a s e d downward. A l l i n d i v i d u a l s have the same d e n s i t y f u n c t i o n of p e r c e p t i o n s , b u t , ex p o s t , through t h e i r s t o c h a s t i c m i s p e r c e p t i o n s , i n d i v i d u a l s may behave d i f f e r e n t l y i n response to a p a r t i c u l a r labour market environment. Since each i n d i v i d u a l ' s behaviour i s s t o c h a s t i c , i t does not d i f f e r s y s t e m a t i c a l l y over time from another i n d i v i d u a l ' s behaviour i n response to the same labour market parameters. The assumption of a f i x e d p e r i o d over which r e t u r n s from search accrue ensures that an i n d i v i d u a l ' s true r e s e r v a t i o n wage, given an unchanging labour market environment, i s constant and independent of h i s d u r a t i o n of search., i f unemployed, and previous o f f e r s , i f employed. Thus, the stock o f unemployed i n d i v i d u a l s i s homogeneous, ex ante, i n t h e i r acceptance b e h a v i o u r , although composed of i n d i v i d u a l s w i t h d i f f e r e n t h i s t o r i e s of unemployment. I f there i s no p e r c e i v e d wage d i s p e r s i o n , i . e . or VW* i s z e r o , then i n d i v i d u a l s accept or r e j e c t an o f f e r a c c o r d i n g to t h e i r p e r c e p t i o n of the r e s e r v a t i o n wage, c^W*. In t h i s case, each f i r m o f f e r s a wage equal t o the p e r c e i v e d r e s e r v a t i o n wage, c^w*. S i n c e the r e s e r v a t i o n wage i s always l e s s than the mean o f f e r , the r e s e r v a t i o n wage f a l l s . Next p e r i o d , each f i r m o f f e r s the new r e s e r v a t i o n wage and so on. In f u l l i n d u s t r y e q u i l i b r i u m , each f i r m o f f e r s a wage equal to the l e v e l of unemployment compensation. There i s no wage d i s p e r s i o n , no 66 search and thus zero unemployment. The d i s t r i b u t i o n of employment over firms i s i n d e t e r m i n a t e . The c r u c i a l element of t h i s model, which d i s t i n g u i s h e s i t from the p e r f e c t l y competit ive model, i s the absence of f u l l s p e c i f i c i n f o r m a t i o n about wage o f f e r s and vacancies on the p a r t of i n d i v i d u a l s . As a r e s u l t , no f i r m i s able to i n f l u e n c e i t s supply of searchers through i t s wage o f f e r and vacancy c r e a t i o n behaviour. I f i t o f f e r s a wage equal to or greater than the r e s e r v a t i o n wage, then a l l s e a r c h e r s , who randomly sample i t , accept the o f f e r . I f the f i r m o f f e r s a wage l e s s than the r e s e r v a t i o n wage, then no searchers w i l l accept a job o f f e r . In i n d u s t r y e q u i l i b r i u m , there i s no search and so no f i r m i s able to commun-i c a t e i t s excess demand f o r l a b o u r . The e q u i l i b r i u m corresponds to the c o l l u s i v e monopsony s o l u t i o n i n which firms c o l l u d e i n t h e i r wage o f f e r s to i n d i v i d u a l s . Firms are unable, however, to choose the j o i n t p r o f i t maximising l e v e l of employment because there i s no s e a r c h . I f a l l i n d i v i d u a l s b e l i e v e there i s no wage d i s p e r s i o n , then e q u i l i b r i u m i n the labour market i s c h a r a c t e r i s e d by zero wage d i s p e r s i o n . The assumptions of t h i s p a r t i c u l a r model are c o n s i s t e n t w i t h the second c o n d i t i o n for zero d i s p e r s i o n o u t l i n e d i n Chapter 1. Equations (9) and (10) show the turnover and acceptance r e l a t i o n s which c o n s t r a i n the f i r m ' s market behaviour. 67 C. Systematic Differences i i i I n d i v i d u a l Behaviour 1. States of Employment In Models I I and I I I , i n d i v i d u a l s d i f f e r systematically i n t h e i r labour market behaviour. Firms make wage and vacancy decisions based on both the l e v e l and composition of employment. Thus, possible states of employment facing the f i r m are enumerated both by number and type of employee i n these models. I t i s assumed that the labour force consists of two types of i n d i v i d u a l . Thus, i f the f i r m can experience l e v e l s of employment 0,1,...,fi, there are M possible employment states where I f the f i r m employs n i n d i v i d u a l s , then i t w i l l be i n any one of n+l d i s t i n c t employment states, depending on the number of each type of i n d i v i d u a l employed. I t i s necessary to impose a mapping which re l a t e s the number of each type of employee to the p a r t i c u l a r state under consideration. Let e ^ be the number of type i i n d i v i d u a l s i n employment state j , and l e t s(e^,e2) denote the state corresponding to e^ type one employees and e„ type two employees. Then, M = n+l E i i = l (n+2)(n+l) . 2 (13) j = sf.e^, e 2 J ) r. (14) 68 I t i s u s e f u l to define the concept of a 'state complement' at th i s point. For any state j ^ , i f 1 = e ^ l . . . . (15) m = . . . . (16) j 2 = s(m,l) . . . . (17) then and J 2 are referred to as 'state complements'. I f L^.U^ ( i = 1»2) denote, the number of type i i n d i v i d u a l s i n the labour force and the number unemployed, r e s p e c t i v e l y , then M U. = L.- N E g.e. J i = 1,2. . . . . (18) 1 1 j = l 2 1 2. Workers' Behaviour i n Model II Firms are not permitted (or are unable) to discriminate against the d i f f e r e n t types of workers i n wage o f f e r and i n h i r i n g i n Model I I . Each firm, however, makes a wage o f f e r and vacancy creation decision which i s dependent both on the s i z e and composition of i t s workforce. This wage o f f e r i s made both to e x i s t i n g employees and to p o t e n t i a l new employees. Thus, while wage d i f f e r e n t i a l s are observed between firms employing the same number but a d i f f e r e n t composition of i n d i v i d u a l s , each i n d i v i d u a l faces the same labour market environment because e x p l i c i t discrimination i s i l l e g a l . The labour market environment i n which i n d i v i d u a l s and firms make decisions may be written as 69 x = (6,6) = ( B 1 , B 2 , . . . , e m B M , 6 1 , 6 2 , . . . , 6 m , . . . , 5 M ) (19) I n d i v i d u a l s s y s t e m a t i c a l l y d i f f e r i n t h e i r labour market behaviour due to t h e i r s y s t e m a t i c a l l y d i f f e r e n t p e r c e p t i o n s o f the true mean o f f e r from s e a r c h , EW*, and thus the r e s e r v a t i o n wage. I n d i v i d u a l s have the same a t t i t u d e s to r i s k , r a t e s of time preference and h o r i z o n s over which they b e l i e v e wage d i f f e r e n t i a l s p e r s i s t . There i s no d i s c r i m i n a t i o n i n h i r i n g , whence a l l unemployed i n d i v i d u a l s have the same p r o b a b i l i t y of a job o f f e r and the same mean o f f e r . T h e r e f o r e , the true r e s e r v a t i o n wage, W*, i s the same for a l l i n d i v i d u a l s . I f and VP^ denote the mean and v a r i a n c e of the d i s t r i b u t i o n of p e r c e p t i o n s of the r e s e r v a t i o n wage f o r each type i i n d i v i d u a l , then M ± = M i ( c 1 1 E W * ) i = 1,2. . . . . (20) V P ± = ^^VW* i = 1,2. . . . . (21) 1 2 cl J cl . . . . (22) °2 = °2 = C 2 . . . . (23) where the argument, c ^ E W * ( i = 1,2) denotes the p e r c e i v e d mean o f f e r f a c i n g a type i i n d i v i d u a l from one search and c^x ( i = 1,2) denotes the corresponding p e r c e p t i o n parameter of the v a r i a n c e of o f f e r s . 70 I f c^ 1 i s unity, then the mean of the d i s t r i b u t i o n of perceptions, M^,equals the true reservation wage, W*. Likewise,if i s unity, the perceived variance of the d i s t r i b u t i o n of perceptions, V P ^ , equals the true variance, VW*. Type i individuals have stochastic misperceptions of the reservation wage represented by a density function whose parameters are JX and V P ^ . Turnover and acceptance functions for the different types of ind i v i d u a l may be written, respectively t ± = t ±(w; x,U; N . D . H . C j 1 , ^ ) i = 1,2 . . . . (24) a^ = a^(w; x,U; N,D,H,c^ 1 ,C2) i = 1,2 . . . . (25) where the argument before the f i r s t semicolon denotes a variable endogeneous to the firm. The variables between the semicolons are endogeneous to the labour market and the remaining arguments are exogeneous. 3. Workers' Behaviour i n Model I I I In Model I I I , the firm can recognise the characteristics of both i t s employees and those searchers who sample i t . A firm can discrim-inate by offering different wages to different types of individuals and creating vacancies f o r one type of in d i v i d u a l and not the other. Again, firms are forced to make wage and vacancy decisions, ex ante. This s t i p u l a t i o n prevents the firm driving out one type of employee i n one period through a low wage of f e r , i n response to a large number 71 of searchers of the other type. In t h i s model, the fir m makes a wage and vacancy creation decision f o r each type of i n d i v i d u a l . Thus, the two types of worker face d i f f e r e n t labour market environ-ments . The labour market environment facing a type i worker ( i = 1,2) may be written x. = ( M 1 ) where denotes the wage o f f e r and vacancy creation decision f o r a type i i n d i v i d u a l from a fir m f a c i n g employment state m. Based on the d i s t r i b u t i o n of wage off e r s and vacancies and the l e v e l of unemployment facing each type of i n d i v i d u a l , the mean of the d i s t r i b u t i o n of perceptions i s computed, as i n IB. The mean of a type i i n d i v i d u a l ' s d i s t r i b u t i o n of perceptions may be written M. = M. (c/EW*) i = 1,2 . . . . (27) 1 x 1 1 ' and the variance of the d i s t r i b u t i o n , VP ± = c2X VW* i = 1,2. . . . . (28) Each type of i n d i v i d u a l has stochastic misperceptions of the reservation wage, which i s represented by a density function with mean, M^, and variance, VP^. Again, i n d i v i d u a l s are assumed to d i f f e r through systematic misperceptions of t h e i r respective reservation wages. Thus, 72 C j 1 4 c ± 2 . . . . . (29) c 2 = c2 = c2- . . . . (30) I t i s easy to model i n d i v i d u a l differences generated by diff e r e n t discount rates, attitudes to r i s k or horizons over which returns from search accrue. Turnover and acceptance rates for the different types of ind i v i d u a l may be written, respectively, t ± = t ±(w; x±,V; N.D.H^CpCp i = 1,2 . . . . (31) a ± = a^w; x ±,U; N.D.H^cJ,^) i = 1,2. . . . . (32) I I . Firms' Wage arid Vacancy Creation Decisions and the Algorithm A. Firms' Behaviour i n Model I Returning to the basic model i n which individuals behave i d e n t i c a l l y , ex ante, i t i s now appropriate to model firm behaviour. Each firm can vary only i t s labour input, whence each has a w e l l defined marginal revenue product function, MRP(n), where n i s the l e v e l of employment. This analysis of firm behaviour i s applicable to the c l a s s i c a l short run. Allowing variation of other factors of production severely com-plica t e s the solution to the problem.^ By assumption, firms are unable to hold inventories. 73 Given an unchanging labour market environment, x, i n d i v i d u a l turnover and acceptance functions facing the firm, which o f f e r s a wage, w, may be written as t = t (w,x) dw < ^ 0 < t < 1. . . . . (33) a = a (w,x) da ^ < 0 dw 0 < a < 1. . . . . (34) Considering a f i r m whose employment l e v e l i s n, l e t denote the p r o b a b i l i t y that exactly i of the n employees do not qu i t i n response to the o f f e r , w. Then, p. = (n) ( 1 - t ) 1 t 1 1 " 1 i < n p ± = 0 i > n . . . . . (35) The flow of new h i r e s to the f i r m i s quite complex. Let there be U searchers and assume each searcher makes one contact per period. Then, any one of the U searchers w i l l contact the f i r m with p r o b a b i l i t y 1/N. During the period, the firm makes off e r s to at most v of the i n d i v i d u a l s who contact i t , where v i s the number of vacancies i t has 74 created. I f denotes the probability of i new hi r e s , then consists of two expressions. In the f i r s t expression, v individuals or less sample the firm. In the second expression more than v individuals, sample the firm. a. = E ( ^ ) ( | ) j ( l - l / N ) U ~ j ( ^ ( l - a ^ - 1 3=1 U . + E (")(1/N)3(1-1/N)U 3 ( Y)a 1(l-a) V \ . . . . (36) j=v+l 3 1 The terms enclosed i n the f i r s t set of parentheses i n each expression compute the probability that j individuals contact the firm. The firm creates v vacancies. The firm makes j offers, i f the number of individuals contacting the firm i s less than or equal to v. The remaining terms i n the f i r s t expression indicate the probability that, of the j offers made to searchers, i are accepted. I f more than v individuals contact the firm, then v offers are made randomly to the searchers. The remaining terms i n the second expression compute the probability that i of the v offers are accepted. This r e l a t i o n may be written more simply as P ( i hires) = P ( i hires | j < v contacts) P(j contacts where j < v) + P ( i hires | j > v contacts) P(j contacts where j > v) . . (37) 75 Relations (35) and (36) determine the p r o b a b i l i t y d i s t r i b u t i o n of employment f o r the fir m next period. The p r o b a b i l i t y d i s t r i b u t i o n of employment i s con d i t i o n a l on the current l e v e l of employment. Let be the p r o b a b i l i t y of employing k i n d i v i d u a l s i n period z+1, given n employees i n period z. J 6 = £ a; p. . nk . T l k - i i=I J = min(k,v) I = k-n i f n+v > k > n = 0 i f k < n , nk = 0 i f k > n+v . . (38) 8 I f k, the number of i n d i v i d u a l s employed i n period z+1, exceeds the l e v e l of vacancy creation, v, then the upper bound for the summation J w i l l be v since, to a t t a i n employment l e v e l k, at l e a s t k-v i n d i v i d u a l s must not q u i t . The maximum number of po s s i b l e employees next period i s n+v. I f k exceeds n+v then the t r a n s i t i o n p r o b a b i l i t y , 8 i s zero. I f k i s less than current employment, n, then the lower bound for the summation i s zero. I t i s consistent to not h i r e anyone but s t i l l have k employees next period. I f k l i e s between the maximum number of 76 potential employees next period, n+v, and the current l e v e l of employment, n, then at least k-n new employees must be hired. Thus, the lower bound for the summation i s k-n. I f n i s the highest employment l e v e l a firm can possible choose to face, then a wage and vacancy decision for each possible l e v e l of employment generates a set of conditional p r o b a b i l i t i e s which describe a Markov process, where levels of employment constitute the states. Let P be the n+l by fi+1 Markovian t r a n s i t i o n matrix, P = [ 9 n k ] n,k = 0,1,...,n. . . . . (39). For any given labour market environment, x, and set of decisions, 6^(n = 0,1,...,n), a Markov t r a n s i t i o n matrix can be generated i n which the i n d i v i d u a l element, 0 v ( 6 ),(n,k = 0,1,... ,n), denotes the probability that a firm with n employees paying a wage rate, wn, and creating v vacancies w i l l have k employees one period l a t e r . Since the evolution of the labour market i s a stochastic process, equilibrium i n the labour market i s necessarily stochastic. I t i s now possible to formulate the firm's objective. Let II(n,k) denote the one period p r o f i t s of a firm which pays a wage rate, w^ , and creates v n vacancies and as a result of the stochastic process occurring i n the labour market ends up with k employees. Then, k n(n,k) = Z MRP (i) - k.w.-c '1=0. n where MRP(O) = 0 and MRP(i) i s f i n i t e i=l,2,...,n. . . . . (40) 77 The f i r s t term indicates t o t a l revenue, while the second represents t o t a l labour cost and c represents fixed non-labour costs of production. Then, the firm's expected one period p r o f i t s are n f(n,S ) = Z 6 . (6 ) n(n,k)... . . . . (41) k=0 A policy i s defined as a wage and vacancy decision for a l l possible levels of employment. Let y r denote a policy then, y r = • ( 6 o , 5 l , * " ' 6 f P * . . . . (42) Let I now denote the number of different p o l i c i e s that a firm could adopt and Y be the set of a l l such p o l i c i e s , then, Y = {y r: r == 1,2,...,I}. . . . . (43) It i s assumed that same po l i c y , y reY, yet to be determined, i s adopted by the firm for ever. Then, the wage and vacancy decision made at any point i n time corresponds to the p a r t i c u l a r l e v e l of employment the firm then experiences. The firm assumes that market parameters are unchanging, that i s the 'labour market environment', x, remains the same. Then, turnover and acceptance are time invariant functions of the wage offer. Then, the firm's state to state t r a n s i t i o n 78 matrix i s time invariant when the poli c y , y , i s adopted for ever. The firm regards the evolution of the labour market to be a stationary Markov process.. This may be written P(y,r) = .[8nk(«n)] n,k = 0 , l , . . . , r i r = 1,2,...,1. . . . . (44) Let F(n,y r) denote the expected p r o f i t over an i n f i n i t e horizon discounted to the beginning of the f i r s t period, when a firm commences with n employees and adopts policy,.y , for ever. I f the labour market environment i s unchanging, then F(n,y r) i s also time invariant. I f the firm wishes to maximise expected discounted p r o f i t s over an i n f i n i t e horizon given an unchanging environment, then i t may be demonstrated that t h i s requires the adoption of a poli c y , say y*eY, for ever. Let F*(n) denote the maximum expected p r o f i t discounted over an i n f i n i t e horizon when the firm commences i n employment state, n. Its functional equation may be written l ^ F*(n) = max [f(n,6*) + ^  S 9 n k ( < S n ) F * ( k ) 1 • • • • <45> Y k=0 where R now denotes the in d i v i d u a l firm's rate of discount. Solving such functional equations i s not easy and i t defies analytic solution. Hadley [1964] has demonstrated that t h i s problem i s i n the class of dynamic stochastic programming problems, however, and can be solved 9 using an algorithm or l i n e a r programming. The algorithm to solve the problem i s outlined below. 79 B. The A l g o r i t h m Let x denote the unchanging l a b o u r market environment i n which i n d i v i d u a l and f i r m d e c i s i o n s are made. Then, r e w r i t i n g (1) and (2) x - (3,6) - ( I6Q,B^ , . . . ,3 ,... ,3^,6Q,5^, . . . , 6 ^ , . . . , 6 ^ ) . . . . (1) ii U = L - E = L - N E & . n . . . . ( 2 ) n=0 where L i s the s i z e of the labour f o r c e , U i s the number of searchers and E the number of i n d i v i d u a l s employed. The turnover and acceptance functions may be c a l c u l a t e d , namely t(w) and a(w). L e t G (n,x) denote expected p r o f i t s discounted over an i n f i n i t e h o r i z o n , when the f i r m adopts p o l i c y f o r ever and commences i n employment s t a t e n . Then, , fi G r ( n , x ) = f ( n , 6 * ) + 3 ^ Z e n k ( 6 * , x ) G r (k,x) . . . . (46) k=0 and x i s i n c l u d e d as an argument to i n d i c a t e the constant environment. W r i t i n g (G r(0 ,5) , G r ( l , x ) , . . . , G r ( i i , x ) ) as G r ( x ) , and ( f(0,sj), f ( l , 6 1 ) , . . . , f ( n , 6 . ) ) as f r ( x ) and d e f i n i n g f ^ x ) as [0 . ( 6 r , x ) ] (n,k = 0,1,...,fi), (460 may be w r i t t e n as G r ( x ) = f r ( x ) + T 5 R ? ( X ) G r ( x ) . . . . . (47) 80 Solving for G (x) yields G r( X) = / i n + i , f t + i - l i i ^ ^ " " 1 f r ( ^ ) . : • • • • w where 1^ + 1 is the fi+1 by n+l identity matrix. The algorithm may be described in the following way (i) Select an arbitrary policy, say y eY, and set the policy s index, j ,at the value zero. Calculate the transition matrix, P (x), and the profit vector, f (x). From (48) —s — 1 — 1 — solve for G (x). Define a vector, HJ (x), and set H J (x) => G (x). Set the value of a scalar r to s. ( i i ) Increment the policy index, so j = j+1. ( i i i ) Assume the firm uses policy, y-, from period two onwards forever and select a new set of decisions, policy r, to be adopted in period one which maximises the present value of expected profits. This requires choosing y = ( { . . { . . I . . J . ) which maximises r 0' 1' ' n f T ^ 6 > lk }n 6 n k ^ > H^Vx). k=0 ^iv) If policy r and policy r are identical,then the problem r r outlined i n (45) has been solved. That i s , i f w = v N ' ' n n and v^ = v^ for a l l n = 0,1,...,n. , the algorithm has converged. If the policies r and r are not identical, set r = r, calculate G r(x) fronts (48), set H J (x) = G r(x) and return to step ( i i ) . 81 Hadley demonstrates that i f there i s a f i n i t e number of possible p o l i c i e s , then the algorithm converges i n a f i n i t e number of steps. In choosing t h e i r optimal policy,firms assume that individuals evaluate the i r offers i n the context of the fixed environment, x = (3,6). Individuals make decisions about turnover and acceptance i n the l i g h t of the d i s t r i b u t i o n of current offers, however. Thus, i f y^ denotes the policy adopted, then individuals respond to the environment, x = (3,y r). Furthermore, i f a l l firms i n the industry-adopt the optimal wage and vacancy decision corresponding to t h e i r current levels of employment, then the stochastic forces i n the labour market w i l l , except i n equilibrium, generate a new environment charac-terised by a different d i s t r i b u t i o n of firms over employment states and thus different levels of aggregate unemployment and vacancies and a new d i s t r i b u t i o n of wage offers. Generally, the old policy w i l l be sub-optimal with respect to the new environment. Thus, the algorithm described can be regarded as solving the p a r t i a l optimisation problem, namely choosing a p r o f i t maximising policy given a constant environment. The feedback of i n d i v i d u a l firms decisions on the labour market environ-ment i s ignored. The approach adopted to solve the f u l l optimisation problem i s to p a r t i a l l y optimise given a p a r t i c u l a r environment, s a y i ^ . Let y^ denote the optimal policy and l e t x denote the environment, (B^,y r). Compute the t r a n s i t i o n matrix, P ( y r ) , where x i s the unchanging labour market environment. Define the new d i s t r i b u t i o n of firms over employment states, 3 +^^ > as h+1 = * i P ( y r } . . . . (49) 82 and the new labour market environment as = ^ i + l ' V * .... (50) Using the partial optimisation algorithm yields a new optimal policy in response to the labour market environment, x^+-^« This new policy i s again used to generate a new environment and so on. Convergence of the algorithm to a f u l l y optimal solution occurs when consecutive, labour market environments d i f f e r by a vector of arbitrary small constants. This convergence criterion i s stringent enough to ensure that consecutive policies are identical, since the vacancy decision i s integer and the wage decision i s considered in fixed increments. The f u l l optimisation algorithm may be written (i) Choose an i n i t i a l labour market environment X Q = (3Q>YQ) • Set the loop index, j , at zero. ( i i ) Solve the parti a l optimisation algorithm as described, subject to the environment, x... Denote the optimal policy by y. ( i i i ) Define x* as (y,3..) and calculate P\ = [ e nk(^ n»x*)], (n,k = 0,1,...,n). Increment the loop index, j , by • 1 and define 3. = 3. ,P. , and x. = (y,3.). 3 3-1 3-1 3 V J " 2 . (iv) Compare x^ . and If Abs|x_.-x^_^| < e, rwhe.xe.-e i s a vector of small arbitrary constants, go to (v). Otherwise, return to ( i i ) . 83 (v) Calculate 8 = 8 Lim P. m-*=° 3 I f Abs|B-3I < e, then f u l l optimisation has been achieved, and so set x* = x. and 3 8* = 6. I f not, return to ( i i ) . The stochastic turnover and acceptance behaviour guarantees that the flows of individuals i n and out of employment w i l l be stochastic and so there w i l l be va r i a t i o n i n the l e v e l of employment over firms. Consequently, equilibrium i s stochastic and i s characterised by a probability d i s t r i b u t i o n of employment, 8* = (3*,3*,...,g|). The d i s t r i b u t i o n , 3*,is conceptually different from the d i s t r i b u t i o n , 3-describing the labour market environment. 3 refers to the actual d i s t r i b u t i o n of firms at a pa r t i c u l a r point i n time. In equilibrium, the actual d i s t r i b u t i o n of firms, 8, exhibits v a r i a t i o n about the distribution,8*, due to the stochastic forces operating i n the market. Since the process described i s a Markov process, the steady state d i s t r i b u t i o n e x i s t s , i f the t r a n s i t i o n matrix i s regular. This d i s t r i b u t i o n i s given by 8 as calculated i n (v). This result i s approximate, since the convergence c r i t e r i a i n (iv) and (v) does not require that 8.. = 8.. Since e > 0 adopting the optimal p o l i c y , y*, does not exactly recreate the same expected labour market environ-ment. The accuracy of the result depends on the magnitude of e. I f y* i s the optimal policy,.then the true steady state d i s t r i -bution 8* i s given by where B j " V l [ e n k ( 6 £ ' V l ) ] - ' (51) 84 I f firms are d i s t r i b u t e d a c c o r d i n g to (3*, then adoption of the optimal p o l i c y e x a c t l y r e c r e a t e s the labour market environment. Thus, the p a r t i a l l y opt imal p o l i c y i s optimal f o r e v e r . The e x p l i c i t formulat ion of f i r m behaviour and d i s c u s s i o n of the a l g o r i t h m to s o l v e Models I I and I I I can be found i n Appendix I . C. An Overview of Model I Having developed the behaviour of market p a r t i c i p a n t s and how the market responds to t h i s b e h a v i o u r , i t i s now p o s s i b l e to summarise the r e l a t i o n s h i p s . T h i s i s a model of the l a b o u r market i n which there are N firms each having the same marginal revenue product f u n c t i o n , MRP(n), and the same r a t e of d i s c o u n t , R. Firms maximise the expected present value of p r o f i t s over an i n f i n i t e h o r i z o n . I n d i v i d u a l workers maximise the expected present value of income over a working h o r i z o n of H p e r i o d s . They b e l i e v e that the wage d i f f e r e n t i a l a s s o c i a t e d w i t h search w i l l p e r s i s t for H^ p e r i o d s , H^ < H . T h e i r r a t e o f discount i s D. In order to generate s p e c i f i c job o f f e r s , workers must engage i n a process of random s e a r c h . Each worker can make one contact per p e r i o d and he r e c o n s i d e r s h i s search d e c i s i o n each p e r i o d . I f unemployed, the worker i s guaranteed unemploy-ment compensation,,w. I n d i v i d u a l s ' perceptions of the r e t u r n s from search are assumed imperfect but not u n r e l a t e d to the a c t u a l r e t u r n s from s e a r c h . The parameters c^ and i n (11) and (12) determine the r e l a t i o n s h i p between the i n d i v i d u a l ' s p e r c e p t i o n of the o f f e r d i s t r i -85 bution and the actual d i s t r i b u t i o n . Each individual's perception of the reservation wage i s drawn from a gamma d i s t r i b u t i o n , p(wc), whose parameters are related to the true reservation wage and the variance of offers. The supply of labour to the industry, L, i s constant. Thus, N, D, R, H^, L, H are exogeneous constants and MRP (n) and p(wc) are exogeneously determined relationships, c^ and can be regarded as s h i f t parameters. I f an in d i v i d u a l searches randomly, then his perception of the mean and variance of the wage d i s t r i b u t i o n i s unbiased. The unknowns i n the model are t n * , (n = 0,1,...,n), the rate of turnover for a firm with n employees. an*> (n = 0,1,...,n), the rate of accession for a firm with n employees. 6 nk*,(n,k = 0,1,...,n), the probability that a firm with n employees In one period w i l l have k employees i n the next period. <$* = (w n*,v n*),(n =_ 0,1,... ,fi) , the wage offer and vacancy creation decision made by a firm with n employees. 8* , (n = 0,1,...,n), the steady state probability of a firm facing n employees. E* ,the l e v e l of employment. U*, the l e v e l of unemployment, and x* = (8*,<5*), the labour market environment. 2 There are (n+l) + 4(fi+l) + 3 unknowns. 86 I n d i v i d u a l search behaviour gives r i s e to the turnover r a t e s to the f i r m which may be w r i t t e n , V = W 5 X * ' U * ; N . D . H ^ C J ^ C J ) n = 0 , 1 , . . . , n . . . . . (52) The turnover r a t e f a c i n g a f i r m i s a f u n c t i o n of i t s own wage r a t e ; the labour market environment and the l e v e l of unemployment; and the number of f i r m s , the i n d i v i d u a l ' s own r a t e of d i s c o u n t , the time h o r i z o n over whichwage d i f f e r e n t i a l s p e r s i s t and the p e r c e p t i o n parameters. L i k e w i s e , the acceptance f u n c t i o n may be w r i t t e n a n * = a n ( w n * ; X * ' U * ; N . ^ I L ^ c ^ ) n = 0 , 1 , . . . , n . . . . . (53) The t r a n s i t i o n p r o b a b i l i t i e s may be w r i t t e n as nk n k ^ n ' n ' n ' ' ' N ) n , k = 0 , 1 , . . . , n . . . . . (54) The steady s t a t e d i s t r i b u t i o n , B n ( n = 0 , 1 , . . . , n ) , i s determined 3 = [ 1 , 0 , . . . , Q : ] . l i m [ 0 r f c ] t f . . . . (55) t-*» 87 The wage and vacancy d e c i s i o n s , (w^*,v^*)(n = 0 , 1 , . . . , f i ) , are the s o l u t i o n to (45), namely F*(n) = max [ f ( n , S * ) + ^ E 9 n k ( < ^ ) F * ( k ) ] . . . . (45.) Y k=0 r r r r where y = (6 ,6,,...,5 , . . . , < S O and there are I p o s s i b l e p o l i c i e s r o 1 n n that c o u l d be adopted. The expected l e v e l of employment i s n E * = N Z n n . g * . . . . (56) n=0 n and the expected l e v e l of unemployment, Tj* = L - E * . . . . . (57) I I I . The C h a r a c t e r i s t i c s of S t o c h a s t i c E q u i l i b r i u m A . The Mean Wage of the Employed I n d i v i d u a l Unemployed i n d i v i d u a l s computing the returns from search c a l c u l a t e the mean wage by e v a l u a t i n g the p r o b a b i l i t y of an o f f e r . A s i m i l a r measure can be computed f o r those c u r r e n t l y employed. I f WB i n d i c a t e s the mean wage b i l l i n c u r r e d by the i n d u s t r y d u r i n g the p e r i o d , and mean employment i s E * , then the mean wage f o r those employed i s MW where (58) 88 Denoting the steady state t r a n s i t i o n matrix by T* where X* = [T„*],(i,j = 0,1,..., n) , then a firm currently facing employ-ment l e v e l n w i l l pay a wage, wn*, to j employees next period with p r o b a b i l i t y , T^*. Then, the mean wage b i l l incurred by a firm i n employment state n i s computed as WBn « Z T n j*.j.w n*. . . . . (59) J=0 There are N.3^* firms facing employment l e v e l n. Thus the mean industry wage b i l l , WB, i s given by n n n WB =N E WB .3 * = E N.3 * £ T .*.j.w *. . . . . (60) „ n n n n . n n i J n n=0 n=0 j=0 J In stochastic equilibrium the l e v e l of mean employment,E*,is constant and E* = N Z n.3 * . . . . (61) ii=0 n • Then, n n N E W B . 3 * EWB.3* n n n n n MW = = . n n N Z n.3 * Z n.3 * „ n n n=0 n=0 T h i s measure of the mean wage i s another summary s t a t i s t i c which may be used f o r comparing s t o c h a s t i c e q u i l i b r i a generated under d i f f e r e n t 89 values of the exogeneous parameters. The mean wage for each type of employed individual can be computed in Models II and III. Comparison of these measures gives some indication of the impact of explicit discrimination on relative wage offers. B. Mean Lifetime Income for the Worker When the labour market i s in stochastic equilibrium, i t i s possible to calculate three different measures of lifetime mean income for an individual. At the beginning of each period an individual i s i n one of three distinct states. He may be unemployed and have just received and accepted a wage offer, w j * ( j = 0 , 1 , . . . , n ) , for next period, from a firm facing employment level, j , states UA = ( 0 , 1 , . . . , n ) . He may be already employed and just about to accept an offer, wj*>(j = 1,2,... ,fi), from his firm which faces employment level j , states EA = (fi+l,n+2, . . . ,2n) . If already employed, he cannot receive a wage offer, w^ *, from his firm. These states are identified then according to whether the individual was previously employed or un-employed and the wage rate accepted. The third category, state 2fi + 1, is that the currently unemployed individual has decided to search further next period and receive unemployment compensation, w. The individual's state to state transition matrix i s closely related to his own probability of receiving an offer, 3 n*Y n*» accepting i t , a^* = a n ( w n * » x*,U*; N,D,H^,c^ , C 2 ), quitting when employed, t * = t (w'*; x*,U*; NJDJH,,C,,C 0 ), and the firm's transition matrix, n n n ' 1 ' 1 ' 2 ' ' = 9nk^ an*' tn*' Vn*'^*'' n' s :*" n c e t* i e f i r m ' s state of employment determines the wage offer. The f i r s t two categories of state, UA and EA, 90 are conceptually different i n that the t r a n s i t i o n vector for a firm, facing employment l e v e l , j , and offering wage, wj*» depends on whether the p a r t i c u l a r i n d i v i d u a l , who has just received and accepted an o f f e r , W j * , for the next period, i s now unemployed or already employed by the firm. I f S denotes the returns from each state then S = ( W Q . W ^ ... ,wfl,w1,w2,... ,wft,w) . . . . . ( 6 3 ) The three measures of l i f e t i m e mean income are : 1. Income gained from the stochastic turnover and accept-ance behaviour modelled i n this chapter, YS; 2. Income derived from never q u i t t i n g a firm, YQ; and 3 . Income gained from turnover and acceptance behaviour based on knowledge of the true reservation wage, YC. I t must be understood that the l a s t two measures are hypothetical i n that they are calculated on the basis of one individual's non-stochastic behaviour i n a world where a l l other individuals and a l l firms behave i n the manner described i n Section I I I . This i n d i v i d u a l i s not recognised by firms. These measures y i e l d interesting insights into the nature of stochastic forces i n the labour market. I f the firm i s subject to s i g n i f i c a n t changes i n employment from one period to the next, then the search behaviour modelled i n I I I w i l l be sub-optimal, because wage d i f f e r e n t i a l s f a i l to p e r s i s t . Expected income from such 91 behaviour, that i s the f i r s t measure, w i l l be less than for some other search rule. Herein l i e s the contradiction, however. Assuming that i n this stochastic environment an in d i v i d u a l i s better o f f i f he quits and accepts offers according to the reservation wage, then i t seems plausible that a l l individuals should adopt t h i s non-stochastic behaviour. I f firms were aware of such behaviour, however, then each would offer s l i g h t l y more than the l e v e l of unemployment compensation, the collusive monopsony solution, and there would be no dispersion and no search. Individuals are homogeneous i n their search behaviour,, then-. Thus, while these measures do y i e l d insights into the ranking of search rules, these measures are p a r t i a l equilibrium measures i n that firms are assumed not to respond optimally to the rules specified. This underlines the importance of the expectations of each set of market participants about the behaviour of the other set of participants. The computation of mean income from stochastic behaviour i s s t r a i g h t -forward. I f an in d i v i d u a l behaves s t o c h a s t i c a l l y , then he accepts a wage offer,w^*, when unemployed, with p r o b a b i l i t y , a^* = a k ( w k * ; x*,U*; N,D,H^, c^j^),and,once employed,he remains employed i n response to a wage off e r , wk*, with probability ( l - t k * ) = 1 - t f c(w k*; x*,U*; N.D,^,^,^) . Consider an in d i v i d u a l who has received and accepted his f i r s t job offer, w^ *, at the beginning of his job horizon. Then, the probability that he receives and accepts an of f e r , wk*, next period i s the product of the probability of accepting the o f f e r , l - t k * , and a conditional p r o b a b i l i t y , namely, that the firm facing employment l e v e l n and offering 92 a wage, w n*» moves to employment level k next period, given that one particular searcher accepted the offer, w^ *. The firm achieves em-ployment level k next period, i f creating v n * - l vacancies and facing h currently employed individuals and U*-l searchers i t can employ k-1 of them. Denoting this transition probability by'TS ^, then TS . = (l-.t. *)6 . ,(a * , t *,v * - l ; U*-l,n; N) n i - k nk-1 n ' n ' n ' ' ' where i e E A n e U A i = k + n'.. . . . . (64a) If an individual, previously unemployed, has just accepted an offer, then in the following period he either becomes unemployed or enters a state denoted by EA. Then TS . = 0 n E EA or n E UA and i e UA . . . . . (64b) ni An individual, already employed with a firm which offers a wage, wn*, say, has a different probability of receiving and accepting an offer, w^ *, next period. The firm has one employee who is certain to remain employed. Thus, facingU* searchers and n-1 other employees and creating v * vacancies the firm must hire k-1 individuals. Then TS' . = ( l - t * ) 6 n i Aa * , t *,v *; U*,n-1,N) mx k n-lk-1 n n ' n ' ' ' where m , i s E A m=n + n i = k. + n . . . . . (64c) 93 From (64a) the probability that an i n d i v i d u a l i n state, n e UA, becomes unemployed next period i s simply T S n , 2 f i + l = 1 " * T S n i n e U A " • • • • <64d> leEA Likewise, from (64c) the probability that an in d i v i d u a l who i s currently employed and receiving a wage, w^ *, w i l l quit i n response to the new wage offer next period i s TS = 1 - . Z _ . TS . me EA, m = n+n.. . . . . (64e) m,2n+l leEA mi ' An i n d i v i d u a l , who i s currently unemployed, w i l l receive an of f e r , w^ *, from a firm with p r o b a b i l i t y , 3n*Yn*« He w i l l accept the offer with p r o b a b i l i t y , a n * = afl(wn*; x*,U*; N,D,H ; L,c 1,c 2). Then T S ? A + I n = Y„* Pj}* n e UA. . . . . (64f) Zn+l-,n n n n T S O A _ I _ I = 0 n e EA. . . . . (64g) zn+1,n An i n d i v i d u a l , who i s currently employed, can only enter one of the UA states or remain unemployed. This explains (64g). From (64f) an i n d i v i d u a l has a prob a b i l i t y T^^^ 2n+l' °^ remaining unemployed where 94 T S2n+l,2fi+l " 1 ~ Z T T A T S2n+l,n * . . . . (64h) neUA ' The i n d i v i d u a l i s assumed to search for one period p r i o r to his job horizon. Then, he has a prob a b i l i t y T^^+i n» °^ securing and accepting a wage of f e r , w^ *, at the beginning of his job horizon and a probability ^2fi+l 2n+l' °^ c o m m e n c : L n 8 n i s work horizon unemployed. He has a zero probability of commencing his work horizon i n one of the th EA states. Denoting the 2ii+l row of TS (the t r a n s i t i o n matrix) by TS2^ +^ then expected income, YS, i s given by YS = T S 2 f i + 1 [S + (TS)S + (TS) 2S + ... + ( T S ) H - 1 S ] . . . . . ( 6 5 ) 1 0 C. Labours' Share of Output I f both sets of participants gain i n response to a change i n an exogeneous parameter, for example, the period length, then a useful summary s t a t i s t i c i s labours' share of t o t a l output. A firm facing employment l e v e l , n, has mean one period p r o f i t s , f ( n , 6 n ) . Denoting t o t a l mean one period industry p r o f i t s by Til, then, fi * Til = N E 6 .f(n,6 ).. . . . . (66) „ n n n=0 From (60), WB denotes the mean industry wage b i l l . Then, i f LS denotes labours' share of t o t a l output, 95 D. The Correlation of Wage Rates Over Time In computing t h e i r returns from job search,individuals assume that, i f they accept a p a r t i c u l a r wage o f f e r , then they w i l l enjoy the same wage over the horizon, H^. I f firms face stochastic turn-over and acceptance behaviour on the part of workers, then they w i l l face d i f f e r e n t levels of employment over time and thus offer different wage rates over time. Of i n t e r e s t , then, i s the correlation of firms' wage offers over time. Evidently, i f the temporal correlation of wage rates i s low then individuals should never quit to search but should remain employed, irrespective of the wage offer. This correlation c o e f f i c i e n t i s computed i n the following way. Let W1 denote the vector of wage rates, defined over states of employ-ment, i n period i . Then, since the labour market i s i n stochastic equilibrium, r T i / i i i f\ W = (w^w^... ,wn,... ,wfi) = w i + k = W* i = 0,1,... k = - i , - i + l ... . . . .(68) _* where W denotes the vector of wage offers characterising stochastic equilibrium. Let g* denote the steady state d i s t r i b u t i o n of employment. Then, the mean wage offer facing an i n d i v i d u a l , i f he i s employed i n period i , i s w1 where 96 n ft1 = E B* w? j=0 J J = w K i = 0 , 1 , . . . k = - i , - i + l , = w*. . . . . (69) L e t v 1 denote the v a r i a n c e of wage o f f e r s , g i v e n that the i n d i v i d u a l i s employed, t h e n , v 1 = E B*(wT)2 - (w 1) 2 j - 0 J J n 2 2 E B* w* - w j - 0 J J i+k = v i = 0 , 1 , . . . k = - 1 , - i + l , = v * . . . . . (70) L e t 6 n k*Xk,n = 0 , 1 , . . . . ,fi) denote the elements of the t r a n s i t i o n m a t r i x , P * , c h a r a c t e r i s i n g e q u i l i b r i u m . Since both the t r a n s i t i o n matrix i s s t a t i o n a r y and the d i s t r i b u t i o n of o f f e r s i s time i n v a r i a n t , then the c o r r e l a t i o n c o e f f i c i e n t i s s t a t i o n a r y . Let P ( w j » w k + ^ denote the j o i n t p r o b a b i l i t y that a f i r m , taken at random, o f f e r s a wage, w j*> i n p e r i o d i and w^* i n p e r i o d i+1. Then, t h i s j o i n t p r o b a b i l i t y may be w r i t t e n 97 P ( w j ' W k ± + 1 ) = p ( w j ' W k } = 3 j * 9 j k * - . . . . (71) Then, the covariance between wage o f f e r s i n period i and period i+1, cov(W"*",W''r'""'") , may be written as _. _. - fi fi „ cov(W ,W ) = E E g * 9* w* w* - w . . . . . (72) j=0 k=0 J j k j k I f cc(W^,W:'"+^) denotes the c o r r e l a t i o n c o e f f i c i e n t , then c c C W 1 , ^ * 1 ) = c ° v ( w % W i + 1 ) = cov(W*,W*) ( 7 3 ) /.i.i+1 v* V V To determine the c o r r e l a t i o n of wage o f f e r s over say, T periods, i t i s necessary to compute p T = [ 0 n k * ] T . . . . (74) and replace the elements, 0 k*(n,k=0,l,... ,fi) , i n the computation -T with the corresponding elements from P . E. The Mean Duration of Employment and the Mean Duration  of Search P r i o r to a Wage Offer Two other summary s t a t i s t i c s , e a s i l y computed, are the mathematical expectation of the duration of unemployment and the mathematical ex-pectation of the duration of search p r i o r to a wage o f f e r . Let PO* denote the p r o b a b i l i t y of re c e i v i n g a job o f f e r i n one search and l e t PA* denote the p r o b a b i l i t y of r e c e i v i n g and accepting a wage o f f e r i n one search. Then, using the normal notation, 98 n PO* = E 0* y* . . . . (75) n=0 n n n PA* = E g* Y*a*.. . . . . (76) n n n n n=0 In stochastic equilibrium these p r o b a b i l i t i e s are time invariant. Then, the mean duration of search p r i o r to an offer, Dg, may be written as D s PO* . . . . (77) and the mean duration of employment, D^ , may be written as D u = PA* ' ' ' ' ' ( 7 8 ) F. Mean P r o f i t s of a Firm I t i s also possible to compute the p r o f i t s of a firm taken at random. I f $* denotes the prob a b i l i t y of a firm attaining employment state n andF*(n)is the mean p r o f i t s discounted over an i n f i n i t e horizon of a firm commencing i n employment state n$ then mean p r o f i t s , EII, are n En = E g *F*(n). . . . . (79) n=0 n This measure can be used as an indication of the industry p r o f i t a b i l i t y under different labour market structures. 