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Schooling, experience, hours of work, and earnings in Canada 1979

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S C H O O L I N G , E X P E R I E N C E , HOURS O F WORK, A N D EARNINGS IN C A N A D A by R I C H A R D D O N A L D S C O T T B . A . , Simon Fraser Un ive rs i t y , 1971 M . A . , Un ivers i ty of Br i t i sh Columbia, 1973 A D I S S E R T A T I O N S U B M I T T E D IN P A R T I A L F U L F I L L M E N T OF T H E R E Q U I R E M E N T S FOR T H E D E G R E E OF D O C T O R O F PHILOSOPHY T H E F A C U L T Y O F G R A D U A T E S T U D I E S D E P A R T M E N T O F ECONOMICS We accept this dissertat ion as conforming to the requ i red standard T H E U N I V E R S I T Y O F BRITISH C O L U M B I A September, 1979 (5) R ichard Donald Scott , 1979 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it 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. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 ii A B S T R A C T T h i s s tudy investigates a broad range of factors which might be thought to influence the employment earn ings of Canadian males. Micro- data drawn from the 1971 census are ana lysed , using as a frame of reference the human-capital model d e r i v e d , and implemented for the United States, by Jacob Mincer . Opening discussion furn ishes a detailed cr i t ique of the model itself, and of the auxi l iary hypotheses required to make it perform empir- ical ly . Part icular emphasis is laid upon the implicit assumption of perpetual long-run equi l ibr ium and upon the neglect of var iables ar is ing on the demand side of the labour market . Genera l ly , it is a rgued that although the human-capital paradigm may serve as a framework for empirical descr ipt ion , it is inadequate as a scientif ic theory because it fails to generate a wide array of hypotheses which are c lear ly susceptible to fa ls i f icat ion. Earn ings funct ions are estimated by ord inary least squares for a sample of almost 23,000 out-of-school males who worked, mainly in the pr ivate sector, at some time d u r i n g 1970. Results y ie lded for Canada by the human-capital specif ication are compared with those reported by Mincer . T h e regress ions are then expanded to include var iables such as i n d u s t r y , reg ion, and occupat ion, together with other personal a t t r ibutes . These are found to r ival the importance of the orthodox human-capital iii var iab les . Con t ra ry to United States resu l ts , the elasticity of earn ings with respect to weeks (or hours) worked is less than un i t y . In l ight of recent analyses which make human-capital investment and labour supply objects of simultaneous decision within a l i fe-cycle context , fu r ther investigation is ca r r i ed out us ing a simplif ied, two- equat ion, l inear model in which earn ings and hours are both endogenous. Estimates performed by the method of three-stage least squares indicate an elasticity of earn ings with respect to hours considerably in excess of un i t y . However, within part icular regional and industrial categories, wages and hours tend to be o f fset t ing . Schooling coeff ic ients, or "rates of r e t u r n , " fall in the 5.25-6.50% range . Terence J . Wales Research Superv i sor iv T A B L E O F C O N T E N T S Page A B S T R A C T ii LIST OF T A B L E S vii LIST OF FIGURES ix A C K N O W L E D G E M E N T x I N T R O D U C T I O N 1 Chapter I. MODELS OF I N V E S T M E N T IN EARNING C A P A C I T Y . . . . 8 Formal Schooling (9) On- the-Job T r a i n i n g (26) General Theor ies of Income Maximization (38) APPENDIX I: T H E E F F E C T O F M A R K E T BIAS ON O P T I M A L I N V E S T M E N T PROFILE 46 N O T E S 51 II. PROBLEMS OF IMPLEMENTATION 60 T h e Schooling Model (61) T h e Postschool Investment Model (90) The General Model (103) APPENDIX 11A : MINCER'S R E G R E S S I O N R E S U L T S . . . 106 A P P E N D I X I IB : BIASES IN T H E EARNINGS F U N C T I O N 1 DUE T O E R R O R S IN T H E M E A S U R E - MENT OF E X P E R I E N C E 108 N O T E S 113 V Chapter P a g e III. T H E EARNINGS F U N C T I O N : S I N G L E - E Q U A T I O N E S T I M A T E S FOR C A N A D A 122 T h e Data, the Sample, and the Var iables (123) Human-Capital Earn ings Funct ions (151) Expanded Earn ings-Funct ion Estimates (165) APPENDIX MIA: T H E WORKING S A M P L E : D I S T R I B U T I O N S OF S E L E C T E D C H A R A C T E R I S T I C S . . . 193 APPENDIX N I B : M I S C E L L A N E O U S REGRESSIONS . . . . 202 N O T E S 203 IV. T H E S I M U L T A N E O U S D E T E R M I N A T I O N OF H U M A N - C A P I T A L I N V E S T M E N T A N D L A B O U R S U P P L Y 209 Theoretical Ana lys i s (212) A n Empirical Model (223) APPENDIX IV: O R D I N A R Y L E A S T - S Q U A R E S E S T I M A T E S O F WORKING HOURS 235 N O T E S . 236 V . EARNINGS A N D H O U R S : S I M U L T A N E O U S - E Q U A T I O N E S T I M A T E S FOR C A N A D A . 240 Estimation Procedure (240) Results (247) APPENDIX V : E S T I M A T E S O B T A I N E D BY I T E R A T I V E T H R E E - S T A G E L E A S T S Q U A R E S 261 N O T E S 262 v i C h a p t e r p a g e V I . S U M M A R Y A N D C O N C L U S I O N S 266 C h a p t e r I (266) C h a p t e r II (270) C h a p t e r III (274) C h a p t e r IV ((279) C h a p t e r V (281) F i n a l R e m a r k s (283) R E F E R E N C E S 286 vii LIST OF T A B L E S Table Page 1. MINCER'S REGRESSION R E S U L T S 106 2. SAMPLING C R I T E R I A 128 3. SUMMARY O F T H E V A R I A B L E S 132 4. E S T I M A T E S FOR T H E O V E R T A K I N G S E T 153 5. F U L L - S A M P L E E S T I M A T E S USING E X P O N E N T I A L E X P E R I E N C E PROFILES 157 6. F U L L - S A M P L E E S T I M A T E S USING Q U A D R A T I C E X P E R I E N C E PROFILES 159 7. V A L U E S OF r X A N D k' C O N S I S T E N T WITH SPECIFIED V A L U E S OF T 1 A N D d ( W E E K S - V A R I A B L E C A S E ) . . . . 164 8. REGRESSION E S T I M A T E S O F T H E E X P A N D E D EARNINGS F U N C T I O N , I 168 9. REGRESSION E S T I M A T E S OF T H E E X P A N D E D EARNINGS F U N C T I O N , II . . . 173 10. REGRESSION E S T I M A T E S OF T H E E X P A N D E D EARNINGS F U N C T I O N , III 176 11. T H E E X P A N D E D EARNINGS F U N C T I O N WITH A V A R I A B L E R A T E O F R E T U R N ( E Q U A T I O N (CP6)) 177 12. T H E E F F E C T S OF O C C U P A T I O N 180 13. T H E E X P L A N A T O R Y POWER AND S I G N I F I C A N C E OF V A R I A B L E S IN T H E E X P A N D E D EARNINGS F U N C T I O N S 183 14. R A T E S O F R E T U R N T O S C H O O L I N G IMPLIED BY V A R I O U S S P E C I F I C A T I O N S OF T H E E A R N I N G S F U N C T I O N 185 15. T H E I N T E R A C T I O N OF S C H O O L I N G A N D E X P E R I E N C E WITH I N D U S T R Y A N D P L A C E OF R E S I D E N C E 190 viii T a b l e P a g e 16. I N D I V I D U A L I N C O M E S B Y S I Z E C A T E G O R Y 193 17. F A M I L Y I N C O M E S O F I N D I V I D U A L S B Y S I Z E C A T E G O R Y 1 9 4 18. S C H O O L I N G B Y A G E G R O U P 1 9 5 19. S C H O O L I N G B Y R E G I O N 196 20 . M E A N E A R N I N G S B Y R E G I O N A N D L E V E L O F S C H O O L I N G 197 21 . S C H O O L I N G B Y I N D U S T R Y 198 22 . M E A N E A R N I N G S B Y I N D U S T R Y A N D L E V E L O F S C H O O L I N G 199 23 . O C C U P A T I O N 200 24. E T H N I C A N D R E L I G I O U S G R O U P 201 25 . P E R I O D O F I M M I G R A T I O N T O C A N A D A 201 26 . S I M U L T A N E O U S E S T I M A T E S : E A R N I N G S 248 27 . S I M U L T A N E O U S E S T I M A T E S : H O U R S 250 ix LIST OF FIGURES Figure Page 1. PHASE DIAGRAM IN (k', h ) -SPACE 221 2. LINEARIZATION OF THE BUDGET CONSTRAINT 231 X ACKNOWLEDGEMENT I should like to thank all the members of my committee, but especially its chairman, Terence Wales, who dispensed much patience and congeniality along with helpful substantive comment. I am equally indebted to Jonathan Kesselman, who supervised the present research at its early and intermediate stages. For excellent program- ming assistance I am grateful to the staff of the University of British Columbia Statistics Centre—in particular, Frank Flynn, Lewis James, and prior to his departure, Keith Wales. My task in preparing this final draft was considerably eased, through the competence and experience of the typist Maryse Ellis. Though the preceding individ- uals contributed a number of improvements, they bear no responsibility for any errors or omissions which may remain. This study was carried out in part while I was in receipt of a Canada Council Doctoral Fellowship. R. D. S. '5 INTRODUCTION Owing to a scarcity of fertile data, Canadian research in the area of human capital has been limited, both in volume and scope.^ As a consequence, we have had to glean, mainly from the American literature, most of what we presently know and teach, about the rates of return to investment in education, and about the complicated web of interaction linking such key variables as schooling, on-the-job training, hours of work and the level of individual earnings. The investigation reported here is an attempt to narrow the current re- search deficit. Results of this work supply a new description of the forces determining employment incomes in Canada, and at the same time, illuminate some important differences between Canadian and American experience.^ The present study selects as a point of departure the human- capital model of income determination, developed over the past two decades by a group of well-known economists, but consistently applied in its most uncompromising form by one member of the school, 3 namely, Jacob Mincer. With the publication of Mincer's recent book. Schooling, Experience, and Earnings, human-capital orthodoxy appears to have reached a major empirical plateau. When fully de- ployed. Mincer's version of the human-capital model succeeds in accounting for just over half the variance of earnings in a large body of microdata drawn from the United States Census. In the 1 2 process, it yields new estimates of the private return to investment in formal education and on-the-job training. Until recently, empirical work of the kind reported by Mincer has been very difficult to pursue in Canada: except in a few special instances,5 researchers have been without access to microdata. The decision by Statistics Canada to issue a large public file of individual observations drawn from the 1971 Census was therefore a welcome advance. Microdata extracted from this new and comparatively rich source, the so-called Public Use Sample, provides an empirical footing for the work reported here.** The initial chapters of this dissertation concern the appli- cation of Mincer's theory and his empirical methods to the Canadian census data. Chapter I introduces the main theoretical arguments of the human-capital school and offers a critical appraisal. It is argued that the human-capital analysis fails to generate an adequate set of testable hypotheses, though it may serve as a convenient framework for empirical description. Chapter II considers various problems of implementation, since empirical measurements, even if only descriptive, may harbour misleading biases. Chapter III exhibits two sets of regression equations. The first set replicates, as nearly as convenience and the data will allow, Mincer's human capital "earnings functions." On the one hand, this exercise furnishes some interesting comparative results for the Canadian economy, and on the other, serves the worthwhile scientific 3 purpose of confronting the human-capital model with new data. 