99 G. S p e c u l a t i v e Vacancy C r e a t i o n I t i s evident that i n t h i s model the l e v e l of vacancy c r e a t i o n i n s t o c h a s t i c e q u i l i b r i u m i s at l e a s t the l e v e l c o n s i s t e n t with s t a t i c p r o f i t maximisation, assuming no c u r r e n t employees q u i t . The quest ion a r i s e s as to whether the f i r m would choose to h i r e more i n d i v i d u a l s i n the c u r r e n t p e r i o d , i n order to e x p l o i t i t s i n t e r t e m p o r a l monopsony power. T h i s hypothesis can be t e s t e d by the f o l l o w i n g procedure. In s t o c h a s t i c e q u i l i b r i u m a f i r m employing n i n d i v i d u a l s creates v * vacancies and o f f e r s a wage, w^*. L e t iUkjW^*), (n,k = 0,1,...,n)} denote s i n g l e p e r i o d p r o f i t s a s s o c i a t e d w i t h employing k i n d i v i d u a l s and paying them a wage, w n * . Using the u s u a l n o t a t i o n , l e t F*(n) denote mean p r o f i t s discounted over an i n f i n i t e h o r i z o n earned by the f i r m , which commences i n s t a t e n , and adopts the o p t i m a l set of d e c i s i o n s , 6*, for ever. Assume the f i r m faces employment l e v e l k next p e r i o d , then i f F ( n , k ) denotes the t o t a l mean p r o f i t s discounted over an i n f i n i t e h o r i z o n of a f i r m , which commences i n s t a t e n and then moves to employment s t a t e k next p e r i o d , F(n,k) = n(k,w * ) . + F*(k) 1+R k < n + v * ., n (80) 100 I f , a f t e r adopting i t s i n i t i a l d e c i s i o n , 8*, the f i r m could n choose i t s l e v e l of employment, next p e r i o d , then i t would choose employment l e v e l , L n, where F ( n , L n ) = max(n(k,w n *) + ) . . . . (81) k Denote the s i n g l e p e r i o d p r o f i t maximising l e v e l of employment a s s o c i a t e d w i t h a wage, w n * , b y n ( w n * ) , then L n - ft(wn*> . . . . (82) s i n c e the f u n c t i o n , F * ( k ) , i s a n o n - d e c r e a s i n g f u n c t i o n of k . T h i s demonstrates that firms may choose to create more vacancies than i s c o n s i s t e n t w i t h s i n g l e p e r i o d p r o f i t maximisation, i n order to e x p l o i t t h e i r i n t e r t e m p o r a l monopsony power. In a d d i t i o n , however, over some s t a t e s , n + v *• > L . . . . (83) n n that i s firms create vacancies i n excess of the l e v e l d e s i r e d h i r e s which are c o n s i s t e n t with the e x p l o i t a t i o n of i n t e r t e m p o r a l monopsony power. Such vacancy c r e a t i o n . c a n be d i r e c t l y a t t r i b u t e d to the s t o c h a s t i c turnover and acceptance behaviour of i n d i v i d u a l s . By 101 d e f i n i t i o n , vacancy c r e a t i o n i s s p e c u l a t i v e . No f i r m i s assured o f f i l l i n g any p o s i t i o n s created due to s t o c h a s t i c search and acceptance behaviour by i n d i v i d u a l s . I n t h i s work, the word w i l l be used to d e s c r i b e those vacancies which can be a t t r i b u t e d to s t o c h a s t i c behaviour by workers. A measure of aggregate s p e c u l a t i v e vacancies can be c o n s t r u c t e d . I f .V denotes aggregate s p e c u l a t i v e v a c a n c i e s , then n V = N Z (n+v * - L )3 * . . . . . (84) s n n n n n=0 The approach adopted to analyse the f i r m ' s vacancy c r e a t i o n d e c i s i o n i s analogous to that d e s c r i b e d by A r c h i b a l d [1954]. He d e r i v e s the optimal wage o f f e r f o r the f i r m p r o f i t maximising i n an imperfect market. He then asks the q u e s t i o n : i f the supply of labour i s s t o c h a s t i c at the wage o f f e r , what l e v e l of vacancy c r e a t i o n would the f i r m create? This i s a s i n g l e p e r i o d model and so the o p t i m a l l e v e l of vacancy c r e a t i o n i s equal to the l e v e l of employment at which the workers' marginal revenue product equals the o p t i m a l wage, minus the current l e v e l of employment. In the i l l u s t r a t i v e model, the problem i s made s l i g h t l y more complicated because the f i r m i s assumed to i n h e r i t a workforce from l a s t p e r i o d , but s t i l l p r o f i t maximises i n a s i n g l e p e r i o d framework. 102 In t h i s model, the f i r m chooses a wage o f f e r and l e v e l of vacancy created to maximise p r o f i t s over an i n f i n i t e horizon subject to stochastic v a r i a t i o n i n the whole labour supply schedule. The question i s then posed : what l e v e l of employment would the fir m choose,if able to influence i n d i v i d u a l s ' behaviour? Does t h i s employment l e v e l exceed the s t a t i c p r o f i t maximising l e v e l of employ-ment? I f so, then the f i r m chooses a l e v e l of employment above the s t a t i c p r o f i t maximising l e v e l of employment, i n order to e x p l o i t i t s intertemporal monopsony power. H. The Co l l u s i v e Reservation Wage I f firms were to collude i n t h e i r wage o f f e r , they would o f f e r a wage ju s t above the l e v e l of unemployment insurance, w. There would be no dispersion, no search and zero unemployment. Analogously, i t i s possible to compute the c o l l u s i v e reservation wage for workers such that t h e i r j o i n t earnings are maximised, under the assumption of p r o f i t maximising behaviour on the part of firms. Assume that each f i r m has an exogeneous marginal revenue product function of the form MRP (n) = a - -n . . . . (85) where n denotes the l e v e l of employment. There are N firms i n the industry and so, i f w denotes the reservation wage, j o i n t earnings, JE, may be written 103 JE = N ( a Q - w)w + (L-N(a - w))\w. . . . . (86) Each firm profit maximises and chooses a level of employment at which the marginal revenue product equals the wage, w. Then, ignoring i n d i v i s i b i l i t i e s , each firm hires ( a Q- w) workers and the remaining (L-N(a o~ w))- individuals earn unemployment compensation, w. Differentiating with respect to w yields Na Q -2Nw + Nw . . . . (87) and so total labour income i s maximised by a collusive reservation wage, w, where N(a + w) a + w o o ,„„. w ^ = ^ — * . . . . (88) Ignoring i n d i v i s i b i l i t i e s , the corresponding l e v e l of employment i s a + w a - w L o ( a Q - w)N = (a Q - -^j— ) N ! = ( ) N.. . . . . (89) An interesting question i s whether the mathematical expectation of the discounted stream of wage offers (including unemployment com-pensation) enjoyed by an individual over his job horizon i s positively related to the mean wage offer enjoyed by an employed worker i n a single 104 period. If, in a labour market i n which information i s perfect, the wage rate exceeds the collusive reservation wage, then the joint earnings of labour i n the single period f a l l below the joint earnings associated with the collusive reservation wage. Then, ignoring the impact of the stochastic forces in the labour market, i t seems plausible that an individual's mean discounted income over his work horizon i s positively correlated with the single mean wage offer up to the collusive reservation wage, ft, and i s negatively correlated to the mean wage offer above the collusive reservation wage.^ 105 CHAPTER 3 THE DISTRIBUTION OF PERCEPTIONS, THE DATA, THE SIMULATION  PROCEDURE AND THE METHOD OF COMPARATIVE STATICS I. A Summary In this chapter the properties of the density function of perception of the reservation wage, which i s adopted i n the simultions, are analysed. The parameter values adopted i n each model are specified. The p a r t i a l optimisation procedure i s out-li n e d and the convergence and cycling c r i t e r i a are examined. The problemsarising from the simulation procedure are discussed, and f i n a l l y the approach adopted i n organizing the comparative s t a t i c results i s described. I I . The Gamma Di s t r i b u t i o n A p r i o r i , the density function of i n d i v i d u a l perceptions of the reservation wage has no p a r t i c u l a r form, except that i t be defined on the p o s i t i v e quadrant. The gamma d i s t r i b u t i o n was chosen because i t has two useful properties, 1. I t i s a two parameter d i s t r i b u t i o n defined on the whole pos i t i v e quadrant; and 2. Through different values of the parameters i t can look Gaussian or skew.L f \ 1 / \ B (~wc) P(wc) = B, AB+1 (wc) exp —r~. . . . . (1) 106 2 This d i s t r i b u t i o n has mean, (B+1)A, and variance, (B+1)A . From Chapter 2 IB, MP and VP represent the mean and variance, respectively, of the density function of in d i v i d u a l perceptions of the reservation wage, where MP = CjW* c1 > 0 . . . . (2) VP = c2EW* c 2 > 0 . . . . (3) and W*, VW* denote, respectively, the true reservation wage and variance of wage offers i n the industry. For consistency, then, i t i s necessary that MP = (B+1)A . . . . (4) and VP = (B+1)A2. . . . . (5) In order to determine the p r o b a b i l i t i e s of turnover and acceptance i n response to a given wage off e r , w, and given labour market parameters, i t i s necessary to evaluate the cumulative of the gamma d i s t r i b u t i o n , P(w), which may be written P(w) = p(wc) dwc.. . . . . (6) Turnover and acceptance p r o b a b i l i t i e s are given by t(w) = l-P(w) . . . . (7) a(w) = P(w) .. . . . . (8) 107 2 Mood [1950] demonstrates that a close approximation to this cumulative probability i s J B •(-) . P(w) = 1.0 - exp(^) Z -4r* . . . . (9) 3 A J=l J ' The integral part of B is used in the summation unit. It i s easy to M demonstrate that the mode, WC , is given by M VP , , i WC = AB = MP - ^  • . . . . (IO)* The general shape of the gamma distribution i s shown i n Figure 5. P WC Figure 5 - The Gamma Distribution Changes i n MP or VP change the parameter values, A and B, and the derivative of the cumulative distribution,P(w), with respect to B i s not defined. Thus, i t i s not possible to determine analytically the impact of changes i n the mean and variance on the shape of the gamma distribution. 108 If increases so that MP increases, however, then from (10) the mode, M WC , shifts to the right and i t i s plausible to assert that the relation ship between the new marginal gamma distribution,p^(wc), and the distribution, p(wc), i s as shown in Figure 6. Figure 6 - Gamma Distributions with Different Means. For values of wc greater than or equal to the new mode, WC^, then p^(wc) exceeds p(wc) and so the marginal rates of turnover and acceptance are higher. Furthermore, for a l l values of w the new cumulative distribution P^(w) i s less than P(w). Then, ceteris paribus, 109 and so, dP(w) dc, dt(w) dc. < 0 • (ID > 0 • (12) da(w) dc. < 0 . . . . . (13) An Increase i n the value of o.^ increases the perceived variance of the d i s t r i b u t i o n of perceptions. I t can be argued that the t a i l s of the d i s t r i b u t i o n became f a t t e r . From above, the mode of the d i s t r i b u t i o n s h i f t s to the l e f t . Then,it can be asserted with con-fidence that an increase i n to changes the d i s t r i b u t i o n i n the manner shown i n Figure 7. p(wc) denotes the new d i s t r i b u t i o n of per-ceptions of the reservation wage. Pl»P WC 1 WC MP wc Figure 7 - Gamma D i s t r i b u t i o n s with D i f f e r e n t Variances 110 Then, for r e l a t i v e l y low offers, the cumulative probability of acceptance increases, while, for r e l a t i v e l y high offers, the cumulative probability of acceptance declines. For values of the wage approximately equal to MP, the change i n the cumulative probability i s indeterminate because the points of intersection of the two curves are indeterminate as are the positions of the respective medians."' I I I . The Data A. Model I The exogeneous constants i n Model I are given the i n i t i a l values as shown i n Table 1. TABLE 1 The - Initial.:Exogeneous- Parameter Values in Model I • , V ~ — ~ : — — " N R D L w H H l n 30 5.26 5.26 120 0.75 10 10 8 N denotes the number of firms," R and D are the respective discount rates for each firm and each i n d i v i d u a l , 7 L i s the labour force s i z e , w i s the l e v e l of unemployment compensation, H, denote the respective horizons of job tenure and the returns from search and fi denotes the maximum l e v e l of employment a firm chooses to face. The marginal revenue product function, MRP(n), i s given the following form MRP(n) = 5 - n . . . . (14) I l l where n i s the l e v e l of employment. F i r m s , p r o f i t maximising i n a p e r f e c t l y competit ive i n d u s t r y , w i l l h i r e i n d i v i d u a l s up to the p o i n t at which m a r g i n a l revenue product equals the wage. Thus, i n ' p e r f e c t l y c o m p e t i t i v e ' e q u i l i b r i u m , each f i r m would h i r e 4 workers and pay a g wage 1.00. The l e v e l of unemployment compensation i s thus s e t below the e q u i l i b r i u m wage. MDI over the work h o r i z o n , H , f o r each 9 i n d i v i d u a l i s approximately 8.028. Discounted p r o f i t s over an i n f i n i t e h o r i z o n f o r each f i r m are 160.0."^ I f i n d i v i d u a l s could c o l l u d e to maximise t h e i r j o i n t e a r n i n g s , by f i x i n g a r e s e r v a t i o n wage,then they would choose the r e s e r v a t i o n wage, w, where a + w « = - £ 5 — = 2.875. . . . . (15) The corresponding l e v e l of employment, L , i s a - w L = ( - 2 2 — ) N = 64 , . . . . ( 1 6 ) 1 1 MDI f o r an employed i n d i v i d u a l i s 23.081 and mean p r o f i t discounted over an i n f i n i t e h o r i z o n i s 45.157. The t o t a l number of employment s t a t e s i s s e t at 9 , s i n c e fl i s set at 8. This value o f fi ensures that no p r o f i t maximising f i r m chooses to create s p e c u l a t i v e vacancies such that there i s a non-zero p r o b a b i l i t y o f the r e s u l t i n g l e v e l of employment exceeding fi. 112 The parameters c^, c^y t n e form of the density function of perceptions of the reservation wage are conceptually d i s t i n c t from the exogeneous labour market parameters shown i n Table 1. The former can be regarded as behavioural parameters while the l a t t e r , although influencing labour market behaviour, are technical para-meters and do not r e f l e c t q u a l i t a t i v e aspects of behaviour. The values of the exogeneous constants specified i n Table 1 and unit values of the perception parameters c^, constitute the set of parameter values from which, using the simulation procedure, the 'basic' solution to Model I i s obtained. Comparative s t a t i c predictions are derived by varying one parameter value and maintaining the remaining parameters at these specified values. Comparative s t a t i c predictions are derived with respect to the perception parameters c^, C 2 » the l e v e l of unemployment compensation, w, the horizon over which returns from search are computed, H^, and the length of the period over which decisions are made. B. Models I I and I I I Table 2 shows the i n i t i a l values of the labour market parameters adopted. TABLE 2 The I n i t i a l Exogeneous Parameter Values i n Models I I and I I I N R D L l L2 w H H l n 50 5.26 5.26 50 50 0.75 10 10 4 113 Using the usual notation, N denotes the number of firms, R and D are the respective discount rates for each firm and each i n d i v i d u a l , and are the sizes of each labour force, w denotes the l e v e l of un-employment compensation, H i s the horizon of job tenure, i s the horizon over which returns from search accrue and n denotes the maximum employment l e v e l the firm chooses to face. In both models, states of employment are enumerated by both number and composition. Given the complexity of the simulation procedures and the increased number of states, i t was decided to reduce the scale of the problem by setting the exogeneous marginal revenue product to MRP(n) = 3.0-n . • . . . . (17) In a 'perfectly competitive' world, each firm hires two individuals and pays a wage of 1.0. MDI i s again 8.028, but mean p r o f i t s over an i n f i n i t e horizon are 40.0. I f individuals could collude to maximise t h e i r j o i n t earnings, then they would choose a reservation wage, ft, where a + w ft = - ° 2 — = 1.875. . . . . (18) The corresponding l e v e l of employment, L, i s given by x a - w 1 9 L = (~ 22—)N = 56. . . . . ( 1 9 ) ^ 114 The corresponding l e v e l of MDI i s 15.053 and p r o f i t s discounted over an i n f i n i t e h o r i z o n are 12.656. Since n i s set at 4, the t o t a l number of s t a t e s , M, i s set at 15. I t i s p o s s i b l e that the r e s t r i c t i o n on the l e v e l of employment faced by the f i r m , imposed by the parameter, fi , i n f l u e n c e s the choice of f i r m s ' wage and vacancy c r e a t i o n d e c i s i o n s i n Models I I and I I I . Comparisons of the g e n e r a l c h a r a c t e r i s t i c s of s t o c h a s t i c e q u i l i b r i u m i n Model I w i t h the c h a r a c t e r i s t i c s of s t o c h a s t i c e q u i l i b r i u m i n these models suggest that the object of f i r m s ' d e c i s i o n s over most employment s t a t e s are not i n f l u e n c e d by t h i s 13 r e s t r i c t i o n on the l e v e l of employment. In Models I I and I I I , over a l l sets of parameter values adopted and over a l l o p t i m a l wage d e c i s i o n s , the l e v e l of employment c o n s i s t e n t w i t h the e x p l o i t a t i o n of i n t e r t e m p o r a l monopsony power c o i n c i d e s w i t h the s t a t i c p r o f i t maximising l e v e l of employment. Thus, having made i t s wage o f f e r , the f i r m never chooses to face a l e v e l of employment above that c o n s i s t e n t w i t h s t a t i c p r o f i t maximisation. Except when c u r r e n t l y employing 4 i n d i v i d u a l s and vacancy c r e a t i o n i s z e r o , the f i r m can always choose t o speculate i n vacancy c r e a t i o n . The parameter values s p e c i f i e d and u n i t values o f the p e r c e p t i o n parameters generate a ' b a s i c ' s o l u t i o n to each model u s i n g the s i m u l a t i o n procedure. In Model I , comparative s t a t i c p r e d i c t i o n s with respect to the t e c h n i c a l parameters, w ^ U and the p e r i o d length 115 and the b e h a v i o u r a l parameters c^, are d e r i v e d . A p r i o r i , the general tenor of such comparative s t a t i c r e s u l t s should be the same i n Models I I and I I I . Consequently, the comparative s t a t i c p r e d i c t i o n s d e r i v e d f o r those models are s o l e l y w i t h r e s p e c t to c ^ ( i = 1 ,2). IV. The S i m u l a t i o n Procedure A. The C a l c u l a t i o n of the P a r t i a l O p t i m i s a t i o n P o l i c y 1. Model I The f i r s t step i n the s o l u t i o n of the f u l l o p t i m i s a t i o n a l g o r i t h m i s the s o l u t i o n of the p a r t i a l o p t i m i s a t i o n a l g o r i t h m , which i s b r i e f l y o u t l i n e d i n Chapter 2 I I B . The f i r m , faced w i t h an unchanging labour market environment, must choose a p o l i c y to be adopted f o r ever which > maximises the present value o f mean p r o f i t s discounted over an i n f i n i t e h o r i z o n . H a d l e y ' s s o l u t i o n procedure i s to s p e c i f y an a r b i t r a r y p o l i c y , s a Y y~» which i s adopted by the f i r m from p e r i o d two onwards. The f i r m , i n employment s t a t e n and f a c i n g the f i x e d labour market e n v i r o n -ment, x , must choose a wage and vacancy c r e a t i v e d e c i s i o n , 5 , i n the f i r s t p e r i o d which maximises mean discounted p r o f i t s over an i n f i n i t e h o r i z o n , namely, V V y - . x ) = f ( n , 6 n ) + 3 ^ £ e n k ( S n , x ) l l ( k , x ) n = 0 , l , . . . , n . • (20) k=0 where f ( n , 6 n ) denotes mean one p e r i o d p r o f i t s for the f i r m which adopts d e c i s i o n ' , ; ; 5 , i n employment s t a t e n , 9 , (6 ,x) denotes the corresponding 116 state to state t r a n s i t i o n vector, R i s the rate of discount and H(k,x) denotes mean p r o f i t s discounted over an i n f i n i t e horizon of a firm, enjoying employment state k, which adopts the arbitrary p o l i c y , y-, for ever i n the labour market environment, x. The set of optimal f i r s t period decisions from the maximisation of Qn(^n»y^»x^ (n=0,l,... ,fi) may be denoted as y . The procedure, as outlined i n Chapter 2 IIB, i s to adopt y r as the arbitrary second period p o l i c y , by se t t i n g r = r , and then choose a new set of optimal f i r s t period decisions which maximises Qn0$n,y.-,x) • Wh e n t n e optimal f i r s t period policy and the arbitrary policy are i d e n t i c a l , the p a r t i a l optimisation problem has been solved. The e f f i c i e n c y of Hadley's algorithm arises from the reduction of the p a r t i a l optimisation problem from the determination of a policy consisting of 2(n+l) unknowns to a sequence of problems each requiring the solution of fi+1 sub-problems involving the choice of a one period decision, consisting of two unknowns, for each state of employment. I wish to outline i n some d e t a i l the solution of the sub-problem namely the choice of 6 to maximise Q (6 ,y-,x) because i t i s i n the J n x n n'^r' solution of this sub-problem where possible inaccuracies i n the simulation procedure appear. This might explain some of the inconsistent comparative s t a t i c r e s u l t s . The form of the maximand precludes the use of t r a d i t i o n a l calculus techniques. The approach adopted i s a non-exhaustive search over a grid of values of the wage offer and the i n t e g r a l vacancy decision. 117 o I n i t i a l l y , the f i r s t period decision adopted, § , i s equal to n' r the corresponding decision of the second period p o l i c y , 6^ . The value A o of the objective function, Q (5 »y~»x), i s computetd. The maximum value of the objective function corresponding to the wage decision, wn , i s found by comparing different values of the objective function for different vacancy creation decisions. The vacancy creation decision 14 i s i n i t i a l l y reduced by unity and the objective function evaluated. I f the objective function has increased, then the l e v e l of vacancy creation i s reduced further and the objective function again evaluated. The optimal vacancy creation decision i s found where the objective function i s maximised, that i s the l e v e l of vacancy creation such that a lower l e v e l of vacancy creation yields a lower value of the objective ~ o function. I f reducing the i n i t i a l l e v e l of vacancy creation from v n reduces the value of the objective function, then this i n i t i a l l e v e l of vacancy creation i s incremented by unity and the objective function re-evaluated. I f the objective function i s increasing, then the l e v e l of vacancy creation i s incremented again, and so on. This procedure yields the optimal vacancy creation decision for ~ o the prescribed wage of f e r , w^  . The wage offer i s reduced by i t s prescribed increment"''"' and the optimal vacancy creation decision i s computed according to the approach elaborated above. I f th i s lower wage offer and corresponding optimal l e v e l of vacancy creation yields a higher value of the objective function, then the wage offer i s reduced further by the prescribed increment, Qtherwise,the wage offer i s incremented above i t s i n i t i a l value, w^ 0, and the optimal vacancy decision computed. 118 The search procedure i s thus non-exhaustive i n that only values of the decision variables i n the neighbourhood of the i n i t i a l decision are considered and only then when the objective function i s increasing. To summarise, then, to find the optimal f i r s t period p o l i c y , when a p a r t i c u l a r policy i s adopted from period two onwards and the labour market environment i s unchanging over time, can be reduced to n+1 problems requiring the choice of a decision for each state of employment. Each such problem i s a two step procedure. (i) Choose the optimal l e v e l of vacancy creation associated with a par t i c u l a r wage offer by incrementing the vacancy creation decision i n the direction of increasing the value of the objective function u n t i l a l o c a l maximum i s determined. ( i i ) Use the same technique to find the wage decision corresponding to a l o c a l maximum by adopting ( i ) and varying the wage decision. This procedure i s computationally more e f f i c i e n t than an exhaustive search over two space,but w i l l only consistently y i e l d the correct solution when the objective function i s a monotonic function of both variables. Otherwise, i t i s possible that the optimal f i r s t period policy constitutes a l o c a l maximum rather than a global maximum of the objective function over a l l states of employment. 2. Model I I The p a r t i a l optimisation procedure i s the same i n Model I I because, while the t r a n s i t i o n vector computations are more complex (see Appendix I) and states of employment are defined over both l e v e l of employment and 119 type of i n d i v i d u a l , the firm s t i l l chooses a single wage and vacancy creation decision for each state of employment, given a p a r t i c u l a r policy i n period two and an unchanging labour market environment, to maximise mean p r o f i t s discounted over an i n f i n i t e horizon. 3. Model I I I The computation of the p a r t i a l optimisation policy i n Model I I I i s more complicated because the firm has to choose a wage and vacancy creation for each type of i n d i v i d u a l . The labour market environment - - -1 -2 x = (3,6 ,6 ) i s unchanging over time. The p a r t i a l optimisation procedure remains b a s i c a l l y the same. The arbitrary second period i s now defined as a wage and vacancy creation decision for each type of i n d i v i d u a l oyer a l l states of employment. The p a r t i a l optimisation procedure reduces to a sequence of problems each requiring the solution of M sub-problems, over 4 variables. Thus, a firm i n employment state m must choose wage and 1 2 vacancy creation decisions, (6 - ,6 ), to maximise mean discounted J m ' m '' pr o f i t s over an i n f i n i t e horizon M Q (6 1,6 2,y-,x) = f(m,6 1,6 2)+ Y, Q . (6 1,6 2,x)H(k,x) xm m'm'-'r' ' m ' m 1+R , . mkv m ' m ' ' k=0 m = 0,1, .. . , M . . . . (21) The solution procedure i s again a non-exhaustive stepwise search procedure. The value of the objective function i s computed when the * 1 " 2 arbitrary decisions (6^ ,6^ ) are adopted i n period one. The wage 120 " 1 ~ 2 offe r s to type one and type two i n d i v i d u a l s , (w ,wm ), are kept constant and the optimal vacancy creation decisions are found by an exhaustive search over the f e a s i b l e non-negative two-space of values of the vacancy creation decision. That i s the vacancy space defined by the requirement that m , *• * f- /ION e^ + e 2 + v^.+ v 2 < n . . . . (22) where v^, v 2 are the vacancy creation decisions. The wage to type one in d i v i d u a l s i s kept constant and the wage to type two i n d i v i d u a l s i s vari e d according to the p r i n c i p l e s elaborated i n A ( l ) . Having chosen the optimal type two wage o f f e r and vacancy creation decisions corresponding to the wage o f f e r to type one i n d i v i d u a l s , the type one wage o f f e r i s var i e d i n the same manner to f i n d the l o c a l maximum. This three step procedure may be summarised ( i ) Given wage o f f e r s to each type of individual,choose the optimal vacancy creation decision to each type of i n d i v i d u a l by the exhaustive search over the g r i d of i n t e g r a l values of vacancy creation decisions i n two space. ( i i ) Adopting step ( i ) , increment the wage o f f e r to type two i n d i v i d u a l s i n the d i r e c t i o n of increasing the value of the objective function u n t i l a l o c a l maximum i s found. ( i i i ) Using ( i ) and ( i i ) , increment the wage o f f e r to type one i n d i v i d u a l s i n the d i r e c t i o n of increasing the value of the objective function u n t i l a l o c a l maximum i s achieved."^ 121 B. Convergence C r i t e r i a As noted i n Chapter 2 I I B , the approach adopted i s , on s e c u r i n g the p a r t i a l o p t i m i s a t i o n p o l i c y , to compare consecutive labour market environments to t e s t f o r f u l l convergence of the a l g o r i t h m . I f the labour market environments d i f f e r by l e s s than a v e c t o r of c o n s t a n t s , o e, set at 0.001, then a steady s t a t e d i s t r i b u t i o n of employment i s computed which i s compared w i t h the employment d i s t r i b u t i o n c o n s t i t u t i n g the labour market environment. I f these two d i s t r i b u t i o n s d i f f e r by l e s s than the v e c t o r , . e , then f u l l o p t i m i s a t i o n of the a l g o r i t h m has been achieved. This convergence c r i t e r i o n was t ightened by r e q u i r i n g that 4 consecutive p a r t i a l o p t i m i s a t i o n p o l i c i e s be i d e n t i c a l before consecutive d i s t r i b u t i o n s of employment (the other v e c t o r s c o n s t i t u t i n g consecutive labour market environments) are compared. I f consecutive p a r t i a l o p t i m i s a t i o n p o l i c i e s are i d e n t i c a l , so that the optimal p o l i c y i n response to an unchanging labour market environment c o i n c i d e s w i t h the p o l i c y c o n s t i t u t i n g that labour market environment, then the labour market i s s a i d to be i n a s t a t e of temporary e q u i l i b r i u m . I f consecutive employment d i s t r i b u t i o n s d i f f e r e d by l e s s than the v e c t o r , e, a f t e r 4 i d e n t i c a l p a r t i a l o p t i m i s a t i o n p o l i c i e s , t h e n consecutive labour market environments are i d e n t i c a l up to an e r r o r , e, and the steady s t a t e employment v e c t o r i s computed. Tests with the procedure demonstrate that such a s t r i n g e n t convergence c r i t e r i o n i s e s s e n t i a l . ^ Otherwise, i t i s p o s s i b l e that 122 a number of p a r t i a l o p t i m i s a t i o n p o l i c i e s are i d e n t i c a l , and so the labour market i s i n temporary e q u i l i b r i u m , despite changing employment d i s t r i b u t i o n s over f i r m s . F i n a l l y , a new labour market environment could be generated which rendered these p a r t i a l o p t i m i s a t i o n s o l u t i o n s s u b -o p t i m a l . ^ I t was hoped that the adoption of an i n t e g r a l vacancy d e c i s i o n and a wage d e c i s i o n def ined i n increments of 0.1 would be s u f f i c i e n t to ensure convergence. U n f o r t u n a t e l y , i n most cases, the s i m u l a t i o n procedure f a i l e d to converge at t h i s degree of accuracy. The approach adopted i s to commence the s i m u l a t i o n by adopting wage increments of 0.1 . When i t appears that the labour market e n v i r o n -ment shows signs of s t a b i l i z i n g , the wage increment i s reduced to 0 . 0 5 . I f the a l g o r i t h m f a i l s to converge a f t e r a s i g n i f i c a n t number of 19 i t e r a t i o n s , then the increment i s reduced to 0.01 and f i n a l l y , where necessary, to 0.005. C. C y c l i n g I t i s f r e q u e n t l y observed that the a l g o r i t h m generates r e s u l t s that c y c l e . T h i s means t h a t for some i > m, where m i s a r b i t r a r i l y set a t 6, (P- x j + i ) < TI . . . . (23) where x^ i s the labour market environment generated a f t e r j i t e r a t i o n s 20 21 of the f u l l o p t i m i s a t i o n a l g o r i t h m and n i s a v e c t o r of constants. This means that the same labour market environment i s b e i n g r e c r e a t e d 123 during the s i m u l a t i o n . Consequently, the optimal p o l i c y a f t e r j + i i t e r a t i o n s i s the same as that a f t e r j i t e r a t i o n s and so on. m i s set at a h i g h number to ensure that the s i m u l a t i o n procedure i s c y c l i n g r a t h e r than converging v i a a sequence of temporary e q u i l i b r i a . I f the s i m u l a t i o n procedure c y c l e s , then the wage increments are again reduced and the procedure i s r e s t a r t e d . I f the s i m u l a t i o n procedure c y c l e s or f a i l s to converge a f t e r a s i g n i f i c a n t number of i t e r a t i o n s at the 0.005 l e v e l , then a new a r b i t r a r y i n i t i a l labour market environment i s i n t r o d u c e d and the procedure r e s t a r t e d w i t h the wage increments s e t at 0.05. The f u l l y o p t i m a l p o l i c y and the a s s o c i a t e d labour market e n v i r o n -22 ment are g e n e r a l l y unique f o r a p r e s c r i b e d increment i n the wage 23 d e c i s i o n . T h i s means that the s i m u l a t i o n procedure, i n a h i g h percentage of cases, generates a f u l l y o p t i m a l p o l i c y and a s s o c i a t e d l a b o u r market environment, independently of the choice o f the i n i t i a l l a b o u r market environment. T h i s suggests that the f u l l y opt imal p o l i c y and a s s o c i a t e d labour market environment are unique, when def ined to an a r b i t r a r y number of decimal p l a c e s . V. Problems A r i s i n g from the S i m u l a t i o n Procedure A . Non-Convergence Most sets of parameter v a l u e s , which do not generate a f u l l y opt imal s o l u t i o n , c y c l e at one or more of the p r e s c r i b e d wage i n c r e -ments. In the absence of i n f i n i t e computing r e s o u r c e s , i t i s i m p o s s i b l e to use the s i m u l a t i o n procedure over a t i g h t g r i d of parameter values to 124 delineate the parameter space between sets of parameter values which generate a solution to the algorithm and those which did not. A p r i o r i , there i s no reason why the problem should be badly behaved for some sets of parameter values and not for others. While i t i s unsatisfactory that some results are missing, so that some comparative s t a t i c comparisons are impossible, this problem i s less s i g n i f i c a n t than would be the case i f I wished to examine the nature of dynamic adjustment i n the labour market by the use of a modified version of the model. Then, the delineation of the parameter space i s necessary. An obvious explanation of the f a i l u r e of the simulation procedure to converge i s simply that the increment i n the wage decision i s too large. The true set of f u l l y optimal wage decision i s defined to a greater degree of accuracy than the minimum wage increment 0.005. A second possible explanation, as already mentioned, i s that the non-exhaustive search procedure for the partial-optimisation problem, i n some cases, yie l d s a policy which i s associated with a l o c a l maximum rather than a global maximum. B. Inconsistent Results Results are considered inconsistent when comparison of the f u l l y optimal p o l i c i e s r e s u l t i n g from different values of a p a r t i c u l a r parameter y i e l d different comparative s t a t i c predictions when, a p r i o r i , the predictions should be the same. Some of the explanations presented 125 to e x p l a i n non-convergence are a l s o r e l e v a n t here. The s o l u t i o n may be i n a c c u r a t e due to i n s u f f i c i e n t l y s m a l l increments i n the wage d e c i s i o n and, i n a d d i t i o n , the non-exhaustive search procedure. A f u r t h e r d i f f i c u l t y a r i s e s through the comparison of s o l u t i o n s which are def ined t o a d i f f e r e n t degree of accuracy i n the wage d e c i s i o n . T h i s occurs because of c y c l i n g and non-convergence. As noted i n I I I B , the convention adopted i s to reduce the wage increment when the s o l u t i o n c y c l e s . Some sets of parameter values may y i e l d f u l l y o p t i m a l s o l u t i o n s a t , say, the wage increment 0.01 and c y c l e at the lower wage increment 0.005,while other sets of parameter values may generate the opposite r e s u l t . Thus, comparative s t a t i c p r e d i c t i o n s are b e i n g made from f u l l y optimal p o l i c i e s which are defined to d i f f e r e n t degrees of accuracy i n the wage d e c i s i o n . E v i d e n t l y n o t h i n g can be done t o cure t h i s problem other than to p e r s i s t i n u s i n g the s i m u l a t i o n procedure w i t h the same sets of parameter values but d i f f e r i n g i n i t i a l l a b o u r market environments. In a d d i t i o n , i t i s p o s s i b l e that s m a l l changes i n c e r t a i n labour market parameters may l e a d to such s m a l l changes i n the true optimal p o l i c y that the combination o f the i n a c c u r a c y of the wage d e c i s i o n and the n o n - g l o b a l p a r t i a l o p t i m i s a t i o n procedure may l e a d to i n c o r r e c t r e s u l t s . For example, a 10% change i n the h o r i z o n over which r e t u r n s from search accrue from 10 to 11 has a s m a l l impact on the computation of the r e t u r n s from search and thus on the o p t i m a l p o l i c y , while a 10% 126 change i n the value of c^, the p e r c e p t i o n parameter, has a s i g n i f i c a n t impact on the nature of s t o c h a s t i c e q u i l i b r i u m and t h i s i s c o n s i s t e n t l y r e v e a l e d i n the r e s u l t s . I t should be emphasised, however, that I am 24 seeking q u a l i t a t i v e and not q u a n t i t a t i v e p r e d i c t i o n s . V I . Comparative S t a t i c Results Comparative s t a t i c r e s u l t s are generated over d i f f e r e n t values of exogeneous c o n s t a n t s , namely the p e r c e p t i o n parameters c^, c^, the l e v e l of unemployment compensation, w, the h o r i z o n over which the r e t u r n s from search are computed, H ^ , and the p e r i o d l e n g t h . The frequency with which a p a r t i c u l a r comparative s t a t i c p r e d i c t i o n of a change i n a summary s t a t i s t i c i s upheld i s presented i n a Table corresponding to the p a r t i c u l a r exogeneous constant. The frequencies r e f e r to 'weak' comparative r e s u l t s . For example, i f a summary s t a t i s t i c , SS, i s p o s t u l a t e d to be a d e c l i n i n g f u n c t i o n of the exogeneous c o n s t a n t , c^, then the exogeneous constant c^ generates one c o r r e c t p r e d i c t i o n i f SSCc^) < S S ^ - A C ; L ) . . . . (24) where Ac^ i s the increment i n the value of the exogeneous c o n s t a n t . I f k d i f f e r e n t values of c^ are adopted i n increments ;' of Ac^, then 25 k-1 p r e d i c t i o n s are generated. •' 127 CHAPTER 4  RESULTS FROM MODEL I I• Introduction A. The Characteristics of Stochastic Equilibrium The following features of stochastic equilibrium are observed for a l l values of the exogeneous constants adopted. Stochastic misperceptions of the reservation wage generate s i g n i f i c a n t wage and employment dispersion i n equilibrium. The d i s t r i b u t i o n of wage rates over employment states l i e s above the perfectly competitive equilibrium wage rate."*" The mean wage offers to searchers and employed individuals exceed the perfectly competitive wage but are less than the average 2 marginal product. Firms exhibit dynamic monopsony power i n that the wage offer i s a declining function of the l e v e l of employment. This relationship between the wage offer and the l e v e l of employment can be demonstrated 3 i n the i l l u s t r a t i v e model by comparative s t a t i c analysis. The l e v e l of vacancy creation i s also a declining function of the l e v e l of em-ployment. I f a l l firms f i l l the vacancies that they create and no workers quit, then a l l firms pay their workforces a wage above t h e i r respective marginal products. In a one period framework, i f behaviour i s non-stochastic, firms hire individuals up to the point at which the worker's marginal product 128 equals the wage o f f e r . In some s o l u t i o n s , over c e r t a i n employment s t a t e s , however, the l e v e l of employment c o n s i s t e n t with the e x -p l o i t a t i o n of i n t e r t e m p o r a l monopsony power (as def ined i n Chapter 2 IIIG) exceeds the s t a t i c , one p e r i o d , p r o f i t maximising l e v e l of employment. B u t , i n a l l s o l u t i o n s , t h e l e v e l of aggregate s p e c u l a t i v e vacancy c r e a t i o n , which i s def ined as aggregate vacancies i n excess of those c o n s i s t e n t w i t h the e x p l o i t a t i o n of i n t e r t e m p o r a l monopsony power, i s a s i g n i f i c a n t component of vacancy c r e a t i o n , exceeding 30% i n every s o l u t i o n . Thus, the s p e c i f i c a t i o n of s t o c h a s t i c turnover and acceptance behaviour causes s u b s t a n t i a l vacancy s p e c u l a t i o n on the p a r t of f i r m s . The aggregate mean l e v e l of vacancy c r e a t i o n exceeds aggregate mean unemployment. E q u i l i b r i u m i n a market i s c h a r a c t e r i s e d by zero excess demand, as c o n v e n t i o n a l l y d e f i n e d . I n a market i n which i n -formation i s p e r f e c t , at the e q u i l i b r i u m wage r a t e , no f i r m wishes to i n c r e a s e i t s l e v e l of employment and unemployment i s z e r o . In t h i s imperfect market, e q u i l i b r i u m i s c h a r a c t e r i s e d by non-zero l e v e l s of aggregate vacancy c r e a t i o n and aggregate unemployment. Aggregate vacancies exceed aggregate unemployment. As Reder [1969] argues, i n a labour market c h a r a c t e r i s e d by imperfect i n f o r m a t i o n , the average l e n g t h of time a vacancy i s open i s not r e l a t e d i n any p a r t i c u l a r way 4 to the average d u r a t i o n of unemployment, a p r i o r i . A necessary, but not s u f f i c i e n t c o n d i t i o n f o r s t o c h a s t i c e q u i l i b r i u m i n the labour market i s that on average the number of i n d i v i d u a l s q u i t t i n g t h e i r jobs equals the number of vacancies f i l l e d . " ' Convergence of the f u l l o p t i m i s a t i o n 129 algorithm demonstrates that st o c h a s t i c equilibrium has been attained i n the labour market. The question remains as to whether a measure of excess demand can be developed which r e g i s t e r s a constant value when the labour market i s i n stocha s t i c equilibrium, i r r e s p e c t i v e of the values of exogeneous parameters. The concept and measure of excess demand i s discussed i n Chapter 6. Comparison of the measures of mean discounted income indicates that, i n th i s environment, an i n d i v i d u a l enjoys the highest mean income by accepting the f i r s t job offered and remaining employed over h i s working horizon, i r r e s p e c t i v e of the wage o f f e r . I n d i v i d u a l s , who quit and accept o f f e r s according to the true reservation wage, earn a lower mean income over t h e i r job horizon and i n d i v i d u a l s , who behave s t o c h a s t i c a l l y , enjoy the lowest discounted income. The f i r s t two measures are hypothetical, since they are com-puted on the basis of one i n d i v i d u a l ' s non-stochastic behaviour, when a l l other i n d i v i d u a l s behave s t o c h a s t i c a l l y . The comparison of these measures, however, does suggest that i n d i v i d u a l s ' s t o c h a s t i c behaviour generates s i g n i f i c a n t v a r i a t i o n i n the l e v e l of employment faces by a fir m over time. Thus, a low current wage enjoyed by an,in d i v i d u a l w i l l not p e r s i s t , i f he remains employed with the same firm. This hypothesis i s confirmed by the observation of the temporal c o r r e l a t i o n of wage o f f e r s over the d i f f e r e n t s o l u t i o n s . The temporal c o r r e l a t i o n i s a d e c l i n i n g function of time and, i n some cases, assumes a negative value which i s small, however, i n absolute terms. 