'The fact that Canadian and American results differ at some key points without invalidating the model supports the present contention that the standard theory is virtually immune from scientific falsification. The second set of regressions in Chapter III explores the consequences of adding to the empirical model variables typically ignored by human-capital theorists. Among the variables inserted are dummies representing region, industry, occupation, urban residence, official language, ethnic and religious group, period of immigration, and family status. The resulting estimates, it is argued, provide a better basis for assessing the contribution of the "orthodox" vari- ables than do Mincer's highly parsimonious specifications. Although the task of replicating Mincer's work, and of explor- ing some alternative hypotheses with Canadian data, is in itself a substantial research undertaking, one seemingly important weakness in the application of the model invites a further stage of inquiry. The difficulty in question arises from Mincer's casual introduction of weeks worked as an exogenous variable in the earnings function. If weeks worked depend on the wage rate, and hence, upon earnings, by way of the individual's labour-supply response, including weeks worked on the right-hand side of a regression in which earnings are the dependent variable will necessarily bias the estimation. Moreover, the coefficients which Mincer and others interpret as rates of return will in fact be complex, displaying the tangled structural effects of both human-capital investment and labour supply (not to mention labour demand). These problems occupy Chapter IV. 4 There, it is observed that a number of economists have lately succeeded in devising theoretical analyses which take into account the simultaneous determination of schooling, on-the-job training, hours of work—and sometimes, consumption—over the life cycle of the utility- maximizing individual or household. Models of this sort yield their results in the form of explicit or implicit solutions which describe optimal lifetime trajectories for the variables under the control of the maximiz- ing agent. As one might expect, these solutions, when they can be derived at all, invariably turn out to be complicated nonlinear functions, involving the rate of time preference, the parameters of the static utility function, and other constants having to do with the production and depreciation of human capital. The implied functional forms present numerous difficulties even under the most favourable circum- stances, but they are practically impossible to estimate with data sets as large as the one examined here. Fortunately, it is possible to implement the general notion of simultaneity using a straightforward procedure, which though some- what lacking in theoretical rigour, may nevertheless prove highly informative. Chapter IV elaborates a two-equation simultaneous system—one linear equation for earnings and one for hours—which appears to capture the essence of the problem. Results, generated by the method of three-stage least squares, are displayed in Chapter V. These may be compared directly with the estimates of Chapter III in order to assess the degree of bias inherent in the single- equation approach. The system estimates, taken on their own, allow ) one to evaluate the structural parameters which govern the income- hours-schooling interaction. Readers primarily interested in empirical results are thus referred to Chapters III and V, or to Chapter VI, where the conclus- ions reached in this dissertation are summarized. Those who wish to review the various theoretical models put forward by the human- capital school may begin with Chapter I. NOTES INTRODUCTION 'In the field of education and training the most important con- tributions have been: Cordon Bertram, The Contribution of Education to Economic Growth, Economic Council of Canada, Staff Study No. 12 (Ottawa: Queen's Printer, 1966); Bruce W. Wilkinson, "Present Values of Lifetime Earnings for Different Occupations," Journal of Political Economy, LXXIV (December, 1966), 556-572; Jenny R. Podoluk, Incomes of Canadians (Ottawa: Dominion Bureau of Statistics, 1968), Chapter 5; David A. Dodge, Returns to Investment in Training: The Case of Canadian Accountants, Engineers, and Scientists (Kingston, Ontario: Industrial Relations Centre, Queen's University, 1972); Canada, Statis- tics Canada, Economic Returns to Education in Canada (Ottawa: Information Canada, 1974). 2 That significant differences do exist was the finding of Jenny R. Podoluk, "Some Comparisons of the Canadian-U .S. Income Distri- butions," Review of Income and Wealth, XVI (September, I970), 279-302, and was earlier hinted in Canada, Economic Council of Canada, Second Annual Review (Ottawa: Queen's Printer, 1965), Chapter 5. 3 His landmark contributions are: "Investment in Human Capital and the Personal Distribution of Income," Journal of Political Economy, LXVI (August, 1958), 281-302; "On-the-Job Training: Costs Returns, and Some Implications," Journal of Political Economy, LXX (October, Supplement, 1962), 50-79; "The Distribution of Labor Incomes: A Survey," Journal of Economic Literature, VII (March, 1970), 1-28. See also "Education, Experience, and the Distribution of Earnings and Employment: An Overview," in Education, Income and Human Behavior, edited by F. Thomas Juster (New York: McGraw- Hill Book Co., 1975). 4 (New York: National Bureau of Economic Research, 1974). 5 Dodge, op. cit., relies on a large private survey directed at individuals in a narrow range of high-level occupations. The study issued by Statistics Canada (op. cit.) used microdata drawn from the Labour Force Survey. 6 7 Another study based on the Public Use Sample appeared as the present draft was undergoing final editing. See Peter Kuch and Walter Haessel, An Analysis of Earnings in Canada (Ottawa: Statistics Canada, 1979), Catalogue No. 99-758E. An unpublished paper by these authors is cited in the following text. CHAPTER I MODELS OF INVESTMENT IN EARNING CAPACITY Human-capital theorists have emphasized two principal means which individuals may invest in earning capacity. One is through formal schooling; the other is through training received on the job. In this chapter, we shall consider in turn models that have been designed to account for the income gains associated with each mode of investment. After reviewing these specific elaborations of human- capital theory, we shall examine the broader approach suggested by Ben-Porath. This well-known model admits formal schooling and on- the-job training as special cases within a general framework of income maximization. At various points in the discussion, we shall turn to existing empirical studies for help in assessing the validity of the human- capital assumptions. We shall not consider in any detail the large body of human-capital research which presupposes the truth of the basic doctrine and seeks only to measure particular parameters, such as the rate of return to education. A selective review of the measurement literature appears in Chapter II. 8 9 FORMAL SCHOOLING r The Model Though simple in appearance, the basic "schooling model" con- tains all the essentials of the human-capital approach.' Individuals who attend school are seen as investing foregone earnings in order to secure additional income during later life. In present-value terms, 2 those who undertake s years of schooling receive V(s) = W(s) T e r tdt = [W(s)/r][e r s - e r T ] = e- r s[W(s)/r][1-e- r ( T- s )] ,• .M> where T indexes the date of retirement, r stands for some appropriate discount rate, and W(s) signifies the annual wage, assumed constant throughout the individual's working life. Similarly, those who under- take (s-d) years of schooling receive V(s-d) = e- r ( s- d )[W(s-d)/r][1-e- r ( T- S + d )] It will be observed that these calculations abstract completely from changes in annual earnings caused by planned or unplanned varia- 3 tions in hours of work. If we now impose the following condition. 10 • . . .(2) 0, we obtain the fraction on the right-hand side being an adjustment for the finite- ness of the working life. If T is large in relation to s, or if T varies in order to make working lives equal whatever the length of H schooling, the preceding expression reduces to the simple form W(s) = W(0)ers , which may also be written conveniently as In W(s) = In W(0) + rs . . . . .(3) Since dW(s)/W(s) = r«ds, we arrive at the conclusion, standard in the human-capital literature, that equal proportionate differences in earnings accompany equal absolute differences in the length of schooling. An Appraisal To assess the usefulness of the preceding result for under- standing real-world economic behaviour, we must now look carefully at V(s-d) = V(s) , and transform the schooling variable so that s-d W(s) = W(0)ers • — 1 " e r T 1 - e- r ( T- s' ' 11 the logic and at the assumptions which underlie it. As a matter of present-value accounting. Equation (1) assumes either that students have no income while attending school or that their earnings just offset tuition and similar direct costs, which are otherwise completely ignored.5 Furthermore, it is assumed that students derive no consumption benefits from their education, either while attending school or during later years. Nonpecuniary aspects of the jobs associated with different levels of schooling are likewise neglected. The errors thus introduced into the cost-benefit arithmetic may be significant; but as this objection to the model is already well known, there is little need to pursue it here . More important to the present study is the interpretation of 6 Equation (2). Mincer invokes the condition without comment, though it is crucial to his analysis. One is left to wonder whether it is an identity or a behavioural postulate. If it is an identity, then r must be an ex post internal rate of return; for as the definition requires, r is the discount rate that equates total benefits, given by V(s), and total opportunity costs, given by V(s-d). If r is indeed an ex post rate of return, what economic information does it convey? Becker has argued^ that when r exceeds the return on com- parably risky investments in physical capital, there is evidence of underinvestment in education. Such reasoning is no doubt correct,/ but from a policy point of view it is regretably superficial. What we really need to know is why the underinvestment occurs. Writers of the human-capital school usually stress the likelihood that imperfec- 12 tions on the supply side of the market restrict the availability of private educational finance. Accordingly, they may favour giving stu- dents various subsidies and loans. It may well be, however, that students fail to invest because they perceive barriers to entry on the demand side. Under such circumstances, distributing subsidies will increase educational attainment and, very probably, cause r to fall; but if r falls, it will not be because inefficient shortages of educated manpower are relieved, but rather because graduates spend additional time queuing for preferred employment, or because they crowd into inferior jobs. Unless steps are taken to counteract the demand-side imperfections, further investment in education may involve considerable social waste. This example merely emphasizes the limitations of ex post measurements. If r is to be interpreted instead as an ex ante rate of return, then Equation (2) must be an equilibrium postulate. As such, it injects into the schooling model a set of implicit hypotheses concerning market behaviour. Although Mincer never really pauses to discuss market processes, it is not very difficult to imagine what a consistent rendering of his model might include. Elaborating slightly upon Equation (1), we obtain V*(s) = e~r's[W*(s)/r.] [1 - e" r' ( T _ s )] , which measures the ex ante lifetime earnings of individual i, whose personal discount rate is r., and whose wage-rate expectations are 13 * 8 summarized by the function W.(s). Let us assume that the individual * behaves so as to maximize V (s). If circumstances permit an interior * maximum, he will then seek to acquire that level of schooling s for * * which dV (s )/ds = 0. The result, omitting a small finiteness correc- tion, is simply dW*(s*) /ds * * W.(s ) Marginal expected returns equal marginal (here average) opportunity * cost. Solving this differential equation for s yields the desired level 9 of schooling. Notice, however, that the preceding condition is irrelevant unless the graph of the function [dW.