130 This demonstrates that there i s l i t t l e information i n a current wage o f f e r about the future stream of wage of f e r s facing an i n d i v i d u a l who accepts the o f f e r . These remarks must be q u a l i f i e d because the f i r m makes decisions on the basis of i t s perceptions of i n d i v i d u a l s ' behaviour. I f a l l i n d i v i d u a l s adopted the same ex ante [0,1] decision r u l e f o r search, then the industry equilibrium would be characterised by the c o l l u s i v e monopsony s o l u t i o n . ^ The term, 'mean discounted income', or MDI w i l l now always r e f e r to l i f e t i m e income from stoc h a s t i c behaviour. Mean discounted p r o f i t over an i n f i n i t e horizon i s a p o s i t i v e function of the i n i t i a l l e v e l of employment. This r e s u l t can be j u s t i f i e d on the grounds that the greater i s the i n i t i a l l e v e l of employment the greater i s the firm's monopsony power. While unable to f i r e employees, a firm, faced with a high l e v e l of employment, can always e x p l o i t i t s monopsony power by o f f e r i n g a r e l a t i v e l y low wage, so that there i s a r e l a t i v e l y high p r o b a b i l i t y of securing a l e v e l of employment consistent with intertemporal p r o f i t maximisation at the p r e v a i l i n g wage. Mean discounted p r o f i t over an i n f i n i t e horizon increases at a decreasing rate as the i n i t i a l l e v e l of employment increases. This r e f l e c t s the diminishing marginal p r o d u c t i v i t y of workers as employment increases. B. An I l l u s t r a t i v e Solution Table 3 shows the f u l l optimisation s o l u t i o n generated by the 131 basic parameter values, that i s those values specified in Table 1 and unit values of perception parameters. TABLE 3 The Basic Solution to Model I Stochastic Equilibrium Mean One Profits over Level of Steady State Wage Vacancy Period an Infinite Employment Distribution Offer Creation Profits Horizon 0 0.020 2.165 4 2.04 80.03 1 0.126 2.085 3 3.26 82.34 2 0.284 1.975 2 4.00 83.61 3 0.317 1.825 2 4.43 84.26 ! 4 0.196 1.710 2 4.69 84.60 i 5 0.051 1.655 1 4.86 84.83 6 0.006 1.580 1 4.98 84.96 7 0.000 1.520 1 5.08 85.06 8 0.000 1.505 0 5.13 85.11 Summary Statistics EW* MW W* VW* LS V V s U PA* po* j En 1.722 1.852 1.609 0.173 0.546 63 45 38 1.561 1.166 j 83.843 Correlation of Offers 1 period • T — „ • • • - — , i • - I... .1 I . . . . . , •• - ~ 2 periods j 3 periods j 4 periods 5 periods 0.379 0.151 1 0.032 1 0.012 0.011 Mean Discounted Income No quitting True Perception 14.446 14.354 Stochastic Behaviour 11.998 132 Steady State Transition Matrix 0.314 0 .369 0 .212 0 .080 0.024 0 . 0 0 . 0 0 . 0 0 . 0 ! 0 .043 0 .333 0 .352 0.197 0.076 0 . 0 0 . 0 0 . 0 0 . 0 1 0.012 0 . 1 2 1 0 .366 0 . 3 4 1 0.160 0 . 0 0 . 0 0 . 0 0 . 0 I 0.009 0 .079 0 .252 0 .360 0 . 2 3 1 0 .069 0 . 0 0 . 0 o o ! 0 .009 0 .070 0.209 0.319 0 . 2 6 1 0 . 1 1 1 0 . 0 2 1 0 . 0 0 . 0 i 0 . 0 0 8 0 . 0 6 1 0 .188 0.306 0 .278 0 .133 0.026 0 . 0 o o ! 0 .010 0 .064 0 . 1 8 1 0 .283 0 .265 0 .147 0 .045 0.006 o .o ! 0 . 0 1 1 0 .067 0 .179 0.269 0 .253 0 . 1 5 1 0 .056 0.012 o.oov 0 . 0 1 1 0 . 0 6 8 0 . 1 7 8 0 .268 0.252 0 .152 0 .057 0.012 0 . 0 0 1 As can be observed the solution exhibits significant wage and employment dispersion. The wage offers l i e i n the ranges 2 .165 to 1 . 5 0 5 . Over 90% of firms face employment levels of between 1 and 4 workers. II. Comparative Static Predictions A. An Introduction In order to organise and analyse the comparative static results, summary st a t i s t i c s are defined. The changes i n the values of these summary st a t i s t i c s describe the impact on stochastic equilibrium in the labour market of a change i n the value of the particular exogeneous parameter. The summary st a t i s t i c s are the following : g (a) the level of the general wage offer distribution; (b) the mean wage offer facing a searcher; (c) the variance of the wage offer distribution facing a searcher; (d) the mean wage enjoyed by an employed individual; (e) mean discounted income enjoyed by an individual; (f) the level of aggregate vacancy creation; 133 (g) the l e v e l of aggregate vacancy speculation; (h) the l e v e l of aggregate unemployment; (i) the mean duration of search p r i o r to an offer; (j) the mean duration of unemployment; (k) mean p r o f i t s discounted over an i n f i n i t e horizon earned by the firm; (1) labour's share of t o t a l output; and (m) the one period correlation of wage offers. B. Systematic Misperceptions of the Reservation Wage 1. The Results Solutions to Model I are obtained for different values of the parameter, c^. The values of c^ adopted l i e i n the range 0.90 to 1.25 i n increments of 0.05. The comparative s t a t i c predictions, generated for di f f e r e n t values of c^, are consistent with respect to the algebraic signs of most of the changes of the summary s t a t i s t i c s computed. An increase i n the parameter, c^, leads to a s h i f t to the right of the wage offer d i s t r i b u t i o n . Indeed, the wage offer over a l l employment states increases. The mean wage offer facing an unemployed searcher and the mean wage rate earned by an employed worker both increase. The variance of the wage offer d i s t r i b u t i o n i s p o s i t i v e l y related to the value of c^, up to the value 1.25. The one period correlation between wage offers decreases as c^ increases. The other temporal correlation coefficients do not show any systematic r e l a t i o n over different values 134 MDI i s p o s i t i v e l y r e l a t e d with the value of up to the value 1.15. The mean wage f a c i n g an employed i n d i v i d u a l only exceeds the c o l l u s i v e r e s e r v a t i o n wage when c^ equals 1.25. Thus, due to s t o c h a s t i c behaviour, the r e l a t i o n s h i p of the one p e r i o d c o l l u s i v e r e s e r v a t i o n wage and the mean wage f o r the employed does not i n d i c a t e the r e l a t i o n s h i p of MDI to the mean wage, over d i f f e r e n t values of c^. An i n c r e a s e i n c^ leads to an i n c r e a s e i n the l e v e l of vacancy c r e a t i o n over some s t a t e s of employment and the same l e v e l of vacancy c r e a t i o n over a l l the other s t a t e s . Aggregating over f irms y i e l d s measures of unemployment and vacancy c r e a t i o n . Both these measures are p o s i t i v e l y c o r r e l a t e d with the value of c^, but there appears to be no simple r e l a t i o n s h i p between these measures. The l e v e l o f aggregate s p e c u l a t i v e vacancies i s a l s o p o s i t i v e l y r e l a t e d to the value of c^. The mean d u r a t i o n of unemployment i s p o s i t i v e l y c o r r e l a t e d to 9 the value of c^, while the mean d u r a t i o n of search p r i o r to an o f f e r shows no systematic r e l a t i o n s h i p to c^. The mean d u r a t i o n of search appears to i n c r e a s e or decrease a c c o r d i n g to whether aggregate u n -employment i n c r e a s e s more or l e s s than aggregate vacancy c r e a t i o n . E i g h t values of c^ are adopted and so i t i s p o s s i b l e to generate seven sets of comparative s t a t i c p r e d i c t i o n s . Table 4 shows the values of the summary s t a t i s t i c s corresponding to each value of c^ and the frequency w i t h which the comparative s t a t i c p r e d i c t i o n s are upheld. The frequencies r e f e r to 'weak' comparative s t a t i c r e s u l t s . TABLE 4 Results arid Comparative Static Predictions  for Misperceptions of the Reservation Wage Summary St a t i s t i c 0.90 0.95 The Perception Parameter, c, 1.20 1.25 Comparative Statics Sign Frequency/7 1.00 1.05 1.10 1715 (a) _ _ _ _ _ _ — — 7 (b) 1.494 1.560 1.722 1.833 1.894 2.099 2.348 2.757 + 7 (c) 0.137 0.163 0.173 0.232 0.291 0.311 0.358 0.331 + 7 (d) 1.614 1.697 1.852 2.005 2.108 2.292 2.549 2.909 + 6 (e) 11.119 11.528 11.998 12.255 12.318 12.568 12.383 11.868 5 (f) 53 53 63 67 68 81 93 119 + 7(-l) (g) 20 20 45 45 43 51 54 95 + 6 (-2) (h) 30 32 38 45 50 56 67 79 + 7 (i) 1.215 1.227 1.166 1.184 1.211 1.164 1.136 1.081 7 (j) 1.484 1.533 1.561 1.708 1.843 1.956 2.298 2.930 + 7 (k) ^100.949 94.668 83.843 73.854 67.425 56.254 42.798 27.039 + 7 (1) 0.488 0.512 0.546 0.575 0.593 0.632 0.678 0.742 + 7 (m) 0.365 0.555 0.379 0.333 0.304 0.310 0.232 0.200 . — 5 136 2. An E x p l a n a t i o n In the i l l u s t r a t i v e model, i t can be demonstrated that i f , c e t e r i s p a r i b u s , a f i r m faces a h i g h e r p r o b a b i l i t y of turnover on the p a r t of i t s c u r r e n t employees and a lower p r o b a b i l i t y of a c c e p t -ance on the p a r t of job seekers i n response to a given wage o f f e r , then a s u f f i c i e n t c o n d i t i o n f o r i t s p r o f i t maximising wage o f f e r to i n c r e a s e i s that the marginal r a t e of turnover (and t h e r e f o r e a c c e p t -13 ance) remains constant or i n c r e a s e s . In Chapter 3 I I , i t i s argued t h a t , c e t e r i s p a r i b u s , i n response to an i n c r e a s e i n the p e r c e p t i o n parameter, c^, firms o f f e r i n g a wage above the mean of the d i s t r i b u t i o n of r e s e r v a t i o n wages, MP, face a h i g h e r absolute and marginal r a t e of turnover by employees and a lower r a t e of acceptance and a h i g h e r marginal acceptance r a t e on the p a r t o f job s e e k e r s . The majority o f f irms i n the i n d u s t r y o f f e r s wages above the mean of the d i s t r i b u t i o n of p e r c e p t i o n s , MP. I f a l l firms i n the i n d u s t r y choose to adopt the same wage and vacancy c r e a t i o n d e c i s i o n s , d e s p i t e the systematic change i n i n d i v i d u a l behaviour, then the steady s t a t e l e v e l of unemployment would be h i g h e r . I f a f i r m r a i s e s i t s wage o f f e r i n response to t h i s systematic change i n i n d i v i d u a l behaviour, then the p r o b a b i l i t y of a c h i e v i n g a h i g h , t h a t i s o p t i m a l , l e v e l of employment (which i s dependent on the wage o f f e r ) i s i n c r e a s e d and thus h i g h e r mean p r o f i t s are earned than by adopting the previous wage decision."''"' 137 I f the majority of firms increase wage off e r s , however, the true reservation wage rises and so there i s a systematic change i n i n d i v i d -uals' behaviour as noted i n Chapter 3. Thus,the higher mean p r o f i t s enjoyed by a firm, i f i t alone raises i t s wage off e r , i s o f f s e t , i f a majority of firms i n the industry increase t h e i r wage offers and thus the general d i s t r i b u t i o n of offers. Firms, who i n i t i a l l y offer a wage below the reservation wage, face a higher probability of maintaining a high l e v e l of employment i n response to the change i n individuals' behaviour. The r i s e i n the general wage d i s t r i b u t i o n , however, caused by decisions of high wage firms, forces these low wage firms to increase t h e i r wage off e r s . Thus, the wage d i s t r i b u t i o n s h i f t s to the ri g h t . The l e v e l of aggregate unemployment rises for two reasons. F i r s t l y , i n a labour market i n which individuals behave s t o c h a s t i c a l l y , given a higher d i s t r i b u t i o n of wage off e r s , firms choose to employ fewer individuals i n the aggregate,due to the declining marginal productivity of labour. I t i s plausible, then, that since firms increase the general d i s t r i b u t i o n of wage offers i n response to the increase i n c^, the mean l e v e l of unemployment would be higher. Secondly, an ind i v i d u a l i s less valuable to the firm because the mean returns over time associated with any given stream of wage offers f a l l s i n response to th i s systematic change i n behaviour. Thus, firms do not increase wage offers s u f f i c i e n t l y to ensure that the previous aggregate l e v e l of employment i s maintained. 138 In a d d i t i o n to o f f e r i n g higher wages to lower the p r o b a b i l i t y of e x i s t i n g employees q u i t t i n g and to increase the p r o b a b i l i t y of searchers accepting o f f e r s , the fir m can increase the number of positio n s offered to searchers by increasing the l e v e l of vacancy creation. An increase i n the l e v e l of vacancy creation r a i s e s the p r o b a b i l i t y of achieving a high l e v e l of employment and thus high p r o f i t s . While there are d i r e c t costs of increasing the wage o f f e r , increased vacancy creation per se i s c o s t l e s s , except that the speculative nature of vacancy creation may lead to firms facing a higher l e v e l of employment than they would choose i n a non-stochastic labour market. There i s a tradeoff, therefore, between increased vacancy creation and an increased wage o f f e r . Both lead to a higher mean l e v e l of employment. I t i s quite consistent for the fir m to increase i t s wage o f f e r and the l e v e l of vacancy creation i n response to t h i s systematic change i n behaviour. The increase i n aggregate unemployment means that the steady state d i s t r i b u t i o n of employment has s h i f t e d towards low employment states. Vacancy creation i s a d e c l i n i n g function of the l e v e l of employment. Thus, even i n the absence of increased vacancy creation over some states, aggregate vacancy creation r i s e s . Since the wage o f f e r d i s t r i b u t i o n s h i f t s to the r i g h t , the l e v e l of employment associated with each wage o f f e r , which i s con-s i s t e n t with the e x p l o i t a t i o n of intertemporal monopsony power, remains constant or f a l l s . In response to an increase i n c^, vacancy creation over each employment state remains constant or r i s e s . Thus, i t i s p l a u s i b l e that aggregate vacancy speculation i s an increasing function of' c. . 139 The increase i n aggregate vacancy creation and the s h i f t i n the wage offer d i s t r i b u t i o n leads to an increase i n the mean wage offer facing a searcher, despite the increase i n unemployment. This result i s plausible but d i f f i c u l t to prove, since the mean wage offer i s dependent on the d i s t r i b u t i o n of vacancies and wage offers over firms and the l e v e l of unemployment. Since the wage offer d i s t r i b u t i o n r i s e s , the mean wage enjoyed by an employed i n d i v i d u a l , MW, r i s e s . Despite the increase i n aggregate unemployment, MDI enjoyed by an in d i v i d u a l r i s e s up to the point that c^ equals 1.15. The aggregate l e v e l of unemployment i n stochastic equilibrium, associated with a p a r t i c u l a r value of MW, always exceeds the hypothetical measure of unemployment, computed under the assumption that a l l firms offer the wage, MW, and p r o f i t maximise i n a labour market, characterised by perfect information. Thus, the comparison on the mean wage, MW, i n this multi-period stochastic model of the labour market with the s t a t i c collusive acceptance wage i s not an accurate indicator of the magnitude of MDI. I t i s consistent that the s h i f t of the d i s t r i b u t i o n of offers to the right should be accompanied by an increase i n the variance of offers. The duration of unemployment rises and again the result i s plausible but d i f f i c u l t to prove. I f individuals faced the same probability d i s t r i b u t i o n of offers whether employed or unemployed, then i t follows that, i f turnover and acceptance behaviour are com-, plementary, a higher rate of unemployment implies a higher probability 140 of q u i t t i n g a job and a lower probability of accepting an offer and a longer duration of unemployment. In this model, the state of un-employment can be regarded as a transitory state for the indiv i d u a l between low wage, high employment states and high wage,low employment states. Thus, while turnover and accpetance behaviour are complementary, searchers have a higher probability of receiving a high offer i f they receive a non-zero of f e r , than individuals currently employed, since both wage offer and vacancy creation are declining functions of the l e v e l of employment. There i s no systematic relationship between the mean duration of search arid c^. The duration of search i s p o s i t i v e l y correlated with the sign of the change of the measure of aggregate vacancies minus aggregate unemployment. This i s quite plausible, since an increase i n the l e v e l of vacancy creation increases, and an increase i n the l e v e l of unemployment decreases the probability of an o f f e r , ceteris  paribus. Over each employment state, firms earn lower p r o f i t s , since they offer higher wage rates. The steady state d i s t r i b u t i o n of firms has shifted to low employment, low p r o f i t firms. Thus,mean industry p r o f i t s are lower. Labour's share of t o t a l output therefore r i s e s . As c^ increases, firms choose to employ less workers i n the aggregate, since they receive higher wage offers and are less valuable. Then the probability that an i n d i v i d u a l accepts a job from a firm offering a wage, w , f a l l s . Likewise, the probability increases that an i n d i v i d u a l , 141 currently employed at a firm offering a wage,wn, quits. Thus, there i s , i n some sense, more stochastic v a r i a t i o n i n the labour market. Then,it i s plausible that the one period correlation c o e f f i c i e n t declines as c^ increases. C. The Level of Unemployment Compensation 1. The Results Comparative s t a t i c predictions are derived for values of unem-ployment compensation, w, ly i n g between 0.60 and 0.95 i n increments of 0.05. In addition, a solution for a zero l e v e l of unemployment compensation i s derived. Six sets of comparative s t a t i c predictions are generated. The results are somewhat ambiguous. An increase i n the l e v e l of unemployment compensation leads to a s h i f t of the wage offer d i s t r i b u t i o n to the right. The mean wage earned by an employed i n d i v i d u a l rises as does the mean wage of a searcher. The variance of offers f a l l . The l e v e l of MDI enjoyed by an i n d i v i d u a l f a l l s . The changes i n the levels of aggregate unemployment and aggregate vacancy creation are not systematic. Both measures simultaneously r i s e twice, f a l l twice and remain constant twice. The measure of aggregate speculative vacancy increases twice, remains constant three times and f a l l s once. 1 Changes i n the mean duration of search are negatively related to changes i n the aggregate measure of vacancies minus unemployment. The mean duration of unemployment generally f a l l s i n response to an increase i n unemployment compensation. The l e v e l of mean industry 142 p r o f i t s generally f a l l s , while labour's share of t o t a l output generally r i s e s . The one period correlation c o e f f i c i e n t of wage offers f a l l s as the l e v e l of unemployment compensation r i s e s . When the model i s solved for a zero l e v e l of unemployment compensation, stochastic equilibrium i s characterised by wage dispersion, vacancy creation and f r i c t i o n a l unemployment. The wage offer d i s t r i b u t i o n l i e s above the perfect competitive wage rate. The average marginal product exceeds the mean wage of employees. The summary s t a t i s t i c s corresponding to a l l these results are shown i n Table 5. 2. An Explanation Ceteris paribus, an increase i n the l e v e l of unemployment compensation reduces the opportunity costs of search, W^. - w, where W-j. denotes the most recent offer received by a searcher or an employed i n d i v i d u a l , and also increases the mean of f e r , EW*, facing a searcher given an unchanging offer d i s t r i b u t i o n . From [7] and [8] i n Chapter 2 IB the true reservation wage i s given by - H l 1 i W* = w + E x ( ^ ) (EW* - W*). . . . . (2) i = l W* - w are the costs of search. The in d i v i d u a l w i l l receive a wage off e r , EW*, after one search, which he w i l l enjoy over a horizon, H . 143 W* = w + Z.EW* 1 + Z (3) where Z (1 + D) 1 i (4) i = l Then dW* 1 + Zcj) 1 + Z > 0 df? (5) where cj> denotes the probability of a zero offer. Rewriting (6) i n Chapter 2, since w constitutes the f l o o r of the d i s t r i b u t i o n of offers. Then, ceteris paribus, an increase i n w, the l e v e l of unemployment compensation, increases the true reservation wage but reduces the true variance of offers. The o v e r a l l impact on the shape of the probability d i s t r i b u t i o n of perceptions of the reservation wage i s unclear. The increase i n the mean of the perceptions d i s t r i b u t i o n implies that individuals have a lower p r o b a b i l i t y , ex ante, of accepting a r e l a t i v e l y high wage o f f e r , while the reduction i n the variance of the perceptions d i s t r i b u t i o n means that individuals have a higher p r o b a b i l i t y , ex ante, of accepting a r e l a t i v e l y high offer. (6) dVW* dw = 2(w - EW*) $ < 0 (7) 16 TABLE 5 Results and Comparative S t a t i c Predictions  for Different Levels of Unemployment Compensation Summary JFJie_L§yel .Qf_Une.mp.lQymen.tL_a, nd. Comp.ensarJ.on^_w Comparative Sta t i c s S t a t i s t i c 0.0 0.65 0.70 } 0.75 0.80 0.85 0.90 0.95 Sign Frequency/6 (a) - -i i 1 — — • — — — + 4 (b) 1.456 1.547 1.689 1.722 1.625 1.675 1.704 1.709 + 5 (c) 0.270 0.206 0.178 0.173 0.178 0.180 0.174 0.153 - 4 .(d) 1.581 1.714 1.821 1.852 1.788 1.841 1.864 1.858 + 4 (e) 9.114 11.185 11.702 11.998 11.880 12.175 12.445 12.591 + 5 . ( f ) . 69 55 63 63 56 57 57 56 + 4 (8) 32 22 36 45 30 30 30 30 + 5 (-3) (h) 33 35 38 38 36 38 38 37 + 4 (-2) ( i ) 1.117 1.237 1.166 1.166 1.242 1.247 1.249 1.246 ? 6(-2) (j) 1.440 1.590 1.563 1.561 1.600 1.646 1.642 1.619 - 4 (k) 99.235 92.655 85.368 83.843 88.759 84.799 83.814 84.621 - 4 (1) \ 0.478 0.510 0.537 0.546 0.529 0.541 0.548 0.548 + 5 ( - l ) (m) 0.441 0.509 0.391 0.379 0.373 0.432 0.449 0.397 . - 4 145 The Increase i n MDI and the mean offer facing a searcher and the decrease i n the variance of offers can be d i r e c t l y attributed to the increase i n the l e v e l of unemployment compensation. The s h i f t of the wage offer d i s t r i b u t i o n to the right and the resultant increase i n the mean wage offer enjoyed by an employed in d i v i d u a l suggest that the impact on the perception d i s t r i b u t i o n of the reservation wage of the increase i n the mean dominates the impact of the decrease i n variance. But, neither the mean duration of un-employment nor the aggregate l e v e l of unemployment increase system-a t i c a l l y , as the l e v e l of unemployment compensation increases. I t must be concluded, therefore, that the impact of an increase i n the l e v e l of unemployment compensation on the nature of stochastic equilibrium i n the labour market i s indeterminate,due to the con-f l i c t i n g influences of increased mean and decreased variance on the form of the perception d i s t r i b u t i o n of the reservation wage. These results c o n f l i c t with the t r a d i t i o n a l position taken by economists. For example, Grubel, Maki and Sax [1973] develop a model to investigate the impact of i n s t i t u t i n g unemployment compensation. They argue that an increase i n unemployment compensation lowers the e x p l i c i t costs of being unemployed. Then, assuming le i s u r e and unemployment are equivalent, the price of le i s u r e i s reduced. More lei s u r e i s consumed because i t i s a normal good. The analysis i s p a r t i a l , because firms' behaviour i n response to t h i s reduced labour-force p a r t i c i p a t i o n i s not analysed. 146 The c r u c i a l d i f f e r e n c e i n formulation i n my model of the imperfect market i s that the p e r c e i v e d v a r i a n c e of o f f e r s i n f l u e n c e s i n d i v i d u a l s 1 turnover and acceptance behaviour. The l e v e l of unemployment compen-s a t i o n i s an element of the d i s t r i b u t i o n of o f f e r s . In c o n t r a s t to the simple model developed by G r u b e l , Maki and Sax [1973], unemployment compensation does not unambiguously i n c r e a s e the ex ante q u i t p r o b a b i l i t y or reduce the ex ante p r o b a b i l i t y of acceptance i n response to a p a r t i c -u l a r wage o f f e r . I n a d d i t i o n , by s p e c i f y i n g f i r m behaviour the model i s set i n a general e q u i l i b r i u m framework. F i r m s , t h e r e f o r e , respond to changes i n i n d i v i d u a l s ' labour market behaviour by a d j u s t i n g t h e i r wage o f f e r and vacancy c r e a t i o n d e c i s i o n s . Consequently, i n t h i s model, the l e v e l of unemployment does not unambiguously r i s e i n response to an i n c r e a s e i n unemployment compensation. The s o l u t i o n to Model I , corresponding to a zero l e v e l of unem-ployment compensation, i s s i g n i f i c a n t i n that i t demonstrates that i t i s not the r e l a t i v e l y h i g h l e v e l of unemployment compensation which accounts f o r both wage d i s p e r s i o n and the wage o f f e r d i s t r i b u t i o n l y i n g above the p e r f e c t l y competit ive wage. Thus, i r r e s p e c t i v e of the l e v e l of unemployment compensation, firms do not choose to create a l a r g e number of vacancies and o f f e r a wage below the p e r f e c t l y competit ive wage such that the l e v e l of employment c o n s i s t e n t w i t h s t a t i c n o n - s t o c h a s t i c p r o f i t maximisation, and, perhaps, a l s o with the e x p l o i t a t i o n of i n t e r t e m p o r a l monopsony power, r i s e s above that c o n s i s t e n t w i t h f u l l employment. 147 D. The Horizon over which Returns from Search Accrue 1. The Results Comparative s t a t i c predictions are derived for values of H^ between 5 and 12 i n increments of unity. The algorithm did not converge for H^ set at 6 despite exhaustive testing. Although solutions to an accuracy of 0.005 i n the wage offer were obtained i n 6 of the 7 solutions, the results obtained are not consistent for one period changes i n H^. This can be confirmed by observation of Table 6. I f comparative s t a t i c predictions are made over 2 period changes i n H^, then the results are more satisfactory. An increase i n the horizon, over which individuals believe the returns from search accrue, leads to a s h i f t to the right of the wage offer d i s t r i b u t i o n . Mean wage offers to both searchers and employed individuals r i s e along with the variance of the offer d i s t r i b u t i o n . The one period correlation of wage offers f a l l s , but not consistently. MDI increases and mean p r o f i t s discounted over an i n f i n i t e horizon f a l l s . Labour's share of t o t a l output r i s e s . Aggregate measures of vacancy creation, speculative vacancy creation and unemployment r i s e . Changes i n the duration of search p r i o r to an offer are not correlated i n a consistent way to changes i n aggregate vacancies minus unemployment. The mean duration of unemployment p r i o r to an offer increases. 148 2. An Explanation Increasing the horizon over which returns from search are believed to accrue increases the perceived returns for an unemployed i n d i v i d u a l . From (3), dW* = rEW*_ (w ••+ ZEW*) , dZ f f ii" 1 + 2 " (1+Z)2 - f > 0 (8) (1+Z) 2 ^ 1 since EW* > w and dZ > 0. Then, ceteris paribus, an increase i n the dH 1 value of increases the mean of the perceptions d i s t r i b u t i o n of the reservation wage. Consequently, the comparative s t a t i c predictions should coincide with results obtained for systematic misperceptions of the reservation wage. There are two related explanations that can explain the incon-s i s t e n t results obtained from unit changes i n H^. (a) The unit change i n does not substantially change the nature of i n d i v i d u a l behaviour and, given the inherent inaccuracy of the solution algorithm,(due to the arbitrary size of the wage increment, the solutions are not consistent. (b) The non-exhaustive search procedure has generated non-optimal solutions. The second explanation i s not too convincing, however, i n that such inconsistent solutions were not generated i n other comparative s t a t i c experiments. TABLE 6 Result3 and Comparative S t a t i c Predictions  for Different Horizons over which Returns from Search Accrue Comparative Stat i c s Summary The Horizon over which Returns Accrue, H. Frequency/5 Frequency/5 S t a t i s t i c 5 7 " 8 9 10 11 12 Sign Unit Period Two Periods (a) i + 3 5 (b) 1.567 1.577 1.712 1.594 1.722 1.607 1.780 + 3 5 (c) 0.168 0.176 0.169 0.182 0.173 0.191 0.190 + 2 5 (d) 1.714 1.734 1.840 1.753 1.852 1.778 1.921 + 3 5 (e) 11.486 11.527 11.934 11.633 11.998 11.632 12.143 + 3 4 (f) 55 55 63 55 63 56 65 + 3 5 (-3) (g) 18 22 36 22 45 30 46 + 3 5(-l ) (h) 34 35 38 35 38 37 41 + -3 5(-2) 1.233 1.166 1.166 1.237 1.166 1.244 1.163 ? 4 2(-l) (j) 1.567 1.583 1.561 1.584 1.561 1.624 1.612 + 2 5(-l) (k) ^ 93.162 91.901 84.438 90.774 83.843 88.636 79.252 - 3 5 (1) 0.512 0.516 0.542 0.521 0.546 0.524 0.559 + 3 5 (m) 0.482 0.832 0.370 0.439 0.379 0.405 0.358 — 3 3 150 E. The Period Length 1. The Results Comparative s t a t i c predictions are derived from Model I for different values of the period length. The parameter, F, denotes the new period length i n terms of the standard period. An increase i n F then implies an increase i n the period length. For consistency, the number of new periods which constitute the work and search horizons are defined as H l H 1 = . . . . (9) H = | . . . . (10) where H^, H denote the new search and work horizons respectively, expressed i n terms of the new period length. Thus, the actual horizons of search, H^F, and work, HF, are equal to the horizons of search and work i n the basic solution. A value of F greater than unity implies that both firms and individuals make decisions less frequently. Likewise, the respective rates of discount and the l e v e l of unemployment compensation are adjusted according to the period length, whence R = RF . . . . (11) D = DF . . . . (12) w = wF . . . . (13) where R and D represent the new rates of discount for firms and 151 individuals,respectively,and w i s the adjusted rate of unemployment compensation. An increase i n F, the length of the decision period, increases the l e v e l of unemployment compensation and also the respective rates of discount for firms and workers. As a re s u l t , future returns are discounted at a. higher rate, but less frequently. The procedure adopted i s to simulate the model given the new period length. The sample s t a t i s t i c s describing stochastic equilibrium are a l l defined i n terms of the new period length. For purposes of comparison,, with the standard period, the mean wage offer for an unemployed i n d i v i d u a l and the mean wage for an employed i n d i v i d u a l are then adjusted by a factor, ^  , the variance of offers i s adjusted r by the factor, — , and both the mean duration of unemployment and the F mean duration of search p r i o r to an offer are adjusted by the factor, F. The measures of aggregate vacancy creation and unemployment are constant over time,once stochastic equilibrium i s achieved,and are compatible, therefore, for different values of F. I t i s necessary to write a simple computer program to convert measures of MDI and mean p r o f i t s discounted over an i n f i n i t e horizon to equivalent measures based on 18 the standard period length and discount rate. These new sample s t a t i s t i c s are compatible and thus comparable. Values of ^ l y i n g between 0.85 and 1.20 i n increments of 0.05 are adopted. An increase i n the period lengthy,which corresponds to a f a l l i n —, leads to a s h i f t to the right of the unadjusted wage offer d i s t r i b u t i o n . Adjusted measures of the mean offer to searchers, 152 the mean wage of employed i n d i v i d u a l s , MDI and the v a r i a n c e of o f f e r s , however, a l l f a l l . The one p e r i o d c o r r e l a t i o n of wage o f f e r s r i s e s . The adjusted measure of mean p r o f i t s r i s e s . Labour's share of t o t a l output f a l l s . Aggregate l e v e l s of vacancy c r e a t i o n , s p e c u l a t i v e vacancy c r e a t i o n and unemployment do not change i n a systematic f a s h i o n , b u t the adjusted measure of the mean d u r a t i o n of unemployment r i s e s . Changes i n the unadjusted measure of mean d u r a t i o n of search p r i o r to an o f f e r are n e g a t i v e l y c o r r e l a t e d to changes i n the aggregate measure of vacancies minus unemployment. Table 7 shows the r e s u l t s and the frequency with which these comparative s t a t i c p r e d i c t i o n s are u p h e l d . 2.. An E x p l a n a t i o n Through the s p e c i f i c a t i o n of a longer d e c i s i o n p e r i o d , market p a r t i c i p a n t s make d e c i s i o n s l e s s f r e q u e n t l y over a p e r i o d of time and thus there i s an i n c r e a s e i n the i m p l i c i t costs of the market imper-f e c t i o n , namely incomplete i n f o r m a t i o n , to both sets of p a r t i c i p a n t s . T h i s can be regarded as a symmetric i n c r e a s e i n the costs of the i m p e r f e c t i o n . The opportunity costs of search f o r an i n d i v i d u a l are i n c r e a s e d and, i n a d d i t i o n , the i n d i v i d u a l r e c o n s i d e r s h i s search d e c i s i o n l e s s f r e q u e n t l y . L i k e w i s e , f o r the f i r m the foregone p r o f i t s a s s o c i a t e d w i t h a suboptimal l e v e l of employment are i n c r e a s e d . I f the i n d i v i d u a l ' s d e c i s i o n p e r i o d remains constant and the f i r m ' s d e c i s i o n p e r i o d i n c r e a s e s , TABLE 7 Results and Comparative S t a t i c Predictions 19 for Different Period Lengths Summary S t a t i s t i c 1.20 1.15 The Inverse of the Period Length 1/F •1 0.90 0.85 Comparative Statics i . 10 1.05 1.00 0.95 Sign Frequency/7 (a) + 7 (b) * 1.768 1.766 1.714 1.753 1.722 1.607 1.591 1.587 - 6 (c) * 0.202 0.173 0.174 0.171 0.173 0.184 0.181 0.181 - 4(-l) (d) * 1.919 1.892 1.850 1.883 1.852 1.767 1.751 1.749 - 6 (e) * 12.137 12.137 11.926 12.011 11.998 11.775 11.613 11.534 - 6(-l) (f) 64 65 63 65 63 55 55 56 - 4 ( - l ) (g) 27 28 27 37 45 30 30 30 + 5(-l) (h) 41 39 39 40 38 34 35 36 - 4(-l) ( i ) 1.180 1.151 1.171 1.155 1.166 1.234 1.237 1.241 ? 5 (j) * 1.356 1.363 1.439 1.514 1.561 1.649 1.759 1.887 + 7 (k) * , 79.341 81.238 83.642 81.421 83.843 90.229 91.000 90.648 + 5 (1) 0.559 0.554 0.544 0.550 0.546 0.528 0.521 0.518 - 6 (m) 0.359 0.360 0.377 0.365 0.379 0.436 0.394 0.415 + 5 154. then the i n c r e a s e i n the i m p l i c i t costs of the i m p e r f e c t i o n are asymmetric and a m u l t i p l e search model of the labour market i s a p p r o p r i a t e . I t i s mistaken, however, to conclude that both sets of p a r -t i c i p a n t s enjoy a s m a l l e r income, i f the d e c i s i o n p e r i o d i s i n c r e a s e d . The two sets of p a r t i c i p a n t s are both attempting to secure the maximum income, given the other set of p a r t i c i p a n t s ' b e h a v i o u r , and so f o r any f i x e d l e v e l of aggregate p r o d u c t i o n the market p a r t i c i p a n t s are p l a y i n g a f ixed.sum game. I f both sets of p a r t i c i p a n t s make d e c i s i o n s more f r e q u e n t l y , i t does not fol low that the l e v e l of employment and thus aggregate p r o d u c t i o n are i n c r e a s e d i n s t o c h a s t i c e q u i l i b r i u m . Indeed, i f F decreases towards z e r o , t h i s d i s c r e t e model approximates a c o n -tinuous model of the labour market. D e c i s i o n making i s instantaneous, but p a r t i c i p a n t s s t i l l face imperfect i n f o r m a t i o n . To demonstrate the p l a u s i b i l i t y of these r e s u l t s , i t i s again necessary to examine the impact on the cumulative f u n c t i o n , P(w), of the change i n the exogeneous parameter, F. I f the wage d i s t r i b u t i o n i s s c a l e d upwards i n accordance with the longer d e c i s i o n p e r i o d and i n d i v i d u a l s make d e c i s i o n s l e s s f r e q u e n t l y , then the p e r c e i v e d r e t u r n s from s e a r c h , R ,^, may be w r i t t e n =1 E 1 x Rp = (3^) (EW*.F-W) . . . . (14) where W i s the current wage o f f e r f a c i n g the i n d i v i d u a l and i s defined i n terms of the new d e c i s i o n p e r i o d , F > 1. 155 Then, i f C_, denotes the opportunity costs of search, r CL, = W - w.F. . . . . (15) r Since F i s the new decision period, individuals expect to receive a . wage, EW*.F, i f they search one period. They w i l l enjoy unemployment compensation, w.F. The reservation wage, W„, associated with the longer decision r period, i s the solution to 1 1 1 Wp - wF = E (p^) (EW*.F-W p). . . . . (16) i = l Hence, Z .EW.F + wF W F " 1+ Z„ ' • ' ' ( 1 7 ) where, 1 1 x Z p = E ( — ) . . . . . (18) i = l 1+5 I t i s easy to demonstrate that Z„ < Z, where Z denotes the value r of a stream of unit returns discounted over the standard horizon at 20 the standard discount rate and F > 1. Then, denoting the reservation wage associated with the standard period by W^., Wp < FWI . . . . . ( 1 9 ) 2 1 156 Then, i f firms raise t h e i r wage offers i n proportion to the longer period length of employment and individuals evaluate the returns from search from adopting the longer decision period, the reservation wage f a l l s r e l a t i v e l y . Ceteris paribus, individuals have a lower p r o b a b i l i t y , ex ante, of q u i t t i n g i n response to a pa r t i c u l a r wage, i f employed, and a higher p r o b a b i l i t y , ex ante, of accepting an of f e r , i f unemployed. Despite less frequent decision making, the firm faces t h i s systematic change i n in d i v i d u a l behaviour. The formal structure of the model i s unchanged so the firm adjusts i t s wage offer. This change i n in d i v i d u a l behaviour which results from lengthening the decision period and r a i s i n g the wage offer d i s t r i b u t i o n proportion-ately i s equivalent to a f a l l i n the perception parameter, c^. In the model of systematic misperceptions a f a l l i n the parameter c^ leads to a f a l l i n the mean of the perception d i s t r i b u t i o n . In response to this systematic change i n behaviour, the wage offer d i s -t r i b u t i o n s h i f t s to the l e f t . The l e v e l of vacancy creation over each employment state remains constant or f a l l s . The increase i n the length of the decision period leads to an increase i n the length of the period i n which individuals are committed to work for a firm p r i o r to reconsidering the i r quit decision. I t i s plausible that firms increase t h e i r l e v e l of wage offers over a l l states of employment, due to the increase i n labour productivity associated with the longer work period, but i n less than proportion to the increase i n the period length. 157 Therefore, the unadjusted measures of mean wages to searchers and employees and the variance of offers rise,but the adjusted measures a l l f a l l . Individuals earn a lower l e v e l of adjusted MDI. The one period correlation of wage offers r i s e s with, a longer decision period,but not consistently. The change i n the correlation c o e f f i c i e n t depends on the r e l a t i v e influence of the increased range of the un-adjusted wage offer d i s t r i b u t i o n on the covariance of successive wage offers as compared with i t s influence on the variance of the wage offer d i s t r i b u t i o n . The comparative s t a t i c predictions, a r i s i n g from systematic mis-perceptions of the reservation wage, show that firms reduce t h e i r l e v e l of wage offers and, i n some cases, the l e v e l of vacancy creation i n response to a decrease i n c^, i n order to reduce the probability of losing p r o f i t s due to an above optimal workforce. Lengthening the period length means that p r o f i t s foregone from a low l e v e l of employment are increased and, likewise, the p r o f i t s l o s t from a high l e v e l of employment are increased. Thus, i t i s not evident that, i n response to a lengthening of the period length, a firm w i l l decrease the l e v e l of vacancy creation i n addition to decreasing the adjusted wage offer. Thus, the l e v e l of vacancy creation over employment states does not change i n a systematic fashion i n response to a change i n the decision period. Aggregate vacancy creation and therefore aggregate 158 unemployment do not change s y s t e m a t i c a l l y . Since the d e c i s i o n p e r i o d i s l o n g e r , the adjusted measure of the mean d u r a t i o n of unemployment r i s e s . The change i n the mean d u r a t i o n of search p r i o r to an o f f e r i s again n e g a t i v e l y c o r r e l a t e d to the change i n the measure of aggregate vacancy c r e a t i o n minus unemployment. When i n f o r m a t i o n i s p e r f e c t , there i s no f r i c t i o n a l unemployment and no s e a r c h . A l l f irms employ four i n d i v i d u a l s and pay the p e r f e c t l y competit ive wage 1.0. Aggregate output and t o t a l p r o f i t s are maximised. Due to the s p e c i f i c a t i o n of a n o n - l i n e a r p r o d u c t i o n f u n c t i o n f a c i n g each f i r m , t o t a l labour income i s maximised when some i n d i v i d u a l s are unem-p l o y e d . U n w i t t i n g l y through imperfect i n f o r m a t i o n , i n d i v i d u a l s earn h i g h e r mean incomes. Although the i m p l i c i t costs of t h i s i m p e r f e c t i o n are i n a sense symmetric to both firms and i n d i v i d u a l s , f irms bear the b r u n t of the c o s t s . When d e c i s i o n p e r i o d s are s imultaneously i n c r e a s e d , the gain can be regarded as symmetric to p a r t i c i p a n t s y e t , due to the s p e c i f i c a t i o n o f the p e r c e p t i o n d i s t r i b u t i o n , firms alone b e n e f i t from l e s s frequent d e c i s i o n making. These r e s u l t s can be j u s t i f i e d on the grounds t h a t , d u e to a longer p e r i o d d u r i n g which employed i n d i v i d u a l s are committed t o work f o r a p a r t i c u l a r f i r m , p r i o r t o r e c o n s i d e r i n g the q u i t d e c i s i o n , f irms enjoy greater i n t e r t e m p o r a l monopsony power. I f i n d i v i d u a l s are a b l e to make d e c i s i o n s more f r e q u e n t l y than f i r m s , then a m u l t i p l e search model i s a p p r o p r i a t e . I f i n d i v i d u a l s can c o l l e c t o f f e r s over the f i r m ' s d e c i s i o n p e r i o d , then i t i s l i k e l y that the new s t o c h a s t i c e q u i l i b r i u m would be c h a r a c t e r i s e d by a h i g h e r 159 general d i s t r i b u t i o n of wage o f f e r s and l e s s d i s p e r s i o n . Given any d i s t r i b u t i o n o f wage o f f e r s , the mean o f f e r i s h i g h e r i n a m u l t i p l e search model, because only the best o f f e r i s c o n s i d e r e d . I t i s p o s s i b l e that i n d i v i d u a l s might enjoy longer p e r i o d s o f unemployment and thus MDI might f a l l . Thus, t h i s asymmetrical r e d u c t i o n i n the i m p l i c i t costs of the market i m p e r f e c t i o n to p a r t i c i p a n t s i s l i k e l y to reduce f i r m s ' p r o f i t s . When s e a r c h becomes exhaustive, however, the p e r f e c t l y competit ive model i s again a p p r o p r i a t e and firms earn maximum p r o f i t s . The i n f o r m a t i o n s t r u c t u r e i s t h e r e f o r e very important i n the m o d e l l i n g of any market. I t appears t h a t i n t h i s labour market workers b e n e f i t unambiguously from imperfect i n f o r m a t i o n and i t appears that i n d i v i d u a l s enjoy r e l a t i v e l y h i g h e r wage o f f e r s when c e r t a i n symmetric or asymmetric changes i n the i m p l i c i t costs of the i m p e r f e c t i o n o c c u r , such as a r e d u c t i o n i n the d e c i s i o n p e r i o d or i f i n d i v i d u a l s engage i n m u l t i p l e s e a r c h . F . Systematic Misperceptions of the Variance 1. The Results Comparative s t a t i c p r e d i c t i o n s are d e r i v e d f o r d i f f e r e n t values of the parameter, c^. The values of l i e i n the range 0.85 to 1.10 i n increments of 0.05. In a d d i t i o n , two other values of c^, namely 0.25 and 0.50, are adopted. F or i n e x p l i c a b l e reasons, the s i m u l a t i o n procedure does not converge w i t h the frequency that occurs w i t h other comparative s t a t i c experiments. T h i s l i m i t s the number of p o s s i b l e comparative p r e d i c t i o n s a v a i l a b l e . 160 Observation of Table 8 shows that the comparative s t a t i c p redictions are not systematic. Individuals and firms enjoy a lower l e v e l of MDI and p r o f i t s r e s p e c t i v e l y , but the measures of mean wages f a l l i n three or less cases out of f i v e . The wage o f f e r d i s t r i b u t i o n does not s h i f t systematically over d i f f e r e n t values of and, furthermore, wage o f f e r s associated with any state do not move i n a systematic way over d i f f e r e n t values of C2« Labour's share of t o t a l output increases i n three cases out of f i v e which cannot be regarded as a s i g n i f i c a n t r e s u l t . Likewise, n e i t h e r the variance of o f f e r s nor the one period c o r r e l a t i o n of wage o f f e r s change i n a systematic fashion i n response to an increase i n the variance perception parameter, The l e v e l s of aggregate vacancy creation and aggregate speculative vacancy creation both remain constant or increase, while the l e v e l of unemployment does not change i n a systematic fashion. Changes i n the duration of search p r i o r to an o f f e r are negatively correlated to changes i n the aggregate measure of vacancies minus unemployment. The duration of unemployment does not change systematically i n response to an increase i n c^. The i n t e r e s t i n g r e s u l t s are obtained when the value of C2 i s reduced s u b s t a n t i a l l y . I t i s argued i n Chapter 2 IB that, i f the parameter C2 i s zero, wage dispersion disappears and stocha s t i c equilibrium coincides with the c o l l u s i v e monopsony equilibrium. Workers' expectations about wage dispersion are s e l f r e a l i s i n g . TABLE 8 Results and Comparative S t a t i c Predictions  for Different Values of the Perception Parameter, c Summary Values of the Variance Paramete r . i C - 1 ! |Comparative S t a t i c s j S t a t i s t i c 0.25 0.50 0.85 0.90 0.95 " 1.00 1.05 1.10 Sign Frequency/5 j (a) j (b) 0.750 1.404 1.618 1.627 1.645 1.722 1.721 1.696 - 3 i i (c). 0.000 0.160 0.194 0.200 0.215 0.173 0.177 0.163 -% 2 \ (d) 0.750 1.605 1.790 1.805 1.833 1.852 1.850 1.821 - 2 | (e) 6.021 11.164 11.768 11.778 11.730 11.998 11.943 11.819 - 3 | (f) 0 40 56 56 58 63 63 63 + 5(-4) j .(g) 0 11 30 30 30 45 36 36 + 4(-3) | ^ (h) 0 27 36 37 40 38 39 38 + 3 j ( i ) 0.000 1.353 1.242 1.244 1.254 1.166 1.171 1.165 ? 5 f (j) 0.000 1.595 1.604 1.620 1.686 1.561 1.587 1.564 + 3 | (k) \ 178.598 103.869 88.669 87.407 84.618 83.843 83.319 85.301 -(1) 0.252 0.487 0.530 0.532 0.535 0.546 0.545 0.536 + 3 j (m) 1.000 0.584 0.464 0.363 0.419 0.379 0.469 0.381 - 3 | ON 162 I f the p e r c e p t i o n parameter i s set at 0 . 5 0 , then s t o c h a s t i c e q u i l i b r i u m i s c h a r a c t e r i s e d by wage d i s p e r s i o n , f r i c t i o n a l unem-22 ployment and v a c a n c i e s . I f i s set at 0.25 or below, then s t o c h a s t i c e q u i l i b r i u m i s c h a r a c t e r i s e d by the c o l l u s i v e monopsony s o l u t i o n . A l l firms pay a wage equal to the l e v e l of unemployment compensation. There i s no f r i c t i o n a l unemployment, wage d i s p e r s i o n , or s e a r c h . The d i s t r i b u t i o n of firms over employment s t a t e s i s i n -determinate. Each f i r m employing l e s s than four workers creates v a c a n c i e s , b u t i s unable to communicate i t s excess demand because there are no s e a r c h e r s . This r e s u l t i s most important. I t demonstrates that i f i n d i v i d u a l s base t h e i r general labour market i n f o r m a t i o n on a sample of four or more wage o f f e r s sampled over time and they compute an unbiased estimate of the v a r i a n c e of the sample mean, then s t o c h a s t i c e q u i l i b r i u m i s c h a r a c -t e r i s e d by zero d i s p e r s i o n . The c r i t i c a l v a l u e , f o u r , c l e a r l y depends on the exogeneous parameters chosen. T h i s r e s u l t demonstrates t h a t , i f i n d i v i d u a l s search e x t e n s i v e l y over time f o r i n f o r m a t i o n about job o f f e r s and wage r a t e s t h e n , although t h e i r labour market behaviour i s s t o c h a s t i c , the e q u i l i b r i u m i s c h a r a c -t e r i s e d by zero d i s p e r s i o n . T h i s i s simply because the v a r i a n c e of a sample mean decreases as the sample i n c r e a s e s . T h i s r e s u l t i s not dependent on i n d i v i d u a l s s e a r c h i n g e x h a u s t i v e l y or i n d u l g i n g i n m u l t i p l e searches per d e c i s i o n p e r i o d . I t simply r e q u i r e s that a l l i n f o r m a t i o n about wage o f f e r s gathered i n the p a s t i s i n c o r p o r a t e d i n t o the i n d i v i d u a l s ' g e n e r a l i n f o r m a t i o n . 163 2. An Explanation An increase i n c^, ceteris paribus, has a non-systematic effect on the ex ante p r o b a b i l i t i e s of turnover and acceptance i n response to any wage offer. From Chapter 3 I I , i t appears that, i f an in d i v i d u a l faces a r e l a t i v e l y high o f f e r , than an increase i n C2 leads to a higher probability of q u i t t i n g , when employed, and a lower probability of acceptance, when unemployed. The converse i s true, i f the in d i v i d u a l faces a r e l a t i v e l y low offer. The results r e f l e c t this non-systematic influence of a change i n C2« Both p r o f i t s discounted over an i n f i n i t e horizon and MDI f a l l as the parameter, C2, r i s e s . In contrast, when c^ increases, individuals earn higher wage rates over a l l employment states and, despite increased search unemployment and vacancy creation i n stochastic equilibrium, they earn higher MDI. Firms do not increase wage offers over a l l employment states and the combination of higher aggregate unemployment and lower wage offers over some states of employment leads to lower MDI for some changes i n C2, i n addition to lower mean p r o f i t s discounted over an i n f i n i t e horizon. G. Conclusion In this Chapter, the general characteristics of stochastic equilibrium i n the labour market have been described for different values of the exogeneous parameters. Comparative s t a t i c predictions, derived with respect to the exogeneous parameters, c. , w, H-, F, and 164 i n d i c a t e the algebraic sign of the changes i n these summary 23 s t a t i s t i c s which describe' equilibrium. I t i s argued that a s u f f i c i e n t condition f o r the existence of wage and employment dispersion i n equ i l i b r i u m i s that workers, although equally productive and i n d i s t i n g u i s h a b l e , behave hetero-geneously. In t h i s model, i n d i v i d u a l s ' labour market behaviour i s stochastic and thus heterogeneous, ex post, i f the wage o f f e r d i s -t r i b u t i o n exhibits non-zero variance i n i t i a l l y , and i f there i s employment dispersion. I f the value of the perception parameter exceeds 0.25, then stochastic equilibrium i s characterised by wage and employment dispersion. I f no wage dispersion e x i s t s i n i t i a l l y then, i r r e s p e c t i v e of whether employment dispersion e x i s t s , firms w i l l be forced by i n d i v i d u a l s ' homogeneous non-stochastic, ex post turnover and acceptance behaviour to o f f e r a wage equal to the perceived reser-v a t i o n wage. Conversely, wage dispersion w i l l only p e r s i s t , i f employment dispersion currently e x i s t s . I f i n d i v i d u a l s incorporate information about past wage o f f e r s i n t o t h e i r 'general labour market information' then dispersion w i l l disappear. Then, i n this model, a s u f f i c i e n t condition f o r hetero-geneous behaviour by i n d i v i d u a l s and thus wage and employment d i s -persion i n equilibrium i s the i n i t i a l existence of wage and employment dispersion and the non-coll e c t i o n of past wage o f f e r s to generate labour market information. 165 Firms exploit their intertemporal monopsony power by making wage offers related to t h e i r current levels of employment. Thus, i t i s not individuals' stochastic behaviour, per se, that causes wage d i s -persion, but rather that firms, facing different levels of employment, choose to offer d i f f e r e n t wage rates and individuals' stochastic behaviour allows them to do so. Firms speculate i n vacancy creation. I f a l l vacancies are f i l l e d , and nobody quits, then the l e v e l of employment attained exceeds that consistent with intertemporal p r o f i t maximisation. This speculation i s based on individuals' stochastic search, acceptance and turnover behaviour. There appears to be a trade-off between the wage offer and the l e v e l of vacancy creation. Rather than create vacancies equal to desired net hires and offer a high wage, whence there i s a high probab-i l i t y that the current workforce i s retained, the firm offers a lower wage and speculates i n vacancy creation. Although the model i s not intended to represent the labour market as behaving as a dynamic process, the explanations of the comparative s t a t i c results are formulated i n a dynamic context. The explanations are necessarily h e u r i s t i c due to the simultaneity of the model. In response to a change i n an exogeneous parameter, both sets of par-t i c i p a n t s ' behaviour changes and the nature of the labour market environ-ment changes. This, i n turn, generates new decisions on the part of market participants and so on. I t i s assumed that the instantaneous 166 changes i n the summary s t a t i s t i c s caused by the change i n the exogeneous parameter are an i n d i c a t i o n of how the nature of s t o c h a s t i c e q u i l i b r i u m i s i n f l u e n c e d by the parameter change. In other words, the d i s c u s s i o n of the r e s u l t s i s merely suggestive because i t attempts to e x p l a i n the nature of the new p a r t i a l e q u i l i b r i u m i n the labour market f o l l o w i n g the parameter change, r a t h e r than the new f u l l s t o c h a s t i c e q u i l i b r i u m . One of the s i g n i f i c a n t t h e o r e t i c a l c o n t r i b u t i o n s of t h i s model i s that the labour market i s modelled i n a general e q u i l i b r i u m context. I n t h i s m o d e l , i n response to a change i n an exogeneous parameter,f irms adopt new d e c i s i o n s . Workers respond to these new d e c i s i o n s and a new labour market environment i s generated, i n which firms again change t h e i r d e c i s i o n s and so on. While the c h a r a c t e r i s t i c s of the p a r t i a l e q u i l i b r i u m s o l u t i o n i n d i c a t e the c h a r a c t e r i s t i c s of the new s t o c h a s t i c e q u i l i b r i u m s o l u t i o n , they are not i d e n t i c a l . For example, the f o l l o w i n g t a b l e shows the s t o c h a s t i c e q u i l i b r i u m s o l u t i o n s . f o r two values o f c^, 1.20 and 1.25, and the p a r t i a l e q u i l i b r i u m s o l u t i o n i n response to an i n c r e a s e of c^ from 1.20 to 1.25. In t h i s example, i t does seem that the comparison of the p a r t i a l e q u i l i b r i u m s o l u t i o n with the i n i t i a l s t o c h a s t i c e q u i l i b r i u m i n d i c a t e s the q u a l i t a t i v e changes i n the summary s t a t i s t i c s d e s c r i b i n g f u l l s t o c h a s t i c e q u i l i b r i u m , a s s o c i a t e d w i t h the i n c r e a s e i n c^. In c o n t r a s t , both Salop and Mortenson analyse f i r m behaviour i n a p a r t i a l e q u i l i b r i u m context. The f i r m ' s d e c i s i o n s over time are assumed to not change the general labour market environment. 167 TABLE 9 A Comparison of P a r t i a l and F u l l E q u i l i b r i u m S t o c h a s t i c E q u i l i b r i u m P e r c e p t i o n Parameter,c^ = 1.20 Mean P r o f i t s One Over an L e v e l of Steady S t a t e Wage Vacancy P e r i o d I n f i n i t e Employment D i s t r i b u t i o n O f f e r C r e a t i o n P r o f i t s H o r i z o n 0 0.127 2.715 4 1.55 42.04 1 0.306 2.650 3 1.98 42.60 2 0.318 2.535 3 2.25 42.95 3 0.182 2.440 3 2.45 43.19 4 0.057 2.365 3 2.60 43.36 5 0.008 2.305 3 2.71 43.50 6 0.000 2.275 2 2.80 43.60 7 0.000 2.260 1 2.84 43.65 8 0.000 2.255 0 2.85 43.66 P a r t i a l E q u i l i b r i u m P e r c e p t i o n Parameter,c = 1.25 o 0.143 | 2.720 4 1.53 40.98 ; 1 0.323 j 2.665 3 1.94 41.52 2 0.313 1 2.550 3 2.21 41.86 3 0.166 I 2.460 3 2.40 42.09 4 0.048 2.385 3 2.54 42.26 5 0.006 1 2.325 3 2.65 42.39 6 0.000 1 2.300 2 2.73 42.48 7 0.000 2.285 1 2.77 42.53 8 0.000 2.280 0 2.78 42.54 I S t o c h a s t i c E q u i l i b r i u m P e r c e p t i o n Parameter, c. = 1.25 0 0.231 3.025 4 1.09 i 26.73 1 1 0.369 2.940 4 1.32 27.01 ' 2 0.263 2.865 4 1.48 27.19 3 0.107 2.800 4 - 1.60 27.33 4 0.026 2.750 4 1.69 27.43 5 0.003 2.720 3 1.76 27.51 ! 6 0.000 2.700 2 1.81 27.56 ! 7 0.000 2.695 1 1.83 27.59 8 0.000 2.690 0 1.83 27.59 168 The c r u c i a l element i n the model i s the p r o b a b i l i t y d i s t r i b u t i o n of perceptions of the reservation wage fa c i n g each i n d i v i d u a l . I t i s the impact of the change i n the exogeneous parameter on t h i s d i s t r i -bution of perceptions which i s the key to analysing the comparative s t a t i c results:. The market i s segregated, so a l l i n d i v i d u a l s are i d e n t i c a l , ex ante. No fa c t o r s u b s t i t u t i o n i s po s s i b l e . Consequently, firms are forced to respond d i r e c t l y through t h e i r wage o f f e r and vacancy creation decision to changes i n the nature of the d i s t r i b u t i o n of perceptions. Otherwise, they w i l l face inadequate or excessive l e v e l s of employment with greater frequency. An important contribution of Model I I , outlined i n Chapter II and Appendix I, i s that firms are able to substitute i m p l i c i t l y between d i f f e r e n t types of i n d i v i d u a l s , although wage and employment discrim-i n a t i o n i s i l l e g a l 2 4 I I I . Vacancy and Unemployment S t a t i s t i c s A. Model I In Model I, firms are t e c h n i c a l l y homogeneous and each member of the workforce i s equally productive. Due to stochastic misperceptions of the reservation wage, however, i n d i v i d u a l s ' turnover and acceptance behaviour, i n response to the same wage and labour market environment, i s heterogeneous, ex post, but t h e i r behaviour does n o t . d i f f e r i n a systematic fashion. E q u i l i b r i u m i s stochastic and i s characterised by wage dispersion and the simultaneous existence of vacancies and f r i c t i o n a l unemploymen t. 169 Although representing p o s i t i o n s r e q u i r i n g the same s k i l l s , the aggregate measure of vacancy creation i s heterogeneous because i t represents u n f i l l e d p o s i t i o n s created at d i f f e r e n t wage rates." Furthermore, the aggregate measure of vacancy creation does not represent desired net h i r e s because i t r e f l e c t s firms' speculation about i n d i v i d u a l s ' turnover and acceptance behaviour. In a non-stochastic world, positions,which are created, are instantaneously f i l l e d . In t h i s stochastic framework, i n s u f f i c i e n t workers may sample the f i r m due to random search, leaving p o s i t i o n s u n f i l l e d , or, since turnover and acceptance decisions are stochastic, some o f f e r s may not be accepted by current employees and job searchers. Reder argues that, a p r i o r i , the duration of unemployment i s not rel a t e d i n any p a r t i c u l a r way to the duration of time a vacancy i s 26; '•> u n f i l l e d . Then, stochastic equilibrium, which i s characterised, by d e f i n i t i o n , by zero excess demand, i s consistent with the simul-taneous existence of vacancies and unemployment, the r e l a t i o n s h i p between them being indeterminate. These r e s u l t s demonstrate that the use of a function of aggregate vacancies and unemployment as a proxy f o r excess demand i n such a 27-homogeneous labour market i s i n c o r r e c t . Thus, any unemployment and vacancy s t a t i s t i c s must be c a r e f u l l y interpreted. B. Heterogeneous Labour Markets In the r e a l world of heterogeneous labour markets, there are sever a l factors which further complicate the i n t e r p r e t a t i o n of vacancy s t a t i s t i c s . 170 F i r s t l y , as noted i n the L i t e r a t u r e Review, Chapter 1 H I E , there i s the conceptual problem of defining and measuring the l e v e l of vacancy creation which i s compatible, i n some sense, with the d e f i n i t i o n of unemployment. An i n d i v i d u a l without work for seven days i s c l a s s i f i e d as unemployed but, while unemployment i s a d i s t i n c t . and recognisable state f o r the i n d i v i d u a l , (ignoring the problem of d e f i n i n g a c t i v e search), the existence of a vacancy i s less w e l l defined. An employer, not a c t i v e l y r e c r u i t i n g , may create a p o s i t i o n f o r a w e l l q u a l i f i e d applicant. Faced with an i n s u f f i c i e n t workforce an employer may adopt a l t e r n a t i v e measures, namely : (a) The i n s t i g a t i o n of overtime; (b) Subcontracting the work elsewhere; and (c) The purchase of a d d i t i o n a l machinery, that i s c a p i t a l deepening, to make h i s labour force more productive. A f i r m may choose to advertise i n a newspaper or tradepaper, but f28' not specify the number of positions a v a i l a b l e . -Secondly, a f i r m i s not l e g a l l y compelled to n o t i f y a manpower 29 o f f i c e (a source of job vacancy s t a t i s t i c s ) of the existence of vacancies. 1 I t may not choose to do so i f , f o r example, i t believes that n o t i f i c a t i o n of u n f i l l e d p o s i t i o n s w i l l a t t r a c t a large number of applicants who w i l l require interviews at considerable cost. Or, i f the vacancy i s not f i l l e d by a l o c a l worker, the f i r m may f e e l committed to o f f e r the p o s i t i o n to an applicant who has t r a v e l l e d a long way to the interview. 171 The firm may regard informal contacts as more e f f i c i e n t . I t may be able to discriminate against the potential employee i n wage offer or conditions, or indeed, choose to not h i r e him, which may be im-possible, i f Manpower i s n o t i f i e d of the characteristics of the job and number of vacancies. Thirdly, the existence of an i n t e r n a l labour market within a firm, as described by Dunlop [1966] and Doeringer and Piore [1971], complicates the interpretation of a vacancy. Employers develop an elaborate set of rules, whether or not subject to c o l l e c t i v e bargaining, r e l a t i n g to promotions, transfers, l a y o f f s , and retirements f o r various job c l a s s i f i c a t i o n s . Entry from outside the organisation i s confined to a few c l a s s i f i c a t i o n s . Each 'port of entry' i s the bottom rungrdf a family of s i m i l a r s k i l l requiring operations. Individuals, on the job, receive t r a i n i n g to enable them to be promoted within a family of jobs. The rules of senior i t y specify the in t e r n a l movement of individuals between these jobs. The concept of the supply of labour i n this market i s thus inappropriate. A vacancy i n a top position may be f i l l e d by a series of promotions throughout the family of jobs, such that the vacancy created i n the external labour market i s for a 'bottom rung' position, whilst maintaining a t r a d i t i o n of in t e r n a l promotion, firms may n o t i f y the external labour market of such a vacancy. I f each such vacancy down the ladder i s advertised, the number of vacancies advertised w i l l exaggerate the number of individuals required. The s k i l l requirements of these 'port of entry' positions may w e l l be higher than 172 formally required, since such applicants w i l l be expected to be promoted within t h i s family of jobs. Fourthly, even within the same industry, the exact s k i l l s required and the non-monetary benefits of the job are not homogeneous over firms. Thus the unemployed worker has imperfect information 30 about a vector of c h a r a c t e r i s t i c s of the job.-Aside from the problem of wage dispersion and speculation i n vacancy creation, demonstrated i n Model I,'; the use of job vacancy and unemployment s t a t i s t i c s to analyse the functioning of the labour market i s subject to considerable d i f f i c u l t i e s , therefore, and con-siderable disaggregation i s required. In addition to the c h a r a c t e r i s t i c s of job vacancies, d e t a i l e d information i s required about the s k i l l s and asp i r a t i o n s of unemployed workers. Otherwise, any loose comparison of vacancy and unemployment s t a t i s t i c s may overestimate the degree bf f r i c t i o n i n the operation of the labour market and disguise the degree 3d1 of s t r u c t u r a l unemployment: 173 CHAPTER 5  RESULTS FROM MODELS I I AMD I I I I. Introduction In t h i s chapter the results from Models I I and I I I are presented. Solutions to each model are generated for d i f f e r e n t combinations of 1 2 values of the perception parameter values, c^ , c^ . 1 2 A r b i t r a r i l y , the value of c^ i s assumed to equal or exceed c^ . In response to the same labour market environment^" and a p a r t i c u l a r wage o f f e r , each type one i n d i v i d u a l has a lower p r o b a b i l i t y of accept-ing a po s i t i o n and a higher p r o b a b i l i t y of turnover,if employed, ex ante, than a type two i n d i v i d u a l . Even when employed, type one in d i v i d u a l s , although equally productive over a single period, are less valuable to the f i r m i n an intertemporal sense than type two individuals because, i n response to the same stream of wage offers over time from the firm, the mean returns from employing a type one i n d i v i d u a l are lower than the mean returns from employing a type two i n d i v i d u a l . For the purpose of explaining both sets of r e s u l t s , i t i s useful to divide the employment states into f i v e d i s t i n c t subgroups. The exogeneous parameters adopted are such that, for a l l values of the perception parameters chosen, the single period p r o f i t maximising l e v e l of employment coincides with the intertemporal p r o f i t maximising 2 l e v e l of employment next period for wage offers l y i n g between 0.75 and 174 1.50. A s i n g l e period p r o f i t maximising f i r m i s i n d i f f e r e n t as to the composition of employment i t a t t a i n s , since, given the wage o f f e r , s i n g l e period p r o f i t s are determined by l e v e l of employment alone. A firm o f f e r i n g a wage i n this range chooses to employ two i n d i v i d u a l s . Two then constitutes the 'optimal' l e v e l of employment. A fi r m , which maximises p r o f i t s discounted over an i n f i n i t e horizon, chooses to employ type two i n d i v i d u a l s who are more valuable, i n the sense described. Thus, the 'optimal workforce' i s defined as two type 3 4 two i n d i v i d u a l s , i f the f i r m o f f e r s a wage between 0.75 and 1.50. Given the choice of the l e v e l and composition Of employment, the firm employs the 'optimal workforce'. Individuals' labour market behaviour i s sto c h a s t i c , however, and the l e v e l and composition of employment the f i r m faces next period i s ne c e s s a r i l y s t o c h a s t i c . An 'outcome' i s defined as the l e v e l and composition of employment that a f i r m faces next period. Mean p r o f i t s are defined as p r o f i t discounted over an i n f i n i t e horizon associated with each outcome, weighted by the p r o b a b i l i t y of a t t a i n i n g that outcome. A firm, which maximises mean p r o f i t s , w i l l make decisions such that there i s a high ex ante probab-i l i t y of a t t a i n i n g an outcome i n the neighbourhood of the 'optimal work-force'.~* The optimal l e v e l of employment i s independent of the composition of employment over the values of the perception parameters adopted. Thus, firms make decisions such that there i s a r e l a t i v e l y high ex ante p r o b a b i l i t y of achieving the optimal l e v e l of employment. A firm, which currently employs 175 solely type two individuals, has the 'optimal' composition of employment, when analysing firms' decisions in both Models II and III, i t proves' easiest to group employment states into subgroups according to their relationship to the 'optimal workforce'. The five conceptually distinct subgroups constitute the elements of 2 x 3 matrix, as shown in Figure 8. BO 00 AO OC b a d SC b c e Figure 8 The Matrix of Employment States In state OC, the firm faces a workforce with the optimal composition of employment, that i s solely type two employees. In state SC, the firm faces a workforce which has a suboptimal composition of employment. Some type one individuals are employed. In state BO, the firm faces a less than optimal level of employment. In state AO, the firm faces an above optimal level of employment. The five distinct subgroups which are denoted by letters (a) to (e) are the following : a. The firm currently employs the 'optimal workforce', that i s state [0,2] 6; b. The firm currently employs a less than optimal level of employment, states [0,0], [1,0], and [0,1]; c. The firm currently employs the optimal level but sub-optimal composition of employment, states [2,0] and [1,1]; 176 d. The f i r m c u r r e n t l y employs an above optimal l e v e l of employment but the ' o p t i m a l workforce' c o n s t i t u t e s a subset of those employed, s t a t e s [ 1 , 2 ] , [ 0 , 3 ] , [ 2 , 2 ] , [1,3] and [ 0 , 4 ] ; and e. The f i r m c u r r e n t l y employs an above o p t i m a l l e v e l of employment which i s suboptimal i n composit ion. The o p t i m a l workforce does not c o n s t i t u t e a subset of c u r r e n t employees, s t a t e s [ 3 , 0 ] , [ 2 , 1 ] , [4,0] and [ 3 , 1 ] . I I . Re su l ts of Model I I A . B a s i c Parameter Values The parameter values s p e c i f i e d i n Chapter 3 IIIB and u n i t values of the p e r c e p t i o n parameters c o n s t i t u t e the ' b a s i c ' set of parameter v a l u e s . Ex ante, i n d i v i d u a l s are i d e n t i c a l i n t h e i r labour market behaviour. Thus, Model I I s i m p l i f i e s to Model I . I n d e e d , i f the values of the p e r c e p t i o n parameters are e q u a l , Model I I s i m p l i f i e s to Model I . F i r m s , f a c i n g the same l e v e l of employment, make i d e n t i c a l wage and vacancy c r e a t i o n d e c i s i o n s , independently of the composition of employment. The steady s t a t e d i s t r i b u t i o n i s symmetric about the s t a t e complements. I f and j 2 are s t a t e complements, then 21 J 2 (1) S o l v i n g the s i m u l a t i o n procedure i n Model I I f o r i d e n t i c a l values of the p e r c e p t i o n parameters c o n s t i t u t e s a t e s t of i t s c o n s i s t e n c y . The b a s i c s o l u t i o n to Model I I i s shown i n Table 10. 177 TABLE 10 The Basic S o l u t i o n to Model I I ; Stochastic E q u i l i b r i u m j i | i Mean One P r o f i t s Over Type One Type Two ! Steady State Wage ; Vacancy Period I n f i n i t e Employment Employment D i s t r i b u t i o n O ffer ; Creation P r o f i t s Horizon. 0 0 j 0.156 1.455 ! 2 0.45 19.13 1 0 0.213 1.420 1 1.01 20.29 0 1 0.213 1.420 i 1 1.01 20.29 2 0 0.087 1.245 1 1.21 20.61 1 1 0.175 1.245 1 1.21 20.61 0 2 0.087 1.245 ! 1 • 1.21 20.61 3 0 0.008 1.185 0 1.29 20.72 2 1 0.025 1.185 ! o 1.29 20.72 1 2 j 0.025 ! 1.185 ! 0 1.29 20.72 0 3 0.008 ! 1.185 0 1.29 20.72 4 0 0.000 1.085 o ! 1.33 20.74 3 1 0.000 1.085 0 1.33 20.74 2 2 0.000 1.085 0 1.33 20.74 1 3 0.000 1.085 0 1.33 20.74 0 4 0.000 i 1.085 0 1.33 20.74 EW MW W* vw* Summary S t a t i s t i c s LS V* V * U* s PA* PO* En 1.323 i 1.145 j 0.081 0.634 54 17 32 1.737 j 1.377 20.250 C o r r e l a t i o n of Offers ;1 Period ! 2 Periods | 3 Periods i 4 Periods ' 5 Periods! ! ! 0.395 ! 0.167 | 0.035 j q. 001 - 0.003 ! Mean Discounted Income • No Q u i t t i n g True Perception Stochastic Behaviour! 10.215 • 10.215 9.070 ! TABLE 10 (continued)  Steady State Transition Matrix 0.570 0.164 0.164 0.091 0.536 0.031 0.091 0.031 0.536 0.069 0.310 0.017 0.069 0.163 0.163 0.069 0.017 0.310 0.061 0.283 0.0 0.061 0.188 0.094 0.061 0.094 0.188 0.061 0.0 0.283 0.081 0.283 0.0 0.081 0.213 0.071 0.081 0.142 0.142 0.081 0.071 0.213 0.081 0.0 0.283 0.025 0.052 0.025 0.171 0.171 0.0 0.0 0.171 0.171 0.384 0.070 0.0 0.035 0.384 0.035 0.0 0.070 0.384 0.434 0.0 0.0 0.145 0.289 0.0 0.0 0.289 0.145 0.0 0.0 0.434 0.372 0.0 0.0 0.186 0.186 0.0 0.062 0.248 0.062 0.0 0.186 0.186 0.0 0.0 0.372 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.075 0.075 0.0 0.0 0.075 0.075 0.0 0.0 0.075 0.222 0.0 0.0 0.0 0.222 • 0.0 0.0 0.0 0.222 0.0 0.0 0.0 0.217 0.0 0.0 0.054 0.163 0.0 0.0 0.108 0.108 0.0' 0.0 0.163 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.075 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.222 0.0 0.0 0.0 0.047 0.0 0.0 0.0 0.047 0.0 0.0 0.0 0.054 0.0 0.0 0.217 0.0 0.0 0.0 0.0 0.0 i 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 o.o : 0.0 0.0 0.0 ; 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 • 0.0 0.0 0.0 0.047 0.0 0.0 : 0.0 0.047 o.o : 0.0 0.0 0.047 oo 179' The general characteristics of stochastic equilibrium with these exogeneous parameters conform to the solutions to Model I . Stochastic equilibrium i s characterised by wage and employment dispersion over levels of employment. The d i s t r i b u t i o n of wage offers l i e s above the perfectly competitive equilibrium wage but the mean wage offers to searchers and employed individuals are less than average marginal product. Firms exhibit dynamic monopsony power i n that the wage offer i s a declining function of the l e v e l of employment and mean p r o f i t s discounted over an i n f i n i t e horizon are an increasing function of the l e v e l of employment. Curiously, although able to create vacancies when facing an employment l e v e l of three, firms do not choose to do so. I f individuals are i d e n t i c a l , and the common perception parameter, c^, exceeds 1.00, however, then firms do create vacancies, when employing three i n d i v i d u a l s . This result r e f l e c t s the trade off between the wage offer and l e v e l of vacancy creation. The firm earns higher mean p r o f i t s by creating a vacancy and increasing the wage offer rather than increasing the wage offer alone, i f c^ increases. The comparative s t a t i c predictions generated by dif f e r e n t sets of equal values of the perception parameters conform exactly to those generated i n Model I. An important feature of a l l solutions of Model I I i s that speculative vacancy creation i s a s i g n i f i c a n t l y smaller percentage of t o t a l vacancy creation than i n Model I. This r e s u l t can be attributed to the reduction i n the number of employment l e v e l s , i n p a r t i c u l a r these exceeding the 180 s t a t i c p r o f i t maximising l e v e l of employment. This r e s t r i c t i o n of the maximum l e v e l of employment, however, does not seem to change the essential nature of firms' decision making. B. Unequal Values of the Perception Parameters 1. The Results Solutions to Model I I are obtained for 7 sets of unequal values 1 2 of the perception parameters, c^ , c^ , l y i n g i n the range 0.90 to 1.15 i n increments of 0.05. Table 11 shows the solution to Model I I for perception parameters values, [1.05,0.95], Stochastic equilibrium i n Model I I i s characterised by wage dispersion and dispersion both i n the l e v e l and composition of em-ployment facing a firm. The d i s t r i b u t i o n of wage offers l i e s above the perfectly competitive wage. Both searchers' and employees' mean offers are exceeded by the average marginal product. The wage offer i s a declining function of the l e v e l of employment faced by the firm, irrespective of the composition of employment. Over each less than optimal l e v e l of employment, the wage offer i s an increasing function of the number of type one individuals employed. The l e v e l of vacancy creation i s independent of the composition of employment over less than optimal levels of employment and i s a declining function of these levels of employment. A firm, currently facing the optimal workforce, creates a vacancy i n one case out of seven. I TABLE 11 The S o l u t i o n to Model I I Corresponding  To P e r c e p t i o n Parameter Values [1 .05,0.95] 181 S t o c h a s t i c E q u i l i b r i u m Type One j Type Two Steady State Wage Vacancy Employment ]. Employment D i s t r i b u t i o n O f f e r C r e a t i o n Mean One P e r i o d P r o f i t s P r o f i t s Over I n f i n i t e H o r i z o n 0 1 0 2 1 0 3 2 1 0 4 3 2 1 0 0 ! 0.143 1.45 : 2 j 0.45 19.77 0 ! 0.180 ; 1.43 1 i 0.98 20.76 1 1 0.247 ; i . 3 8 1 1.08 21.34 0 ! 0.079 j 1.26 1 i 1.17 j 21.04 1 I 0.158 | 1.22 1 1.26 21.49 2 ! 0.147 : 1.29 : 0 1.32 21.92 0 • 0.008 j 1.14 ' 1 1.25 21.12 1 0.022 | 1.11 ! 1 1.33 21.49 2 0.014 1.14 :' 0 1.39 21.83 3 0 . 0 1.12 0 1.43 22.09 0 0.0 1.10 0 1.30 21.12 1 0.001 1.08 0 1.36 21.46 2 0 . 0 1.05 0 1 1.42 21.74 3 0 . 0 1.03 . 0 | 1.45 21.96 4 0 . 0 1.01 j 0 j 1.48 22.13 Summary S t a t i s t i c s EW MW1 MW2 W* VW* L S 1 1.146 1.323 1.310 1.100 0.089 0.274 0.351 j v * ' V * I s j V * u 2 * P A 1 * P A 2 * 1 PO* En |49 13 J 20 12 ; 2.058 1.783 J . 1-532 21.105 1 C o r r e l a t i o n of Offers Mean Discounted Income 1 P e r i o d i 2 P e r i o d s j 3 P e r i o d s 4 P e r i o d s 5 P e r i o d s 0.422 0.187 0.048 0.005 - 0.008 No Q u i t t i n g True P e r c e p t i o n "1 S t o c h a s t i c Behaviour Type 1 10.015 9.629 8.682 JType 2 10.143 10.143 I 9.287 TABLE 11 (continued)  Steady State Transition Matrix i 0.573 0.197 0.130 0.108 0.531 0.030 0.072 0.027 0.558 0.080 0.324 0.017 0.063 0.113 0.219 0.031 0.0 0.289 0.090 0.291 0.015 0.076 0.144 0.153 0.049 0.050 0.215 0.039 0.0 0.227 0.093 0.302 0.0 0.075 0.164 0.120 0.069 0.085 0.188 0.059 0.032 0.217 0.052 0.0 0.228 0.036 0.050 0.015 0.199 0.133 0.0 0.0 0.204 0.139 0.371 0.065 0.0 0.025 0.391 0.041 0.0 0.0 0.681 0.330 0.046 0.0 0.079 0.288 0.022 0.0 0.216 0.234 0.0 0.0 0.444 0.367 0.0 0.0 0.118 0.260 0.0 0.026 0.230 0.128 0.0 0.118 0.264 0.0 0.0 0.374 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.083 0.061 0.0 0.0 0.084 0.064 0.0 0.0 0.0 0.148 0.046 0.0 0.010 0.156 0.038 0.0 0.0 0.235 0.0 0.0 0.0 0.198 0.0 0.0 0.029 0.188 0.0 0.0 0.070 0.156 0.0 0.0 0.144 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.019 0.015 0.0 0.0 0.019 0.0 0.0 0.0 0.291 0.0 0.0 0.0 0.040 0.0 0.0 0.0 0.046 0.0 0.0 0.0 0.108 0.0 0.0 0.273 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.016 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.048 0.0 0.0 ' 0.0 0.058 0.0 j 0.0 0.0 0.075 M CD ro 183 Also, a firm, currently employing three type one indi v i d u a l s , does not create a vacancy i n one case out of seven. In some cases, firms employing a mixed workforce of three employees w i l l also create vacancies. Thus, the vacancy creation decisions are reasonably consistent over different sets of the perception parameters. Then, over the optimal or above optimal levels of employment, the vacancy creation decision i s not independent of the composition of employment. These exists a tradeoff between the wage offer and vacancy creation decision and i f , i n p a r t i c u l a r , a firm employs the optimal workforce, i t creates zero vacancies and offers a higher wage than a firm facing an optimal l e v e l , but suboptimal composition of employment. Type one employees' mean wage exceeds the mean wage of type two employees but type two individuals enjoy a higher MDI and a larger share of output. Type one individuals enjoy both a higher l e v e l and a longer duration of unemployment than type two individuals. For each l e v e l of employment, mean p r o f i t s discounted over an i n f i n i t e horizon are a declining function of the number of type one individuals employed. Discounted p r o f i t s are not, however, an i n -creasing function of the l e v e l of employment independently of the composition. This i s quite plausible and i s shown i n Table 11 where higher discounted p r o f i t s are enjoyed by a firm i n state [0,1] than a firm i n state [2,0]. 184 The values of these summary s t a t i s t i c s over the seven s o l u t i o n s and the frequency with which the s p e c i f i e d r e l a t i o n s h i p between the summary s t a t i s t i c s corresponding to each type of i n d i v i d u a l i s upheld are shown i n Table 12. 2. An E x p l a n a t i o n A g a i n , i t i s p o s s i b l e to demonstrate the p l a u s i b i l i t y of these r e s u l t s by an a p p l i c a t i o n of the i l l u s t r a t i v e model. I f h i g h - t u r n o v e r type one i n d i v i d u a l s have a h i g h e r marginal turnover r a t e than low-turnover type two i n d i v i d u a l s , then i t can be shown t h a t , f o r a given i n i t i a l l e v e l of employment, the wage o f f e r i s an i n c r e a s i n g f u n c t i o n of the number of type one i n d i v i d u a l s employed. Most f irms i n the 9 i n d u s t r y o f f e r a wage above the mean of the d i s t r i b u t i o n of p e r c e p t i o n s and thus face a h i g h e r marginal turnover r a t e on the p a r t of type one i n d i v i d u a l s , c e t e r i s p a r i b u s . In the i l l u s t r a t i v e model, however, firms do not make e x p l i c i t vacancy c r e a t i o n d e c i s i o n s . Firms maximise p r o f i t s a s s o c i a t e d w i t h the mean l e v e l of employment corresponding to the wage o f f e r . By t a k i n g the mathematical e x p e c t a t i o n , t h e r e f o r e , the flow supply of labour to the f i r m i s reduced to a n o n - s t o c h a s t i c flow supply of labour dependent on the wage o f f e r . Thus, the i l l u s t r a t i v e model only sheds l i g h t on the r e s u l t s , when f i r m s , f a c i n g the same l e v e l but d i f f e r e n t composition of employment, make the same vacancy c r e a t i o n d e c i s i o n . In t h i s case, each f i r m faces the same s t o c h a s t i c flow supply of workers who r e c e i v e an o f f e r but the p r o b a b i l i t y of acceptance d i f f e r s between f i r m s . TABLE 12 Summary S t a t i s t i c s for Unequal Values  of the Perception Parameters 1 c l . 2 C l MW2 MDI^ MDI2 U l U2 PA* PA* ! L S i LS 2 0.95 0.90 1.237 1.214 8.524 8.913 16 9 1.890 1.752 i | 0.271 0.321 1.00 0.90 1.244 1.235 8.534 8.968 17 10 1.987 1.791 ! 0.268 0.329 1.00 0.95 1.288 1.308 8.776 9.087 17 14 1.917 1.869 | 0.292 0.328 1.05 0.95 1.323 1.310 8.682 9.287 20 12 2.058 1.783 | 0.274 0.351 1.05 1.00 1.353 1.366 8.852 9.154 20 17 2.051 1.979 ! 0.303 0.337 1.10 0.90 1.355 1.328 8.658 9.559 22 10 2.181 1.746 : 0.264 0.370 1.10 1.00 1.424 1.392 8.982 9.452 22 16 2.035 1.723 ' 0.303 0.360 Freq/7 5 . 7 7 7 7 \ 186 As i n Model I, the wage i s a d e c l i n i n g function of the l e v e l of employment the fir m currently faces, independently of the composition of employment."