*(s)/ds]/W*(s) = d • In W*(s)/ds intersects r. from above. In other words, the individual's expected rate of return must decline with s. 1 0 If not, or if no intersection occurs, the optimal level of schooling will be zero, as high as possible, or indeterminate, depending on the particular circumstance. Now, to reach the market level of aggregation, we may think of r. as being drawn from a frequency distribution with mean r and variance Var(r). Given information on this distribution, on the distribution of expected wages, and on the process linking expected 14 and observed wages, we can determine, at least in principle, the supply of enrollees as W(s) varies, and ultimately, the total stock of workers at each level of schooling.^ We thus have a set of long-run supply curves. Presumably, there exists a matching set of demand curves 12 based on the profit-maximizing behaviour of employers. In equilibrium, the curves achieve intersections which enforce an equalization of present values, as Equation (2) requires. The discount rate which makes these present values equal will be that of the marginal investor in formal schooling. The equilibrium structure of wages (earnings) will, finally, be implicit in Equation (3). By concentrating entirely upon equilibrium positions. Mincer, and Becker as well, avoid the complicated question of disequilibrium adjustment. This tactic achieves great elegance and simplicity, but it leaves in darkness the basic functioning of the labour economy. As Schultz says. What we want to know is the relative rates of return to investment opportunities and what determines the change in the pattern of these rates over time. To get on with this analytical task, we must build models that reveal the very inequalities that we now conceal and proceed to an explanation of why they occur and why they persist under particular dynamic conditions. 1 3 These "inequalities"—the imperfections and disequilibria which seem to pervade labour markets—have been the concern of many labour econom- ists, especially those writing before the rise of modern human-capital 14 theory; but in the schooling model such disturbances are deemed unimportant. 15 If the model is to provide anything more than ex post description (however useful that might be for some purposes), one must assume that dynamic forces succeed in equating present values, and that they do so, within tolerable limits of approximation, not just "in the long run," but at any moment one might happen to select for empirical study. With- out this auxiliary dynamic hypothesis, implementation of the static theory embodied in Equation (3) becomes impossible. Unfortunately, prima facie evidence against the equalization assumption is both strong and abundant. Early studies by Houthakker, Hansen, and Hanoch in the United States,'5 and by Wilkinson in Canada"* show wide variation in the present values of lifetime earnings across schooling groups. Subsequent research in North America and elsewhere has reinforced this finding.'^ One must therefore approach the equalization assumption with some skepticism. Meanwhile, it is interesting to note that Mincer's preoccupation with equilibrium loci has the effect of suppressing completely the demand side of the labour market. Near the end of Schooling, Experience, and Earnings he warns that " . . . the earnings function in this study is a 'reduced form' equation, in which both demand conditions and supply responses determine the levels of investment in human capital, rates 18 of return, and time worked." Yet, no exogenous demand variables actually appear in Equation (3). This supply-side approach to earnings determination contrasts sharply with earlier research. As Bluestone, Murphy, and Stevenson observe: 16 Labour market investigation in the 1950's was oriented toward the "demand" side, or industry side, of wage determin- ation. During this period, labour economists concentrated on researching interindustry and interregional wage differentials and developing models to measure the effects of unionization, profits, concentration, and capital intensity on industry rates The 1960's saw a major shift from industry studies to research on human capital. . . . Abstracting from the effect of industry and institutional structure, the human-capital- oriented research focused on the education, skills, training, health, mobility, and attitudes of the labour force . . . In a "vulgar" or extreme human capitalist approach, all industries are treated as though operating in the same labour market, labour mobility is assumed perfect within skill categories, and because of competition, all industries have the same set of economic and institutional conditions. In this model, all variance in wages, including "equalizing" differences, can be explained by the "supply" characteristics of individual workers. 19 In view of the strong assumptions needed to guarantee long-run equilibrium, and thereby purge the schooling model of demand-side influences, it would appear wise to consider the weaker, yet more easily defensible analytic notion of short-run or "temporary" equilibrium. In a temporary equilibrium, stocks of human capital—that is to say, the number of workers at each level of schooling—need not "fit" the wage structure implied by Equation (3), given local conditions of demand within regions or industries. Demand conditions then determine the actual wage structure, given the stocks of human capital, which though possibly evolving toward long-run equilibrium, are nevertheless fixed in the short run. The result will generally be some departure from long-run equilibrium, which can be explained only by permitting demand-side variables to surface in an expanded reduced-form earnings function. 17 An expanded model, admitting both demand and supply variables, will be derived and tested in Chapter III. This model may be viewed as an attempt, albeit a crude one, to synthesize the alternative approaches to wage determination discussed by Bluestone, Murphy, and Stevenson. Supporting Analysis and Extensions To provide a deeper rationale for the schooling model, Becker has suggested that we view its lone constant r as the outcome of equilibrium, not in the market for labour, but in a set of individual 20 "markets" for human capital. The student-investor, who is the decision-making agent in each market, faces an upward sloping supply of educational finance and a downward sloping demand for educational investment. The supply schedule portrays the marginal interest cost of each dollar committed to schooling, and the demand schedule, 21 the marginal expected yield. By equating these values, the individual maximizes net lifetime earnings. He thus determines the optimal amount to invest in schooling and the equilibrium return on his total investment, much as suggested in the preceding subsection. This equilibrium return might appear to explain the 11 r" of Mincer's analysis, except that in Becker's framework the rate in question is a marginal one, based on the dollar cost of schooling, whereas, in Mincer's own explicit formulation of the problem it is essentially an average, based on the time cost of schooling evaluated 18 at some constant opportunity wage W(0). Mincer's "macro" model, un- like Becker's microeconomic rationale, admits no interim rise in the opportunity wage as schooling progresses, nor does it take into account any possible rise in the interest charges that individuals may have to bear. It treats r as a constant rather than as an equilibrating variable. Any distinction between average and marginal rates of return is therefore unnecessary: the two are the same by assumption. However, as we shall observe in Chapter II, Mincer does not always impose this strong restriction in his empirical work. It is worth noting that Becker—and Mincer too, for that matter- develop their models without considering the rate of time preference. They focus upon the maximization of earnings, not utility. Thus Becker, most paradoxically, mimics the neoclassical theory of investment in physical capital by assuming, implicitly, that consumption and investment in human capital can be made analytically independent. The individual undertakes whatever investment is needed to maximize earn- ings, and then, treating maximized earnings as a constraint, spreads consumption optimally over his life cycle in accordance with the market 22 rate of interest and his rate of time preference. The trouble with this approach in Becker's case is that it requires the market for consumption loans to be isolated, somewhat implausibly, from the market for investment finance. Otherwise, the amount an individual borrows for the purpose of consumption spreading will affect the terms under which he may borrow for the purpose of investment. An individual who is an efficient maximizer (of utility) 19 will therefore plan his consumption and investment simultaneously. Perfect loan markets, with perfect arbitrage between them, would re- store independence; but Becker has assumed the contrary. As we shall see in Chapter IV, models based on utility maximization are capable of handling such an assumption in principle, although they 23 typically shy away from the very great complexities involved. The chief use of Becker's model, flawed or not, has been to analyze cross-sectional relationships between rates of return and the level of schooling. For Becker and fellow human capitalists, the demand curves of the model measure individual ability, and the supply curves, opportunity. If the variance of ability within the population exceeds the variance of opportunity, the resulting scatter of individual equilibria will tend to describe a positively sloping line; the more volatile demand curve will "identify" the supply schedule. We shall then observe a positive association between schooling and the rate of return. In the reverse case, we shall witness a negative associa- tion, and in the case of equal variances, no correlation whatever. The model is thus capable of accommodating any empirical outcome. In light of the remarks already directed toward Mincer's version of the schooling model, it should come as no surprise to find Becker interpreting the demand side of his own analysis solely as a means of portraying the personal characteristics of individuals. Though Becker deals only with "ability" (a composite of various personal attributes), the demand curves which he postulates must 20 surely depend not only upon this factor but also upon (the individual's perception of) general labour-market conditions. Nevertheless, individuals of equal ability always face identical demand curves. "In- equality of opportunity" cannot occur through unequal access to high- paying jobs in favoured regions or industries, but only through un- equal access to investment finance. In an interesting attempt to apply Becker's demand-and- 25 supply framework, Haessel and Kuch postulate an explicit reduced- form equation for r., namely, K r i = a o + k l } a k A \ k > • • • •(*) where the a's are reduced-form coefficients, and the A's stand for 26 personal attribute variables. Substituting (4) into (3) yields K In W.(s) = In W.(0) + (a 0 + \ a k A i k ) s j In W.(0) a 0s. + J ^ s . A ^ ) (5) Given the form of the K additional variables X.̂  = s.A.̂  appended to the basic equation, one might label (5) the "interactions model." On ad hoc grounds for the most part, Haessel and Kuch select seven characteristics—religion, ethnicity, occupation, class of worker (salaried or self-employed), period of immigration, marital status, and place of schooling—to define the A.^. In so doing, they explore a number of worthwhile hypotheses, but they do not exhaust the possibilities of the model, given the available data. In particular, the authors do not consider the effects that region and industry of employment might have on the rate of return, as measured in the short run or under conditions of sustained market imperfection. Hypotheses pertaining to these factors will be tested, within an inter- actions framework, in Chapter III. Although we have so far dealt with the schooling model, strictly speaking, as a theory of earnings determination, it has actually been applied in its purest and simplest form as a theory of 27 earnings distribution. Observe that if we take variances on both 28 sides of (3) and assume W(0) to be independent of r and s, the general result is Var (In W) = Var[ln W(0)] + Var(rs) = Var[!n W(0)] + r 2 • Var(s) +s 2 -Var(r) + 2rs Cov(r,s) + R(r,s), . . .(6) where R(r,s) is a function involving certain expected values and 29 Cov(r,s). However, if r and s are also independent of one another (6) reduces to Var(ln W) = Var[ln W( 0) ] + f 2 • Var( s) +s 2«Var(r) + Var(r) • Var(s) (7) In both cases, the left-hand side turns out to be an already familiar measure of earnings inequality; hence, the distributional implications of the model appear immediate and direct. One should of course remember that Var(ln W) is by no means the only plausible measure of inequality, and that its adoption for policy purposes must ultimately 30 rest upon normative considerations. Writers of the human-capital school—Becker, Chiswick, and Mincer—adhere consistently to the assumption that r and s behave as in- dependent random variables, and so are content to apply (7) in attempt- ing to analyse distributional questions. They obtain the unambiguous result that inequality depends in positive fashion upon the means and the variances of r and s. This prediction with respect to s is somewhat surprising, in view of the levelling effect popularly credited to education. One must bear in mind, however, that policies designed to raise s will seldom leave Var(s) unchanged; it is unlikely, in other words, that all groups will receive equal increments of schooling. The practical out- come will depend on who gets the additional education. Furthermore, it is difficult to think that r would remain constant in the face of an increase in s. Ceteris paribus arguments based on (7) may thus prove misleading. As we have seen, the independence assumption, which ul- timately supports the preceding results, implies in the context of Becker's analysis that the dispersion of "abilities" and the dispersion of "opportunities" throughout the population must be roughly equal. Mincer contends: "There are no a priori reasons for specifying which dispersion is greater, and the empirical evidence suggests there is little if any correlation between rates of return and quantities invested 31 across individuals." As a matter of fact, evidence for the United States of a significant relationship between r and s is rather widespread. 32 33 The work of Hansen and of Hanoch, and Mincer's own findings, taken at face value, reveal an apparent negative association, but Mincer dismisses these results as the effect of not holding hours of labour 34 constant. We shall examine this argument carefully in Chapter II and test it by alternative methods in Chapters III and V. For the time being, it is sufficient to note that what seems true of the United States may not be true of Canada. If years of schooling and the rate of return are, in fact, negatively correlated, then (6) rather than (7) is the appropriate formula. Since by hypothesis Cov(r,s) < 0, the relationship between Var(ln W) and s is no longer unambiguously positive: an increase in the general level of education need not generate an increase in inequality. Using Hanoch's rate-of-return estimates, Marin and Psacharopoulos produce simulations which do exhibit a decline in inequality as the 35 result of such an increase. The popular view of education thus receives some comfort. When we come to consider the entire distribution of earnings 36 rather than merely its variance, inspection of (3) is enough to show that if schooling is normally distributed, the distribituon of earnings will be lognormal, or more significantly, that the distribution of earn- ings will not be lognormal (as is sometimes supposed) unless schooling is normally distributed. In general, the distribution of earnings will be skewed to the right—a customary finding—as long as the distribu- tion of schooling is not radically skewed to the left. Oulton, in particular, finds this yield of theoretical predic- 37 tions unimpressive. The problem, he says, is that the human- capital approach to distribution theory is incomplete: "The distribution of income is made to depend on the distribution of education (or 38 training in general), but the latter is unexplained." Proceeding out of skepticism, Oulton looks for the end of the analytical chain in the area of marginal productivity theory. He postulates an aggregate CES production function which makes distinct inputs—that is to say, imperfect substitutes—of 39 workers who differ by level of education. Here, Q stands for real out- put, and L g for the number of workers with s years of schooling (s - 0, 1, n); the a g reflect such workers' "inherent productivity"; and below, o = 1/(1 + b) will be used to denote the constant elasticity of substitution. Physical capital is ignored. If workers are paid their marginal products, it is easy to show that W s = W 0 ( a s / a 0 ) ( L s / L 0 r 1 / O ' • • • -(9> Subst i tut ing (9) into (3) and solving reveals 25 L s = L c r V a o ) a e " r S O ' • • • -HO) Final ly , if we assume for expositional convenience that (a / a j takes s 0 Y S the form e ' , where y is possibly a function of s. Equation (10) becomes L s ~ L 0 e • • • • - ( I D T h i s express ion implies the form of the schooling d is t r ibut ion . If the latter is to display the humped character requ i red by (3) to explain the observed d istr ibut ion of earn ings , inspection of Equation (11) suggests that y must f i r s t exceed and then fall below r as s r i ses . In other words, the a g must conform to a part icular pa t te rn . Oulton concludes that . . . there are no a pr ior i reasons for expect ing this par t i c - ular pattern of ' inherent product iv i ty ' to be found in the real wor ld . If, therefore, the model is thought to be an adequate descr ipt ion of real i ty, it would be for essential ly accidental reasons. . . .1° Owing to the somewhat restr ict ive nature of the product ion specification advanced in (8), it is perhaps a little unwise to accept this statement without fu r ther ana lys is . One might at least cons ider the possibi l i ty that, in the long r u n , technology may be endogenous. If the a g eventual ly adjust to accommodate a schooling d istr ibut ion determined, say, by ability or socio-economic background, the result- ing pattern of coefficients will be far from "accidental." To confirm this speculation here, within a rigourous maximizing framework, would unfortunately require a major disgression. Therefore, let us simply accept Oulton's essential point: that in the short run, most certainly, and perhaps also in the long run, human-capital theory is suspect because it ignores the demand side of the earnings-distribution problem. ON-THE-JOB TRAINING Mincer's Theory The schooling model we have just examined actually arises as a special case within the more general framework offered by human- capital theorists to account for on-the-job training and other forms of postschool investment. Mincer's current approach to on-the-job train- ing is a straightforward elaboration of the model suggested originally 41 by Becker and Chiswick. This treatment rests on the distinction between an individual's actual earnings after p years of work experience, W.(p), and his earning capacity, E.(p). The latter equals Wj(p) plus C.(p), the income foregone in order to attain further skills or earning capacity. If we now think of each increment of foregone earnings as yielding some rate of return r , we may write (in discrete form) the P accounting identity 27 EP ' E0 + X r t C t ~= W p + C p (12) where the subscript relating to individuals has been dropped for con- venience. The next step is to make investment C a function of earn- P ing capacity; that is. C P  = KP EP • 0 S K P - • • One may interpret k p as the proportion of total "market time" devoted to skill acquisition during year p. The logic of (12) then implies E p = Ep-1 + V I S " ! = E p - 1 ( , + r p - 1 k p - l } By successive substitution, we obtain p-1 E = E l (1 + r t k t ) , P " t=0 which is approximately equivalent to In E = In E + J. r k , . . . .(14) t=0 as long as r tk t is small. Since E p = Wp/(1 - k^), we finally arrive at In W = In E + I r k + In (1 - k ) . . . . .(15) P U 1 * P During formal schooling, individuals may be thought to specialize in the production of human capital, and thus for t = 0, 1, s, k = 1. In this case, if the rate of return is the same in each period, (15) reduces to (3), the basic schooling model, with E Q redefined to mean earning capacity in the absence of both education and experience (that is Eo = w o K Allowing separate, though constant rates of return (denoted here by r and r , respectively) to each of these investment modes. Mincer partitions (15) in the necessary manner to obtain (as will be discussed later), this model implies that measured earnings Wp rise steadily until retirement. To explain the slight "hump" some- times detected in age-earnings profiles, one must introduce the concept In W (16) However, if k t declines monotonically over the individual's working life of depreciation. 42 If human capital depreciates at some constant rate d, then E = E p-1 + r - dE P p-1 which leads eventually to In W = In WQ+(r'e - d) s + r x £ (k' - d/r x) + In (1 - k' ) t=0 . . .(16') 29 e e One may think of r = ( r 1 - d ) H as the net rate of re turn to schooling and of k = (k 1 - d / r ) as the net propensi ty to invest in human P P cap i ta l . Primes denote the correspond ing gross va lues . Because elements of the summation on the r ight -hand side of (16') may turn out to be negat ive, it is now possible for W p to decline over some i n t e r v a l — presumably near the end of the individual 's working life, when he is unable to amortize large gross investments. Whatever the precise empirical resu l t . Equation (16') stands as the culmination of Mincer's theoretical ana lys is : it is the model for which he attempts to der ive an operational l ikeness. B y recogniz ing opportunit ies for postschool investment. Mincer and his fellow human capital ists prov ide a convenient rationale for the observed tendency of indiv idual earn ings to rise over most (if not all) of the life c y c l e . Moreover , as long as k p decreases with time, the expanded model implies that earn ings prof i les , even in the absence of variat ions in labour supp ly , will appear concave from below. The model thus "explains" one of the styl ized facts connected with l i fe- cyc le earn ings . The final important implication of Mincer's analysis has to do with his controvers ia l notion of "over tak ing . " Because postschool investors sacrif ice potential income, they at f i rs t earn less than h y p o - thetical non investors , whose earn ings profi les are assumed to remain hor izonta l . Later , as re turns accrue and as commitments of potential income decl ine, investors earn more. If we focus momentarily upon dollar costs and r e t u r n s , then , at the overtak ing year of exper ience p. 30 p-1 W~ = W + r X Y C 4 - C~ = W f17) p s t= 0 * P s ' ' ' * if and only if Pr 1 t=0 1 1 If annual dollar investments were constant at C during the first p years X after school leaving, we should obtain r pC, which means that p = 1/r . On the other hand, if dollar investments decline as we expect, it is easy to show that p < 1/rx. Hence, we can place an upper x x bound on p, provided we know r . Mincer assumes that r " . . . is not very different from the rate of return as usually calculated [for 43 education] . . . ," thus making p "a decade or less." Now, if rates of return and the detailed pattern of invest- ment, as opposed to the total planned accumulation, do not vary in- ordinately across individuals, overtaking will occur in practice within a relatively narrow band of years after school leaving. In other words, the earnings profiles of large and small postschool investors, and of noninvestors, if