^ This r e f l e c t s the firm's intertemporal monopsony power. The wage o f f e r d i s t r i b u t i o n l i e s above the p e r f e c t l y competitive equilibrium wage. This r e s u l t i s p l a u s i b l e , because, since the l e v e l of f r i c t i o n a l unemployment i s non-zero, a majority of firms i n the industry face l e v e l s of employment below that consistent with p r o f i t maximization and payment of the s t a t i c equilibrium wage. By o f f e r i n g a wage above the s t a t i c equilibrium wage, the f i r m r a i s e s the ex ante p r o b a b i l i t y of increasing i t s workforce and thus increasing i t s p r o f i t s S i n c e a majority of firms o f f e r a wage above the s t a t i c e q uilibrium wage, high employment firms are forced to o f f e r a wage above the s t a t i c equilibrium wage because the true reservation wage i s r e l a t i v e l y high. I t i s now appropriate to analyse the firms' wage decisions by considering each of the f i v e sets of employment states defined i n I : (a) The firm currently employs the 'optimal workforce'. I t s decisions are such that there i s a high ex ante p r o b a b i l i t y of r e t a i n i n g that workforce. I t does not wish to employ any more workers so, assuming that i t creates vacancies, i t o f f e r s a r e l a t i v e l y low wage, speculating about stochastic turnover and acceptance behaviour by current employees and searchers. Then, i t faces the p o s s i b i l i t y of l o s i n g e i t h e r or both valuable type two employees. Given stochastic search and acceptance 187 behaviour by unemployed i n d i v i d u a l s , any vacancy c r e a t e d may n o t be f i l l e d . More type one i n d i v i d u a l s than type 12 two i n d i v i d u a l s are unemployed. Then, there i s a h i g h e r p r o b a b i l i t y that a type one searcher w i l l r e c e i v e an o f f e r than a type two s e a r c h e r . Although a type one searcher has a h i g h e r p r o b a b i l i t y ^ ex a n t e , of r e f u s i n g the o f f e r , he has a h i g h e r p r o b a b i l i t y of r e c e i v i n g and a c c e p t i n g an o f f e r than a type two i n d i v i d u a l (see (b) ) . Hence, i f a f i r m creates v a c a n c i e s , i t faces a h i g h p r o b a b i l i t y of a t t a i n i n g a suboptimal l e v e l or suboptimal composition o f employment. In other words, the f i r m faces a greater v a r i a n c e i n the l e v e l and composition o f employment next p e r i o d , and 13 thus lower mean p r o f i t s . Then, to maximise mean p r o f i t s , the f i r m creates zero vacancies and o f f e r s i t s c u r r e n t workforce a r e l a t i v e l y h i g h wage, so that there i s a s m a l l p r o b a b i l i t y of employees q u i t t i n g . T h i s hypothesis i s supported by o b s e r v a t i o n o f the elements of the row corresponding to the ' o p t i m a l workforce' i n the t r a n s i t i o n m a t r i x , shown i n Table 11. The element, which i n d i c a t e s the p r o b a b i l i t y that a f i r m f a c i n g the ' o p t i m a l workforce' r e t a i n s the workforce, assumes the greatest value i n the m a t r i x . The f i r m faces a l e s s than optimal l e v e l of employment. T h e r e f o r e , i t creates vacancies and o f f e r s a h i g h e r wage than h i g h e r employment f i r m s , so that i t has a h i g h probab-188 i l i t y of i n c r e a s i n g the s i z e of i t s workforce. The l e v e l s o f vacancy c r e a t i o n f o r these three s t a t e s of employment are c o n s i s t e n t w i t h i n t e r t e m p o r a l p r o f i t maximization, as defined i n Chapter 2 IIIiG, and thus s p e c u l a t i v e vacancy c r e a t i o n i s z e r o . A f i r m i n employment s t a t e [1,0] o f f e r s a h i g h e r wage than a f i r m i n s t a t e [ 0 , 1 ] , s i n c e type one i n d i v i d u a l s have a h i g h e r p r o b a b i l i t y of q u i t t i n g i n response to a p a r t i c u l a r wage o f f e r , ex ante, than type two i n d i v i d u a l s and these firms wish to r e t a i n t h e i r e x i s t i n g workforces. The f i r m i n employment s t a t e [0,0] o f f e r s the h i g h e s t wage i n the i n d u s t r y . This conforms to the r e s u l t obtained i n Model I . The f i r m enjoys zero i n t e r t e m p o r a l monopsony power. These f irms c o u l d choose a lower wage o f f e r and i n c r e a s e the l e v e l of vacancy c r e a t i o n . Type one i n d i v i d u a l s have a h i g h e r p r o b a b i l i t y of sampling the f i r m and a c c e p t i n g an 14 o f f e r than type two i n d i v i d u a l s . There i s an extremely low p r o b a b i l i t y , however, of say, even two i n d i v i d u a l s sampling the same f i r m . In a d d i t i o n , a f i r m c u r r e n t l y employing a more v a l u a b l e type two i n d i v i d u a l faces a h i g h p r o b a b i l i t y of a t t a i n i n g a suboptimal composition as w e l l as a suboptimal l e v e l of employment i n response to vacancy s p e c u l a t i o n and a low wage o f f e r . 189 Then, summarising,a firm, currently facing an under-size workforce, creates vacancies equal to desired net hi r e s and o f f e r s a higher wage to a current type one employee than a type two employee i n the state complement. The firm faces the optimal l e v e l but suboptimal composition of employment. In each of the sol u t i o n s , the fir m creates a s i n g l e vacancy for both states of employment. Observation of the relevant rows of the t r a n s i t i o n matrix shows that the f i r m has a r e l a t i v e l y high p r o b a b i l i t y of maintaining the same composition of employment, despite some quits and accessions. Ex ante, a type one i n d i v i d u a l has a higher p r o b a b i l i t y of q u i t t i n g i n response to a p a r t i c u l a r wage o f f e r but a type one i n d i v i d u a l also has a higher p r o b a b i l i t y of receiving and accepting an o f f e r . Thus, as i n Model I, there e x i s t s a trade o f f between the wage o f f e r and vacancy creation decision. Firms w i l l not o f f e r a r e l a t i v e l y high wage and create zero vacancies, so that there i s a high p r o b a b i l i t y of maintaining the e x i s t i n g workforce which has suboptimal composition, because there i s a r e l a t i v e l y low ( i n the case of state [2,0], zero) p r o b a b i l i t y of achieving a worse composition of employment. Indeed, there i s a low p r o b a b i l i t y of achieving a b e t t e r composition of employment. Then, to summarise, firms who face the optimal l e v e l but sub-optimal composition of employment create a s i n g l e speculative 1 9 0 vacancy. This r e f l e c t s the trade o f f between the wage o f f e r and vacancy de c i s i o n , noted i n Model I, and also the small p r o b a b i l i t y that such an o f f e r may lead to a su b s t i t u t i o n i n employment between a type one i n d i v i d u a l and a type two i n d i v i d u a l through one quit and one accession. The wage o f f e r over these two states of employment i s an increasing function of the number of type one i n d i v i d u a l s employed. The f i r m employs an oversize workforce, but the 'optimal' workforce constitutes a subset of that workforce. For the reasons s p e c i f i e d i n (a),the f i r m creates zero vacancies. I t s wage decision i s such that there i s a high p r o b a b i l i t y of a t t a i n i n g the optimal l e v e l of employment. Thus, at a given current l e v e l of employment, the wage o f f e r i s an increasing function of the number of type one i n d i v i d u a l s employed. Individuals' labour market behaviour, i n some sense, i s compatible with the firm's decisions. Less valuable type one i n d i v i d u a l s have a higher p r o b a b i l i t y , ex ante, of q u i t t i n g i n response to a p a r t i c u l a r wage o f f e r , and so there i s a r e l a t i v e l y high p r o b a b i l i t y of achieving the optimal composition of employment. The f i r m faces an above optimal l e v e l and a suboptimal composition of employment. I t s decisions are such that there i s a high p r o b a b i l i t y of a t t a i n i n g the optimal 191 employment l e v e l . I f the firm currently employs four workers, then i t i s constrained to create zero vacancies and i t s wage offer i s , therefore, an increasing function of the number of type one individuals employed. The results associated with an employment l e v e l of three are not consistent. A firm, i n employment state [3,0], creates a speculative vacancy and offers a r e l a t i v e l y low wage i n six cases out of seven. Creating a speculative vacancy enables the firm to offer a lower wage. There i s a non-zero prob a b i l i t y of achieving a superior composition of employment and a zero probability of attaining a worse composition of employment. In employment state [2,1] the results are not systematic, since the firm creates a vacancy i h three cases out of seven. A p r i o r i , there i s no reason to believe that the vacancy decision should be consistent over the seven sets of per-ception - parameters. The choice of wage and vacancy decisions may w e l l be influenced by the magnitude of the perception parameters and the difference between them.^ 3. A Summary of Firm's Decisions i n Model I I The key to analysing the results i s to note the significance of the composition of employment and the supply of searchers on firms' decisions. F i r s t l y , the prob a b i l i t y of retaining the exi s t i n g workforce i n response to a p a r t i c u l a r offer depends on the composition of employment. Thus, 192 over a l l employment states which constitute a p a r t i c u l a r l e v e l of employment, i f the vacancy creation decision i s the same, then the wage o f f e r i s an i n c r e a s i n g function of the number of type one i n d i v i d u a l s employed. Secondly, by contrast to Model I, firms are not i n d i f f e r e n t between r e t a i n i n g current employees and, say, facing one q u i t and one accession. Thus,speculative vacancies are created and lower wage 16 o f f e r s made by firms i n the states corresponding to optimal or above l e v e l s of employment i n which there i s a low or zero p r o b a b i l i t y of achieving a worse composition of employment. Firms are e s s e n t i a l l y speculating on the basis of type one i n d i v i d u a l s ' turnover and acceptance behaviour alone. L^ Firms do not create vacancies when currently employing the optimal workforce. T h i r d l y , vacancy decisions are influenced by the supply of searchers. Speculative vacancies are not created by firms i n states corresponding to low l e v e l s of employment because there i s a low p r o b a b i l i t y that s u f f i c i e n t searchers w i l l sample the firm. These three factors explain firm decisions which appear non-systematic i n comparison to decisions i n Model I. In Model I, firms' wage and vacancy creation decisions demonstrate t h e i r intertemporal monopsony power and i n d i f f e r e n c e between current employees and searchers. 4. An Explanation of the Summary S t a t i s t i c s Type one employees enjoy a higher d i s t r i b u t i o n of wage o f f e r s than type one employees. This r e f l e c t s t h e i r higher ex ante p r o b a b i l i t y of turnover i n response to a p a r t i c u l a r o f f e r and, i n addition, the greater 193 supply of type one s e a r c h e r s . Both the composition o f the workforce and the composition of the unemployed i n f l u e n c e the f i r m ' s wage o f f e r . Thus, type one employees earn a h i g h e r mean wage than type two employees. Since each searcher faces the same d i s t r i b u t i o n o f o f f e r s , a type one i n d i v i d u a l has a longer mean d u r a t i o n o f unemployment. I t i s p l a u s i b l e , then, that a type one i n d i v i d u a l , d e s p i t e e a r n i n g a h i g h e r mean wage, when employed, earns a lower MDI than a type two i n d i v i d u a l , due to a longer mean d u r a t i o n of unemployment. Type one i n d i v i d u a l s are l e s s v a l u a b l e to the f i r m and earn h i g h e r mean wages. Thus, f irms i n the aggregate w i l l employ fewer type one i n d i v i d u a l s . Thus, type one i n d i v i d u a l s enjoy a h i g h e r l e v e l o f u n -employment than type two i n d i v i d u a l s . Although type one employees earn a h i g h e r mean wage, they earn a lower share of output due to t h e i r h i g h e r l e v e l of unemployment. Type one i n d i v i d u a l s are l e s s v a l u a b l e than type two i n d i v i d u a l s . Then, f o r any given l e v e l of employment, mean p r o f i t s discounted over an i n f i n i t e h o r i z o n are a d e c l i n i n g f u n c t i o n o f the number of type one i n d i v i d u a l s i n i t i a l l y employed. I n d i v i d u a l s ' turnover and acceptance behaviour i s s t o c h a s t i c and so firms face d i f f e r e n t l e v e l s of employment over time and make d i f f e r e n t wage and vacancy c r e a t i o n d e c i s i o n s . Consequently, the temporal c o r r e l a t i o n of wage o f f e r s i s a d e c l i n i n g f u n c t i o n of t ime. Since employment d i s p e r s i o n e x i s t s i n i t i a l l y and i n d i v i d u a l s have imperfect i n f o r m a t i o n about wage o f f e r s , turnover and acceptance 194 behaviour i s stochastic and f r i c t i o n a l unemployment i s non-zero i n stochastic equilibrium. Search i s random and therefore stochastic equilibrium cannot be characterised by segregated workforces. Even i n Model I I I , when firms can e x p l i c i t l y discriminate between individuals i n wage offer and employment, mixed workforces 7 characterise equilibrium, although individuals evaluate wage offers i n r e l a t i o n to t h e i r own wage offer d i s t r i b u t i o n . 5. The Concepts of Vacancy Creation and Excess Demand In Model I , although a s i g n i f i c a n t component of vacancy creation represents desired net h i r e s , firms speculate i n vacancy creation. By increasing the number of in d i v i d u a l s , i n t o t a l , to whom offers are made the firm i s able to offer a lower wage rate. The firm i s i n d i f f e r e n t between retaining current employees and facing, say, one accession and one quit, because i n d i v i d u a l s , ex ante, are i d e n t i c a l i n thei r labour market behaviour and productivity, irrespective of employment status. In Model I I , the composition of the current workforce and the li k e l i h o o d of achieving a superior composition through new hires assumes key importance. Speculative vacancy creation i s only undertaken by firms who currently employ an optimal or above optimal sized workforce, and who have a low probability of attaining a worse composition of em-ployment. Below the optimal l e v e l of employment, vacancy creation decisions are not speculative and represent desired net hi r e s . 195 This non-systematic nature of vacancy creation over some levels of employment further undermines i t s use as a proxy for firms' excess demand for labour. I t seems unlikely that, even i n a labour market i n which vacancies represent positions requiring i d e n t i c a l s k i l l s , a homogeneous measure of vacancies can be created. Vacancies are heterogeneous,since they represent job offers at different wage rates. 6. Theoretical Contribution of Model I I Both Salop and Sanborn i n f e r that a necessary condition for wage d i f f e r e n t i a l s i s that firms recognise different types of in d i v i d u a l and are allowed to discriminate i n wage offer and h i r i n g . The results of Model I I indicate the contrary. Inter-firm wage d i f f e r e n t i a l s at the same l e v e l of employment e x i s t , even when the firm'" i s not allowed to discriminate. In response to the prevailing wage offer d i s t r i b u t i o n , type one individuals have a higher probability of q u i t t i n g , ex ante, and a higher ex ante marginal probability of q u i t t i n g than type two individuals. Firms, unable to discriminate e x p l i c i t l y between individuals i n wage offer and h i r i n g , make wage offer and vacancy creation decisions which are dependent on both the l e v e l and composition of employment. Over those states of employment which correspond to the same l e v e l of employment and the same vacancy creation decision, the wage offer i s an increasing function of the number of type one individuals employed. Low-turnover type two indi v i d u a l s , who also have a low marginal pro b a b i l i t y of turnover, are discriminated against i n the wage off e r . This i s essen-196 t i a l l y Salop's argument. A low-turnover, type two i n d i v i d u a l , however, i s more valuable to the firm, because the mean returns associated with h i r i n g him and then offering him a stream of wage rates over time are higher than the returns associated with h i r i n g a type one ind i v i d u a l and offering him the same stream of wage rates. Not only does the type one i n d i v i d u a l have a higher probability of q u i t t i n g , ex ante, but he has a lower probability of acceptance i n response to a given wage off e r . Consequently, i f the firm faces the 'optimal workforce', then i t attempts to retain this optimal workforce by offering a r e l a t i v e l y high wage rate, as argued by Sanborn. This wage rate i s higher than any wage rate offered by a firm facing an optimal or above l e v e l of employment. Thus, Sanborn's and Salop's arguments are p a r t i a l l y correct when firms cannot e x p l i c i t l y discriminate i n wage offer and employment. In the aggregate, however, firms' i n a b i l i t y to discriminate e x p l i c i t l y i n wage offers becomes c r u c i a l . Although type two individuals are more valuable to a firm, the mean wage enjoyed by a type two in d i v i d u a l i s 18 less than the mean wage enjoyed by a type one employed i n d i v i d u a l . By contrast, i n Model I I I , more valuable type two workers earn a higher mean wage than type one workers. C. Comparative S t a t i c Predictions of Model I I 1. An Introduction There are twenty summary s t a t i s t i c s whose changes describe the impact on stochastic equilibrium i n the labour market of a change i n the value of perception parameters. They are : 197 (a) The l e v e l of the general wage o f f e r d i s t r i b u t i o n ; (b) The mean of the o f f e r d i s t r i b u t i o n f a c i n g a s e a r c h e r ; (c) The v a r i a n c e of a s e a r c h e r ' s d i s t r i b u t i o n of wage o f f e r s ; (d) The mean wage earned by a type one employed i n d i v i d u a l ; (e) The mean wage earned by a type two employed i n d i v i d u a l ; (f) MDI enjoyed by a type one i n d i v i d u a l ; (g) MDI enjoyed by a type two i n d i v i d u a l ; (h) Aggregate unemployment; ( i ) Aggregate type one unemployment; (j) Aggregate type two unemployment; (k) Aggregate vacancy c r e a t i o n ; (1) Aggregate s p e c u l a t i v e vacancy c r e a t i o n ; (m) The mean d u r a t i o n of search p r i o r to an o f f e r for a s e a r c h e r ; (n) The mean d u r a t i o n of unemployment for a type one i n d i v i d u a l ; (o) The mean d u r a t i o n of unemployment f o r a type two i n d i v i d u a l ; (p) Mean f i r m p r o f i t s ; (q) Type one i n d i v i d u a l s ' share of t o t a l output; (r) Type two i n d i v i d u a l s ' share of t o t a l output; (s) L a b o u r ' s t o t a l share of output; and (t) The one p e r i o d c o r r e l a t i o n c o e f f i c i e n t of wage o f f e r s . 198 2. The Results E l e v e n d i f f e r e n t combinations of the p e r c e p t i o n parameters, 2 c^ are adopted i n the range 0.90 to 1.15. A l l other exogeneous 1 2 parameters remain constant, c^ equals or exceeds c^ i n a l l combin-a t i o n s of the p e r c e p t i o n parameters chosen. These s o l u t i o n s y i e l d ten 19 weak comparative s t a t i c p r e d i c t i o n s , f i v e a s s o c i a t e d w i t h an i n c r e a s e 1 2 i n c^ and f i v e a s s o c i a t e d w i t h an i n c r e a s e i n c^ . While the r e s u l t s 2 are more systematic i n the case of an i n c r e a s e of c^ , there i s no evidence to suggest that the q u a l i t a t i v e change i n labour market behaviour would be d i f f e r e n t when c ^ i s i n c r e a s e d as compared with an i n c r e a s e i n 2 c^ . The a n a l y s i s of the r e s u l t s r e l a t e s to an i n c r e a s e i n the parameter 1 c 1 , but a s i m i l a r d i s c u s s i o n i s a p p r o p r i a t e i n the case of an i n c r e a s e 2 i n c^ An i n c r e a s e i n the value of the p e r c e p t i o n parameter, leads to a non-systematic s h i f t i n the wage o f f e r d i s t r i b u t i o n . There i s a general tendency f o r wage o f f e r s to i n c r e a s e over s t a t e s of employment, 20 unless the p a t t e r n of vacancy c r e a t i o n d e c i s i o n s changes. The mean and v a r i a n c e of the o f f e r d i s t r i b u t i o n f a c i n g searchers g e n e r a l l y r i s e and the mean wages earned by both employed type one and type two i n d i v i d -uals always i n c r e a s e . MDI of type one i n d i v i d u a l s g e n e r a l l y f a l l s but MDI o f type two i n d i v i d u a l s r i s e s . Both t o t a l unemployment and type one i n d i v i d u a l s ' unemployment r i s e w h i l e type two i n d i v i d u a l s ' unemployment f a l l s . Aggregate vacancy c r e a t i o n r i s e s . 199 The mean duration of search p r i o r to an offer i s again negatively correlated to the change i n the aggregate measure of vacancies minus unemployment. The mean duration of unemployment for type one individuals r i s e s , while changes i n the duration of unemployment of type two i n d i v i d -uals are not related i n a systematic way to changes i n the perception parameter. The mean l e v e l of p r o f i t s discounted over an i n f i n i t e horizon f a l l s . Type one i n d i v i d u a l s 1 share of output f a l l s , but type two individuals' share of t o t a l output r i s e s . Labour's share of t o t a l output r i s e s . The one period correlation of wage offers f a l l s when c ^ r i s e s . The values of the summary s t a t i s t i c s over a l l combinations of the perception parameters adopted and the frequency with which the comparative 1 2 s t a t i s predictions are upheld for increases i n c^ and c^ are shown i n Table 13. 3. An Explanation (a) An Introduction I t seems that the absolute magnitudes of the parameter values are unimportant i n the explanation of the re s u l t s . Some r e s u l t s , however, have to be explained i n the context of the r e l a t i v e parameter 1 2 values chosen. In p a r t i c u l a r , i f c^ and c^ are equal, then a l l individuals are i d e n t i c a l , ex ante, so Model I i s appropriate and firms make the same wage and vacancy creation decisions over each l e v e l of employment,irrespective of the composition of employment. In this case, TABLE 13 Comparative Static Results for Model II 1 2 Different Combinations of cx and c\ 0.95 | 0.95 1.00 1.00 1.00 1.05 1.05 1.05 1.10 1.10 1.10 Sign Freq/ Sign Freq/ j Freq/ 1 10 0.90 0.95 0.90 0.95 1.00 0.95 1.00 1.05 0.90 1.00 1.10 (cJ+> 5 (c2"0 5 ' j a) } I i i + i 1 + i i 2 b) 1.079 1.169 1.085 1.133 1.196 1.146 1.169 1.249 1.160 1.257 1.283! + 3 + 8 c) 0.067 0.071 0.073 0.085 0.081 ' 0.089 0.100 0.091 0.099 J 0.101 0.112| + 5 + 3 8 d) 1.237 1.276 1.244 1.288 1.323 1.323 1.353 1.386 1.355 1.424 1.455 + 5 + 5 10 e) 1.214 1.276 1.235 1.308 1.323 1.310 1.366 1.386: 1.328 1.392 1.455 + \ 5 + 5 10 f) 8.524 8.988 8.534 8.776 9.070 8.682 8.852 9.133 8.658 8.982 9.143 1 I 3 + 8 g) 8.913 8.988 8.968 9.087 9.070 9.287 9.154 9.133 9.559 9.452 9.143 + 5 - 3 8 h) 25 28 27 31 32 32 37 38 32 38 | 44 + 5 5 10 !*> 16 1.4 \ 17 j 17 16 20 20 19 22 22 I 22 + 5 5 (-2) 10(-2) 9 14 1 10 14 16 12 17 19 10 16 |22 3(-l) i ; + • 5 8(-l) k) 44 52 I 43 ; 48 54 49 51 59 48 58 j 61 + 2 | + 5 7 1) 14 19 ! | 13 17 13 12 18 13 17 \ 16 - 4(-l) 1 " 2(-l) 6 (-2) m) | 1.570 1.369 j 1.608 1.541 1.377 1.532 1.541 1.335 1.553 1.368 }1.374 ? 4 i ? S 5 9 n) ! 1.890 1.654 I 1.987 | 1.917 1.737 I 2.058 2.051 1.825 2.181 2.035 12.022 + 4 i j i ~ 5 9 o) \ 1.752 ; 1.654 1.79li 1.869 1.737 1.783 1.979 1.825 1.746 1.723 j2.022 - \ 2 i j + 2 4 p) 24.674 (22.020 24.147J21.526 20.250 21.105 19.182 17.912 20.592 .7.439 15.612 J -5 { _ 5 10 q) 1 0.271 i 0.310 1 0.268 ! 0.292 0.317 0.274 0.303 0.327 0.264 0.303 I 0.337 t - 5(-l) | + 5 10(-1) r) \ 0.321 0.310 | 0.329 j 0.328 0.317 0.351 0.337 0.327 0.370 0.360 j0.337 + 5 | - 5 10 s) \ 0.592 0.620 ! 0.597 0.620 0.634 0.625 0.640 0.654 0.634 0.663 |0.674 + 5(-l) + 5 10(-1) t) 0.455 0.452 | 0.485 0.437 0.395 0.422 0.361 0.342 0.481 0.386 j0.300 i. - . . . 3 5 8 201 1 2 an i n c r e a s e i n c^ above c^ leads to a d i f f e r e n t p a t t e r n o f vacancy c r e a t i o n and wage d e c i s i o n s . Some of the corresponding changes i n the summary s t a t i s t i c s d i f f e r from those a r i s i n g from the comparison of two s o l u t i o n s when both sets o f p e r c e p t i o n parameters are unequal. (b) P r e d i c t i o n s a r i s i n g from an i n c r e a s e i n Given any future stream of wage o f f e r s , the mean r e t u r n s to a f i r m a s s o c i a t e d w i t h h i r i n g a type one i n d i v i d u a l f a l l , because, c e t e r i s p a r i b u s , he has a h i g h e r p r o b a b i l i t y of t u r n o v e r , ex ante. Although type one i n d i v i d u a l s are as p r o d u c t i v e , when employed, they are both l e s s v a l u a b l e to the f i r m , once employed,and have a lower p r o b a b i l i t y of acceptance, when unemployed. I f f irms make the same d e c i s i o n s as b e f o r e , then the r a t e o f unemployment of type one i n -d i v i d u a l s r i s e s . More firms face l e v e l s of employment which are suboptimal , i n the sense d e f i n e d . Firms counter t h e . i n c r e a s e i n type one unemployment by an i n c r e a s e i n wage o f f e r s over some s t a t e s o f employment. Since type one i n d i v i d u a l s are l e s s v a l u a b l e , firms i n the aggregate w i l l n o t choose to h i r e as many at h i g h e r wage r a t e s and t h i s d e c i s i o n i s r e i n f o r c e d by type one i n d i v i d u a l s ! behaviour. Thus, d e s p i t e the i n c r e a s e i n the g e n e r a l wage o f f e r d i s t r i b u t i o n , aggregate type one unemployment r i s e s . Type two i n d i v i d u a l s now face a h i g h e r d i s t r i b u t i o n of o f f e r s and are l i k e l y to have a h i g h e r p r o b a b i l i t y of acceptance and a lower p r o b a b i l i t y of q u i t t i n g i n response to an o f f e r from a f i r m i n a 21 p a r t i c u l a r employment s t a t e . Thus, aggregate unemployment of type two i n d i v i d u a l s g e n e r a l l y f a l l s . The analogy of a simple s t a t i c two 202 factor model of firm behaviour i s appropriate. A s h i f t to the l e f t of the supply of one factor leads to both a substitution effect and a scale effect. Precisely the same phenomenon i s observed i n this dynamic stochastic model of firm behaviour. Firms' wage and vacancy decisions reinforced by each type of individual's labour market behaviour leads i n the aggregate to substitution i n employment between type one and type two individuals. The scale effect of the systematic change i n type one individuals' market behaviour i s measured by the increase i n aggregate unemployment. At this higher l e v e l of wage offers firms choose to employ fewer individuals i n the aggregate. This increase i h unemployment means that there i s a s h i f t i n the di s t r i b u t i o n of employment to low employment, high wage firms. In addition, firms' wage offers over employment states generally increase and, i n some cases, vacancy creation as w e l l . Thus, the mean offer enjoyed by searchers generally r i s e s . The mean offers enjoyed by each type of employed i n d i v i d u a l always increase. The higher l e v e l of wage offers over some employment states and the unchanged l e v e l of unemploy-ment compensation lead to an increase i n the variance of the wage offer d i s t r i b u t i o n . Type two individuals facing a higher d i s t r i b u t i o n of wage offers and, enjoying a lower l e v e l of unemployment, earn a higher MDI. The influence on the MDI of type one individuals of an increase i n c ^ i s not systematic. In contrast to Model I , i n which an increase i n c^ i s analysed, MDI of type one individuals f a l l s i n three cases out of f i v e . 203 These r e s u l t s are q u i t e p l a u s i b l e . In the segregated market analysed i n Model I , type one i n d i v i d u a l s alone enjoy a h i g h e r d i s t r i b u t i o n of o f f e r s a r i s i n g from t h e i r systematic change i n behaviour. Thus, d e s p i t e an i n c r e a s e i n the i n c i d e n c e of unemploy-ment, they enjoy a h i g h e r MDI up to and i n c l u d i n g the value of c^ of 1.15. In Model I I , however, i n response to the h i g h e r d i s t r i b u t i o n of o f f e r s , type two i n d i v i d u a l s enjoy a h i g h e r l e v e l of employment and thus s t e a l some of the p o t e n t i a l gains a c c r u i n g to type one i n d i v i d u a l s from h i g h e r wage o f f e r s . Thus, these non-systematic r e s u l t s are c o n s i s -tent i n that they r e f l e c t the indeterminate e f f e c t on type one i n d i v i d u a l s ' MDI of an i n c r e a s e d mean d u r a t i o n of unemployment o f f s e t by a g e n e r a l l y h i g h e r d i s t r i b u t i o n of wage o f f e r s . There i s a s h i f t i n employment to low wage, h i g h vacancy c r e a t i o n f i r m s . Thus, the l e v e l o f aggregate vacancy c r e a t i o n g e n e r a l l y i n c r e a s e s . S p e c u l a t i v e vacancy c r e a t i o n i s zero at below o p t i m a l l e v e l s of employment, and so s p e c u l a t i v e vacancy c r e a t i o n g e n e r a l l y f a l l s . The change i n the d u r a t i o n of search p r i o r to an o f f e r i s again n e g a t i v e l y r e l a t e d to the change i n the aggregate measure of vacancies minus unemployment. The mean d u r a t i o n of unemployment f o r type one i n d i v i d u a l s g e n e r a l l y i n c r e a s e s . The mean d u r a t i o n of unemployment enjoyed by type two i n d i v i d -u a l s decreases i n two cases out of f i v e . A f a l l i n the mean d u r a t i o n o f type two i n d i v i d u a l s ' unemployment r e s u l t s from the r i s e i n aggregate vacancy c r e a t i o n and the r e d u c t i o n i n type two unemployment which reduces the mean d u r a t i o n o f search and the h i g h e r d i s t r i b u t i o n of wage o f f e r s which i n c r e a s e s the p r o b a b i l i t y of acceptance. 204 In response to the systematic change i n type one individuals' behaviour, wage offers r i s e and aggregate unemployment ri s e s and so a firm's mean p r o f i t discounted over an i n f i n i t e horizon f a l l s . Despite the increase i n mean wage off e r , type one individuals' share of t o t a l output f a l l s , due to t h e i r lower l e v e l of employment. By contrast, type two employees enjoy higher wages and employment and a larger share of t o t a l output. Unsurprisingly, labour's t o t a l share of output increases. The one period correlation of wage offers generally f a l l s . This can be explained by the greater stoc h a s t i c i t y associated with type one individuals' turnover and acceptance behaviour. (c) Predictions a r i s i n g from Equal Values of the  Perception Parameters The counter i n t u i t i v e results can be explained by noting that two of the fi v e comparative state predictions arise from an increase i n 1 2 c^ above equality with c^ . In each case, the pattern of vacancy creation changes. When the perception parameters are equal, firms create the same number of vacancies over the optimal employment l e v e l and zero 1 2 vacancies above the optimal l e v e l of employment. When c^ exceeds c^ , firms create zero vacancies for the optimal workforce and a vacancy for the employment state corresponding to three type one individuals. This results i n a f a l l i n the l e v e l of aggregate vacancy creation. Together with the r i s e i n unemployment, this leads to a f a l l i n the mean offer a r i s i n g from search. In both cases, the mean duration of unemployment for type two individuals increases. 205 The remaining i n c o n s i s t e n t r e s u l t s are n o t r e a d i l y comprehensible, except i n the case of p e r c e p t i o n parameter v a l u e s , [1.10,1.00] and [ 1 . 0 5 , 1 . 0 0 ] . The r i s e i n c^" i s accompanied by s i g n i f i c a n t vacancy s p e c u l a t i o n , both i n employment s t a t e [ 0 , 2 ] , the ' o p t i m a l workforce' and above o p t i m a l l e v e l s o f employment. The i n c r e a s e i n type one unemployment i s matched by a s i g n i f i c a n t r i s e i n the l e v e l of aggregate vacancy c r e a t i o n . T h i s r e d u c t i o n i n the d u r a t i o n of search p r i o r to an o f f e r o f f s e t s the tendency f o r the mean d u r a t i o n o f unemployment of type one i n d i v i d u a l s to i n c r e a s e . Consequently, MDI of type one i n d i v i d u a l s r i s e s , s i n c e they enjoy a s h o r t e r d u r a t i o n o f unemployment and h i g h e r 22 wage o f f e r s when employed. 2 (d) P r e d i c t i o n s a r i s i n g from an i n c r e a s e i n c^ 2 When c^ i n c r e a s e s the r e s u l t s g e n e r a l l y accord w i t h those corresponding to an i n c r e a s e i n again with the exception o f those r e s u l t s a s s o c i a t e d with equal values o f the p e r c e p t i o n parameters. I n 2 two of the three cases i n which c^ i n c r e a s e s and becomes equal w i t h C j \ the v a r i a n c e of wage o f f e r s f a l l s . T h i s r e s u l t i s p l a u s i b l e , because t h e r e i s no wage d i s p e r s i o n over s t a t e s of employment corresponding to the same l e v e l but d i f f e r e n t composition o f employment, when the p e r -c e p t i o n parameters are e q u a l . The change i n the p a t t e r n of vacancy c r e a t i o n over employment s t a t e s 2 1 a r i s i n g from the i n c r e a s e i n c^ i n t o e q u a l i t y w i t h c^ generates an i n c r e a s e i n aggregate vacancy c r e a t i o n and aggregate s p e c u l a t i v e vacancy c r e a t i o n i n each case. The r e s u l t i n g f a l l i n the d u r a t i o n o f search p r i o r 206 to an o f f e r o f f s e t s the tendency f o r the d u r a t i o n o f unemployment of type two i n d i v i d u a l s to r i s e . T h i s r e s u l t occurs i n a l l three cases. MDI of type two i n d i v i d u a l s r i s e s for p e r c e p t i o n parameter values of [0.95,0.95] and [0.95,0.90] and [1.00,0.95] and [ 1 . 0 0 , 0 . 9 0 ] . Such a r e s u l t i s not i m p l a u s i b l e , because type two i n d i v i d u a l s earn h i g h e r wages when employed and, i n the f i r s t case, have a s h o r t e r d u r a t i o n o f 23 unemployment. I I I . Results of Model I I I A. B a s i c Parameter Values 1. An I n t r o d u c t i o n In Model I I I , e x p l i c i t d i s c r i m i n a t i o n i n wage o f f e r s and h i r i n g by f irms i s l e g a l . The exogeneous parameters s p e c i f i e d i n Chapter 3 IIIB and u n i t values o f the p e r c e p t i o n parameters c o n s t i t u t e the b a s i c set of parameter values for Model I I I . Although i n d i v i d u a l s are i d e n t i c a l , ex ante, the firms have the economic i n c e n t i v e to d i s c r i m i n a t e between them. For example, i f a f i r m c u r r e n t l y employs a below o p t i m a l workforce s o l e l y of a p a r t i c u l a r type of i n d i v i d u a l , then i t w i l l create vacancies and o f f e r a higher wage to t h i s type of i n d i v i d u a l than the other type. T h i s r e f l e c t s the h i g h e r value to the f i r m of i n d i v i d u a l s c u r r e n t l y employed than s e a r c h e r s , who may or may not sample the f i r m and accept an o f f e r . I f the s o l u t i o n corresponding to the b a s i c s e t o f parameter v a l u e s 24 i s unique, then i t i s evident that the s o l u t i o n i s symmetric. 207 Then, f o r any employment s t a t e j , i f w 3 , v | , w 3 , v 3 denote the wage and vacancy c r e a t i o n d e c i s i o n s f o r each type of i n d i v i d u a l , r e s p e c t i v e l y , and, i f employment s t a t e k i s the s t a t e complement, = w 2 J . . . . (2) = v 2 j . . . . (3) w 2 = w i • . . . . (4) v2 = V .... (5) 3 k - Bj . . . . (6) where k = s ( e 2 J , e ^ ) . . . . . (7) N e i t h e r s o l u t i o n corresponding to equal values of the p e r c e p t i o n parameters i s symmetric. There are both conceptual and computational reasons to e x p l a i n these r e s u l t s . (a) I f the f i r m faces a mixed workforce w i t h the same number of workers of each type, i t i s p o s s i b l e t h a t , due to the i n t e g r a l vacancy c r e a t i o n d e c i s i o n , i t might choose to make a d i f f e r e n t wage o f f e r and vacancy c r e a t i o n d e c i s i o n f o r each type of i n d i v i d u a l i n response to a p a r t i c u l a r labour market environment. Regarding the determination of these d e c i s i o n s as a step i n the s i m u l a t i o n procedure, t h i s means that the parameters of each i n d i v i d u a l ' s d i s t r i b u t i o n of p e r c e p t i o n s w i l l d i f f e r . Consequently, i n d i v i d u a l s w i l l d i f f e r s y s t e m a t i c a l l y i n t h e i r 208 labour market behaviour i n subsequent i t e r a t i o n s and the s o l u t i o n generated w i l l be non-symmetric, given that there i s some degree of inaccuracy i n the s i m u l a t i o n procedure. (b) L i k e w i s e , a computational problem may a r i s e because the wage d e c i s i o n i s only def ined to an accuracy of 0.005. For example, i f i n s t a t e j , where e ^ = and the f i r m makes the same vacancy c r e a t i o n d e c i s i o n s , i t i s p o s s i b l e that the optimal wage d e c i s i o n l i e s between two s u c c e s s i v e p o s s i b l e wage d e c i s i o n s . The optimal wage d e c i s i o n s produced by the s i m u l a t i o n , w^ and w 2 , then, may be such that w- < w < w 2 . . . . (8) and w 2 - w 1 = 0.005 . . . . (9) where w denotes the optimal s o l u t i o n for both types of i n d i v i d u a l . Such a non-symmetric p a r t i a l o p t i m i s a t i o n s o l u t i o n , generated by the s i m u l a t i o n procedure, w i l l r e s u l t i n i n d i v i d u a l s f a c i n g d i f f e r e n t o f f e r d i s t r i b u t i o n s . A g a i n , i n subsequent s i m u l a t i o n s , the s y s t e m a t i c a l l y d i f f e r e n t behaviour of type one and type two i n d i v i d u a l s may be r e i n f o r c e d by f i r m d e c i s i o n s , given the i n a c c u r a t e s i m u l a t i o n procedure. (c) The non-symmetric search procedure, adopted i n the s o l u t i o n of Model I I I , may be the cause of the non-symmetric r e s u l t , a l t h o u g h , i n each case, the o p t i m a l d e c i s i o n generated by the search procedure i s compared w i t h the optimal d e c i s i o n corresponding to the s t a t e complement. 209. 2. The Results Stochastic equilibrium i s characterised by dispersion both i n the l e v e l and composition of employment. Intra-firm wage d i f f e r e n t i a l s e x i s t , although individuals are i d e n t i c a l , ex ante, and there i s d i s -persion i n each wage offer d i s t r i b u t i o n both over the l e v e l and com-position of employment. Over each l e v e l of employment, the wage offer to each type of in d i v i d u a l i s generally an increasing function of the number of that type employed. Firms create vacancies at less than optimal levels of employment which, i n t o t a l , exceed desired net hires. Facing the optimal l e v e l of employment consisting of one type of in d i v i d u a l , the firm creates speculative vacancies for the other type of i n d i v i d u a l at a low wage offer. Vacancy creation i s zero for above optimal levels of employment. Table 14 shows the basic solution to Model I I I . 3. An Explanation In analysing firms decisions, i t i s easiest to group employment states according to whether the employment l e v e l i s less than, equal to, or greater than the optimal l e v e l of employment. (a) The firm faces an undersize workforce. The firm wishes to increase the size of i t s workforce. Thus, i t creates vacancies. Although able to discriminate i n h i r i n g , the firm creates vacancies for both types of i n d i v i d u a l . I f the firm creates vacancies for type one individuals alone, say, then, due to stochastic search, the firm faces a r e l a t i v e l y high probability of not being sampled by a type one searcher. Then, i t faces a high probability of continuing to face a TABLE 14. The Basic Solution to Model III Stochastic Equilibrium Mean One P r o f i t s Over Type One Type Two Steady State Type One Type Two Type One Type Two Period I n f i n i t e Employment Employment Di s t r i b u t i o n Wage Wage Vacancy Vacancy P r o f i t s Horizon 0 0 0.106 1.270 1.270 2 2 0.502 24.069 1 0 0.210 1.260 1.090 1 1 1.241 25.817 0 1 0.209 1.090 1.260 1 1 1.238 25.786 2 0 0.116 1.180 0.750 0 2 1.510 26.360 1 1 0.211 1.165 1.165 0 0 1.515 26.269 0 2 0.115 0.755 1.180 2 0 1.506 26.313 3 0 0.0 1.045 0.745 0 0 1.579 . 26.461 2 1 0.017 1.060 1.005 0 0 1.582 26.420 1 2 0.016 0.940 1.085 0 0 1.580 26.388 0 3 0.0 0.745 1.050 0 0 1.569 26.415 4 0 0.0 0.955 0.745 0 0 1.590 26.443 3 0.0 1.015 0.750 0 0 1.590 26.448 2 2 0.0 0.990 0.910 0 0 1.600 26.396 1 3 0.0 0.745 1.050 0 0 1.569 26.415 0 4 0.0 0.745 0.955 0 0 1.591 26.395 211 TABLE 14 (continued)  Summary Sta t i s t i c s EW1 EW2 MW1 MW2 W* 0.937 W* 2 VW* vw* LSj^ LS 2 0.962 0.959 1.189 1.200 0.935 0.053 0.053 0.282 0.283 V* 1 v* V1 S ._ U* U* PA* PA* PO* PO* En 43 43 12 12 14 15 2.402 2.435 1.703 1.711 25.861 Correlation of Offers 1 Period 2 Periods 3 Periods 4 Periods j 5 Periods' Type One 0.658 0.443 0.188 - 0.053 - 0.217 | Type Two 0.729 0.567 ! 0.388 0.242 0.155 Mean Discounted Income No Quitting True Perception Stochastic Behaviour Type One 9.094 9.064 8.270 Type Two 9.125 9.080 8.268 TABLE 14 (Continued) Steady State T r a n s i t i o n M a t r i x 0.590 0.152 0.163 0.065 0.584 0.015 0.067 0.015 0.581 0.026 0.262 0.001 0.035 0.155 0.150 0.028 0.001 0.269 0.038 0.224 0.0 0.038 0.172 0.056 0.041 0.041 0.203 0.035 0.0 0.217 0.054 0.231 0.0 0.045 0.221 0.010 0.053 0.151 0.084 0.035 0.0 0.217 0.057 0.0 0.238 0.019 0.042 0.022 0.160 0.137 0.0 0.0 0.128 0.171 0.659 0.014 0.000 0.0 0.660 0.0 0.000 0.014 0.653 0.444 0.0 0.0 0.195 0.253 0.0 0.0 0.205 0.254 0.0 0.0 0.444 0.374 0.0 0.0 0.358 0.049 0.0 0.107 0.239 0.033 0.0 0.0 0.444 0.0 0.0 0.375 0.0 0.005 0.006i 0.0 0.038 0.0 ' 0.0 0.0 0.038 0.0 0.036 0.000 0.0 0.0 0.0 0.0 0.000 0.034 0.293 0.0 0.0 0.0 0.286 0.0 0.0 0.0 0.256 0.0 0.0 0.0 0.269 0.0 0.0 0.193 0.080 0.0 0.0 0.170 0.095 0.0 0.0 0.0 0.0 0.0 0.0 0.0 j 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.304 0.0 0.0 0.0 0.073 0.0 0.0 0.0 0.043 0.0 0.0 0.0 0.304 0.0 0.0 0.262 0.0 0.0 0.001 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.001 0.0 0.0 0.0 0.0 o.oo 0.001 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.067 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.069 ro 213 below optimal l e v e l of employment. Therefore, i t creates vacancies for both types of searcher. The l e v e l of vacancy creation for each type of i n d i v i d u a l equals t o t a l desired net hires. Then, i n t o t a l , the firm's vacancy creation decision does not represent desired net hir e s . Search i s stochastic, however, so there i s a low probability that a l l vacancies w i l l be f i l l e d . I f the firm currently has no employees, then i t makes the same wage offer to each type of searcher. I f the firm faces a non-zero l e v e l of employment, then the wage offer to each type of i n d i v i d u a l i s an increasing function of the number currently employed. This r e f l e c t s the value to the firm of a current employee. I f the firm chooses to offer a r e l a t i v e l y low wage to a current employee, then there i s a high probability that the ind i v i d u a l quits. Then the firm i s forced to increase the l e v e l of vacancy creation for each type of i n d i v i d u a l , so that there i s a high probability of securing the optimal l e v e l of employment. The r e l a t i v e l y low wage to an employee i s also a wage offer to a searcher. Thus, despite increased vacancy creation, the firm faces a high probability of facing a below optimal workforce next period. The variance of the l e v e l of employment the firm faces next period increases, because of increased vacancy creation. In short, due to the stochastic elements of job search, the i m p l i c i t costs of vacancy creation, which are measured by foregone p r o f i t s , have risen to the firm, despite i t s a b i l i t y to discriminate i n h i r i n g . Consequently, current employees are more valuable to the firm i n Model I I I , i n some sense, than i n Model I and earn a higher wage than that offered to searchers alone. 