there are any, will be observed to intersect at roughly the same point. The experience cohort thus identified should exhibit less inequality than others in the working population, although strictly speaking, such an inference depends on the further assumption that there exists an appropriately small correlation between potential earnings at school leaving and the propensity to engage in postschool 31 investment. The cross-cohort patterns of inequality found by Mincer actually display the expected minima only in the case of high-school graduates, leading him to conclude in the contrary instances that the 45 correlation just named must not be sufficiently small. Thus, again, the human-capital approach proves capable of accommodating any con- ceivable result. An Appraisal One may surely be forgiven for remarking that just a single unambiguous prediction—that earnings profiles are concave—does not seem a very substantial dividend with which to repay the preceding analysis. Consistency with stylized fact is comforting but inconclusive, particularly in the face of competing explanations. One of these holds that concave earnings profiles are largely the result of biological factors connected with aging. If this hypothesis is true, age should figure at least as prominently as experience in the determination of cross-sectional earnings. The rare data sets which supply infor- mation on both of these independent variables unfortunately generate mixed qualitative results, although the weight of quantitative evidence seems to rule out extreme versions of the age hypothesis. Malkiel and Malkiel find that age is not significant when included in a 46 regression along with experience. However, studies of the engin- eering profession, by Cain, Freeman, and Hansen, and by Klevmarken and Quigley, uncover a small but not unimportant effect of age on 47 48 earnings. Lazear encounters a relatively strong age effect, and Psacharopoulos, observ ing a backward economy, reports that even 49 i l l iterate, unski l led workers exhibit concave earnings prof i les . One must conclude that investment behav iour , represented empirical ly by years of work exper ience, may not be the sole determinant of concav i ty . A stronger objection to the postschool investment model ar ises from the potential s ignif icance of costless learning by d o i n g . A s Blaug observes , ". . . any psychological theory of ' learning c u r v e s , ' in which appreciation over time is part ly but only part ly offset by depreciation and obsolescence, will likewise account for concave age- earnings prof i les . If learning by doing predominates over forms of t ra in ing which use real resources or sacrif ice output , the investment interpretat ion of earn ings prof i les appears to lose much of its appeal , since an act iv i ty which is costless and as inexorable as the passage of time cannot be the subject of an investment dec is ion. However, in Human Capital, Becker argued that labour mobility and competition for jobs would effect ively eradicate costless opportunit ies for l e a r n i n g . 5 1 If such opportunit ies ever arose, workers would crowd into them, forc ing wage rates to adjust unti l p roduct iv i ty -constant and product iv i ty -enhanc ing employment y ie lded the same present value of lifetime earn ings . In equi l ibr ium, the r is ing income prof i les again intersect the horizontal ones, and workers must make a choice . A s in the case of the schooling model, the human-capital interpretat ion of on-the-job tra in ing depends completely on the belief that competition succeeds in equating present va lues . Whether competitive forces in real-world labour markets actually possess such power is clearly open to debate. In general, the objections raised against the schooling model seem to apply with equal force to the expanded theory. If anything, market processes and the role of demand appear more deeply submerged in the latter than in the former. Equations (12)-(16') might very easily be regarded as identities with no direct behavioural significance. The model contains, in a sense, too many "degrees of freedom"; because potential income is unobservable, so is the crucial investment parameter kp. Though, as we shall see in the next section, the income maximiz- ation models put forward by some human-capital writers do make one or two predictions concerning the time path of k , the restrictions placed, P by inference, upon observable quantities like measured income are normally too weak to generate a very powerful or discriminating test of the theory. Supporting Arguments To the extent that human capitalists concern themselves at all with market functioning and firm behaviour, it is usually in order to explain the mechanism through which workers undertake investment "expenditures" while on the job. That full-time workers, like full- time students, forego income, and do so to a planned degree (given by kp), m a Y n o t be immediately obvious. In the case of foregone income invested in generally marketable skills, Becker's well-known r e s p o n s e " was to argue that because a trained worker could always obtain his marginal product in a competitive labour market, that worker would receive the ent ire re turn on any investment made b y him, a n d would, if necessary , be willing to pay its full cos t . A n employer who had to bear the cost initially but who could guarantee himself none of the re tu rn (because the worker might quit) would requ i re compensation for any tra in ing p r o v i d e d . Untra ined workers pay the needed compen- sation by accept ing a wage which falls short of their marginal p roduc t . 53 In the model ingeniously devised by Rosen, such workers choose the amount of their investment by selecting a job with the appropr iate character i s t i cs . Rosen states: T h e nature of the market is such that workers have their choice among a l l -or -noth ing bargains or 'package dea ls , ' in which they simultaneously sell the services of their ski l ls and 'purchase' a job of fer ing a f ixed opportuni ty to l earn . B y the same token firms purchase serv ices of ski l ls and at the same time 'sell' jobs o f fer ing learning possibi l i t ies. T h e labor market prov ides a broad range of choice in these matters. . . . . . . Pr ices of jobs could be either expl ic it or implicit, but the dist inction is of no analytical importance. . . . O rd ina r i l y , investment costs are simply subtracted from gross pay and no expl ic it pr ice need be q u o t e d . 5 4 In Rosen's model it makes no d i f ference whether f irms supp ly cost ly forms of tra in ing or costless learning by d o i n g . Both in the market for exist ing ski l ls and in the market for skill development, competition assures a simple, determinate resu l t . Firms offer a prof it-maximizing menu of learning opportuni t ies , and over the life cyc le , workers move from job to job ( v a r y i n g k Q ) in pursu i t of their investment goals . 35 It must be conceded that this view of on-the-job training and life-cycle investment places a rather heavy information burden upon both parties to the learn-and-earn bargain. Workers and employers must be able to predict, within tolerable limits, the training characteristics of a great many jobs. Whether they can do so with sufficient accuracy to make the theory realistic is a difficult question. Further- more, it might appear to some that the notion of workers' having to change jobs continually in order to fulfill their investment plans seriously misrepresents the nature of occupational mobility in the labour market. As Blaug says skeptically, ". . . it is . . . doubt- ful that all interoccupational, and even more intraoccupational, move- ments of labor can be reduced to the action of sowing and reaping the 55 advantages of labor training. . . . " That workers remain in essentially the same occupation and "ride" a fixed learning curve seems, all in all, a simpler explanation for what we observe in the labour market. In the case of training which is valuable only to the firm 56 which provides it, Becker's argument was that employers could collect the entire return and would therefore be willing to pay the entire cost, but that they would more likely share costs and returns with workers in order to discourage turnover. By promising workers a rising experience profile of wage rates, employers could reduce quits and, hence, the loss of investment in "specific training." The wage profile which kept such losses to a minimum would implicitly determine the equilibrium sharing of costs and returns. 36 In a recent ar t ic le , however, Donaldson and E a t o n 5 7 contend that the idea of shared investment is mistaken. Accord ing to their def in i t ion, "shar ing" occurs only if the wage profi le of fered to the worker makes him better off in present-va lue terms than he would be in a l te rn - ative employment. It is immediately obvious that competition among workers will never permit shar ing in this sense. Super ior opportunit ies will always be e roded . The firm will manipulate the wage profi le in order to minimize the loss of exper ienced workers ; but since its wage bill (in present-va lue terms) is f i xed , it must ultimately collect the total net benefit of any specif ic t ra in ing it decides to under take . Grant ing the important point with regard to shar ing , one should not however be misled by the Donaldson-Eaton analysis into th ink ing that specif ic t ra in ing does not pose an investment problem from the worker's viewpoint. When of fered a r i s ing wage prof i le , as opposed to a flat one in alternative employment, the worker must still decide which to accept ; and for this purpose he must perform an investment ca lcu la- t ion. The Dona Id son-Eaton ana lys is , a l though suff ic ient to make its point, suf fers to a certain extent from its fai lure to elaborate the worker's decision problem. One may also question whether it is appropr iate to assume competitive behaviour in modelling the relationship between f irms and their employees. A s Reder commented in his review of Human Cap i ta l , . . . an individual employee c a n , by qu i t t ing , impose a loss on an employer of h is (the employer's) whole share of the re turn on t ra in ing . Hence, any share of the return that a worker lets an employer keep makes that employer better off than he 37 would have been if the worker had quit. On the other hand, it is obvious that by discharge, the employer can impose an analogous loss on the worker. Thus is generated the zone within which bargaining power, strategic skill, institutional rules, etc., determine wage rates. 5 8 However, if workers (and firms) accurately foresee these bargaining possibilities, the gains or losses which flow from them will presumably affect the initial decision of whether or not to accept employment (or hire) at a given starting wage. Competition for opportunities to bargain should negate any advantages or disadvantages which bargaining might otherwise entail. As far as Mincer is concerned, the analytical differences be- tween general and specific training are of little ultimate consequence, since their separate influences upon age-earnings profiles are empirically indistinguishable, given the available data. Both imply, very simply, that earnings (exclusive of depreciation) rise with work experience. In the absence of detailed information on learning curves and on the direct and indirect expenditures of firms, experience must serve as a proxy for all the various modes of on-the-job training. In fact, as we shall see in the next chapter, experience can itself be estimated from census data only by means of a further proxy. 