214 (b) The f i r m faces the optimal l e v e l of employment. I n s t a t e [1 ,1] , the f i r m o f f e r s the same r e l a t i v e l y h i g h wage to each i n d i v i d u a l and creates zero v a c a n c i e s . I f i t o f f e r s a low wage to one i n d i v i d u a l , then i t faces a h i g h p r o b a b i l i t y of l o s i n g an employee. I t i s again forced to create vacancies for each type of i n d i v i d u a l and face the s t o c h a s t i c elements of job s e a r c h . In s t a t e s [ 2 , 0 ] , [ 0 , 2 ] , the f i r m o f f e r s c u r r e n t employees a r e l a t i v e l y h i g h wage and creates zero v a c a n c i e s . The f i r m has a h i g h p r o b a b i l i t y of r e t a i n i n g i t s workforce. I t o f f e r s the other type of i n d i v i d u a l a wage equal or s l i g h t l y above the l e v e l of unemployment compensation and creates v a c a n c i e s . Such vacancy c r e a t i o n i s s p e c u l a t i v e , i n the sense defined i n Chapter 2 I I I G , and there i s a low p r o b a b i l i t y 26 t h a t such p o s i t i o n s w i l l be f i l l e d . Over these employment s t a t e s , t h e n , the wage o f f e r to a c u r r e n t employee i s an i n c r e a s i n g f u n c t i o n of the number employed. (c) The f i r m c u r r e n t l y employs an o v e r s i z e workforce. The f i r m creates zero vacancies and adopts wage d e c i s i o n s such that there i s a h i g h p r o b a b i l i t y of a t t a i n i n g the o p t i m a l l e v e l of employment. A f i r m c u r r e n t l y employing one type of i n d i v i d u a l i n i t s o v e r s i z e workforce makes a wage o f f e r to those i n d i v i d u a l s , which i s a d e c l i n i n g f u n c t i o n of the number employed. The wage o f f e r to the other type of i n d i v i d u a l i s a r b i t r a r y because none of these i n d i v i d u a l s are employed and vacancy c r e a t i o n i s z e r o . A f i r m employing a mixed workforce of three i n d i v i d u a l s , s t a t e s [1,2] and [ 2 , 1 ] , makes wage o f f e r s such that there i s a r e l a t i v e l y h i g h 215 p r o b a b i l i t y of r e t a i n i n g a mixed workforce, but optimal l e v e l of employment. By c o n t r a s t , a f i r m employing a mixed workforce of four i n d i v i d -uals with an unequal number of each type of i n d i v i d u a l , s t a t e s [3,1] and [ 1 , 3 ] , o f f e r s a wage approximately e q u a l l i n g the l e v e l of unem-ployment compensation to the m i n o r i t y group. In these cases, the f i r m i s d i r e c t l y e x p l o i t i n g i t s a b i l i t y to d i s c r i m i n a t e i n wage o f f e r s i n order to d r i v e out one type of i n d i v i d u a l and reduce the v a r i a n c e of employment next p e r i o d . The f i r m has the opportunity to make s i m i l a r wage o f f e r s , when employing a mixed workforce of three i n d i v i d u a l s , b u t , i n t h i s case, a low wage o f f e r to the type of i n d i v i d u a l c o n s t i t u t i n g the m i n o r i t y of the workforce leaves the f i r m w i t h the optimal l e v e l of employment, and so i t i s forced to o f f e r a very h i g h wage to m a i n t a i n t h i s l e v e l of employment. L i k e w i s e , a f i r m i n employment s t a t e [ 2 , 2 ] , which o f f e r s one type of i n d i v i d u a l a low wage, forces out these employees and has to o f f e r a r e l a t i v e l y h i g h wage to r e t a i n other employees. I n a l l cases, the wage o f f e r to the type of i n d i v i d u a l who p r e -dominates i n the workforce exceeds the o f f e r to the other type of i n d i v i d u a l . T h i s r e f l e c t s the value to the f i r m of r e t a i n i n g some or a l l of the type of i n d i v i d u a l s who predominate i n the workforce, r a t h e r 27 than o f f e r i n g low wages and c r e a t i n g s p e c u l a t i v e v a c a n c i e s . 4. A Summary of F i r m s ' D e c i s i o n s I f the p e r c e p t i o n parameters are e q u a l , then i n d i v i d u a l s are i d e n t i c a l , ex ante, i n t h e i r labour market behaviour. Firms d i s c r i m i n a t e 216 i n h i r i n g and wage o f f e r between the type of employee who predominates i n the workforce and the other type of i n d i v i d u a l , who i s a c u r r e n t employee or s o l e l y a s e a r c h e r . Such d i s c r i m i n a t i o n i s p r a c t i s e d at l e s s than o p t i m a l l e v e l s of employment because of the h i g h i m p l i c i t costs of vacancy c r e a t i o n a r i s i n g from the a b i l i t y to d i s c r i m i n a t e i n h i r i n g . To attempt to h i r e a s i n g l e i n d i v i d u a l , t h e f i r m i s forced to create vacancies for both types of i n d i v i d u a l . T h i s leads to a greater v a r i a n c e of employment i n the next p e r i o d and thus lower mean p r o f i t s . Thus, the f i r m attempts to r e t a i n c u r r e n t employees, when f a c i n g a l e s s than optimal l e v e l of employment. When f a c i n g a segregated workforce of the o p t i m a l s i z e , the f i r m uses i t s d i s c r i m i n a t o r y power to c r e a t e s p e c u l a t i v e vacancies at a low wage o f f e r f o r the type of i n d i v i d u a l not employed. F a c i n g an above optimal l e v e l of employment, the f i r m ' s a b i l i t y to d i s c r i m i n a t e i n wage o f f e r allows i t to d r i v e out employees, who c o n s t i t u t e a m i n o r i t y of the workforce, w i t h a low wage. Thus, the a b i l i t y to d i s c r i m i n a t e i n wage o f f e r leads t o a reduced v a r i a n c e of employment next p e r i o d . The f i r m ' s a b i l i t y to d i s c r i m i n a t e i n wage o f f e r and h i r i n g manifests i t s e l f , t h e n , i n d i f f e r e n t ways, a c c o r d i n g to the l e v e l of employment. B. Unequal Values of the P e r c e p t i o n Parameters 1. An I n t r o d u c t i o n Seven s o l u t i o n s to Model I I I corresponding to unequal p e r c e p t i o n 1 2 parameter v a l u e s , c 1 and c^ are generated. These j o i n t values l i e 217 i n the range 0.90 to 1.15 i n increments of 0.05. In order to organise and analyse the results associated with different combinations of parameter values, eleven summary s t a t i s t i c s associated with each in d i v i d u a l are computed for each solution. Then, the consistency of the solutions i s checked by comparing the summary s t a t i s t i c s corres-ponding to each solution. The summary s t a t i s t i c s are : (a) The mean of the wage offer d i s t r i b u t i o n facing a searcher; (b) The variance of the wage offer d i s t r i b u t i o n facing a searcher; (c) The mean wage offer enjoyed by an employee; (d) MDI; (e) Aggregate unemployment; (f) Aggregate vacancy creation; (g) Aggregate vacancy speculation; (h) The mean duration of search p r i o r to an o f f e r ; (i ) The mean duration of unemployment; (j) The share of t o t a l output; and (k) The one period correlation of wage offers. 2. Results 1 2 Again, c^ i s set a r b i t r a r i l y above c^ . Then, type one individuals have a higher pr o b a b i l i t y of rejecting a p a r t i c u l a r o f f e r , i f unemployed, and a higher probability of q u i t t i n g , i f employed, than type two i n d i v i d -uals, assuming they face the same labour market environment. Each type of i n d i v i d u a l , however, evaluates an o f f e r i n the context of his own di s t r i b u t i o n of offers. 218 S t o c h a s t i c e q u i l i b r i u m i n a labour market, i n which wage and employment d i s c r i m i n a t i o n i s p r a c t i s e d , i s c h a r a c t e r i s e d by d i s p e r s i o n both i n the l e v e l and composition of employment. I n t r a - f i r m wage d i f f e r e n t i a l s e x i s t , and there i s d i s p e r s i o n i n each wage o f f e r d i s t r i b u t i o n both over the l e v e l and composition of employment. The wage o f f e r d i s t r i b u t i o n f a c i n g type one i n d i v i d u a l s i s not s y s t e m a t i c a l l y h i g h e r than that enjoyed by type two i n d i v i d u a l s . Indeed, the t r u e mean o f f e r f a c i n g a type two searcher and the mean wage earned by a type two employee both exceed the corresponding o f f e r s f a c i n g type one i n d i v i d u a l s . The p e r c e i v e d mean wage o f f e r of a type one s e a r c h e r , though, exceeds the p e r c e i v e d mean o f f e r f a c i n g a type two s e a r c h e r . The v a r i a n c e of the wage o f f e r d i s t r i -b u t i o n f a c i n g a type one searcher exceeds the v a r i a n c e of o f f e r s f a c i n g a type two s e a r c h e r . Then, ex ante, type one i n d i v i d u a l s have a lower p r o b a b i l i t y of a c c e p t i n g a p a r t i c u l a r o f f e r than a type two s e a r c h e r . Type two i n d i v i d u a l s earn a h i g h e r MDI than type one i n d i v i d u a l s . Aggregate unemployment i s h i g h e r and aggregate vacancy c r e a t i o n i s lower f o r type one i n d i v i d u a l s than f o r type two i n d i v i d u a l s . Type one i n d i v i d u a l s enjoy a longer mean d u r a t i o n of unemployment and a longer mean d u r a t i o n of search p r i o r to an o f f e r than type two i n d i v i d u a l s . Type two employees enjoy a l a r g e r share of t o t a l output than type one employees. The c o r r e l a t i o n of type one wage o f f e r s over one p e r i o d i s h i g h e r i n four cases out of seven. 219 Over each employment l e v e l except f o u r , p r o f i t s discounted over an i n f i n i t e h o r i z o n are an i n c r e a s i n g f u n c t i o n of the number of type two i n d i v i d u a l s employed. The s o l u t i o n to the s i m u l a t i o n procedure f o r p e r c e p t i o n parameter v a l u e s [1.05,0.95] i s shown i n Table 15. Table 16 shows the values of the summary s t a t i s t i c s f o r each type of i n d i v i d u a l , and the frequency w i t h which the p r e d i c t e d r e l a t i o n s h i p between corresponding summary s t a t i s t i c s f o r each i n d i v i d u a l i s upheld i n the seven cases. 3. An E x p l a n a t i o n The a n a l y s i s of the r e s u l t s i s complicated by the endogeneity of each d i s t r i b u t i o n of perceptions of the r e s e r v a t i o n wage so that each i n d i v i d u a l evaluates a wage o f f e r i n the context of h i s own d i s t r i b u t i o n of wage o f f e r s . Thus, while a type two i n d i v i d u a l has a h i g h e r probab-i l i t y of acceptance, i f unemployed, and a lower p r o b a b i l i t y of t u r n o v e r , i f employed, i n response to a p a r t i c u l a r wage o f f e r and the same labour market environment, type two i n d i v i d u a l s may face a h i g h e r d i s t r i b u t i o n of wage o f f e r s . I t i s i m p l a u s i b l e , however, t h a t over a l l s t a t e s type two i n d i v i d u a l s should enjoy a h i g h e r wage o f f e r than type one i n d i v i d u a l s r e c e i v e i n the s t a t e complement. I t i s p r e c i s e l y that type two i n d i v i d u a l s can be o f f e r e d a lower stream of wage o f f e r s over time than type one i n d i v i d u a l s , but w i t h the same mean d u r a t i o n of employment, which i s the source of t h e i r g r e a t e r v a l u e . Type two searchers do enjoy a h i g h e r t r u e mean wage o f f e r but compute a lower mean o f t h e i r d i s t r i b u t i o n of p e r c e p t i o n s of TABLE 15 The Solution to Model I I I  with Perception Parameters, [ 1 . 0 5 , 0 . 9 5 ] Stochastic Equilibrium 1 i i Mean One j P r o f i t s Over Type One Type Two Steady State Type One Type Two ! Type One | Type Two Period | I n f i n i t e Employment Employment Distri b u t i o n Wage Wage i Vacancy ] Vacancy P r o f i t s | Horizon 0 0 0 . 0 9 5 1 . 3 1 5 1 . 2 7 5 i ! 2 1 2 0 . 4 7 8 2 3 . 5 4 6 1 0 0 . 1 8 2 1 . 2 9 0 1 . 1 0 0 1 1 ; 1 1 . 1 9 7 ! 2 5 . 2 2 3 ! 0 1 0 . 2 2 9 1 . 0 8 5 1 . 2 8 5 i 1 ! I 1 . 2 3 4 | 2 5 . 4 9 7 2 0 0 . 1 2 3 I 1 . 2 0 5 0 . 8 3 0 i o 1 2 1 . 4 5 8 1 2 5 . 7 0 3 1 1 0 . 1 5 0 i 1 . 1 4 5 1 . 1 7 0 ! 0 i 1 1 . 4 6 9 | 2 5 . 8 7 0 0 2 0 . 1 8 4 1 0 . 7 4 5 1 . 2 1 5 1 o i 0 ! 1 . 4 8 1 | 2 6 . 1 3 1 3 0 0 . 0 1 . 0 7 0 0 . 7 5 0 I 0 1 1 ! 1 . 5 1 9 i 2 5 . 7 9 4 2 1 0 . 0 1 4 | 1 . 0 2 0 1 . 1 4 5 ; 0 ! o i 1 . 5 3 9 i 2 5 . 9 4 7 1 1 2 I 0 . 0 2 4 0 . 7 5 0 1 . 1 9 0 •= • I I 0 ! 1 . 5 0 7 ; 2 6 . 1 5 3 i 0 3 0 . 0 ; 0 . 7 4 5 1 . 0 6 5 ! o I o ' 1 . 5 4 4 | 2 6 . 2 1 9 | 4 0 ;' 0 . 0 0 . 9 8 0 , 0 . 7 4 5 o 1 o ! 1 . 5 3 8 : 2 5 . 7 7 8 j 3 i • - 1 0 . 0 0 . 9 0 5 • 1 . 1 7 5 o i o j 1 . 5 3 9 j 2 5 . 9 4 0 i 2 I \ 2 ! 0 . 0 0 0 0 . 7 5 0 1 1 . 1 6 0 i o i o \ 1 . 5 3 5 ! 2 6 . 1 5 5 ! 1 ! 1 3 I 0 . 0 0 . 7 5 0 \ 1 . 0 5 5 0 0 : 1 . 5 1 2 ! 2 6 . 1 8 6 0 t 4 0 . 0 0 . 7 4 5 ! 0 . 9 7 0 i o 0 ; 1 . 5 5 3 i ' 2 6 . 1 7 9 ro o r 221 TABLE 15 (continued)  Summary S t a t i s t i cs EW 1 E W2 MW1 MW„ j . 2 . . . W* W l VW* VW* j L S 1 L S 2 0.951 1.039_ 1.210 1.221 0.970 , 0.960 ,0.056 0.049 | 0.252 0.326 V* v* V2 V 1  V S v 2 V S U* U* ; PA* PA* PO* PO* En 31 49 i ' 1 20 18 9 2 . 7 8 6 1.793 2.168 1.362 25.482 C o r r e l a t i o n of Offers 1 P e r i o d 2 P e r i o d s 3 P e r i o d s Type One 0.474 0.305 0.161 Type Two 0.735 0.571 0.376 Type One Type Two . 4.Periods 0.047 0.182 5 P e r i o d s : - 0.053 0.021 Mean Discounted Income |No Q u i t t i n g 8.807 9.545 True P e r c e p t i o n 8.909 9.501 S t o c h a s t i c Behaviour 8.118 8.869 TABLE 15 (Continued) Steady State T r a n s i t i o n M a t r i x 0.609 0.068 0.054 0.028 0.034 0.016 0.036 0.033 0.018 0.026 0.051 0.036 0.022 0.026 0.042 0.203 0.594 0.014 0.272 0.111 0.0 0.217 0.094 0.004 0.0 0.225 0.071 0.009 0.005 0.0 0.102 0.010 0.626 0.002 0.175 0.223 0.001 0.135 0.203 0.187 0.0 0.183 0.207 0.169 0.204 0.035 0.213 0.0 0.648 0.0 0.0 ;0.432 0.068 0.000 0.0 0.373 0.047 0.001 0.0 0.0 0.034 0.085 0.166 0.014 0.565 0.0 0.006 0.390 0.049 0.0 0.0 0.362 0.079 0.032 0.0 0.008 0.0 0.111 0.000 0.027 0.761 0.0 0.0 0.578 0.441 0.0 0.0 0.475 0.372 0.369 0.0 0.0 0.0 0.0 0.0 0.0 0.287 0.0 0.0 0.0 0.275 0.010 0.0 0.0 0.0 0.006 0.030 0.0 0.034 0.0 0.0 0.012 0.280 0.002 0.0 0.0 0.238 0.008 0.0 0.0 0.003 0.0 0.029 0.000 0.088 0.0 0.0 0.0 0.141 0.0 0.0 0.0 0.182 0.071 0.0 -0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.346 0.0 0.0 0.0 0.272 0.29.6' 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.076 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.008 0.0 0.0 0.0 0.0 0.052 0.0 0.0 0.0 0.000 0.0 0.0 0.001 0.0 0.0 0.0 0.0 0.006 0.0 0.0 0.0 0.017 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0*0 0.0 0.0 0.0 0.0 0.052 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.089 N 5 N 3 TABLE 16 Summary S t a t l 8 t l c a f o r Unequal V a l u e s o f t h e P e r c e p t i o n Parameters I n Model I I I Perception * * * * Parameters 0^ tMj tt^ V » 2 Mt^ K » 2 MDIj MDI 2 U 2 Vj^ V j POj P 0 2 PAj^ P A 2 LSj^ L S 2 CC^ CCj 1.00 0.90 0.934 1.033 0.054 0.049 1.190 1.207 8.104 8.849 16 8 35 48 1.795 1.359 2.639 1.702 0.256 0.320 0.722 0.751 1.05 0.95 0.951 1.039 0.056 0.049 1.210 1.221 8.118 8.869 18 9 31 49 2.168 1.362 2.786 1.793 0.252 0.326 6.474 0.735 1.10 0.90 0.950 1.083 0.063 0.052 1.257 1.246 8.220 9.284 19 6 27 42 2.446 1.354 2.971 1.588 0.249 0.345 0.484 0.666 1.10 0.95 1.001 1.108 0.076 0.066 1.286 1.301 8.426 9.233 19 11 49 51 1.537 1.342 2.602 1.801 0.271 0.345 0.729 0.694 1.10 1.00 1.007 1.109 0.076 0.067 1.289 1.309 8.454 9.192 19 12 50 51 1.531 1.343 2.574 1.863 0.276 0.341 0.721 0.700 1.10 1.05 1.029 1.057 0.080 0.077 1.313 1.321 8.633 8.935 18 15 47 45 1.616 1.628 2.545 2.288 0.294 0.324 0.696 0.667 1.15 1.10 1.050 1.078 0.089 0.086 1.356 1.363 8.744 9.067 19 16 49 46 1.551 1.590 2.607 2.347 0.301 0.334 0.708 0.684 Frequency/7 224 28 the r e s e r v a t i o n wage than type one i n d i v i d u a l s . The determination of r e l a t i v e o f f e r s to each type of i n d i v i d u a l i s made more complicated by the a b i l i t y of firms to d i s c r i m i n a t e i n wage o f f e r and h i r i n g and i n d i v i d u a l s ' heterogeneous labour market behaviour. In a market i n which a l l i n d i v i d u a l s are i d e n t i c a l , ex  ante, and firms are unable to d i s c r i m i n a t e , a f i r m enjoys c o n s i d e r a b l e monopsony power. I t i s i n d i f f e r e n t between r e t a i n i n g i t s e x i s t i n g workforce and l o s i n g c u r r e n t employees and simultaneously h i r i n g new w o r k e r s . . I t o f f e r s a wage which i s a d e c l i n i n g f u n c t i o n o f i t s l e v e l of employment. The b a s i c s o l u t i o n to Model I I I demonstrates t h a t the a b i l i t y to d i s c r i m i n a t e has a non-systematic impact on wage and vacancy c r e a t i o n d e c i s i o n s over employment s t a t e s . The i m p l i c i t costs of f i l l i n g new p o s i t i o n s has i n c r e a s e d , due to s t o c h a s t i c s e a r c h , but the a b i l i t y to d i s c r i m i n a t e i n wage o f f e r s means that i n segregated workforces s p e c u l a t i v e vacancies can be created a t low wage r a t e s and, i n a mixed f o r c e , i f necessary, one type of employee can be e f f e c t i v e l y dismissed by a low wage o f f e r . I t proves e a s i e s t to analyse f i r m s ' d e c i s i o n s by comparing i t s c u r r e n t l e v e l and composition of employment to the o p t i m a l workforce. (a) I f the f i r m c u r r e n t l y employs the ' o p t i m a l w o r k f o r c e ' , then i t s d e c i s i o n s are such that there i s a h i g h p r o b a b i l i t y of r e t a i n i n g the ' o p t i m a l workforce' (see the a p p r o p r i a t e row o f the t r a n s i t i o n m a t r i x i n Table 15). I f the f i r m creates vacancies and o f f e r s a lower 225 wage, then, as argued i n the explanation of the results to Model I I , there i s a high probability that the firm faces a suboptimal l e v e l or composition of employment. Although firms can discriminate i n h i r i n g , vacancies have to be created for both types of worker,due to the stochastic elements of job search. I t follows then that the firm w i l l not create vacancies, because i t currently faces a l e v e l and composition of employment which i s optimal i n both a single period and multiperiod framework. Thus, i t offers a r e l a t i v e l y high wage to type two employees and creates no vacancies. (b) The firm faces a less than optimal l e v e l of employment. I t s decisions are such that there i s a high probability of achieving the optimal l e v e l of employment. This can be observed i n Table 15. Thus, i t creates vacancies. Due to the stochastic elements of job search, the firm i s forced to create vacancies for each type of searcher. The l e v e l of vacancy creation for each type of in d i v i d u a l i s equal to the l e v e l of desired net hires consistent with p r o f i t maximisation. These vacancies, i n t o t a l , are speculative. I f the firm faces zero employment, then i t wishes to hire 2 workers. Type one individuals have a lower p r o b a b i l i t y , ex ante, of accepting a pa r t i c u l a r offer. Given stochastic search, there i s a low probability that type two searchers w i l l sample the firm. Thus, the firm offers a higher wage to type one searchers than type two searchers. A type one in d i v i d u a l i n employment state [1,0], receives a higher wage than a type two in d i v i d u a l receives i n the state complement [0,1]. 226 These decisions r e f l e c t the heterogeneous labour market behaviour of different types of individuals and the higher i m p l i c i t costs of vacancy creation and thus the greater value of current employees resulting from the a b i l i t y to discriminate i n h i r i n g . I t i s i n the wage offer to searchers alone that firms are able to demonstrate t h e i r preference for type two employees. A firm i n state [1,0], i s prepared to of f e r a type two searcher a higher wage than a type one searcher receives i n the state complement. Type one individuals are less valuable when hired and this i s reflected i n the r e l a t i v e offers to searchers alone over these states. Despite the higher o f f e r to type two searchers, there i s a lower probability that a type two searcher, w i l l sample a part i c u l a r firm and accept an offer than a type one searcher i n the state complement, because type two individuals have a lower incidence of unemployment. Over suboptimal levels of employment, then, the firm creates vacancies for both types of i n d i v i d u a l . The wage offer to each type of worker i s an increasing function of the number employed at each l e v e l of employment. (c) The firm faces an optimal l e v e l but suboptimal composition of employment. The firm employs at least one type one worker. Again, because the i m p l i c i t costs of f i l l i n g vacancies are high, the firm makes wage offer decisions such that there i s a high ex ante probability of retaining i t s workforce, (see Table 15). Thus,the wage offer to both type one and type two employees over this l e v e l of employment i s an increasing function of the number employed. 227 At this optimal l e v e l of employment, firms create speculative vacancies for type two individuals. The l e v e l of vacancy creation corresponds to current employment of type one individuals and thus represents desired net hires of type two individuals consistent with attaining the optimal workforce. In Model I , there i s a substitution between the wage offer and vacancy creation decision. Ceteris paribus, a higher l e v e l of vacancy creation e n t a i l s a lower wage offer. I f a firm currently employs a mixed workforce, state [1,1], and creates zero vacancies, the wage offer to type one individuals can be expected to exceed the wage offer to type two individuals. This r e f l e c t s type one individuals higher ex ante probability of turnover. By creating vacancies for type two individuals, however, there exists a non-zero pro b a b i l i t y of attaining the 'optimal workforce'. Consequently, i n a mixed workforce,the offer to type one individuals i s less than the offer to type two individuals. Employing a segregated workforce, state [2,0], the firm's wage offer to type two searchers i s r e l a t i v e l y low. The probability of f i l l i n g such vacancies i s small, however, due to the stochastic elements of job search. Thus, the firm offers r e l a t i v e l y high wages to type one individuals, so that there i s a high probability that they do not turn-over. The a b i l i t y to discriminate enables speculative vacancies to be created for type two searchers at lower wage rates than are offered to current type one employees. (d) The firm faces an above optimal l e v e l of employment but the 'optimal workforce' constitutes a subset of the current workforce, 228 s t a t e s [ 1 , 2 ] , [ 0 , 3 ] , [ 2 , 2 ] , [1,2] and [ 0 , 4 ] . The r e s u l t s are not independent of the p e r c e p t i o n parameter v a l u e s . For j o i n t values of the p e r c e p t i o n parameters, [1 .00,0.90] and [ 1 . 1 0 , 0 . 9 0 ] , the f i r m creates zero vacancies and makes o f f e r s to each type of i n d i v i d u a l such that there i s a h i g h p r o b a b i l i t y of s e c u r i n g the optimal l e v e l of employment. The f i r m o f f e r s a wage equal to or 29 below the l e v e l of unemployment compensation over a l l these states to type one employees. The wage o f f e r to type two employees over each employment l e v e l i s a d e c l i n i n g f u n c t i o n of the number employed. The f i r m wishes to r e t a i n two of the c u r r e n t type two workers. For h i g h e r j o i n t values of the p e r c e p t i o n parameters, the f i r m creates a vacancy f o r a type one searcher i n s t a t e [ 1 , 2 ] , and o f f e r s 30 a wage equal to or exceeding the l e v e l of unemployment compensation. The o f f e r to type one i n d i v i d u a l s i n s t a t e [2,2] equals or exceeds the l e v e l of unemployment compensation, but the o f f e r i n s t a t e [1,3] i s equal t o or l e s s than the l e v e l of unemployment compensation. The p a t t e r n of o f f e r s to type two employees over the f i v e states i s u n -changed. For low values of the p e r c e p t i o n parameters, the low wage o f f e r to type one employees, zero vacancy c r e a t i o n d e c i s i o n and r e l a t i v e l y h i g h o f f e r to type two employees are c o n s i s t e n t with there b e i n g a low v a r i a n c e of employment next p e r i o d and h i g h p r o b a b i l i t y of a t t a i n i n g the ' o p t i m a l w o r k f o r c e ' . 229 Higher j o i n t values of the perception parameters correspond to 2 higher values of the perception parameter, c^ . Then, i n order to retain two type two employees, the firm must offer a higher wage to type two workers. I f the firm currently employs only two type two individuals, however, rather than offer type two individuals a r e l a t i v e l y high wage, i t chooses to offer type one individuals a wage equal to or exceeding unemployment compensation to exploit t h e i r higher marginal probability of acceptance, and, i n addition, i f not constrained by the employment level,.create a speculative vacancy for type one ind i v i d u a l s . This enables the firm to offer a s l i g h t l y lower wage to type two individuals than otherwise. I f the firm currently employs three or more type two indi v i d u a l s , then a low wage w i l l drive out any current type one employee, but the wage offer to type two individuals i s lower than i n a workforce of two type two ind i v i d u a l s , because the firm only wishes to re t a i n two type 31 two individuals. Thus, the firm does not choose to create speculative vacancies or attempt to retain i t s current type one employee. (e) The firm faces an above optimal l e v e l and a suboptimal compos-i t i o n of employment. The wage decisions by firms r e f l e c t their desire to a t t a i n the optimal l e v e l of employment. In most cases, the wage offer to type one and type two individuals i s an increasing function of the number employed for any given t o t a l l e v e l of employment. I f the firm currently employs a mixed workforce, then i t s wage offer to type one individuals i s less than the offer to type two individuals i n eleven 230 32 cases out of a p o s s i b l e f o u r t e e n . The f i r m wishes to r e t a i n the type two i n d i v i d u a l and l o s e a l l but one of the type one i n d i v i d u a l s . I n a segregated type one workforce, by c o n t r a s t , the f i r m wishes to r e t a i n two of the workforce and o f f e r s a h i g h e r wage. In the other three cases, the f i r m d r i v e s out type two i n d i v i d u a l s from employment s t a t e [3,1] with a low wage o f f e r . T h i s reduces the v a r i a n c e of em-ployment. In s i x cases out of seven, the f i r m creates a s p e c u l a t i v e vacancy for type two i n d i v i d u a l s at a wage equal to or j u s t above the l e v e l of unemployment compensation i n employment s t a t e [ 3 , 0 ] . Such behaviour i s q u i t e p l a u s i b l e , s i n c e the f i r m would p r e f e r to employ type two i n d i v i d u a l s and the vacancy i s created at a low wage o f f e r . In a mixed workforce of three i n d i v i d u a l s , the f i r m does not create v a c a n c i e s , because any wage o f f e r made i s an o f f e r to c u r r e n t employees as w e l l . The p r o b a b i l i t y of a type two searcher sampling the f i r m i s low. The o f f e r to c u r r e n t type one employees would n e c e s s a r i l y be lower due to t h i s vacancy c r e a t i o n . Consequently, the v a r i a n c e of the l e v e l and composition of employment next p e r i o d would be h i g h e r and mean p r o f i t s lower. 4. A Summary of F i r m s ' Decis ions I t i s important to note that, i n a n a l y s i n g f i r m behaviour, to argue that firms are attempting to a t t a i n the ' o p t i m a l workforce' i s m i s l e a d i n g , although the c l a s s i f i c a t i o n of employment s t a t e s i s u s e f u l . F i r m s ' wage 231 and vacancy creation decisions r e f l e c t the value of r e t a i n i n g current employees and h i r i n g new employees i n the l i g h t of t h e i r turnover and acceptance behaviour and the r e l a t i o n s h i p between the current l e v e l and optimal l e v e l of employment. Individuals' behaviour may be such that there i s a high p r o b a b i l i t y that the 'optimal workforce' i s indeed attained. The seemingly non-systematic wage and vacancy creation decisions for each i n d i v i d u a l over employment states occur, because of a number of c o n f l i c t i n g factors a r i s i n g from the firms' a b i l i t y to discriminate i n biring- and wage o f f e r . The costs to the f i r m of creating and f i l l i n g vacancies, which are measured i n terms of foregone p r o f i t s , have r i s e n due to h i r i n g discrimination and stocha s t i c search. Thus, at les s than optimal l e v e l s , vacancy creation for each type of i n d i v i d u a l i s not speculative> but i s speculative i n t o t a l . As i n the case when the perception parameters are equal, i t i s the a b i l i t y of firms to discriminate between the type of employee who constitutes the majority of the workforce and the other type of worker who i s i n the minority or i s s o l e l y a searcher, which assumes key im-portance. The a b i l i t y to discriminate gives the f i r m extra degrees of freedom i n i t s decision making, but i t suff e r s a loss of intertemporal and, indeed, s t a t i c monopsony power because, due to stocha s t i c search, i t i s forced to make separate vacancy creation decisions f o r each type of i n d i v i d u a l . At les s than the optimal l e v e l of employment, f or example, 232 each vacancy c r e a t i o n d e c i s i o n i s non-zero and the f i r m faces a r e l a t i v e l y h i g h v a r i a n c e of employment next p e r i o d . Consequently, u s i n g i t s d i s c r i m i n a t o r y power, i t o f f e r s r e l a t i v e l y h i g h wages to the type of i n d i v i d u a l who predominates i n i t s workforce and reduces i t s vacancy c r e a t i o n . The l e v e l of vacancy c r e a t i o n for each type of i n d i v i d u a l i s c o n s i s t e n t w i t h o b t a i n i n g the optimal l e v e l of employment by f i l l i n g e i t h e r type of vacancy and assuming no q u i t s . I n c o n t r a s t to Models I and I I , t h e r e f o r e , c u r r e n t employees, who predominate i n the workforce, are r e l a t i v e l y more v a l u a b l e to firms because of the h i g h e r i m p l i c i t cost of vacancy c r e a t i o n . ' Hence, i n Model I I I , i t can be argued that the f i r m enjoys l e s s i n t e r t e m p o r a l monopsony power. At equal or above o p t i m a l l e v e l s of employment, the nature of f i r m s ' behaviour changes. Now d e s i r e d net h i r e s are zero and, i n some cases, f irms wish to d i s c a r d workers. In segregated workforces, firms s t i l l o f f e r a h i g h e r wage to c u r r e n t employees than to s e a r c h e r s , i f s p e c u l a t i v e vacancies are c r e a t e d . In mixed workforces, however, type two i n d i v i d u a l s earn a h i g h e r wage than type one i n d i v i d u a l s . I n some employment s t a t e s , the o f f e r to type one i n d i v i d u a l s a c t u a l l y equals the l e v e l of unemployment com-p e n s a t i o n or l e s s , because the f i r m wishes to d i s c a r d these employees. Thus, as compared w i t h the b a s i c s o l u t i o n to Model I I I , at o p t i m a l and l e s s than optimal l e v e l s of employment, d i f f e r e n c e s between i n d i v i d -uals are r e f l e c t e d by the comparison of r e s p e c t i v e o f f e r s i n s t a t e 233 complements, and the firm's a b i l i t y to discriminate between employees and searchers remains important. In mixed workforces, i n d i v i d u a l differences dominate firms' a b i l i t y to discriminate between searchers and employees. Consequently, over these states, type two employees earn higher wages than type one employees. 5. An Explanation of the Summary S t a t i s t i c s Generally, type one searchers only receive offers from firms currently employing undersize workforces. Then, the probability that a type one searcher receives a zero wage offer i s higher than for a type two searcher. Thus, the variance of offers facing a type one searcher exceeds the variance of offers facing a type two searcher. Type two individuals enjoy a higher mean wage, when employed, and have a lower p r o b a b i l i t y , ex ante, of q u i t t i n g i n response to a pa r t i c u l a r wage offer than type one individuals. Thus, type two individuals enjoy a higher MDI than type one indi v i d u a l s , although, over some states of employment, type one employees enjoy a higher wage rate. In Models I and I I , i n d i v i d u a l s , who do not quit, enjoy the largest MDI. In this model, firms can discriminate i n wage of f e r s , whence, i f a type one ind i v i d u a l i s a member of an oversize workforce, the firm e f f e c t i v e l y lays him off with a low wage offer. I f , hypo-t h e t i c a l l y , the in d i v i d u a l never quits, then i t i s possible that he also receives the same low wage o f f e r , since the firm wishes to retain 234 the other employees. I f he quits because the offer i s lower than his reservation wage, then he receives unemployment compensation, searches and faces a non-zero pro b a b i l i t y of receiving an offer i n excess of his reservation wage. Thus, i n f i v e out of the seven cases, the measure of MDI associated with turnover and acceptance according to the true reservation wage i s higher than that associated with never q u i t t i n g . Most vacancies for type one individuals are created by firms currently facing undersize workforces. Thus, aggregate vacancy creation for type one individuals i s lower than aggregate vacancy creation for type two individuals because speculative type two vacancies are created at optimal and above optimal employment l e v e l s . Type one individuals are less valuable than type two individuals at a p a r t i c u l a r wage offer and thus fewer are employed i n stochastic equilibrium, since type one individuals do not receive systematically lower offers. Type one individuals enjoy a higher unemployment rate and fewer vacancies. Then type one searchers have a longer mean duration of search p r i o r to an o f f e r . Type one individuals wait longer before receiving an offer and have a lower probability of accepting a p a r t i c u l a r o f f e r . Then, i t i s plausible that type one individuals enjoy a longer duration of un-employment. Type two individuals enjoy a higher mean wage when em-ployed and a higher rate of employment. Thus, they enjoy a larger share of t o t a l output than type one individuals. A p r i o r i , the respective correlations of each individual's wage offer over one period have no p a r t i c u l a r r e l a t i o n . 235 6 i . The Concept of Vacancy C r e a t i o n and Excess Demand In Models I and I I , two motives behind vacancy c r e a t i o n can be i d e n t i f i e d . F i r s t l y , vacancies are created which represent d e s i r e d net h i r e s of firms c u r r e n t l y f a c i n g a l e s s than optimal l e v e l of employment. Secondly, s p e c u l a t i v e vacancies are created at low wage o f f e r s which demonstrate the f i r m s ' monopsony power. Such vacancies r e f l e c t s p e c u l a t i o n about i n d i v i d u a l s ' turnover and acceptance behaviour and f i r m s ' d e c i s i o n s represent an attempt to a t t a i n the o p t i m a l l e v e l of employment at a r e l a t i v e l y low wage. In Model I I , these vacancies a l s o r e p r e s e n t the d e s i r e to s u b s t i t u t e between type one and type two i n d i v i d -u a l s . Although vacancy s p e c u l a t i o n i n c r e a s e s the v a r i a n c e of the l e v e l of employment, there i s a low p r o b a b i l i t y of a c h i e v i n g an i n f e r i o r com-p o s i t i o n of employment next p e r i o d . In Model I I I , t h e nature of f i r m s ' d e c i s i o n making i s more complicated, due to i t s a b i l i t y to d i s c r i m i n a t e i n wage o f f e r and h i r i n g . As a l r e a d y n o t e d , vacancies r e p r e s e n t i n g d e s i r e d net h i r e s are s p e c u l a t i v e i n t o t a l , due to s t o c h a s t i c s e a r c h , r a t h e r than s t o c h a s t i c turnover and acceptance behaviour. The f i r m i s c o n s t r a i n e d by the r e s t r i c t i o n of the maximum l e v e l of employment from o f f e r i n g a lower wage and c r e a t i n g s p e c u l a t i v e vacancies f o r each, type of i n d i v i d u a l . Given the r e s u l t i n g i n c r e a s e i n the v a r i a n c e of employment, however, i t i s u n l i k e l y that the f i r m would create these v a c a n c i e s . Aggregate vacancy c r e a t i o n at below o p t i m a l employment l e v e l s then represents twice the aggregate measure of d e s i r e d n e t h i r e s . 236 At the optimal l e v e l of employment and above, s p e c u l a t i v e v a c a n c i e s , g e n e r a l l y f o r type two searchers, are created at low wage o f f e r s . Such vacancy c r e a t i o n demonstrates the a b i l i t y to d i s c r i m i n a t e i n wage o f f e r between searchers and c u r r e n t employees r a t h e r than the f i r m s ' monopsony power a l o n e . T h i s vacancy c r e a t i o n behaviour u n d e r l i n e s the problem of c o n s t r u c t i n g a homogeneous measure of d e s i r e d net h i r e s . In Models I and I I , vacancies are created at d i f f e r e n t wage r a t e s a c c o r d i n g to the c u r r e n t employment l e v e l b u t , w i t h the exception of the f i r m f a c i n g a zero l e v e l of employ-ment, these o f f e r s are a l s o made to c u r r e n t employees. By c o n t r a s t , i n Model I I I , some s p e c u l a t i v e vacancies are accompanied by wage o f f e r s which are made to searchers alone or to an i n d i v i d u a l type who c o n -s t i t u t e s a m i n o r i t y of the workforce and whom the f i r m may wish to d r i v e out of i t s employment. Since vacancies r e p r e s e n t i n g d e s i r e d net h i r e s can be regarded as s p e c u l a t i v e i n t o t a l , due to s t o c h a s t i c s e a r c h , r a t h e r than s t o c h a s t i c turnover and acceptance behaviour, a more a p p r o p r i a t e measure of s p e c u l a t i v e vacancies may be those vacancies created by f irms which are unequal between i n d i v i d u a l s . That i s those vacancies created by firms f a c i n g the o p t i m a l or above optimal l e v e l of employment f o r one 33 type of i n d i v i d u a l which c l e a r l y do not represent d e s i r e d net h i r e s . 7. The T h e o r e t i c a l C o n t r i b u t i o n of Model III The c o n t r i b u t i o n of t h i s model i s i n the e x p l i c i t f o r m u l a t i o n of f i r m s ' d i s c r i m i n a t o r y behaviour. I t i s p o s s i b l e to s e t t l e some of the debates between economists as to which type of worker earns the h i g h e r wage. 237 Salop argues t h a t , i f one type of employee has a h i g h e r m a r g i n a l q u i t r a t e than another, then he earns a h i g h e r wage, i f the f i r m d i s c r i m i n a t e s i n i t s wage o f f e r . U n f o r t u n a t e l y , he does not s p e c i f y how i n d i v i d u a l s ' q u i t p r o b a b i l i t i e s , are determined. The assumption i n Model I I I i s that an i n d i v i d u a l ' s ex ante p r o b a b i l i t y of a c c e p t i n g an o f f e r i s based on the e v a l u a t i o n of the o f f e r i n the context of the o f f e r d i s t r i b u t i o n f a c i n g him. I n M c C a l l ' s model d e s c r i b e d i n the Chapter I VD, i f the e x p l i c i t costs of search and the r e t u r n s from b e i n g unemployed and not s e a r c h i n g are the same f o r both white and b l a c k workers, more b l a c k i n d i v i d u a l s would drop out of the l a b o u r f o r c e , i f they evaluate t h e i r o f f e r s i n the context of t h e i r own d i s t r i b u t i o n of wage o f f e r s . In a s t r i c t sense, Salop i s i n c o r r e c t . The wage o f f e r d i s t r i b u t i o n f a c i n g a type one i n d i v i d u a l i s not s y s t e m a t i c a l l y h i g h e r than the wage o f f e r d i s t r i b u t i o n f a c i n g a type two i n d i v i d u a l . The reason i s s i m p l e . Under c e r t a i n circumstances, i t pays firms to d i s c r i m i n a t e i n wage o f f e r between i n d i v i d u a l s on the b a s i s of t h e i r d i f f e r e n t mean r e t u r n s from employment. In p a r t i c u l a r , i n making wage o f f e r s to s e a r c h e r s , a f i r m o f f e r s a h i g h e r wage to a type two searcher i n s t a t e [1,0] than to a type one searcher i n the s t a t e complement o r , i f the f i r m c u r r e n t l y employs a mixed workforce which i s equal to or above the optimal em-ployment l e v e l , then, again?, i t o f f e r s more v a l u a b l e type two workers a h i g h e r wage. I t i s when the f i r m wishes to ensure that c u r r e n t type 34 one employees are r e t a i n e d that S a l o p ' s arguments are c o r r e c t . 238 Although the wage offer di s t r i b u t i o n s are such that type one individuals do indeed have a higher marginal quit rate, i n response to a p a r t i c u l a r o f f e r , than type two individuals i n the state complement, the mean wage enjoyed by a type two employee i s higher than that enjoyed by a type one i n d i v i d u a l . In an ove r a l l sense, then, Salop i s incorrect. Sanborn argues that, i n the absence of discrimination through ignorance or prejudice, low quit individuals w i l l enjoy a higher wage offer than high quit individuals because thei r costs of turnover are amortised over a longer period of time. In the aggregate, employed type two individuals do, indeed, enjoy a higher mean wage offer but, as indicated, type two individuals do not enjoy systematically higher wage offers than type one individuals. C. Comparative S t a t i c Predictions of Model I I I 1. An Introduction Nine d i f f e r e n t combinations of the perception parameters, c ^ 2 and c^ are adopted i n the range 0.90 to 1.15. A l l other exogeneous constants are kept constant. c^^> a r b i t r a r i l y , i s assumed to exceed 2 c^ . Six comparative s t a t i c predictions are generated, four associated 2 1 with an increase i n c^ and two with an increase i n c^ . Most q u a l i t a t i v e predictions are independent of the absolute or r e l a t i v e values of the perception parameters and, indeed, independent of which perception parameter increases. There are twenty-five summary s t a t i s t i c s whose changes describe the impact on stochastic equilibrium of a change i n the value of an exogeneous parameter. They are the following : (a) The wage offer d i s t r i b u t i o n facing type one individuals; (b) The wage offer d i s t r i b u t i o n facing type two individuals; (c) The mean wage offer enjoyed by a type one searcher; (d) The mean wage offer enjoyed by a type two searcher; (e) The variance of offers facing a type one i n d i v i d u a l ; (f) The variance of offers facing a type two i n d i v i d u a l ; (g) The mean wage earned by a type one employed i n d i v i d u a l ; (h) The mean wage earned by a type two employed i n d i v i d u a l ; ( i ) MDI enjoyed by a type one i n d i v i d u a l ; (j) MDI enjoyed by a type two i n d i v i d u a l ; (k) Aggregate unemployment; (1) Aggregate type one unemployment; (m) Aggregate type two unemployment; (n) Aggregate vacancy creation for type one searchers; (o) Aggregate vacancy creation for type two searchers; (p) Aggregate type one speculative vacancy creation; (q) Aggregate type two speculative vacancy creation; (r) The mean duration of search p r i o r to an offer for a type one searcher; (s) The mean duration of search p r i o r to an offer for a type two searcher; (t) The mean duration of type one unemployment; (u) The mean duration of type two unemployment; (v) Mean firm p r o f i t s discounted over an i n f i n i t e horizon; 240 (w) Type one labours' share of t o t a l output; (x) Type two labours' share of t o t a l output; and (y) Labours' t o t a l share of output. The values of the summary s t a t i s t i c s over combinations of the perception parameters are shown i n Table 17. The analysis of the 2 results relates to an increase i n c^ . 