38 G E N E R A L T H E O R I E S OF INCOME MAXIMIZATION In the Becker -Ch iswick-Mincer ana lys is , individuals decide upon the amount and timing of their investment in human capital by choosing a sequence of values for k^. If the foregoing model is to be understood as something more than a tautoloqy in which k' = C IE p p p ex post, one must supply a behavioural theory to predict the course of this variable over the individual 's life c y c l e . The f i rst to approach 59 the task was B e n - P o r a t h . His model, and the extension prov ided b y H a l e y , ^ may be termed "general" insofar as they treat schooling and on-the-job tra in ing as special cases within a choice-theoretic f ramework. That framework is nevertheless one of income rather than util ity maximization. In the present context both y ie ld the same resul t , since the authors continue to assume a single good, ignor ing le isure. Ben-Porath 's essential contr ibut ion to the analysis was the idea of an individual product ion function for human capi ta l . A p p l y i n g this device, one assumes that the individual "manufactures" increments Qj_l of human capital by b r i n g i n g together purchased inputs D and a portion of some exist ing capital stock H. The production funct ion, in its most general form, may be written Q,_l = F (k», H, D) .(18) 39 However, Ben-Porath invokes the so-cal led "neutral ity assumption" to obtain Q H = f ( k ' H , D) . . . .(18') Here, human capital is treated as an augmenting factor , and k'H represents effective investment time. If this time were sold in the labour market, it would b r i n g earn ings of w ( k ' H ) , where w signif ies the f ixed rental pr ice of human cap i ta l . "Neutral i ty" hinges on the assumption that effective investment time and effect ive work time incorporate the same augmenting factor , H . T h u s , human capital increases earn ing potential and the abi l i ty to generate fu r ther earn ing potential in exactly the same propor t ion . In Haley's somewhat simplified vers ion of the model, purchased inputs d isappear , and the product ion function becomes Q H ( t ) = a l ( t ) y , . . . .(19) where l(t) = k'(t) H(t) Al l var iables are treated as cont inuous funct ions of time. T h e f i rs t parameter, a , measures indiv idual ef f ic iency in human-capital p r o - duct ion , and the second u, denotes the level of re turns to scale. Unless re turns to scale are decl in ing (0 < JJ < 1), the model will not y ie ld an acceptable solut ion. 40 In view of depreciat ion, the individual 's stock of human capital must evolve according to the differential equation H(t) = Q H ( t ) - dH(t) . . . . .(20) Earn ing capacity is simply E(t) = w H ( t ) , and "disposable earn ings" are given by W(t) = wH(t) - wl(t) = [1 - k '(t)] wH(t) (21) The problem for the individual is to choose k'(t) in order to maximize f T J = W(t)e r t d t , . . . .(22) 0 subject to (20) and (21) . Together with the boundary restr ict ions H(t) ^ 0 , l(t) ^ 0 , H(t) - l(t) > 0 , and some initial condition H(0) = , Equations (20)-(22) define a relatively simple problem in control theory . A s usua l , the solution procedure generates a set (more specif i- ca l ly , a continuum) of shadow pr ices for human capital , namely: A(t) = [w/(r + d)] [1 - e ( r + d ) ( t T ) ] , 0 < t ^ T 41 These decline over the life cyc le because of the dwindl ing opportuni ty to amortize new investment pr ior to the f ixed retirement date. The reasonable supposit ion that the stock of human capital becomes worthless at retirement justif ies the t ransversa l i ty condition A(T) H ( T ) = 0 . . . . .(25) Wherever the individual attains an interior solution, he optimizes by choosing k ' ( t ) , and hence Q H ( t ) , so that the marginal cost of p r o - duc ing the des ired amount of human capital equals the ru l ing shadow pr ice , X ( t ) . Since A(t) falls cont inuously over time, and since marginal cost is perforce assumed to be a r is ing function of human-capital out- 61 put , the increments Q|_j(t) added to the human-capital stock must decline monotonicaily over the life c y c l e . Effect ive investment time 62 l(t) must also decline monotonicaily; to be speci f ic , _ f r + H l J r + d ) ( t - T ) l ( t ) = ,(t) . I ^ - g 2 - e ^ ( r 4 d ) ( r _ T ) < o (26) T h e behaviour of k'(t) is more di f f icult to estab l i sh . From the definit ion l(t) = k ' ( t ) H ( t ) , and from Equation (20), one may deduce that • • • k'(t)/k'(t) = l ( t )/ l ( t ) - H(t)/H(t) , . . . .(27) or k'(t) = k'(t)[? ( t )/ l ( t) - Q H ( t ) / H ( t ) +d] . 42 T h e sign of the bracketed express ion appears indeterminate, unless d = 0. T h e n , without quest ion , k'(t) < 0 . In genera l , it would seem that fulfil lment of the optimal plan might require k'(t) to increase over some interval late in the indiv idual 's life c y c l e , when H( t )/H(t) < 0 . However, this conclusion cannot be accepted without f i rst subs t i tu t ing , for the endogenous var iables in (27), their equivalents in terms of the model parameters, r, d , a , u , T , and H^. T h e resul t ing express ion for k'(t) is v i r tua l ly impossible to deal with analyt ica l ly . Instead, k'(t) was simulated numerical ly for a wide range of parameter combin- at ions. In every case, k'(t) decl ined monotonically. T h e s imula t ions also confirm Haley's assert ion that k'(t) must d isplay an inflection point . Results ve r i f y that the funct ion decl ines f i rs t at a decreas ing , 63 and later at an increasing rate. A t ret irement, of course , k'(t) = k ' (T) = 0 At the opposite end of the age scale, the foregoing analysis may not a p p l y , for indiv iduals typica l ly appear not to achieve interior maxima. When X(t) is h igh because of the long amortization period in prospect at the beginning of the economic life cyc le , optimization accord ing to the rule MC(t) = X ( t ) may requ i re the investment of more human capital than the individual c u r r e n t l y owns. A t such times, the boundary condit ion H(t) - l( t) ^ 0 holds with equal i ty , and the individual - specializes in the product ion of human capi ta l , sett ing k'(t) equal to one. T h o u g h it is natural to identify the per iod of specialization with that of formal school ing, the two need not be coextens ive . Specialization may v e r y well cease before schooling f in ishes; indeed, many "full-time" students devote a considerable number of hours to market work. Such behaviour is consistent with the theory , since the optimal plan may dictate k'(t) < 1 for some t < s . The length of the specialization per iod , whether or not it falls short of s, is determined endogenously as part of the optimization 64 programme. Haley shows that the length depends posit ively upon a, the individual 's personal eff ic iency parameter, and negatively upon r, d , and H^. The latter is of course the individual 's initial endowment of human capi ta l . That a an H n , which may be posit ively corre lated , should have opposite effects on the period of specialization is a part icu lar ly intr igu ing outcome of the ana lys is . Unfortunate ly , the broad implications of the model stand up rather poorly in the face of ex ist ing ev idence . A second-der ivat ive test conducted by B e n - P o r a t h 6 5 makes use of the fact that 3 C1/D/3t _ r + d . . r , R , : ' ( r + d H t - T ) * ' * " ' [ } ( l/ l) 1 - e l r a m 1 ' T h i s equation predicts "the rate at which the decline in investment over the life cyc le should a c c e l e r a t e . 1 , 6 6 Employing the data from Mincer's 1962 s tudy of on-the-job t ra in ing , Ben-Porath f inds that investment ( in ferred from age-earn ings profi les) falls much more rapidly than one would expect on the basis of Equation (28). Moreover, estimates of y , obtained b y combining (28) and (26), suggest that re turns to scale are 44 nearly constant (u = 1.0). T h i s result tends to contradict the cruc ia l assumption upon which the model r e s t s . One explanation may be that the neutral i ty hypothesis is f a l s e . 6 7 If human capital is biased towards the market, and if the bias increases with time, investment will in fact decl ine more rapid ly than Equation (26) p red i c t s . Whether the decline will accelerate nevertheless appears 68 uncer ta in . St i l l , there does not seem to be any weaker or more general hypothesis which preserves testabi l i ty . One cannot use an equation like (28), for example, to identify a fu r ther set of bias parameters. On the other hand , if the only conceivable s t ructure one may impose upon the model—the neutral i ty h y p o t h e s i s — i s rejected by the ev idence, the chief advantage of Ben-Porath's expl ic it maximization approach d isappears . One might just as well employ the simpler, ad hoc analysis put forward by Mincer . Other problems may of course account for the apparent fai lure of the Ben-Porath model. Three that have been d iscussed in the l i terature a r e : vintage effects that may d istort cross-sect ion age-earn ings p r o f i l e s ; 6 9 l i fe-cycle variation in hours of w o r k ; 7 0 and the use of c o n - tradictory assumptions in the construct ion of investment pro f i l es . 7 ^ 72 Brown proposes remedies for all three , but his resul ts are not wholly encourag ing . Though he obtains plausible estimates of u, the values implied for r appear unreasonably low. 73 In another s t u d y , Heckman once again encounters constant returns to scale. Upon estimating k ' ( t ) , he f inds an initial segment of the funct ion that is posit ively s loped, and second-order propert ies that are the reverse of those forecast by Haley. On the other h a n d , 74 Haley's own research , us ing grouped data and a complicated non- linear estimation procedure , s trongly supports the Ben-Porath theory . Parameter estimates fall within reasonable limits and display relat ively small var iances . One is therefore left with an indecis ive result and a need for f u r t h e r , detailed research . APPENDIX I T H E E F F E C T O F M A R K E T BIAS ON T H E O P T I M A L I N V E S T M E N T P R O F I L E We have seen in the foregoing text that if neutral i ty holds, it is possible to entertain a human-capital product ion function of the form Q H = a ( k ' H ) y = a l y Marginal cost is thus g iven by MC = w / ( 3 Q H / 3 l ) = (w/ay) I 1 y Optimization accord ing to the rule MC = X implies that ( w / a y ) l 1 _ y = [ w / ( r + d ) ] [ 1 - e ( r + d ) ( t _ T ) ] . ( r + d ) ( t - T ) 1 l 1 / { 1 - * i ) { ^ L B . I i - . ( r * d ) ( t - T ) ] y (A .1 .1) Now, to insert the notion of market b ias, we may rewrite the product ion function in the following manner: Q H = a ( b l ) y = y l y , . . . . ( A . 1 . 2 ) 46 *7 where Y = a b y and b = b(t) . If b, the bias parameter, equals one, we have neutra l i ty . If 0 < b < 1, human capital is biased towards the market: the cur rent increment adds Q M to earning capacity but only bQ^ to potential investment input . If b > 1, human capital has an "investment b ias . " We may suppose that b is an exogenous function of time (age) . It should be obvious from the preceding derivat ion that yy M _ ( r+d)( t -T) I < i r + d 1 1 e 1 | J if b < 1 (A .1 .3) At all points d u r i n g the nonspecialization phase of the life cyc le , market bias reduces the level of investment in human capi ta l . Market bias also reduces the length of the specialization phase. Both effects are due to the increase in marginal cost . Dif ferentiat ing (A.1 .3) in logarithmic form yields - ( r + d ) e ( r + d ) ( t - T ) Y ( , ^ } Y ( 1 - y ) [ 1 - e ( r * , < t - T > ] (A .1 .4) which is unambiguously negative if y < 0 —that is , if market bias increases with age . One might reasonably expect this condition to ho ld . If so. comparison of ( A . l . t ) and (26) demonstrates that |I7I '| > | f / l | . Market bias causes investment to decl ine more rap id ly (in proport iona terms) than under condit ions of neutra l i ty . However, if market bias is constant (y = 0) , l '/l ' = l/l ; and the rate of decline is unaf fected . For convenience in what follows, let us now implicitly define some new notation by re -express ing ( A . 1.1) as il_ = _L_ + - R X . 11 z y z(1 - X) Dif ferent iat ing once more, we obtain Y Y - Y 2 + - R 2 X z Y 2 z(1 - X ) 2 ( Y Y - Y 2 ) ( 1 ~ X ) 2 - y 2 R 2 X z y 2 ( 1 - X ) 2 where y = d 2 y / d t 2 . We wish to d iv ide the preceding express ion by \J_ = y(1 - X) - yRX I' z y ( l - X ) dCI'/l'l dt The result is d( l ' / l ' )/d t = ( Y Y - Y ) H - X ) 2 - y 2 R 2 X | i*'/1• Y d - X ) [ Y ( 1 - X ) - Y R X ] We must f inally compare ( A . 1.5) and (28). In our present notation the latter is simply R/(1 - X ) . Market bias will increase the relative rate of deceleration if ( Y Y - Y 2 )(1 - X ) 2 - Y 2 R 2 X > R Y d ~ X ) [ Y ( 1 - X ) - Y R X ] 1 - X or ( Y Y - Y 2 ) d " X ) 2 - Y 2 R 2 X < Y tyd-x ) - Y R X ] R , since the quant i ty in brackets is negat ive. Cont inu ing , we f ind (YY - Y 2 ) d - X ) < YVR Y Y 1 - X JL_ _ _X > _R_ > 0 Y Y 1 - X It is not clear why this condition should hold in genera l . If V > 0, the left side may even be negat ive. We must conclude that weak hypotheses concerning market bias are not suff ic ient to explain B e n - Porath's f i nd ings . A s a matter of fact, the present inequality becomes increasingly di f f icult to satisfy (ceteris paribus) with advanc v - ( r+d)( t -T) v . .. . . ., ing age, s ince X = e r i ses . Yet , it is only in the upper age range that the market-bias explanation is needed. N O T E S C H A P T E R I 'Or ig ins of this doctr ine may be traced back as far as Adam Smith, A n Inquiry into the Nature and Causes of the Wealth of Nations, edited by Edwin Cannan (New Y o r k : Modern L i b r a r y , 1937), p. 101, and beyond Smith, to S i r William Petty in the late seventeenth c e n t u r y . See Bernard F. K iker , "The Historical Roots of the Concept of Human C a p i t a l , " Journal of Political Economy, L X X I V (October, 1966), 481-499. Its modern f lowering must be credi ted to Theodore Schultz and C a r y Becker . The seminal art ic les were: Theodore W. Schu l tz , "Capital Formation by Educat ion ," Journal of Political Economy, L X V l l l (December, 1960), 571-583, and "Investment in Human Cap i ta l , " American Economic Review, LI (March, 1961), 1-17; C a r y S. Becker , "Investment in Human Cap i ta l : A Theoret ical A n a l y s i s , " Journal of Political Economy, L X X (October, Supplement, 1962), 9-49. 2 T h e der ivat ion which follows is the work of Mincer , "The Distr ibut ion of Labor Incomes: A S u r v e y . " T h i s vers ion of the model d i f fers from the one employed by Becker mainly in its use of continuous rather than discrete time. C f . C a r y S. Becker , Human Capital (New Y o r k : National Bureau of Economic Research , 1964), Chapter III. 3 For the moment we may thus regard earn ings and wage rates as interchangeable. 4 Accord ing to Mincer , the latter condition is satisf ied approximately in the case of American males. See School ing, Exper ience and Earn ings , p. -8, n . 2. 5 B e c k e r ' s ear ly estimates imply that if college students earn approximately one-quarter as much as non-s tudents , the income received will in fact just balance d i rect cos ts . See Human Cap i ta l , p p . 74-75. Dodge found that, on average, the part-t ime earnings of Canadian students great ly exceeded direct costs (Returns to Investment in Un ivers i ty T r a i n i n g , Tab le 5.1 and 5.2, p p . 77-78). Since students sacrif ice leisure as well as earn ings to attend school , valuing their opportuni ty cost presents fu r ther problems. See Donald O. Parsons, "The Cost of School T ime, Foregone Earn ings , and Human Capital Format ion," Journal of Political Economy, LXXXII (march/Apr i l , 1974), 251-266. 51 52 "School ing, Experience, and Earn ings , p. 10. 7We consider here only the f i rs t moments of any probabi l i ty d is t r ibut ions connected with W. ( s ) . We thus ignore the question of r i s k . On this point see John C . Hause, "The Risk Element in Occupational and Educational Cho ices : Comment," Journal of Political Economy, LXXXII (July/ A u g u s t , 1974), 803-805. 8 * T h e requi red initial condit ion is W.(0) = W. n . 9 * Otherwise , the second-order condit ion d 2 In W j ( s ) / d s 2 < 0 will not be fu l f i l led . 1 0 S e e his "Under investment in College Educat ion ," American Economic Review, L (May, 1960), 347, or Human Cap i ta l , Chapter V . ^Some initial steps have been taken by R ichard B. Freeman, T h e Market for Co l lege-Tra ined Manpower(Cambr idge, Massachusetts : Harvard Un ivers i ty Press , 1971), Chapter I and Chapter II. In addition to enrol lment, of course , one must take into account such things as labour- force part ic ipat ion, deaths , ret irements, and net migrat ion. 12 A model which incorporates demand has been tested by John F. O 'Connel l , "The Labor Market for Eng ineers : A n A l ternat ive Methodology," Journal of Human Resources , VII (Winter, 1972), 71-86. I J T h e o d o r e W. Schu l tz , "The Reckoning of Education as Human Cap i ta l , " in Educat ion, Income, and Human Capital , , edited by W. Lee Hansen (New Y o r k : National Bureau of Economic Research , 1970), p. 301. 14 A classic example is C lark K e r r , "The Balkanization of Labor Markets" in E. Wight Bakke et a l . , Labor Mobility and Economic Oppor tun i ty (New Y o r k : Technology Press and John Wiley and Sons , Inc . , 1954); but see as well L loyd C . Reyno lds , T h e S t ruc tu re of Labor Markets (New Y o r k : Harper and B r o t h e r s , I n c . , 1951). A more recent work in this tradit ion is A lber t Rees and George P. Shu l tz , Workers and Wages in an Urban Labor Market (Ch icago: Un ivers i ty of Chicago Press , 1970). 1 5 Hendrik, S Houthakker , "Education and Income," Review of Economics and Stat ist ics , XLI ( F e b r u a r y , 1959), 14-17. W. Lee Hansen , "Total and Private Rates of Return to Investment in Schoo l ing , " Journal of Political Economy LXXI ( A p r i l , 1963 ) , 128-141. Giora Hanoch, "An Economic Ana lys i s of Earn ings and Schoo l ing , " Journal of Human Resources , II (Summer, 1967), 310-329. 53 1 6 " P r e s e n t Values of Lifetime Earn ings for Dif ferent Occupat ions ." 1 7 A useful su rvey is George Psacharopoulos and Keith H i n c h - l i f fe. Returns to Educat ion: A n International Comparison (Amsterdam: Elsevier Scientif ic Publ ishing Company, and San Franc isco : Jossey- Bass Inc . , 1973). 18 School ing, Exper ience , and Earn ings , p. 137. 19 B a r r y Bluestone, Willis M u r p h y , and Mary Stevensen, Low Wages and the Working Poor (Ann A r b o r : Institute of Labor and Industrial Relations, Un ivers i ty of Michigan, 1973), p p . 19 f f . 20 See Gary S . Becker , Human Capital and the Personal Distr ibut ion of Income (An A r b o r : Un ivers i ty of Michigan, 1967. 21 Accord ing to Becker , y ields decline for a number of reasons: (1) the cont inuing addition of a variable factor , school ing, to a f ixed factor , mental and physical abi l i ty , leads to diminishing re tu rns ; (2) foregone earnings rise (faster than product iv i ty in learning) as education accumulates; (3) the amortization period shortens; (4) the mar- ginal ut i l i ty of additional earn ings fa l ls ; (5) r isk avers ion may r ise as human capital increases . These last two arguments seem rather out of place in an income maximizing framework. A s for the interest cost , it r ises because of segmentation in the loans market and the need for students to resort to increasingly expensive source . 22 See Dale W. Jorgenson, "The Theory of Investment B e h a v i o r , " in Determinants of Investment Behavior edited by Robert Ferber (New Y o r k : Columbia Un ivers i ty Press for the National Bureau of Economic Research , 1967). Note that separable ut i l i ty , def ined over leisure and consumption, is not suff ic ient to make earnings and uti l ity maximization co inc ide. 23 For the moment, however, note T . D . Wallace and L . A . Ihnen, "Ful l -T ime Schooling in L i fe -Cyc le Models of Human Capital Accumulat ion ," Journal of Political Economy, LXXXIII ( F e b r u a r y , 1975), 137-156. These authors explore the extreme imperfection of no borrowing for investment purposes . 2H Mincer adopts this orthodox interpretat ion, though he does br ie f ly acknowledge the possible impact of labour-market fac tors . See School ing, Exper ience and Earn ings , p. 138. 25 Walter Haessel and P . J . K u c h , "An Ana lys is of the Deter- minants of the Size Distr ibut ion of Earn ings in C a n a d a , " Un ivers i ty of Western Ontar io , unpub l i shed , 1976. 54 26 We shall cons ider here only an exact specif ication of the model, with schooling the only form of human cap i ta l . The problems encountered when a stochastic term is present will be d i scussed , along with other quest ions of implementation, in Chapter III. 27 See Gary S . Becker and B a r r y R. Ch iswick , "Education and the d istr ibut ion of E a r n i n g s , " American Economic Review, LVI (May, 1966), 358-369; and Jacob Mincer, "Time-Series Changes in Personal Income Inequality in the United States from 1939, with Projections to 1989," Journal of Political Economy, L X X X (May- June , 1972, Supplement) , S34-S66; Ch iswick , Income Inequality: Regional Ana lys is within a Human Capital Framework (New Y o r k : Columbia Un ivers i ty Press for the National Bureau of Economic Research, 1974); Mincer , School ing, Exper ience , and Earn ings , Chapter II and Chapter VI. 28 In the present context W(0) way be interpreted as represent- ing the individual 's initial endowment of abil ity and human capi ta l . Whether it is in fact uncorre lated with r and s is therefore somewhat dub ious . " T o be prec ise , R ( r , s ) E 2 l E [ r - r ) 2 ( s - s ) ] + 2r E [ ( r - r ) ( s - s ) 2 ] + E [ r - r ) 2 ( s - s ) 2 ] - [ C o v ( r , s ) ] 2 , where E is the expectations operator . T h e theorem is due to Leo A . Goodman, "On the Exact Var iance of P r o d u c t s , " Journal of the American Statistical Assoc iat ion, LV (December, 1960), 708-713. 3 0 S e e the well-known paper by A . B . A tk inson , "On the Measurement of Inequal i ty ," Journal of Economic T h e o r y , VI (September, 1970), 244-263, and R. Love and M . C . Wolfson, Income Inequal ity: Statistical Methodology and Canadian Il lustrations (Ottawa: Statist ics Canada , 1976), Catalogue 13-559. A defect of the var iance-of- logar i thms measure is that it does not necessar i ly sat isfy "Dalton's cond i t ion ," which states that any t ransfer from a r ich to a poor individual must register as a decline is inequal i ty , p rov ided the amount of the t rans fer is not so large as to reverse the part ies' rank ing in the income d i s t r i - but ion . It should also be recognized that the present d iscussion re fers only to contemporaneous cross-sect ional and not to lifetime inequal i ty . Within the restr ic ted framework of the schooling model where age- earning prof i les (after graduation) are hor izontal , this dist inct ion is 55 unimportant; but such is not always the case . See Har ry C . Johnson , "Some Micro-Economic Reflections on Income and Wealth Inequal i t ies," Annals of the American Academy of Political and Social Sc ience, CDIX (September, 1973), 54-59, or Morton Pagl in , "The Measurement and T r e n d of Inequal i ty: A Basic Rev i s i on , " American Economic Review, L X V (September, 1975), 598-609. 31 School ing, Exper ience, and Earn ings , p. 27. 32 Both , op . c i t . 33 School ing, Exper ience and Earn ings , p. 53, Table 3.3 and p. 92, Table 5.1. T h i s material is reproduced for convenience in Append ix 11A - 34 School ing, Exper ience , and Earn ings , pp . 54-55. 35 t 332-338. 35 Alan Marin and George Psacharopoulos, "Schooling and Income D is t r ibu t ion , " Review of Economics and Stat ist ics, LVIII (Augus t , 1976), 36 We shall ignore the d istr ibut ion of Wg. 3 7 N i c h o I a s Ou l ton , "The Distr ibut ion of Education and the Distr ibut ion of Income," Economica, XLI (November, 1974), 387- 402. oo I b i d . , p p . 388-389. 39 The simpler and more common assumption has been that workers in d i f ferent educational categories are perfect ly interchange- able, accord ing to the number of "eff iciency units" they supp ly , along linear product ion isoquants . See, for example, Zvi Gr i l i ches , "Notes on the Role of Education in Production Funct ions and Growth A c c o u n t i n g , " in Educat ion, Income, and Human Capi ta l , edited by W. Lee Hansen (New Y o r k : National Bureau of Economic Research, 1970). T h e more general form of (8) still rules out complementer- it ies. C f . Samuel Bowles, "Aggregat ion of Labor Inputs in the Economics of Growth and Planning Experiments with a Two-Leve l C . E . S . F u n c t i o n , " Journal of Political Economy, LXXVIII ( January- F e b r u a r y , 1970), 68-81. 