2. The Results 2 In response to an increase i n c^ , the wage offer d i s t r i b u t i o n s facing both type one and type two individuals generally s h i f t to the rig h t . For some parameter changes, however, there i s a change i n the pattern of vacancy creation over employment states and so not a l l wage offers over a l l states increase. The mean wage offer facing each type of unemployed i n d i v i d u a l and the variances of the respective distributions a l l increase. The mean wage for both type one and type two employed individuals r i s e . Type one individuals enjoy an increase i n MDI, but MDI of a type two in d i v i d u a l f a l l s . Total mean unemployment r i s e s . Type two unemployment ri s e s while type one unemployment f a l l s . Aggregate mean vacancy creation for both types of in d i v i d u a l generally r i s e s , but there i s no systematic change i n aggregate speculative vacancy creation for either type of i n d i v i d u a l . The mean duration of search p r i o r to an offer i s again negatively related to the aggregate measure of vacancies minus unemployment. Consequently, type one individuals' mean duration of search f a l l s , TAJoLE 1/ Comparative S t a t i c Results f o r Model I I I 3 5 ' 3 6 ' - - 1.2.. _• Different Combinations of c l s c x 1 Comparative S t a t i c s " " | 1.00 0.90 1.00 1.00 1.05 0.95 1.10 0.90 1.10 0.95 j 1.10 i ! 1.00 1.10 1.05 ; l . i o 1.10 • 1.15 i l . i o [ Sign 1 (CT+) Freq/ 2 Sign 2 (cit) 1 Freq/ l ! 4 j Freq/ i 6 ' a) 1 j j + ! 1 + 2 1 3 b) j ; + i 2 i + 1 ! 3 c) 0.934 0.962 0.951 0.950 1.001 | 1.007 | 1.029 i 1.030 1.050 | + i 2 \ + 4 6 d) 1.035 0.959 1.039 1.083 1.108 \ 1.109 ' 1.057 I 1.050 1.078 j + i •  | 2 j + j 2 1 ! 4 e) 1 0.054 0.053 • 0.056 0.063 0.076 | 0.076 1 0.080 \ 0.084 0.089 1 + i 2 + 4 ( - l ) ! 6(-l) j f > 3 0.049 0.053 1 0.049 0.052 0.066 \ 0.067 0.077 1 0.081 0.086 t | + 2 } + 4 i 6 !g) 1.190 1.189 1.210 1.257 1.286 j 1.289 1.313 \ 1.319 1.356 + | 2 j + 4 6 h) 1.207 1.200 1.221 1.246 1.301 ; 1.309 1.321 . 1.325 1.363 + 2 | + 4 6 i ) 8.104 8.270 8.118 8.220 8.426 j 8.45,4 8.633 8.810 8.744 — - ! 1 1 + 4 5 j ) 8.849 9.268 8.869 9.284 9.233 I 9.192 8.935 : 8.862 . 9.067 ; ' + 2 s 4 6 ? k) 24 29 27 25 30 I 31 33 \ 32 35 ! + 2 • I + 3 5 i« l f i 14 18 t 19 j 19 l 19 ! 18 • 16 19 + 1 2 I 4(-2) 6(-2) ; m) 8 15 ; 6 11 j 12 15 16 16 - K - i ) \ + 4 5(-l ) n) 35 43 \ 31 27 49 \ 50 47 45 49 + ! 2 \ 2 4 i o) 48 43 ! 49 | 42 51 1 51 ' 45 39 46 i 2 I f + j 2 ( - l ) | 4 ( - l ) | P) 9 : 12 j i 1 1 • o 17 i ie 11 11 12 _ ! o ( j 3 ( - l ) j 3(-l) ! q) 16 12 i 20 9 : 18 1 1 8 9 5 10 t 1 1 i 3 ( - l ) j 4 ( - l ) [ r) 1.795 1.703 j 2.168 2.446 j 1.537 I 1.531 1.616 1.635 1.551 ? • 2 j I 4 6 1 s) 1.359 1.711 I i 1.362 1.354 i 1.342 j 1.343 1.628 1.686 1.590 - 2(-l) j 7 l 4 6( - l ) j t) 2.639 2.402 | 2.786 2.971 2.602 | 2.574 2.545 2.462 2.607 + 1 j - J 4 j 5 I u) \. 702 2.435 j • 1 1.793 1.588 j 1.801 1 1.863 2.288 2.383 2.347 1 i i i + ; 4 ! 5 | v) 26.802 | 25.861 j 25.482 125.418 1 22.428 j22.068 21.307 21.483 19.983 2 1 \ 3 ! . 5 j w) 0.256 | 0.282 0.252 j 0.249 ; 0.271 1 0.276 1 i i 0.294 0.309 0.301 — 1 j I + j 4 . i 5 x) 0.320 0.283 | 0.326 j j 0.345 j 0.345 i 0.341 i ' 0.324 | 0.313 0.334 + 2 { 4 ( - l ) j 6 ( - l ) ; y) 0.576 0.565 j 0.578 j 0.594 j 0.616 j 0.617 1 0.618 ! 0.622 j 0.635 + i 2. .1 ? + I 4 1 6 242 while type two i n d i v i d u a l s ' mean d u r a t i o n of search does not change s y s t e m a t i c a l l y . Type one i n d i v i d u a l s ' d u r a t i o n of unemployment g e n e r a l l y f a l l s , but type two i n d i v i d u a l s ' d u r a t i o n o f unemployment r i s e s . Mean f i r m p r o f i t s discounted over an i n f i n i t e h o r i z o n f a l l . Type two employed i n d i v i d u a l s ' share o f t o t a l output f a l l s , but type one i n d i v i d u a l s ' share r i s e s , as does l a b o u r ' s t o t a l share of output. 3. An E x p l a n a t i o n I f firms make the same wage o f f e r and vacancy c r e a t i o n d e c i s i o n s , 2 d e s p i t e the i n c r e a s e i n the p e r c e p t i o n p a r a m e t e r , . c ^ , then type two unemployment w i l l r i s e and there w i l l be a s h i f t i n the d i s t r i b u t i o n of f irms to low employment s t a t e s i n which type one i n d i v i d u a l s are employed. To counter t h i s systematic change i n type two i n d i v i d u a l s ' behaviour, firms i n c r e a s e the general wage o f f e r d i s t r i b u t i o n to type two i n d i v i d u a l s . Type two i n d i v i d u a l s are l e s s v a l u a b l e to the f i r m , 2 r e l a t i v e l y , than p r i o r to the i n c r e a s e i n c^ , so firms i n the aggregate do not choose to h i r e so many type two i n d i v i d u a l s . T h i s d e c i s i o n i s , i n a sense, r e i n f o r c e d by the systematic change i n type two i n d i v i d u a l s ' turnover and acceptance behaviour. Type two i n d i v i d u a l s enjoy a h i g h e r mean wage o f f e r , whether em-p l o y e d or unemployed. The v a r i a n c e of t h e i r o f f e r d i s t r i b u t i o n r i s e s , because the unemployment compensation i s constant, w h i l e wage o f f e r s g e n e r a l l y i n c r e a s e and there i s a s h i f t to low employment, h i g h wage f i r m s . 243 The i n c r e a s e i n unemployment of type two i n d i v i d u a l s i s accom-panied by an i n c r e a s e i n aggregate type two vacancy c r e a t i o n , due to the s h i f t i n the employment d i s t r i b u t i o n . Consequently, the change i n the mean d u r a t i o n of search p r i o r to an o f f e r i s i n d e t e r m i n a t e , but the mean d u r a t i o n of unemployment r i s e s , due to the systematic change i n behaviour. These p r e d i c t i o n s match those generated i n Model I from systematic over or under e s t i m a t i o n of the r e s e r v a t i o n wage. In t h i s model, however, f irms are able to s u b s t i t u t e d i r e c t l y between type one and type two i n d i v i d u a l s i n employment due to t h e i r a b i l i t y to d i s c r i m i n a t e i n wage o f f e r and employment. The r e s u l t s conform to the p r e d i c t i o n s of a simple two f a c t o r model of f i r m behaviour. Firms s u b s t i t u t e one f a c t o r f o r the other f a c t o r whose p r o d u c t i v i t y has f a l l e n or whose wage has r i s e n . Aggregate employment f a l l s . In order to h i r e more type one i n d i v i d u a l s i n the aggregate, firms i n c r e a s e the d i s t r i b u t i o n 38 of o f f e r s to type one i n d i v i d u a l s . The mean wage o f f e r s f a c i n g both employed and unemployed type one i n d i v i d u a l s r i s e along with the v a r i a n c e of o f f e r s . Type one employment r i s e s . Type one vacancy c r e a t i o n g e n e r a l l y i n c r e a s e s , d u e to the o v e r a l l s h i f t i n the d i s t r i b u t i o n of employment. The two cases i n which aggregate type one vacancy c r e a t i o n a c t u a l l y f a l l s can be explained by arguing that the i n c r e a s e i n vacancy c r e a t i o n a r i s i n g from the s h i f t i n the d i s t r i b u t i o n of employment to low employ-ment, h i g h vacancy c r e a t i o n states i s o f f s e t by the simultaneous s h i f t 244 to states of employment i n which more type one individuals are em-ployed and type one vacancy creation i s lower. The mean duration of search of type one individuals f a l l s along with the mean duration of unemploymen t. Type one individuals earn a higher MDI and a large share of output. In contrast to Model I , type two individuals earn a lower MDI and lower share of output. In Model I I I , the a b i l i t y of firms to discriminate between individuals means that, i n the aggregate, substitution occurs between individuals i n employment. In the aggregate, however, the results generally conform to the results i n Model I , namely labour's t o t a l share of output and t o t a l MDI both increase i n response to an 2 increase i n the perception parameter, c^ . The general tenor of these comparative s t a t i c predictions i s sim i l a r to those generated i n Model I I . Since type two individuals have a higher probability of turnover and a lower probability of acceptance, ex ante, i n response to a p a r t i c u l a r wage off e r , they are less valuable to the firm than before. Consequently, firms wish to substitute away from them i n employment. In Model I I , firms are forced to raise the wage offer d i s t r i b u t i o n to both types of i n d i v i d u a l , i n the absence of discrimination and type one individuals, whose behaviour has not changed, are the direct beneficiaries of the systematic change i n type two individuals' behaviour. Even when firms are able to discriminate i n wage offers between the two types of individuals, type one individuals benefit d i r e c t l y from type two individuals' behaviour. Firms wish to increase the employment of type one individuals and so are forced to increase t h e i r d i s t r i b u t i o n of wage offers. 245 IV. A Comparison of Models I I and I I I A. An Introduction There are s i x sets of perception parameters over which direct comparison of the two models i s possible. There are twenty-one sample s t a t i s t i c s whose comparison r e f l e c t the differences i n the nature of stochastic equilibrium i n the labour market, when firms can and cannot discriminate d i r e c t l y i n wage offer and employment. These changes i n the summary s t a t i s t i c s are the key of analysing the impact of f a i r employment laws on the incomes enjoyed by non-preferred, less valuable type one workers. The summary s t a t i s t i c s are the following : (a) The mean wage offer facing a type one searcher; (b) The mean wage offer facing a type two searcher; (c) The variance of the d i s t r i b u t i o n of offers facing a type one searcher; (d) The variance of the d i s t r i b u t i o n of offers facing a type two searcher; Ce) The mean wage earned by a type one employee; (f) The mean wage earned by a type two employee; (g) MDI enjoyed by- a type one i n d i v i d u a l ; (h) MDI enjoyed by a type two i n d i v i d u a l ; ( i ) The r a t i o of MDI for each type of i n d i v i d u a l ; (j) Aggregate mean unemployment of type one individuals; (k) Aggregate mean unemployment of type two individuals; (1) Aggregate mean vacancy creation; (m) The mean duration of search p r i o r to an off e r for type one individuals; 246 (n) The mean duration of search p r i o r to an offer for type two individuals; (o) The mean duration of unemployment for a type one i n d i v i d u a l ; (p) The mean duration of unemployment for a type two i n d i v i d u a l ; (q) Mean fir m p r o f i t s ; (r) Type one labour's share of output; (s) Type two labour's share of output; (t) Labour's t o t a l share of output; and (u) The r a t i o of the shares of output for each type of i n d i v i d u a l . B. The Results For equal values of the perception parameters, the offers to both types of in d i v i d u a l are higher over a l l states of employment i n Model I I than i n Model I I I . The comparison of stochastic e q u i l i b r i a corres-ponding to the basic parameter values i n Models I I and I I I i s shown i n Table 18. For unequal values of the perception parameters, the offers to type one individuals are higher i n Model I I than i n Model I I I . over 39 a l l states of employment. The relationship between respective offers to type two individuals i s less systematic. Generally,the wage offers to type two individuals from firms facing a l e v e l of employment optimal 40 or less than optimal are higher i n Model I I . The offers from firms employing a mixed workforce,which exceeds the optimal l e v e l of employment, 41 are generally higher i n Model I I I . The comparison of stochastic e q u i l i b r i a i n Models I I and I I I corresponding to parameter values [1.05,0.95] i s shown i n Table 19. TABLE 18 Comparison of Stochastic E q u i l i b r i a i n Models I I and I I I  Corresponding to the Basic Parameter Values Model I I j ; Model I I I j Type One j Employment ! Type Two Employment Steady State Dist. W V Mean One\ Period :' P r o f i t s ; P r o f i t s Over an I n f i n i t e Horizon I Steady; State j Dist. : i i w i j w2 | V l V2 Mean | One | Period i P r o f i t s ; P r o f i t s | Over an j I n f i n i t e \ Horizon ! * — ' i o i 0 0.156 1.455 2 0.45 ! 19.13 0.106 j 1.27 i 1.27 | 2 2 • 0.50 i . 24. i 07 j 1 1 0 0.213 1.420 1 1.01 : 20.29 | 0.210 1.26 1.09 ; 1 1 1.24 25. 82 | o s 1 0.213 1.420 1 1.01 : 20.29 j 0.209 1.09 1.26 i 1 1 1.24 25. 79 j 2 | 0 0.087 • 1.245 1 1.21 20.61 j 0.116 1.18 0.75 | 0 2 1.51 26. 36 | 1 . 1 1 0.175 1.245 1 1.21 20.61 i 0.211 1.165 1.165 j 0 0 1.51 26. 27 j 0 2 0.087 1.245 1 1.21 20.61 j 0.115 0.755 1.18 j 2 0 1.51 26. 31 | 3 0 0.008 1.185 0 1.29 20.72 0.0 1.045 0.745 ; 0 0 1.58 26. 46 j 2 1 0.025 1.185 0 ! 1.29 20.72 1 0.017 1.06 1 . 0 0 5 ; . 0 o 1.58 26. 42 ! 1 2 0.025 1.185 0 1.29 20.72 \ 0.016 j 0.94 1.085 i 0 0 1.58 26. 39 | 0 \ 3 0.008 1.185 0 1.29 20.72 j 0.0 : 0.745 1.050 | 0 0 1.57 26. 41 4 0 0.0 1.085 0 1.33 20.74 1 0.0 } 0.955 ;0.745 ; 0 0 1.59 26. 44 | 3 1 . 0.0 1.085 0 1.33 20.74 I 0.0 ; 1.015 0.75 j 0 0 1.59 26. 45 | 2 2 0.0 1.085 j 1.33 20.1k \ 0.0 ] 0.99 0.91 i 0 0 1.60 26. 40 | 1 3 0.0 1.085 i 0 20.74 | 0.0 j 0.745 1.05 \ 0 0 1.57 26. 41 | i 0 4 0.0 1.085 0 i 1.33 i . 20.74 | I 0.0 | 0.745 0.955 j 0 0 1.59 26. 40 j TABLE 19 Comparison of Stochastic E q u i l i b r i a In Models I I and I I I  Corresponding to Parameter Values [1.05,0.95] M°del 1 1 Model I I I Type One Employment Type Two Employment Steady State Dist. i W f 1 v ; Mean One Period P r o f i t s P r o f i t s Over an I n f i n i t e Horizon ; Steady ; State W-i Dist. W 2 V l ! V2 . Mean One Period P r o f i t s P r o f i t s Over an j I n f i n i t e Horizon 0 i o 0.143 1 1.45 ! 2 j 0.45 ; 19.77 0.095 j 1.315 1.275 i 2 1 2 ; 0.478 23. 546 | 1 ! ° 0.180 j 1.43 ! 1 ! 0.98 j 20.76 \ 0.182 | 1.29 1.10 i i | 1 j 1.197 ! 25 223 0 i 0.247 \ 1.38 I j 1.08 | 21.34 : 0.229 | 1.085 1.285 i i i 1 j 1.234 ! 25-497 2 ! o 0.079 i 1.26 l ! 1.17. 1 21.04 I 0.123 ! 1.205 0.830 | 0 2 j 1.458 j 25. 703 1 i 1 0.158 j 1.22 i | 1.26 ; 21.49 j 0.150 j 1.145 1.170 ! o • 1 j 1.469 ! 25-870 0 ! 2 0.147 • 1.29 0 1.32 ! 21.92 0.184 I 0.745 1.215 0 0 \ 1.481 | 26. 131 3 | ; o 0.008 | 1.14 i | 1.25 21.12 0.0 | 1.07 0.75 0 1 ! 1.519 I '25. 794 2 1 ' I i 0.022 j 1.11 l j 1.33 21.49 0.014 ; 1.02 1.145 0 0 \ 1.539 i 25. 947 | 1 2 0.014 • i i . i 4 0 j j 1.39 j 21.83 0.024 0.75 1.19 1 , 0 1.507 | 26. 153 ! 0 3 0.00 [1.12 ; o i 1.43 ; 22.09 0.0 0.745 0.065 o; 0 1.544 j 26. 219 j 4 \0 0.0 1.10 i o •! 1.30 | 21.12 0.0 0.098 ; 0.745 0 I 0 1.538 ! 2 5 -778 j 3 1 0.001 1.08 ! 0 ! 1.36 | 21.46 0.0 0.905 j 1.175 o; 0 1.539 1 25. 940 j 2 2 0.0 1.05 j 0 1 1.42 i 21.74 0.0 0.75 j 1.16 0 : 0 15.35 26. 155 | 1 3 0.0 1.03 | 0 ! 1.45 ; 21.96 0.0 0.75 1.055 o ; 0 1.512 26. 186 j 0 4 0.0 1.01 j i o 1 1 1.48 j i 22.13 0.0 0.745 0.97 o j 0 1.553 26. 179 | to oo 249 A l l measures of mean wage offers and income are higher i n Model I I than i n Model I I I over both types of in d i v i d u a l . The r a t i o of MDI for each type of i n d i v i d u a l i s higher i n Model I I than i n Model I I I , i n which e x p l i c i t discrimination i s practised. Aggregate levels of mean unemployment for each type of in d i v i d u a l are lower i n Model I I I , but aggregate vacancy creation i s higher. The duration of search p r i o r to an offer i n Model I I I i s higher for type one individuals but lower for type two individuals. The mean duration of unemployment i s higher for both types of ind i v i d u a l i n Model I I I . Firms enjoy s i g n i f i c a n t l y lower p r o f i t s discounted over an i n f i n i t e horizon over a l l states i n Model I I and mean discounted p r o f i t i s lower. Both types of i n d i v i d u a l enjoy a larger share of t o t a l output i n Model I I and the r a t i o of labour shares i s higher i n Model I I . Table 20 shows the values of the summary s t a t i s t i c s and the frequency with which the relationship between them i s upheld i n Models I I and I I I over a l l perception parameter v a l u e s . 4 2 C. An Explanation Again, the endogeneity of turnover and acceptance behaviour by each type of in d i v i d u a l complicates the explanation of r e l a t i v e wage offers and the summary s t a t i s t i c s associated with Models I I and I I I . In p r a c t i s i n g wage and employment discrimination i n Model I I I , the firm i s not discriminating on the basis of ignorance or prejudice about the productivity of each type of i n d i v i d u a l , but on the basis of TABLE 20 Comparison o f Summary S t a t i s t i c s o f Models I I and I I I MDI, P e r c e p t i o n »«, ' ' MDI.. LS Parameters ™2 W l W 2 roi W 2 ^ l  mI2 iSf^ D l D 2 T P 0 1 P 0 2 P A 1 P A 2 E n " l ' U 2 1 8 is" 1.00 I I 1.085 1.085 0.073 0.073 1.244 1.235 8.534 8 .968 0.952 17 10 43 1.608 1.608 1.987 1.791 24.147 0.268 0.329 0.597 0.814 0 .90 I I I 0 .934 1.035 0 .054 0.049 1.190 1.207 8 .104 8.849 0.916 16 8 83 1.795 1.359 2.639 1.702 26.802 0.256 0.320 0.576 0.800 1.00 I I 1.196 1.196 0.081 0.081 1.323 1.323 9.070 9.070 1.0 16 16 54 1.377 1.377 1.737 1.737 20.250 0.317 0.317 0.634 1.00 1.00 I I I 0.962 0.959 0.053 0.053 1.189 1.200 8.270 8.268 1.0 14 15 86 1.703 1.711 2.402 2.435 25.861 0.282 0.283 0.565 0.995 1.05 I I 1.146 1.146 0.089 0.089 1.323 1.310 8.682 9.287 0.935 20 12 49 1.532 1.532 2.058 1.783 21.105 0.274 0.351 0.625 0.780 0 .95 I I I 0.951 1.039 0.056 0.049 1.210 1.221 8 .118 8.869 0.915 18 9 80 2.168 1.362 2.786 1.793 25.482 0.252 0.326 0.578 0.773 1.10 I I 1.160 1.160 0.099 0.099 1.355 1.328 8.658 9.559 0.906 22 10 48 1.553 1.553 2.181 1.746 20.592 0.264 0.370 0.634 0.713 0.90 I I I 0.950 1.083 0.063 0.052 1.257 1.246 8.220 9.284 0.885 19 6 69 2.446 1.354 2.971 1.588 25.418 0.249 0.345 0.594 0.724 1.10 I I 1.257 1.257 0.101 0.101 1.424 1.392 8.982 9.452 0.950 22 16 58 1.368 1.368 2.035 1.723 17.439 0.303 0.360 0.663 0.841 1.00 I I I 1.007 1.109 0.076 0.067 1.289 1.309 8.454 9.192 0.919 19 12 101 1.531 1.343 2.574 1.863 22.068 0.276 0.341 0.617 0.809 1.10 I I 1.283 1.283 0.112 0.112 1.455 1.455 9.143 9.143 1.0 22 22 61 1.374 1.374 2.022 2.022 15.612 0.337 0.337 0.674 1.00 1.10 I I I 1.030 1.050 0.084 0.081 1.319 1.325 8.810 8.862 0.993 16 16 84 1.635 1.686 2.462 2.383 21.483 0.309 0.313 0.622 0.987 I I -»• I I I - - - - _ _ _ _ + Freq / 6 6 6 6 6 6 6 6 6 6 6 6 6 U l o 251 individuals' labour market behaviour which d i f f e r s systematically. These differences manifest themselves i n the turnover and acceptance behaviour of workers and thus t h e i r value to the firm. The a b i l i t y to discriminate i n wage offer and employment i n Model I I I , therefore, gives the firm extra degrees of freedom i n i t s decision making. Thus, despite the increase i n the i m p l i c i t costs of creating and f i l l i n g a vacancy, i t i s plausible that firms i n the aggregate enjoy higher employment and higher p r o f i t s . At optimal and less than optimal employment l e v e l s , i t i s the a b i l i t y of firms to discriminate between current employees and searchers i n Model I I I , irrespective of the perception parameter values, which i s the key to explaining the r e l a t i v e wage offers between the two models. The firm i s able to offer the i n d i v i d u a l type, who constitutes a minority of the workforce or i s solely a searcher, a r e l a t i v e l y low wage. These low wage offers constitute part of the wage offer d i s t r i -bution from which the i n d i v i d u a l computes h i s perception of the reserv-ation wage. Consequently, offers to this type of i n d i v i d u a l from firms i n employment states i n which they form a majority of the workforce while higher are not as high as the corresponding offers to a l l employees and ' 43-searchers i n Model I I . *' Thus, wage offers to both types of individuals are lower i n Model I I I than corresponding offers to individuals i n Model I I , at optimal and less than optimal employment l e v e l s , irrespective of 44;' the perception parameter values. -252 For equal values of the perception parameters, at above optimal levels of employment, firms also discriminate i n wage offer i n favour of the individuals who constitute a majority of the workforce i n Model I I I . This wage discrimination reduces the variance of employment next period. Again, the low offers to individuals constituting a minority of the workforce force down other wage offers. Thus, over a l l states of employment, wage offers are higher i n Model I I , i f the perception parameter values are equal. When perception parameters are unequal, however, the pattern of wage offers changes at above optimal employment l e v e l s . In a mixed workforce*firms offer type one individuals a lower wage rate than type two individuals. This r e f l e c t s the r e l a t i v e values of the different types of in d i v i d u a l i n an oversize workforce. Thus, over a l l states of employment, offers to type one individuals i n Model I I I are below corresponding offers i n Model I I . By contrast, type two individuals earn a higher wage i n these mixed workforces than i n the corresponding states i n Model I I . In Model I I , the firm's wage offer to a l l current employees r e f l e c t s i t s desire to at t a i n the optimal l e v e l of employment. In Model I I I , firms are making offers to individuals according to thei r r e l a t i v e value and these offers reinforce the prob a b i l i t y that type two individuals do not q u i t . Thus, at optimal and less than optimal em-ployment l e v e l s , offers to type two individuals i n Model I I exceed corresponding offers i n Model I I I and,for above optimal employment lev e l s , the converse i s true. 253 Since most firms are i n low employment states, the respective mean offers to searchers and employees i n Model I I exceed corresponding mean offers i n Model I I I . The wage offer d i s t r i b u t i o n has a smaller range i n Model I I I and,since the absolute difference between these offers and the l e v e l of unemployment compensation i s lower i n Model I I I , the variance of the offer d i s t r i b u t i o n i s lower for both types 45 of individuals i n Model I I I . Since firms are able to fine tune t h e i r wage decisions according to the number of each type of in d i v i d u a l i n the workforce and the t o t a l size of the workforce, rather than the ove r a l l composition of the work-force, i t i s plausible that i n Model I I I there are lower levels of unem-ployment of both types of in d i v i d u a l than i n Model I I . Despite low levels of unemployment, individuals earn a lower MDI i n Model I I I . This result i s quite satisfactory i n the l i g h t of com-46 parative s t a t i c results i n Model I , associated with an increase i n c^. " The r a t i o of the l e v e l of MDI for each i n d i v i d u a l i s higher i n Model I I than i n Model I I I . This r e f l e c t s the i n a b i l i t y of firms to discriminate i n wage offers between individuals i n Model I I . Their wage decisions are based on the o v e r a l l composition of employment and so type one employees, i n contrast to Model I I I , earn a higher mean offer than type two employees. The l e v e l of aggregate vacancy creation i s higher i n Model I I I than i n Model I I , due to the stochastic elements of job search and the creation of speculative vacancies at low wage offers i n Model I I I . 254 The mean d u r a t i o n of search p r i o r to an o f f e r f o r type one i n d i v i d -uals i s h i g h e r i n Model I I than i n Model I I I . T h i s r e s u l t i s q u i t e c o n v i n c i n g , given the f i r m s ' a b i l i t y to d i s c r i m i n a t e i n h i r i n g . In four cases out of s i x , the d u r a t i o n of search p r i o r to an o f f e r for type two i n d i v i d u a l s i s lower i n Model I I I than i n Model I I . T h i s r e s u l t i s q u i t e p l a u s i b l e , because vacancy c r e a t i o n f o r type two i n d i v i d -u a l s i n Model I I I i s almost equal to aggregate vacancy c r e a t i o n i n Model I I . T h i s r e f l e c t s the a b i l i t y of f irms to d i s c r i m i n a t e i n h i r i n g . The two exceptions correspond to equal v a l u e s of the p e r c e p t i o n parameters. The d u r a t i o n of unemployment of type one i n d i v i d u a l s i s h i g h e r i n Model I I I and t h i s a g a i n corresponds w i t h i n t u i t i o n . The r e s u l t s f o r type two i n d i v i d u a l s , however, are not s y s t e m a t i c . One f a c t o r i n f l u e n c i n g the r e s u l t s i s t h a t , w h i l e vacancy c r e a t i o n f o r type two i n d i v i d u a l s i s r e l a t i v e l y h i g h , some of these p o s i t i o n s are created a t r e l a t i v e l y low wage r a t e s s o , although the d u r a t i o n of search p r i o r to an o f f e r i s r e l a t i v e l y low, the d u r a t i o n of unemployment i s r e l a t i v e l y h i g h . I n a l l cases, firms enjoy h i g h e r mean p r o f i t s i n Model I I I , than i n Model I I . Each type of i n d i v i d u a l enjoys a h i g h e r share of output i n Model I I than i n Model I I I . The r a t i o of l a b o u r shares o f output i s h i g h e r i n Model I I than i n Model I I I , due to the f i r m s ' a b i l i t y to d i s c r i m i n a t e i n Model I I I . B . F a i r Employment Laws Of p a r t i c u l a r i n t e r e s t i n the comparison of these models i s the impact of ' f a i r employment' laws. I n Model I I I , type one i n d i v i d u a l s 255 are discriminated against i n wage offer and employment, i n the aggregate, because, although equally productive, they have a higher p r o b a b i l i t y of turnover, when employed, and a lower probability of acceptance, when unemployed, i n response to a given wage offer than type two individuals. A type one ind i v i d u a l earns a lower mean wage and lower MDI over his job horizon. I f e x p l i c i t wage and h i r i n g discrimination i s i l l e g a l , then firms are forced to make wage and vacancy creation decisions, ex ante, 4 7 and make offers randomly. Firms' employment a c t i v i t i e s are thus non-discriminatory i n Model I I and conform to the ' f a i r employment' laws. Firms' decisions are based on the o v e r a l l composition of employment. Under f a i r employment laws, both sets of participants gain increased mean wage offers and MDI and,indeed,the MDI of a type one worker increases proportionately more than the MDI of a type two in d i v i d u a l . The share of output earned by type one individuals increases proportionately more than the share of output earned by type two i n d i v i d -uals. Fair employment laws do have the impact of increasing the mean offers and l i f e t i m e incomes of non-preferred individuals proportionately more than preferred individuals. Fair employment laws were introduced, however, to outlaw discrim-ination against p a r t i c u l a r groups of individuals based on ignorance or prejudice. I t would seem unjust, perhaps, to allow discriminatory firm behaviour when a l l i n dividuals, i f employed, are equally productive. 256 On the other hand, i n d i v i d u a l s ' i n t e r t e m p o r a l p r o d u c t i v i t y d i f f e r s s y s t e m a t i c a l l y but randomly due to d i f f e r e n c e s i n turnover and 48 acceptance behaviour. ' By banning e x p l i c i t d i s c r i m i n a t i o n , as represented by Model I I I , f irms are s u b j e c t i n g type two i n d i v i d u a l s , who are o b j e c t i v e l y more v a l u a b l e , to d i s c r i m i n a t i o n on the b a s i s o f t h e i r lower marginal and absolute p r o b a b i l i t y of turnover. T h i s i s represented by t h e i r lower mean wage when employed. But, under f a i r employment laws, both types of i n d i v i d u a l enjoy h i g h e r mean wage r a t e s and MDI and type two i n d i v i d u a l s s t i l l earn a higher MDI than type one i n d i v i d u a l s . Furthermore, i f each type of i n d i v i d u a l belonged to a segregated labour market, then type one i n d i v i d u a l s would earn s y s t e m a t i c a l l y h i g h e r wages (and MDI over c e r t a i n v a l u e s of c^) than type two 49 i n d i v i d u a l s . V. Conclusion Most of the questions posed i n Chapter I IIIB have been already answered. Despite systematic d i f f e r e n c e s i n i n d i v i d u a l b e h a v i o u r , s t o c h a s t i c e q u i l i b r i u m i n Model I I I i s not c h a r a c t e r i s e d by segregated workforces because i n d i v i d u a l s ' behaviour i s s t o c h a s t i c . Type one i n d i v i d u a l s are l e s s v a l u a b l e on the b a s i s of t h i s systematic d i f f e r e n c e i n behaviour and, i n the aggregate, enjoy lower mean wages and MDI and a higher l e v e l o f unemployment than type two i n d i v i d u a l s , as a r e s u l t of f i r m s ' d i s c r i m -i n a t i o n i n wage o f f e r s and h i r i n g , as argued by Sanborn. 257 I f f a i r employment laws are enacted, mean p r o f i t s i n the industry f a l l , w h i l e i n d i v i d u a l s ' MDI r i s e along with labour's share of output. Non-preferred type one i n d i v i d u a l s increase t h e i r share of output proportionately more than type two i n d i v i d u a l s , but both types of worker face higher rates of unemployment. I n t e r - f i r m wage d i f f e r e n -t i a l s p e r s i s t over firms with the same l e v e l but d i f f e r e n t composition of employment, however, because firms are behaving competitively and recognise the c h a r a c t e r i s t i c s of t h e i r workforces. F a i r employment laws do achieve the objective of outlawing e x p l i c i t d i s crimination against l e s s preferred and le s s intertemporally productive type one i n d i v i d u a l s , but at the cost of the e x p l o i t a t i o n of type two i n d i v i d u a l s , who are more valuable because firms can o f f e r them lower wate rates. The existence of type one i n d i v i d u a l s i n the labour force does have the impact of r a i s i n g the general l e v e l of wage rates i n the industry above that made by firms h i r i n g from a type two labour force. 258 CHAPTER 6 THE CONCEPT OF VACANCY CREATION AND  EXCESS DEMAND IN AN IMPERFECT MARKET I. Introduction In the thesis, I have formulated and solved simultaneous models of imperfect labour market. Due to incomplete s p e c i f i c information about job of f e r s , unemployed individuals are forced to search randomly. Individuals evaluate a wage offer on the basis of the labour market information gained through search. This labour market information i s modelled by postulating stochastic misperceptions of the reservation wage. Consequently, individuals' turnover and acceptance behaviour i s stochastic. Firms make wage offers and vacancy creation decisions to maximise discounted p r o f i t s over an i n f i n i t e horizon. Equilibrium i n these imperfect labour markets i s stochastic and characterised by wage dispersion, vacancies and f r i c t i o n a l unemployment. In Model I , individuals are equally productive and are, thus, i d e n t i c a l , ex ante, and so the firm i s i n d i f f e r e n t between retaining current employees and h i r i n g new ones. Due to random search and stochastic misperceptions of the reservation wage, however, individuals' turnover and acceptance behaviour i s stochastic and thus non-homogeneous, ex post. In Models I I and I I I , individuals' behaviour i s also stochastic, but, i n addition, type one individuals' behaviour d i f f e r s systematically from type two individuals' behaviour. 259 In this Chapter the concept of a vacancy i s examined i n the three models and the measures of aggregate vacancies are interpreted. The functioning of a labour market i n which information i s perfect i s compared with the workings of these models of imperfect markets. The concept of excess demand i s examined and an index of market pressure i s developed. F i n a l l y , the contribution of the thesis i s outlined. I I . The Vacancy Concept A. Speculative Behaviour In a l l three models, individuals' labour market behaviour i s stochastic and so a firm's vacancy creation behaviour i s , i n a s t r i c t sense, speculative. No firm i s assured that any vacancy created w i l l be f i l l e d . In the discussion of the r e s u l t s , however, the term 'speculative' i s used to describe vacancy creation which i s not consistent with attaining the l e v e l of employment associated with intertemporal p r o f i t maximisation, as defined i n Chapter 2 IIIG. I f the l e v e l of vacancy creation exceeds the corresponding l e v e l of desired net h i r e s , then firm behaviour i s speculative. B. Model I In Model I , stochastic equilibrium i s characterised by wage d i s -persion, vacancies and functional unemployment. The wage offered by the firm and the l e v e l of vacancy creation are both declining functions of the l e v e l of current employment. 260 The firm i s i n d i f f e r e n t between r e t a i n i n g current employees and h i r i n g i n d i v i d u a l s currently unemployed. Rather than o f f e r a r e l a t i v e l y high wage and create vacancies equal to desired net h i r e s , the f i r m e x p l o i t s i t s monopsony power and o f f e r s a lower wage, creates more costless vacancies and speculates about i n d i v i d u a l s ' s t o c h a s t i c behaviour. The aggregate measure of vacancies represents posi t i o n s a v a i l a b l e at d i f f e r e n t o f f e r s . Even though i n d i v i d u a l s are equally productive and a l l p o s i t i o n s require the same s k i l l s , the aggregate i s a measure over heterogeneous elements. C. Model I I In Models I I and I I I , the labour force consists of two types of i n d i v i d u a l who systematically d i f f e r i n t h e i r labour market behaviour. In Model I I , firms are unable to discriminate between i n d i v i d u a l s , e i t h e r i n wage o f f e r or i n h i r i n g . I f a f i r m faces an undersize workforce, then i t s l e v e l of vacancy creation equals desired net h i r e s . This i s consistent with many solutions to Model I. Eirms do not create speculative vacancies at below optimal l e v e l s of employment because, due to stochastic search, there i s a low p r o b a b i l i t y that s u f f i c i e n t i n d i v i d u a l s w i l l sample the firm. At the optimal l e v e l of employment, vacancies are created i n the states [2,0] and [1,1], where there i s a low or zero p r o b a b i l i t y that a worse composition of employment w i l l be achieved. At above optimal l e v e l s of employment, firms generally do not create vacancies. 261 To conclude then, firms' vacancy speculation i n Model I I i s constrained at less than optimal l e v e l s of employment by the stochastic supply of i n d i v i d u a l s sampling the fir m and, at optimal l e v e l s of employment, by the p o s s i b i l i t y of achieving an i n f e r i o r composition of employment. Again,the aggregate measure of vacancy creation i s non-homogeneous, because i t represents p o s i t i o n s at d i f f e r e n t wage o f f e r s and i t does not measure aggregate desired net h i r e s . D. Model I I I In Model I I I , firms discriminate against i n d i v i d u a l s i n both wage o f f e r and employment. Each firm makes a wage o f f e r and vacancy creation decision f o r each type of i n d i v i d u a l . Stochastic equilibrium i s characterised by wage dispersion, vacancies and f r i c t i o n a l unem-ployment for both types of i n d i v i d u a l . I n t r a - f i r m wage d i f f e r e n t i a l s are observed. The a b i l i t y to discriminate i n wage o f f e r and h i r i n g has two s i g -n i f i c a n t implications i n Model I I I . F i r s t l y , the fir m i s forced to create vacancies f o r both types of i n d i v i d u a l s at l e s s than optimal l e v e l s of employment, due to stocha s t i c search. For t h i s reason, speculative vacancy creation i s redefined i n Model I I I (see Chapter 5 I I I B5). Secondly, the firm can create speculative vacancies at low wage rates, due to i t s a b i l i t y to discriminate between the type of i n d i v i d u a l s who constitute a majority of the workforce and the other type. Thus, at optimal l e v e l s of employment, speculative vacancies are created i n some 262 states f o r the type of i n d i v i d u a l who constitutes a minority of the workforce at a wage as low as the l e v e l of unemployment compensation. I I I . The Concept and Measure of Excess Demand In a p e r f e c t l y competitive labour market, excess demand at any wage o f f e r i s measured by the difference between firms' desired net hi r e s associated with p r o f i t maximisation and the l e v e l of unemployment. I f excess demand i s p o s i t i v e , the market mechanism operates through an increase i n the p r e v a i l i n g wage o f f e r . Equilibrium i s characterised by zero excess demand. At the equilibrium wage rate, unemployment i s zero. No f i r m wishes to h i r e any more labour. In the labour market- structures considered, equilibrium i s characterised by the simultaneous existence of unemployment and vacancies, but excess demand, however defined, i s zero. P a r t i c i p a n t s i n such labour markets have imperfect information, however, and, by contrast to a per f e c t labour market, the demand f o r a d d i t i o n a l workers on the part of the firm, represented by a r e l a t i v e l y high wage o f f e r , i s not communicated to a l l p a r t i c i p a n t s . Only current employees and searchers, who sample the fir m and receive a job o f f e r , know the wage rate. Consequently, excess demand cannot be defined. Several authors use a function of the aggregate measures of vacancies and unemployment as a proxy f o r excess demand.'*" A glance at the tables i n Chapters 4 and 5 suggests that there appears to be no systematic r e l a t i o n s h i p between these measures i n equilibrium under the labour 263 market s t r u c t u r e s c o n s i d e r e d . Aggregate vacancy c r e a t i o n does not represent d e s i r e d net h i r e s . Furthermore, firms are unable to l a y o f f i n d i v i d u a l s , when f a c i n g an o v e r s i z e workforce. Then, i r r e s p e c -t i v e of whether firms s p e c u l a t e , aggregate vacancy c r e a t i o n exceeds aggregate d e s i r e d net h i r e s . Some firms wish to i n c r e a s e t h e i r workforces w h i l e other firms wish to reduce t h e i r workforces, but i n the aggregate, both the wage o f f e r d i s t r i b u t i o n and the steady s t a t e d i s t r i b u t i o n of employment are constant. Since the concept o f excess demand i s i n a p p r o p r i a t e i n these models, I w i l l attempt to c o n s t r u c t an index of market p r e s s u r e i n the context of Model I . Reder [1969] argues t h a t , a p r i o r i , there i s no p a r t i c u l a r r e l a t i o n between the mean d u r a t i o n o f unemployment and the mean 3 time d u r i n g which a vacancy i s u n f i l l e d . Then, i n e q u i l i b r i u m , there i s no p a r t i c u l a r r e l a t i o n between the aggregate magnitude o f unemployment and v a c a n c i e s . Let U , V denote the aggregate measures of unemployment and vacancy c r e a t i o n , r e s p e c t i v e l y . Let D^, D denote the mean d u r a t i o n of unemployment and the mean time a p o s i t i o n i s u n f i l l e d , r e s p e c t i v e l y . A necessary c o n d i t i o n f o r s t o c h a s t i c e q u i l i b r i u m i s that U_ = V_ D D * . . . . (1) u v An e q u i v a l e n t c o n d i t i o n i s that aggregate h i r e s equal aggregate q u i t s . N e i t h e r c o n d i t i o n i s s u f f i c i e n t , however, s i n c e s t o c h a s t i c e q u i l i b r i u m r e q u i r e s that the d i s t r i b u t i o n of firms over employment s t a t e s i s constant. 264 The aggregate measure of unemployment i s homogeneous. Each i n d i v i d u a l , ex ante, has the same p r o b a b i l i t y of r e c e i v i n g and a c c e p t i n g a wage o f f e r . The aggregate measure of vacancy c r e a t i o n , however, as i n d i c a t e d , i s heterogeneous. Thus the d u r a t i o n over which a vacancy i s open i s an average d u r a t i o n . I t i s not easy to see how such a measure might be computed, s i n c e the model i s s t o c h a s t i c and the f i r m may create s e v e r a l v a c a n c i e s , some of which may seem to be f i l l e d given the mean l e v e l of employment next p e r i o d , or perhaps e x i s t i n g employees do not q u i t . I t i s a more f r u i t f u l approach to c o n s i d e r the r e l a t i o n s h i p between aggregate h i r e s , H , and aggregate q u i t s , Q. E q u a t i o n (1) may be r e w r i t t e n Q = H . . . . . (2) L e t x^ = (S^ ,^ ) denote the l a b o u r market environment at the end of p e r i o d j . The f i r m makes d e c i s i o n s , f o r p e r i o d j+1. The acceptance and turnover functions f a c i n g the f i r m may be w r i t t e n as i+1 i+1 -1 n = a n ^ n ' X ^ n = 0, 1 , . . . , n . . . . . (3) t J + 1 = t ( w j + 1 , x j ) n = 0, l , . . . , n . . . . . (4) n n n Ex a n t e , mean q u i t s from a f i r m , employing n i n d i v i d u a l s , are n t n + 1 * T h e n > aggregate q u i t s at the b e g i n n i n g of p e r i o d j+1 are 265 n . Q,,_ =" N E f ^ . n t 3 " ^ . . . . . (5) J + 1 n = O n n L e t a J ^ denote the p r o b a b i l i t y that k• i n d i v i d u a l s are h i r e d by a f i r m i n employment s t a t e n , i n p e r i o d j+1. Then, from Chapter 2, Equation (36) U TT i J U " J M • , i k ' M " k M = m i n ( v ^ + 1 , J ) ( n = Q , 1 , . . . , n ) . . . . . (6) Then t o t a l mean h i r e s , H 3 + \ by a f i r m employing n i n d i v i d u a l s may be w r i t t e n as v ^ 1 . n . j+1 H 3 - • Z k a . ( n = 0 , l , . . . , n ) . . . . (7) and so n , „ nk k=0 j+1 n . n . n H... =N E g J H J X = N E e J E k a J , . . . . (8) J + 1 n=0 n n n=0 n k=0 n k where H j + ^ denotes aggregate h i r e s i n p e r i o d j+1. The e q u a l i t y of aggregate h i r e s and aggregate q u i t s , ex ante, i s not a s u f f i c i e n t c o n d i t i o n for e i t h e r s t o c h a s t i c e q u i l i b r i u m or constant aggregate employment, s i n c e i n d i v i d u a l s respond to c u r r e n t o f f e r s . The a c t u a l turnover and acceptance functions f a c i n g a f i r m employing n i n d i v i d u a l s are t j + 1 = t ( w j + 1 , x ) . . . . (9) n n n ' a-J + 1 = a (w-j + 1 ,x) . . . . (10) n n n 266 where x = ( 6 j + 1 , g j ) , . . . . (11) Then, denoting the corresponding aggregate, e x . p o s t , measures by Qj +-^ and a s u f f i c i e n t c o n d i t i o n f o r constant employment i s v. - v. •(12) H _ . + ^ may be r e w r i t t e n as H . , , = U. Z g j Y J + 1 a j + 1 . . . . (13) J+1 3 n = 0 n n n where XL denotes the number of i n d i v i d u a l s unemployed a t the end of - i+1 p e r i o d j and Y N denotes the p r o b a b i l i t y of r e c e i v i n g an o f f e r a t a p a r t i c u l a r f i r m , employing n i n d i v i d u a l s , g iven that the f i r m i s sampled. From Chapter 2, Equation (3) . + 1 V1 U . - l _ U . - l - J v ^ + 1 Y„ = Z ( J J ) ( 1 / N ) J (1-1/N) 3 m i n ^ ,1) (14) n J=0 Aggregate h i r e s then are simply the l e v e l of unemployment m u l t i p l i e d by the p r o b a b i l i t y of r e c e i v i n g and a c c e p t i n g a job o f f e r . An a d d i t i o n a l c o n s t r a i n t on t h i s measure of market p r e s s u r e i s that i t assumes a constant value only when d i s t r i b u t i o n of employment i s constant . While the aggregate, ex p o s t , measure of h i r e s minus q u i t s i n d i c a t e s whether market pressure i s p o s i t i v e or n e g a t i v e , i t i s evident that t h i s measure can r e g i s t e r a zero value when the labour market i s not i n e q u i l i b r i u m . 4 267 The equality of hires and quits i s , therefore, a necessary but not sufficient condition for stochastic equilibrium. The measures of hires and quits alone then are insufficient to construct an index of market pressure which registers a constant value in stochastic equilibrium. Additional information is required. If the researcher knows the steady state transition matrix, then he can determine whether the labour market is i n equilibrium, but, again, i t i s not evident that he can construct an index which assumes a unique and constant value i n equilibrium. Thus,it i s both a problem of definition and construction. The researcher can define the labour market to be in equilibrium, i f the steady state distribution of firms over employment states i s constant, but he cannot construct a measure which w i l l assume a constant value in stochastic equilibrium. The ina b i l i t y to construct a measure i s an aggregation problem. IV. Conclusion By modelling simultaneously the behaviour of participants, i t i s possible to examine the characteristics of equilibrium in a market characterised by imperfect information. Firms make wage offer and vacancy creation decisions to maximise mean profits discounted over an i n f i n i t e horizon. Individuals adopt search rules to maximise mean income discounted over their job horizons. The behaviour of both sets of participants i s dependent on the labour market environment and how they perceive the behaviour of the other set of participants. Equilibrium 268 i s characterised by wage dispersion,.vacancies and f r i c t i o n a l unem-ployment. In contrast to other models of imperfect markets, wage dispersion i s endogeneous. By analysing the model, i t i s possible to determine the necessary and s u f f i c i e n t conditions for wage and employment dispersion i n equilibrium. In this general equilibrium framework, i t has been possible to examine the concept of a vacancy and, more p a r t i c u l a r l y , the concept of excess demand i n an imperfect market. I t appears to be impossible to define excess demand i n a meaningful sense out of equilibrium, but i t i s possible to construct a crude index of market pressure."' This r e s u l t has important implications for empirical studies of the labour market. Aggregate vacancy creation by firms i n the same industry i s heterogeneous and does not represent desired net hires. Comparison of stochastic equilibrium i n Model I with perfectly competitive equilibrium shows that firms bear the i m p l i c i t costs of the market imperfection. F r i c t i o n a l unemployment characterises equilibrium, due to individuals' stochastic behaviour. Consequently, firms pay wages above the perfectly competitive equilibrium wage, but the mean offer i s less than the average marginal product. Despite the incidence of unemployment, individuals enjoy a higher MDI than that enjoyed i n perfect competition,but a lower MDI than that associated with workers' collusion. 269 If both sets of market participants make decisions more frequently, then individuals enjoy higher incomes although, in a sense, there has been a symmetric reduction in the cost of the market imperfection. Since employees make the quit decision more frequently, firms enjoy less monopsony power. In the limit, market participants make decisions continuously, but equilibrium w i l l be characterised by wage dispersion, f r i c t i o n a l unemployment and vacancies, i f information is imperfect. In the ab-sence of collusion, firms earn maximum profits, i f individuals behave identically, ex post. These results underline the significance of the information structure in analysing a market. A reduction in the implicit costs of imperfect information leads workers, alone, enjoying larger incomes. It i s in the area of labour market discrimination that the important contribution i s made. The labour force consists of two groups of individ-uals who systematically differ in their labour market behaviour. Thus firms' discriminating behaviour i s based on objective differences between individuals and is motivated by profit maximisation. An important result i n Model II i s that inter-firm wage dispersion persists in equilibrium, even when firms behave competitively and cannot exp l i c i t l y discriminate in wage offer and employment. In this framework i t has been possible to evaluate the arguments presented by Sanborn and Salop concerning individuals' relative wage offers. It turns out that, i f firms cannot expli c i t l y discriminate i n 270 wage offer and hiring, individuals who have a lower absolute and marginal probability of turnover are discriminated against in wage offer. This can be observed by noting that firms in states employing predominantly type two individuals generally make lower offers than firms in the state complement. On the other hand, the general level of wage offers enjoyed by type two individuals i s higher due to the presence of type one individuals in the workforce than would be en-joyed by type two individuals in a segregated workforce. When firms can expli c i t l y discriminate,they can make wage offers to each type of individual based on his mean returns and labour market behaviour. As a result, low turnover individuals, who are more valuable to the firm, earn a higher mean wage when employed than high turnover individuals. It has also been possible to analyse the impact of 'fair employment' laws on the incomes of the less valuable type one individuals. If explicit wage and hiring discrimination is i l l e g a l , then firms do behave f a i r l y i n that a l l unemployed individuals do have an equal probability of receiving a job offer. As a result, type one individuals enjoy a proportionately larger increase i n income than type one individuals and so the 'fair employment' laws do aid the 'non-preferred.' In a l l the models, comparative static predictions have been gener-ated with respect to the exogeneous parameters. The prediction gener-ated by a change in the level of unemployment compensation i s interesting. Since the variance of the wage offer distribution has changed, individuals do not unambiguously choose more unemployment. 271 FOOTNOTES CHAPTER 1 "''All models of search, notably Mortenson [1970], Salop [1973a] and Gronau [1971], e x p l i c i t l y assume that individuals have imperfect information about wage offers. Wage dispersion i s assumed to e x i s t . Except i n Mortenson's paper, individuals must be employed to search. S t i g l e r , i n his seminar paper, argues that i n product markets too 'price dispersion i s a manifestation, and indeed, i s the measure of ignorance i n the market.' ( S t i g l e r [1961], p.214). 2 The word 'expected'denotes the mathematical expectation rather than workers' or firms' subjective evaluation. The term 'mean' i s also used to denote mathematical expectation. 3 In h i s model of intertemporal firm behaviour, Salop argues that r a t i o n a l behaviour on the part of firms i s s u f f i c i e n t to generate 'very short run (wage) differences from (possibly) i d e n t i c a l firms being at different points along t h e i r adjustment paths,' (Salop [1973b], p. 342-343). He argues that long run d i f f e r e n t i a l s arise due to e x p l i c i t differences i n firm costs or production functions. Mortenson [1970] e x p l i c i t l y assumes the existence of wage dispersion. He models firm behaviour, aggregates thei r decisions and determines a 'natural' un-employment rate. 4 S t i g l e r [1961], p.214 and Gronau [1971], p.290. 5Rothschild [1973], p.1285. In these models the behaviour of only one set of market par-ticipants i s formally specified. For example, S t i g l e r and Gronau analyse optimal search rules for individuals assuming that price or wage dispersion exi s t s . Mortenson, Salop and Archibald analyse firm behaviour given non-stochastic behaviour on the part of workers. In a l l these models, the description of the operation of the market i s unconvincing. 7 Using the notation i n Chapter 1 TI;;^ th_en, ( i f • C denotes the t o t a l costs of employing e ind i v i d u a l s , C = wS(e,w). . . . . (1) Then marginal costs, MS(e,e), may be written as 272 MS(i ,e) - § - g __1 + y_S_ dw de dw de = -r^- + w . . . . (2) b 2 where S 2 denotes the d e r i v a t i v e of the supply of labour w i t h respect to the wage o f f e r , w. To determine the impact of a change i n e on the m a r g i n a l cost s c h e d u l e , d i f f e r e n t i a t e the m a r g i n a l c o s t f u n c t i o n with respect to w, keeping the supply of labour constant . Sjde + S2dw = 0 . . . . (3) Then, L 9 dw _ 1^ de s 2 < 0 . . . . (4) and so dMS(e,e) _ _1 dw S , dw. dw de S 9 de ~ 2^ 12 b 22 d e ; de l b 2 5 1 S S l 5 2 _ 2 L & 12 b 22 S 2 J 2 2 W h e r e S 12 = ^ a n d S 2 2 = T T ' • ' • ' ( 5 ) dw I f turnover and acceptance behaviour are complementary and a(w) denotes the r a t e of acceptance, then the expected flow supply of labour can be w r i t t e n S(e,w) = a(w)[S+e] . . . . (6) where S denotes the expected number of searchers who sample the f i r m . 273 Then, S 1 2 = a*(w) > 0 S 2 2 = a"(w) [S+e] < 0 where a'(w) da dw a"(w) = ^ f . . . . (7) dw2 Then from (5) (8) dMSC^e) < Q de Hence the f i r m enjoys dynamic monopsony power. g An almost e q u i v a l e n t i n t e r p r e t a t i o n of s t a t e RC i s t h a t firms are unable to recognise i n d i v i d u a l s p r i o r to h i r i n g b u t , once employed for one p e r i o d , d i f f e r e n t types of i n d i v i d u a l s can be r e c o g n i s e d . I f the f i r m c o u l d d i s c r i m i n a t e , however, then i t would be prepared to d i s c r i m i n a t e against i t s workforce i n wage o f f e r although unable to recognise the behaviour o f i n d i v i d u a l s s e a r c h e r s . 9 R o t h s c h i l d p o i n t s out that ' a person who r i g i d l y fol lows a f i x e d sample s i z e r u l e w i l l , even i f he gets a p r i c e q u o t a t i o n l e s s than the cost of s e a r c h , keep on sampling u n t i l h i s quota of p r i c e quotations i s f u l f i l l e d 1 , R o t h s c h i l d [1973], p.691. "^1 am indebted to an anonymous r e f e r e e for demonstrating t h i s with a simple n u m e r i c a l example. The optimal r u l e of search i s d e r i v e d and d i s c u s s e d i n Chapter 2 IB. 1 : L R o t h s c h i l d [1973], p.710_. 1 2 R e d e r [1969], p . 7. 13 T h i s r e s u l t i s i n c o r r e c t . Bergman assumes that group unemployment r a t e s can be d e r i v e d from the average l e n g t h of unemployment over a l l groups. Furthermore, she does not c o n s i d e r the p o s s i b l y d i f f e r e n t rates o f acceptance from the two groups. Assume a labour market i s i n s t o c h a s t i c e q u i l i b r i u m w i t h no wage d i s p e r s i o n . There are two groups of i n d i v i d u a l s with d i f f e r e n t r a t e s of turnover and acceptance. Firms do not d i s c r i m i n a t e . Then each group's labour market behaviour may be represented as a Markov p r o c e s s . 274 E i u. X E . X 1-d. X d . X U . X a . X 1-a. X Then the p r o b a b i l i t y of q u i t t i n g a p o s i t i o n i s d i ( i = 1,2) and the p r o b a b i l i t y of a c c e p t i n g a p o s i t i o n i s a^ ( i = 1,2). I f the labour market i s i n s t o c h a s t i c e q u i l i b r i u m t h e n , E„ = ( l - d . ) E . + a . U . . . . . (1) i x x x x and so E . = - r U . . . . . (2) i d ± x where ~E^, IL a r e the l e v e l s of employment and unemployment r e s p e c t i v e l y . Then, U i d i U i = E . + U , = 7 7 T d 7 ' * ' * ( 3 ) x i x x Hence, the r a t i o of r e s p e c t i v e unemployment r a t e s , i f the r e s p e c t i v e r a t e s o f acceptance are the same,is d^ a + d 2 — = - , , —j . . . . (4) u 2 a + d x d 2 where a denotes the common r a t e of acceptance. T h i s r a t i o exceeds u n i t y i f d, > d 2 « The r a t e o f acceptance i s given by the p r o b a b i l i t y of a job o f f e r m u l t i p l i e d by the p r o b a b i l i t y that the o f f e r i s accepted. 1 4 R o t h s c h i l d [1973], p.1286. ^ A n important d i s t i n c t i o n i s made here between m a r g i n a l and a b s o l u t e q u i t r a t e s . In my model, the a b s o l u t e ex ante p r o b a b i l i t y of q u i t t i n g a job i s given by the cumulative of the p r o b a b i l i t y d i s t r i b u t i o n of p e r c e p t i o n s . The m a r g i n a l p r o b a b i l i t y of q u i t t i n g i s g i v e n by the m a r g i n a l p r o b a b i l i t y d i s t r i b u t i o n o f p e r c e p t i o n s . 275 "^The s p e c i f i c a t i o n of h i r i n g and f i r i n g costs modify these p r e d i c t i o n s . 1 7 F r e e m a n [1973], p.17. 18 M c C a l l provides no j u s t i f i c a t i o n f o r t h i s statement. 19 In my f o r m u l a t i o n of i n d i v i d u a l s ' job search b e h a v i o u r , t h e costs of search are simply the wages foregone by being unemployed. Given that the l e v e l of unemployment compensation c o n s t i t u t e s the f l o o r to the wage o f f e r d i s t r i b u t i o n , no i n d i v i d u a l s drop out of the labour f o r c e . 20 S t r i c t l y speaking, these p r e d i c t i o n s are comparative dynamic p r e d i c t i o n s because e q u i l i b r i u m i n t h i s labour market i s s t o c h a s t i c and thus dynamic. 276 FOOTNOTES CHAPTER 2 I f a l l firms offer a wage equal to the l e v e l of unemployment compensation, there are no e x p l i c i t costs of search, and a l l workers are i n d i f f e r e n t between employment and unemployment. 2 I assume an i n d i v i d u a l always accepts an offer equal to his reservation wage although, s t r i c t l y speaking, he i s i n d i f f e r e n t between employment and further search. 3 The true reservation wage, W*, i s the same irrespective of whether the in d i v i d u a l i s employed or unemployed. 4 The anonymous referee, mentioned i n Chapter 1, Footnote 10, defines f(w) as the d i s t r i b u t i o n of wage offers and r as the reservation wage. Then, the.probability of finding an acceptable wage off e r , p, may be written as p = / f(w)dw . . . . (1) and the mean duration of search, s, i s s = i . . . . (2) The mean wage off e r , given that search has terminated, We, i s 1 oo W = £ / wf (w)dw . . . . (3) e p- r I f H^ denotes the horizon over which the wage d i f f e r e n t i a l i s assumed to per s i s t and D the rate of discount, expected gross returns from search, V, gained over this horizon are s+H r . 1 3 - 1 V = Z (T^) (We-W_). . . . . (4) j=s+l 277 T h i s e x p r e s s i o n represents the present value of the wage d i f f e r e n t i a l enjoyed over the h o r i z o n , H^. The costs o f s e a r c h , V_ are given by the income foregone d u r i n g the d u r a t i o n of s e a r c h , s, namely s i J - 1 V I = E (1+D ) ( W I " w ) ' . . . . (5) An i n d i v i d u a l chooses to q u i t and search i f max(V-V_) > 0, . . . . (6) r He searches i f the expected r e t u r n s a s s o c i a t e d with the optimal r e s e r v a t i o n wage, r * , over the h o r i z o n , H ^ , exceed the costs borne d u r i n g the p e r i o d o f s e a r c h . T r i v i a l l y , i f s=l then p=l and so r=w and We=EW*. S u b s t i t u t i n g i n (4) and (5) y i e l d s the same d e c i s i o n r u l e as d e r i v e d i n the main t e x t . The referee argues t h a t , i f s ^ l , then the o p t i m a l d e c i s i o n r u l e d e r i v e d i n (6) i s g e n e r a l l y not e q u i v a l e n t to the one p e r i o d r u l e d e r i v e d i n the t e x t . A mean p r e s e r v i n g s h i f t of the d i s t r i b u t i o n f(w) i n c r e a s e s the r e s e r v a t i o n wage a s s o c i a t e d w i t h t h i s optimal r u l e but the r e s e r v a t i o n wage a s s o c i a t e d with the s i n g l e search remains constant. T h i s o p t i m a l r u l e i s again a s e q u e n t i a l r u l e i n that an i n d i v i d u a l compares the r e s e r v a t i o n wage, r * with h i s c u r r e n t wage throughout h i s d u r a t i o n of s e a r c h . ^Telser [1973] p . 4 3 . H i s work i s d i s c u s s e d i n the L i t e r a t u r e Review, Chapter 1 IVB. T h i s i s a s i m p l i f y i n g assumption, s i n c e an i n d i v i d u a l , who i s employed, has d i f f e r e n t labour market experiences from one who i s unemployed and s e a r c h i n g . I f an i n d i v i d u a l ' s general labour market i n f o r m a t i o n i s accumulated over a long p e r i o d of time, then such d i f f e r e n c e s d i s a p p e a r . Then, the v a r i a n c e of the sample mean wage tends to zero. 7 I t would be i n t e r e s t i n g to al low the f i r m to vary i t s c a p i t a l s t o c k . Some c a p i t a l labour s u b s t i t u t i o n i s p o s s i b l e i n response to a change i n an exogeneous parameter. 278 " i n d i v i d u a l s have no systematic preferences f o r f i r m s . Thus the terms c u , P k _ ^ a r e independent. 9 H a d l e y [1964], p.454. "^The other measures of expected income are e a s i l y computed. An i n d i v i d u a l , who never q u i t s , i s assumed to accept the f i r s t o f f e r he r e c e i v e s and remain employed, whence, a^* = 1 k = 0 , 1 , . . . , n . . . . . (1) t f c * = 0 k = 1 , 2 , . . . , n . . . . . (2) The remaining computations are the same. I f an i n d i v i d u a l ' s turnover and acceptance behaviour i s based on h i s p e r c e p t i o n o f the r e s e r v a t i o n wage, say WC, then a ^ = 1 i f w f c* > WC k = 0 , 1 , . . . , n , = 0 otherwise. . . . . (3) t k * = 0 i f w f c * > WC k = l , 2 , . . . , n = 1 otherwise. . . . . (4) The remaining computations a r e the same. 11 A l l these measures are computed i n Models I I and I I I w i t h the exception of the. measure of s p e c u l a t i v e vacancy creation, .In Model I I I . 279 FOOTNOTES CHAPTER 3 ''"By c o n t r a s t , the r e c t a n g u l a r d i s t r i b u t i o n has the same b a s i c shape i r r e s p e c t i v e of the parameter v a l u e s . 2Mood [1950], p.114. 3 The cumulative gamma d i s t r i b u t i o n i s homogeneous of degree zero f o r p r o p o r t i o n a l changes i n the wage o f f e r d i s t r i b u t i o n . I f a l l wage o f f e r s , i n c l u d i n g unemployment compensation, are i n c r e a s e d by a p r o p o r t i o n , T , then MP i n c r e a s e s by a f a c t o r , T , and VP by a f a c t o r , T 2 . From (4) and (5) i n the main t e x t , B i s unchanged and A r i s e s by a f a c t o r , T , which leaves w unchanged. From ( 9 ) , P(w) i s unchanged. A 4 The mode occurs where the marginal d i s t r i b u t i o n , p(wc) assumes i t s maximum v a l u e . Then d i f f e r e n t i a t i n g p(wc) y i e l d s dp (wc) 1 , sB-1 / - w c N / T , wc, " d w c " • —B+T ( W C ) e x p ( — ) ( B - r ) . . . . (1) B! A M The mode occurs at WC = AB. ^Compare, f o r example, the normal d i s t r i b u t i o n which i s both symmetric and a continuous f u n c t i o n of both i t s arguments, namely the mean and v a r i a n c e . 6 I t must be emphasised that the number of f i r m s , N , i s f i x e d both d u r i n g the s i m u l a t i o n procedure and i n s t o c h a s t i c e q u i l i b r i u m . The s i m u l a t i o n procedure does not represent the dynamic changes i n the labour market. The problem of firms f a c i n g bankruptcy through employing zero i n d i v i d u a l s i s not examined. Firms enjoying zero employment do not leave the i n d u s t r y because the e q u i l i b r i u m i s s h o r t r u n . T h i s model does not determine, t h e r e f o r e , the optimal number of firms i n the i n d u s t r y . ^As i n any t h e o r e t i c a l s i m u l a t i o n , the values of the parameters which are chosen are a r b i t r a r y . R and D are given the values 5.26 so that S- = = ° - 9 5 - . . . . (1) 1 + —^— 1 +'—=— 100 100 T h i s eases the computation of f i r m s ' p r o f i t s discounted over an i n f i n i t e h o r i z o n under p e r f e c t c o m p e t i t i o n . 280 g Given the r e l a t i v e l y "small" number of firms i n the industry, however, i t i s l i k e l y that firms would enjoy some degree of monopsony power, even i f both firms and individuals had perfect information about wage rates, vacancies and unemployment. 9 I f w denotes the wage rate, mean discounted income for each employed i n d i v i d u a l may be written H-1 . IL MDI = E w r 1 = W e ( 1 ~ r } i=0 e (1-r) where r = 1 n .. . . . . (1) 1 +i§o wg i s 1.00 i n perfect competition and so MDI = ° Q 4 ^ 4 = 8.028. . . . . (2) ^ I f n denotes the equilibrium l e v e l of employment, and w the corresponding wage, then p r o f i t discounted over an i n f i n i t e horizon, ITD, may be written 0 0 n 2 . (5n - - n (5-n ) IID = E (5n - -| - n w ) r 1 = 6 2 e - 6 . n e 2 e e i=0 n 1 - r 2 n 2(1-r) since p r o f i t maximising entrepreneurs set WQ = 5-n^. Equilibrium oyn 11 E See Footnote 11. empl ment ,n g, i s 4 and so IID i s 160.00. e 6 LSee Chapter 2 IIIH for a derivation of these results. 12-281 13 In Model I,stochastic equilibrium i s characterised by vacancy speculation over some, i f not a l l , employment states. There are nine possible employment states. In employment state nine, the firm i s constrained to create zero vacancies. Then, i t can be argued that the l e v e l of expected p r o f i t s discounted over an i n f i n i t e horizon associated with commencing i n employment state nine i s lower than i f vacancy creation i s unrestricted. Firms i n employment state eight, however, s t i l l create one vacancy although there i s a non-zero prob a b i l i t y of entering state nine i n which p r o f i t s are lower. Thus, despite the constraint on the number of employment states, firms s t i l l speculate i n vacancy creation. Thus, i t can be argued that the constraint on the l e v e l of vacancy creation does not s i g n i f i c a n t l y influence firm behaviour i n states one to eight. In Model I I , firms do not speculate i n vacancy creation i f they face a low l e v e l of employment. I f speculation were p r o f i t a b l e , they would choose to speculate and the constraint on employment would not prevent t h i s . Thus, there i s no reason to suggest that i n Model I I the r e s t r i c t i o n on the number of employment states changes the nature of firms' vacancy creation decisions at higher levels of employment. In Model I I I , firms do speculate i n vacancy creation, i f they face a low l e v e l of employment. Such speculative behaviour can be j u s t i f i e d at low levels of employment, but not at high levels of employment. Thus, there i s no reason to suggest that the object of firms' decisions i s influenced by the r e s t r i c t i o n on the number of employment states. This i s explained further i n Chapter 5. 14 The search over vacancy space and thus the choice of the optimal vacancy creation decision are constrained by a non-negativity constraint and the inequality v < n - n . . . . (1) where v denotes the l e v e l of vacancy creation. "^The choice of wage increment i s discussed i n B. ^ T h i s p a r t i a l optimisation procedure i s non-symmetric i n that the type one wage offer i s kept constant while the type two wage offe r i s varied. This problem i s examined i n Chapter 5. X^It i s simple to observe i f the convergence c r i t e r i o n i s too stringent. I f a large number of consecutive p a r t i a l optimisation p o l i c i e s are i d e n t i c a l , but consecutive employment distrib u t i o n s are not converging, then, perhaps, e i s too small. 282 18 I t i s probable t h a t , i f a sequence of temporary e q u i l i b r i a do n o t l e a d t o f u l l e q u i l i b r i u m , then i t would be d i s c o v e r e d by f a i l u r e of the employment d i s t r i b u t i o n to converge to i t s steady s t a t e v a l u e . The procedure i s more r i g o r o u s and perhaps more e f f i c i e n t . 19 The number o f i t e r a t i o n s always exceeds 500. 20 * In Model I I I , x~* i s i n t e r p r e t e d as the complete l a b o u r market environment a f t e r j i t e r a t i o n s of the f u l l o p t i m i s a t i o n a l g o r i t h m , 1 1 1 1 2 2 2 2 The s u b s c r i p t denotes the s t a t e of employment and the s u p e r s c r i p t , the type of i n d i v i d u a l . a c t u a l t e s t used i n Models I and I I i s to assume c y c l i n g i f , | EW j - E W 1 + : j | < Y . . . . (1) | E W j - 1 - E W i + j _ 1 | < Y . . . . (2) where EW^ denotes the t r u e mean wage o f f e r a f t e r j i t e r a t i o n s and y i s set a t 0.00001. In a d d i t i o n to b e i n g c o m p u t a t i o n a l l y e f f i c i e n t , t h i s c r i t e r i o n i s s u f f i c i e n t l y s t r i n g e n t to ensure t h a t the model c y c l e s . In Model I I I , due to systematic d i f f e r e n c e s i n behaviour and the r e s u l t i n g wage and employment d i s c r i m i n a t i o n , the t r u e mean wage o f f e r i s d i f f e r e n t f o r each type of i n d i v i d u a l . I t i s s u f f i c i e n t , however, to t e s t the convergence of the t r u e mean wage o f f e r of type two i n d i v i d u a l s alone to i d e n t i f y c y c l i n g r a t h e r than b o t h mean wage o f f e r s . • 22 In a s m a l l number of cases t e s t e d non-unique s o l u t i o n s were o b t a i n e d . T h i s suggests that the procedure i s generating a l o c a l r a t h e r than a g l o b a l maximum. These cases c o i n c i d e w i t h i n c o n s i s t e n t comparative s t a t i c r e s u l t s . The ' c o s t ' of a few i n a c c u r a t e r e s u l t s must be t r a d e d o f f a g a i n s t the c o n s i d e r a b l y h i g h e r computing costs a s s o c i a t e d w i t h an exhaustive search p r o c e d u r e . 23 The d i s t r i b u t i o n o f f irms over employment s t a t e s i s ' u n i q u e ' to the a c c u r a c y , e, and the wage d e c i s i o n to the accuracy of the p r e s c r i b e d increment. 283 24 The computer programs a s s o c i a t e d w i t h the three models are long and complex and i n the i n t e r e s t s of economy are not i n c l u d e d i n the t h e s i s . A copy of the programs may be obtained from the author on request. 25 A ' s t r o n g ' comparative s t a t i c i s the requirement t h a t , SSG^ hCj) < SSCc^) < S S ^ - A c ^ . . . . . (1) I n t h i s case K-2 comparative s t a t i c p r e d i c t i o n s a r e generated. A s i n g l e i n c o n s i s t e n t r e s u l t g e n e r a l l y reduces the number of c o r r e c t comparative s t a t i c p r e d i c t i o n s by 2. For example, i f the consecutive values of the summary s t a t i s t i c s are ( 7 , 6 , 4 , 5 , 3 , 2 , 1 ) , two of the r e s u l t s a r e i n c o r r e c t . A s t r o n g e r comparative s t a t i c t e s t i s the ' g l o b a l ' t e s t , namely that S S O ^ ) f .SSCcj) f o r a l l ^ < and S S ( c 1 ) < S S ^ ) for a l l ^ < . . . . (2) Again k-2 comparative s t a t i c p r e d i c t i o n s are generated from k d i f f e r e n t values of the exogeneous constant and a s i n g l e i n c o r r e c t r e s u l t may s u b s t a n t i a l l y reduce the frequency w i t h which the comparative s t a t i c p r e d i c t i o n s a r e u p h e l d . 284 FOOTNOTES CHAPTER 4 """This r e s u l t i s p l a u s i b l e because there i s f r i c t i o n a l unemployment i n s t o c h a s t i c e q u i l i b r i u m , and so some firms face l e v e l s of employment lower than the l e v e l of employment a s s o c i a t e d w i t h e q u i l i b r i u m under p e r f e c t c o m p e t i t i o n . By o f f e r i n g a wage above the p e r f e c t l y competit ive e q u i l i b r i u m wage, such firms face a h i g h e r p r o b a b i l i t y of r a i s i n g t h e i r l e v e l of employment and thus i n c r e a s e mean p r o f i t s . 2 The average marginal product i s def ined as the marginal product corresponding to the average l e v e l of employment i n each f i r m . The average product i s MRP(n), where n = L - U . N 3 Using the n o t a t i o n adopted i n the i l l u s t r a t i v e model, Chapter 1 I I , S(e,w) denotes the expected supply of workers prepared to accept an o f f e r at wage r a t e , w. e denotes the l e v e l of employment i n h e r i t e d from l a s t p e r i o d . Assuming turnover and acceptance functions are complementary, then S(e,w) = a(w)S + ( l - t ( w ) ) e = a(w)(S+e) = e . . . . (1) i s the mean flow supply of searchers and a l l searchers r e c e i v e Then a(w) > 0. . . . . (2a) a 1 (w)(S+e) > 0. . . . . (2b) S 9 1 = a ' (w) > 0. . . . . (2c) a"(w)(S+e) < 0, . . . . (2d) where S o f f e r s . dS(e,w) _ de " b l dS(e,w) _ dw " b 2 d 2 S(g,w) = _ dedw 12 d 2 S(e,w) T r i v i a l l y , S S 1 2 = S ^ 285 I f MC(e) denotes m a r g i n a l c o s t , then M R V N _ d(wS(e,w)) _ S , . MC(e; = ^~r~1 = w + — • . . . . (3) Denoting the m a r g i n a l revenue product by MRP(e), the f i r m maximises p r o f i t s at the employment l e v e l at which m a r g i n a l revenue product equals m a r g i n a l c o s t . Then, MRP (e) = w + - r ^ . . . . . (4) 2 To o b t a i n the impact on w of a change i n the l e v e l of c u r r e n t employ-ment, e, d i f f e r e n t i a t e w i t h respect to e. dMRP(e) _ , dMRP(e) _ dw _ ... S S 2 2 . dw . S l S S 1 2 de Sl + ~di S 2 d f " ( 2 "72" ) d f + S 7 " 7 T " * * ( 5 ) b 2 . S 2 D i m i n i s h i n g r e t u r n s to l a b o u r are assumed, so dMRP(e) < 0. Then, s i n c e = S S 1 2 , d e , dMRP(e) S = de b l < 0. . . . . (6) •• CO i . & b 2 2 dMRP(e) _ ! 1 ~ 0 2 " de b 2 i J Then, the o p t i m a l wage o f f e r i s a d e c l i n i n g f u n c t i o n o f the f i r m ' s c u r r e n t l e v e l o f employment. 4 R e d e r [1969], p . 7 . "*The e q u a l i t y o f aggregate h i r e s and q u i t s i s c o n s i s t e n t w i t h a changing d i s t r i b u t i o n of employment over f i r m s . The s u f f i c i e n t c o n d i t i o n f o r s t o c h a s t i c e q u i l i b r i u m i s that the d i s t r i b u t i o n o f employ-ment over f irms i s c o n s t a n t . ^These r e s u l t s suggest that a more s o p h i s t i c a t e d s e a r c h r u l e f o r unemployed i n d i v i d u a l s may be a p p r o p r i a t e . The model i s t r u l y s i m u l -taneous, however, so f i r m s ' behaviour i n response to t h i s new s e a r c h r u l e would have to be s p e c i f i e d t o evaluate the new r u l e . 286 'In some cases the steady state d i s t r i b u t i o n w i l l be degenerate due to approximating the p r o b a b i l i t i e s to three s i g n i f i c a n t figures. 8 A s h i f t to the right of the wage offer d i s t r i b u t i o n i s defined as a non-negative change i n wage offers over a l l states of employment. 9 This contrasts the results obtained by Bergman i n her simulation model. She finds that an increase i n the turnover rate, given a fixed number of jobs, leads to a f a l l i n the duration of unemployment, since the burden of unemployment i s spread over more people. A separation immediately opens up a position for an unemployed searcher. Unemploy-ment increases s i g n i f i c a n t l y . In her model there i s no e x p l i c i t firm behaviour. The pr o b a b i l i t y of accepting a wage offer i s solely depend-ent on the individual's duration of unemployment and does not change when the separation rate changes. In the absence of increased vacancy creation or wage offers by firms, an increase i n the separation rate must necessarily increase the l e v e l of unemployment. ^See Chapter 3 VI for a discussion of the comparative s t a t i c test used. "'"''"The question mark refers to the predicted change i n the mean duration of search p r i o r to an o f f e r . This change appears dependent on the r e l a t i v e changes i n the aggregate levels of unemployment and vacancy creation and not d i r e c t l y on c^. 12 The right hand column indicates the frequency with which the summary s t a t i s t i c remains constant or changes i n the direction predicted by the adjacent column. The negative numbers i n brackets after these frequencies indicate the number of times the summary s t a t i s t i c remains constant. A l l summary s t a t i s t i c s are defined to three s i g n i f i c a n t figures and thus some of the results arise from th i s approximation. 287 13 Using the notation adopted before (Footnote 3) firms w i l l choose to employ individuals up to the point at which MRP (e) = S + (1) Consider the introduction of a s h i f t parameter c i n t o the acceptance function. Then replace the acceptance function a(w) by a(w,c) where da - j - = a < 0. dc c Then the supply of labour function i s now written S(e,w,c) = a(w,c) [S+e] and so d S ( e > > c ) = S = (S + 5)5 < 0 dc c c d ^ ( l > w > c ) - s , = (s + 5)5 . dcdw 2c cw . . . . (2) . . . . (3) . . . . (4a) . . . . (4b) D i f f e r e n t i a t i n g the p r o f i t maximisation condition with respect to c y i e l d s dMRP(e) dw dMRP(e) _ _ dw f c S , dw • . de b2 dc de c Zdc S 9 " _ 2^b22dc b 2 c ; * L b2 . • . . (5) C o l l e c t i n g terms, r dMRP(e) de SS 2c dMRP(e) de (6) A s u f f i c i e n t condition for -r~ > 0 i s > 0. From (4b) t h i s requires that a ^ > 0. This,in turn, requires that the marginal acceptance rate i s an increasing function of c i n the range of w considered. 14 This r e s u l t can be derived from a simple model of the labour market. Assume that the labour market i s i n equilibrium and there i s no wage dispersion. Then the labour market behaviour of the homogeneous group con s t i t u t i n g the labour force may be represented as a Markov process E U E 1-d d U a 1-5 288 where E , U denote the s t a t e s of employment and unemployment r e s p e c t i v e l y . The p r o b a b i l i t y of q u i t t i n g a p o s i t i o n i s d and the p r o b a b i l i t y of sampling a f i r m , r e c e i v i n g an o f f e r and a c c e p t i n g i t i s a . S t o c h a s t i c e q u i l i b r i u m r e q u i r e s that E = (1 - d)E i + aU . . . . (1) where E , U denote the e q u i l i b r i u m l e v e l s of employment and unemployment r e s p e c t i v e l y . Then, U = - ^ . . . . (2) and _ _ _ _ J _ • •. • • (3) u U+E a+3 where u i s the f r a c t i o n of the l a b o u r f o r c e unemployed. T r i v i a l l y , > 0. dd "'""'it should be p o i n t e d out t h a t t h i s method of e x p l a i n i n g comparative s t a t i c r e s u l t s i s a t r i f l e m i s l e a d i n g because the argument i s couched i n dynamic terms. The d i r e c t i o n of change of the summary s t a t i s t i c s i s i n f e r r e d from a s h o r t run response o f market p a r t i c i p a n t s to the change of an exogeneous parameter r a t h e r than a c o n v i n c i n g d i s c u s s i o n of the nature of the new s t o c h a s t i c e q u i l i b r i u m . Due to the endogeneity of p a r t i c i p a n t s ' behaviour, i t i s remotely p o s s i b l e that the new s t o c h a s t i c e q u i l i b r i u m may be c h a r a c t e r i s e d by values of the summary s t a t i s t i c s which c o n t r a d i c t the d i r e c t i o n of change p r e d i c t e d by the s h o r t run dynamic response. Almost, a l l the r e s u l t s , however, can be j u s t i f i e d on the b a s i s of t h i s short run response. 16 T h i s r e s u l t may be demonstrated t h u s , dVW* _ „ n . . dEW* , ^ X - T P , dEW* 1 —r=- -2 E (w - EW*) § — - = — h 2(w - E W * H [ 1 7=— J dw „ n Yw dw Y dw n=0 n = ~ 2 d - W * ( E w <j) + w* - EW*( E <j> + c}>)) + 2(w - EW*)* dw _ n w T v _Tw Y / / v / r n=0 n n=0 n = 2(w-EW*) <J> < 0, 289 ^The solutions compared for two period changes i n are those solutions with H.. equal to 5 and 7, 7 and 9, 8 and 10, 9 and 11, and 10 and 12. 18 I f , for example, F > 1 then the new decision period exceeds the standard period. To calculate adjusted MDI, the proportion, JL, of expected income enjoyed over the new decision period i s F discounted by /TTjy.» t n e standard rate of discount, because i t i s F - l enjoyed i n the f i r s t standard period, and the remaining proportion, —p— i s enjoyed during the second standard period and i s discounted by Q^})2» Subsequent new periods have to be allocated between standard periods. 19 An asterisk indicates an adjusted measure-20_ _ Z_ - Z may be written S l / 1 1 \ H l 1 Z F - z - A L T ^ i - — -1 " E- — - < 0 i=l Ul+D) 1 (l+D) 1/ i - H j+l (1+D)1 (1) since D > D and < B^. 21 From (17) i n the main text, and using Footnote (20) F(Z_.EW* + w) F(Z.EW* + w) U - FW = -F I 1 + Z_ 1 + Z r F(EW*((1+Z)Z_ - (l+Zpz'j + w(Z-Z-)) (1+Z_)(1+Z) EW*(Z_-Z) + w(Z-Z_)^ (1+Z_)(1+Z) F(EW*-w)(Z -Z) < 0 . . . . (1) (1+Z )(1+Z) since EW* > w and Z_ < Z. r 290 22 I f the i n c r e a s e i n from 0.50 to 0.85 i s regarded as an e x t r a comparative s t a t i c p r e d i c t i o n , then the o v e r a l l r e s u l t s do n o t become any more s y s t e m a t i c . MDI, the mean wage o f f e r to a s e a r c h e r , the mean wage enjoyed by an employee and l a b o u r ' s share of t o t a l output a l l r i s e along w i t h the v a r i a n c e of the o f f e r d i s t r i b u t i o n . Measures of vacancy c r e a t i o n , unemployment and d u r a t i o n of unemployment a l l i n c r e a s e , w h i l e discounted p r o f i t s and the c o r r e l a t i o n of wage o f f e r s a l l f a l l . 23 A complete l i s t i n g of the r e s u l t s a s s o c i a t e d w i t h Model I can be obtained from the author on request . 24 Regarding the wage o f f e r d i s t r i b u t i o n c h a r a c t e r i s i n g s t o c h a s t i c e q u i l i b r i u m as a random sample of o f f e r s from N f i r m s , the hypothesis that the sample came from a p o p u l a t i o n w i t h a gamma d i s t r i b u t i o n was t e s t e d u s i n g the Kolmogorov-Smirnov T e s t . Over a l l s o l u t i o n s to Models I and I I the hypothesis was r e j e c t e d w i t h 95% confidence. The wage o f f e r d i s t r i b u t i o n s corresponding to Model I I I were t r e a t e d as independent and t e s t e d s e p a r a t e l y . A g a i n , the hypothesis that the wage o f f e r d i s t r i b u t i o n came from a p o p u l a t i o n w i t h the gamma d i s t r i b u t i o n was r e j e c t e d w i t h 95% c o n f i d e n c e . D e t a i l s of the s t a t i s t i c a l t e s t used can be obtained from the a u t h o r , on request . 25 E v i d e n t l y , given the n o n - l i n e a r i t y of the p r o d u c t i o n f u n c t i o n and b i n o m i a l p r o b a b i l i t i e s of s e a r c h , any homogeneous measure of aggregate vacancy c r e a t i o n based on a mean wage o f f e r i s meaningless, and i n a p p r o p r i a t e . 26 Reder [1969] p . 7 . Reder's work i s d i s c u s s e d i n Chapter 1 H I D . 27 In Chapter 6 the concepts of vacancy c r e a t i o n and excess demand are f u r t h e r examined. The c o n s t r u c t i o n of an index of market p r e s s u r e i s attempted. 28 The d e f i n i t i o n of a c t i v e recruitment i s examined by A r c h i b a l d [1954], Burgess [1969] and Devine and Marcus [1967], Chapter 1 H I E . 29 M a k i ' s study [1971] i n d i c a t e s that firms r e q u i r i n g white c o l l a r or p r o f e s s i o n a l workers are l e s s l i k e l y to seek Manpower's a s s i s t a n c e than firms employing blue c o l l a r workers. The reason i s that Manpower i s more e f f i c i e n t i n the d i s s e m i n a t i o n of extensive i n f o r m a t i o n , namely data or vacancies and wage rates over firms and workers seeking j o b s . P r o f e s s i o n a l jobs r e q u i r e s c r e e n i n g and thus i n t e n s i v e i n f o r m a t i o n i s r e q u i r e d , t h a t i s , d e t a i l e d s u b j e c t i v e opinions about both job and p o t e n t i a l employee. P r i v a t e agencies and i n f o r m a l contacts are more e f f i c i e n t i n the generation of such i n f o r m a t i o n . My three Models a r e a p p l i c a b l e t o a b l u e c o l l a r labour market i n which i n d i v i d u a l s are homogeneous i n the r e q u i r e d s k i l l s and the a d m i n i s t r a t i v e costs of vacancy c r e a t i o n to the f i r m are assumed z e r o . 291 ""imperfect i n f o r m a t i o n about the non-monetary b e n e f i t s may lead an employee to subsequently q u i t the j o b , d e s p i t e a r e l a t i v e l y h i g h money wage. 31 A loose comparison of vacancy and unemployment may l e a d researchers to b e r a t e the i n e f f i c i e n c y of Manpower o f f i c e s . Weinberg [1966] argues that job vacancy data, generated to shed l i g h t on most aspects of the labour market, i s u s e l e s s . For placement, he p o i n t s o u t , extremely f i n e and accurate d e t a i l of job requirements, wages, whether the job i s seasonal or temporary, p a r t - t i m e or f u l l - t i m e , by l o c a l area i s needed. T h i s combination of i n t e n s i v e or extensive i n f o r m a t i o n i s generated by l o c a l Manpower o f f i c e s . T r a i n i n g requirements are not measured by v a c a n c i e s , p a r t i c u l a r l y when there e x i s t s an i n t e r n a l labour market and the formal requirements of a vacancy do not r e f l e c t the s k i l l s r e q u i r e d . He suggests that employment trends or employers' p r o j e c t i o n s of future requirements would be a more a p p r o p r i a t e i n d i c a t o r . For economic f o r e c a s t i n g , the primary need i s f o r n a t i o n a l data which could be p r o v i d e d by Manpower, i f vacancy n o t i f i c a t i o n was compulsory. An enormous sample would be r e q u i r e d to r e v e a l demand and supply r e l a t i o n s h i p s by area and o c c u p a t i o n . 292 FOOTNOTES CHAPTER 5 "''In Model I I I , f irms d i s c r i m i n a t e i n wage o f f e r and h i r i n g and so each type of i n d i v i d u a l does not face the same labour market environment. T h i s d i s c u s s i o n provides i n s i g h t s , however, i n t o the r e l a t i v e wage o f f e r s enjoyed by each type of i n d i v i d u a l . 2 I n Chapter 2 I I I G the i n t e r t e m p o r a l p r o f i t maximising l e v e l of employment, a s s o c i a t e d w i t h employment s t a t e m and wage o f f e r , w*, i s defined as L , where m' F ( L ,w*) = max (II(k,w*) + ^ S T O m' m , ' m 1+R k and n(k,w*) are one p e r i o d p r o f i t s a s s o c i a t e d w i t h employment s t a t e , k , and wage o f f e r , w . F*(k) denotes mean p r o f i t discounted over an i n f i n i t e h o r i z o n a s s o c i a t e d w i t h adopting a p o l i c y , 6*, f o r ever and commencing i n s t a t e k. R i s the r a t e of d i s c o u n t . 3 1 2 I f the p e r c e p t i o n parameters c^ and c^ d i f f e r s u b s t a n t i a l l y , i t i s p o s s i b l e that a f i r m chooses to employ three type two i n d i v i d u a l s i n preference to two type one i n d i v i d u a l s at a wage i n the s p e c i f i e d range and so the s i n g l e p e r i o d p r o f i t maximising l e v e l of employment may n o t c o i n c i d e w i t h the i n t e r t e m p o r a l p r o f i t maximising l e v e l of employme