40 Oul ton, o p . c i t . , p. 394. 41 "Education and the Distr ibut ion of E a r n i n g s . " 56 42 One must do so, at least, within the context of the present model, which abstracts from all variation in time worked . Deprec ia- tion of human capital receives part icu lar ly detailed treatment in Sherwin Rosen, "Measuring the Obsolescence of Knowledge," in Educat ion, Income, and Human Behav ior , edited by F. Thomas Juster (New Y o r k : McGraw-Hi l l Book C o . , 1975). 43 School ing, Exper ience , and Earn ings , p. 49. 44 Formally, observe that In W s = In Es + l n ( 1 - k 0 ) ; In W? = In E : where In W p = In E s + r x K p ' + l n ( 1 - k ~ ) , p < p' * T , V = X k t • t=0 There fore , V a r ( l n W ) = V a r ( l n E g ) + Var [ ln( 1-k Q) ] + 2 Cov[ ln E s , l n ( 1 - k 0 ) ] ; V a r ( l n W~ ) = V a r ( l n E ) ; p s V a r ( l n W~, = V a r ( l n E ) + r 2 • V a r ( K ,)+2r • Cov( ln E ,K ,) p' s x p' x s p 1 + 2Cov[ln E s , l n ( 1 - k p l ) ] + 2 r x [ K ~ , l n ( 1 - k p , ) ] . If the covar iances are small, V a r ( l n W~) will constitute the minimum. P C f . School ing, Exper ience , and Earn ings , p. 102. 4 5 j b i d . , p. 103. 46 Burton G . Malkiel and Judi th A . Malkiel , "Male-Female Pay Dif ferent ia ls in Professional Employment," American Economic Review, LXIII (September, 1973), 693-705. 57 47 Glen G . Ca in , R ichard B. Freeman, and W. Lee Hansen, Labor Market Ana lys is of Engineers and Technica l Workers (Balt imore: Johns Hopkins Press , 1973); A n d e r s Klevmarken and John M. Quig ley , "Age , Exper ience , Ea rn ings , and Investments in Human Cap i ta l , " Journal of Political Economy, L X X X I V ( F e b r u a r y , 1976), 47-72. 48 Edward Lazear , "Age , Exper ience and Wage Growth," American Economic Review, L X V (September, 1976), 548-559. 49 Presumably, such workers do not receive any on-the- job t r a i n i n g . See George Psacharopoulos, "School ing, Exper ience , and Ea rn ings : The Case of an L D C , " Journal of Development Economics, IV (March , 1977), 39-48. 5 ^Mark B laug , "The Empirical Status of Human Capital T h e o r y : A Sl ight ly Jaundiced S u r v e y , " Journal of Economic L i terature , XIV (September, 1976), 837. 5 1 Ibid ., p p . 45-47. 52 Human Cap i ta l , p p . 11-18. 53 Sherwin Rosen, "Learning and Exper ience in the Labor Market , " Journal of Human Resources , II (Summer, 1972), 326-345. 5 4 l b i d . , p. 328. 5 5 B l a u g , "Human Capital T h e o r y , " p. 837. 56 Human C a p i t a l , p p . 18-29. 5 7 D a v i d Donaldson and B. C u r t i s Eaton, "F i rm-Speci f ic Human Cap i ta l : A Shared Investment or Optimal Entrapment?" Canadian Journal of Economics, IX ( A u g u s t , 1976), 462-472. 58 Melvin W. Reder , "Gary Becker 's Human Cap i ta l : A Review A r t i c l e , " Journal of Human Resources , II (Winter, 1967), 100. 59 Yoram Ben-Pora th , "The Production of Human Capital and the Life Cyc le of E a r n i n g s , " Journal of Political Economy, L X X V ( A u g u s t , 1967), 352-365. 58 William J . Haley, "Human Cap i ta l : T h e Choice Between Investment and Income," American Economic Review, LXIII (Decem- b e r , 1973), 929-944. Note also the fol lowing: Eytan Shesh insk i , "On the Individual 's Lifetime Allocation Between Education and Work," Metroeconomica, X X ( January , 1966), 42-29; Y . Comay, A . Melnik, and M . A . Pollaschek, "The Option Value of Education and the Optimal Path of Investment," International Economic Review, XIV (June, 1973), 421-435. 61 T h e marginal cost funct ion is g iven by MC = w / ( 3 Q H / 3 l ) 1-Y = ( w / a y ) l = (W/Y) a " 1 / Y Q H { 1 - Y ) / y . The condition 0 < y < I ensures that 9MC/9Q, , > 0 . If y > 1, X(t) will intersect the marginal cost function from below, and the second- order condit ion for a maximum will not ho ld . In this s i tuat ion, the individual would never wish to devote any time to market work. 62 Th i s expression is easily der ived by setting MC(t) = X ( t ) , and di f ferent iat ing in logs . 63 Haley, "Human Cap i ta l : T h e Choice Between Investment and Income," p. 937. 6 t t l b i d . , p p . 937-938. 65 Yoram Ben-Pora th , "The Production of Human Capital and T i m e , " in Educat ion, Income, and Human Capi ta l , edited by W. Lee Hansen (New Y o r k : National Bureau of Economic Research, 1970). 6 6 l b i d . , p. 139. 67 Mincer especial ly has emphasized this problem. See his "Comment," in Educat ion, Income, and Human Cap i ta l , edited by W. Lee Hansen (New Y o r k : National Bureau of Economic Research, 1970). 68 A r igourous proof may be found in Append ix I. Ben-Poroth argues for the l ikelihood of increasing market bias in s tat ing: The market does not make it possible to get something for noth ing , so that neutral improvement in human capacity costs more than special ized improvement. . . . When there is still a large investment program ahead, it is advisable to emphasize devices that . . . make the individual a more eff icient producer of human capi ta l . Later , . . . the fraction of investment outlays devoted to ski l ls that are for purposes of fu r ther investment will b e smaller. ["The Production of Human Capital and T ime ," p. 143]. Ben-Porath thus rever ts to the notion of heterogeneous human capital Such an idea seems notably out-of-joint with orthodox human capital theory , which emphasizes the homogeneous value of se l f - investment. 6 9 S e e Thomas Johnson and Freder ick J . Hebein, "Investment in Human Capital and Growth in Personal Income, 1956-1966," American Economic Review, L X I V (September, 1974), 604-615. 7 S v e shall of course be dealing ful ly with this problem in Chapters IV and V . T h e f i r s t to raise it seriously appears to have been Lester Thurow, "Comment," in Educat ion, Income, and Human Capi ta l , edited b y W. Lee Hansen (New Y o r k : National Bureau of Economic Research , 1970), p . 154. 7 1 S h e r w i n Rosen has pointed out that Mincer's investment series implicitly assume constant returns to scale. See "Income Generat ing Funct ions and Capital Accumulat ion ," Harvard Institute for Economic Research , Discussion Paper No. 306 (unpub l i shed) , 4 f\ •» ~» 72 Char les Brown, "A Model of Optimal Human-Capital Accumu lation and the Wages of Young High School Graduates , " Journal of Political Economy, L X X X I V ( A p r i l , 1976), 299-316. 73 James J . Heckman, "Estimates of the Human Capital Production Function Embedded in a L i fe -Cyc le Model of Labor S u p p l y , ' in Household Production and Consumpt ion, edited by Nestor E. Ter lecky j (New Y o r k : National Bureau of Economic Research , 1975). Notwithstanding the title, labour supply does not enter the c i ted estimates in an essential way. 74 William J . Haley, "Estimation of Earn ings Profiles from Optimal Human Capital Accumulat ion ," Econometrica, X L I V (November, 1976), 1223-1238. C H A P T E R II PROBLEMS O F IMPLEMENTATION Studies which seek to apply the preceding models in some way to available earn ings data now make up a vast body of research . Even by 1964, ef forts to compute the rates of re turn to var ious forms of education had prol i ferated to such an extent that Becker found it necessary to caution against "excesses" in the use of the human-capital c o n c e p t . 1 T h e outpour ing of work has cont inued, though undoubtedly with some important ref inements. For present purposes , there is little value in attempting to survey the quantitat ive results of this immense l i terature. Specif ic attention will be given to the few signif icant pieces of Canadian research that have appeared , and to the f ind ings of Mincer, whose work prov ides a basis of comparison for the empirical results reported later in this s t u d y . Mainly, however, this chapter will examine the assorted problems of estimation and interpretation that arise in implementing the models just s u r v e y e d . Such problems must be faced, even if one holds the under ly ing analysis to be beyond falsif ication and therefore deficient as a scientif ic theory of individual behav iour . In the absence of fur ther qual i f icat ion, the human-capital paradigm 60 , 61 may prove misleading even in its other , more mundane role as a frame- work for ex post measurement and descr ip t ion . A s in the preceding chapter , we shall look f i rs t at the school- ing model and then at the analysis of on-the-job t ra in ing . We shall consider implementation of the "general model" very br ie f l y , since the data and methods used are of minor relevance to the cu r ren t s t u d y . T H E S C H O O L I N G MODEL Implementation of the schooling model appears s t ra ight forward . One has merely to add a conventional d is turbance term u. to Equation (3) , so that with W.(0) = W Q for all i. In W. = In W Q + r e S j + u. . . . .(29) Regress ing In W on s over any des i red cross-sect ion of indiv iduals then e prov ides an estimate of r , the rate of re turn to school ing. Equation e (29) assumes that r is the same for all members of the chosen popula- t ion . In a tr iv ia l sense, therefore, the simple regression estimate por t rays the mean. Equation (29) does permit individual variat ion in In W Q through the addit ive d is turbance u ; but the latter, in a d s o r b - ing such var iat ion, must remain uncorrelated with s. We shall explore in the next subsection the consequences of violating the two preced ing condi t ions . 62 When Mincer appl ies Equation (29) to census microdata on American males, the model explains only 7% of the var iance in the log- 2 arithm of annual (1959) earn ings . T h e apparent rate of re turn to schooling is also 7%. Th i s value of r e is well below the estimates of earl ier American studies , which compute rates of re turn d i rect ly by 3 comparing average or f itted age-earn ings prof i les . Direct estimates for the United States typical ly fall in the 10-16% range . Podoluk's results for Canada indicate re turns of 16.3% to a high school diploma and 19.7% to a un ivers i ty d e g r e e . 5 In the face of such evidence, the low f igure y ie lded by the s imple-regression approach casts immed- iate doubt upon the val idity of the schooling model. 2 T h e unimpressive value of R registered by (29) is not in itself v e r y d i s t u r b i n g . No one could reasonably expect the schooling model to fu rn i sh a complete descr ipt ion of the earnings generation p rocess : var iables other than schooling are obviously important. T h e simple model may nevertheless contribute to an adequately formulated earn ings funct ion . We must therefore look closely at the problems sur round ing its implementation. The suspected bias in the s imple-regression estimate of r may stem from a number of econometric d i f f icu l t ies . These may be grouped under the following f ive head ings : (1) individual variation in the rate of r e t u r n , (2) endogeneity of school ing, (3) expectations and economic growth , (4) omission of abi l i ty and family b a c k g r o u n d , (5) omission of other var iab les . We shall now examine each set of problems in deta i l . 63 Individual Variation in the Rate of Return e The assumption that r is the same for all individuals certa in ly places a v e r y strong a pr ior i restr ict ion upon the schooling model. More genera l ly , one might argue that individual rates of return contain e - e a personal component v . . Hence, we may write r. = r + v. , as in Chapter I. For completeness, one might also recognize a personal factor wj, govern ing initial earning capac i ty . In this case, let us write W._ = W.w!, so that In W.„ = In W. + w., where w. = In w! . 10 0 I 10 0 I I I Modify ing (29) appropr iate ly , we obtain In W. = (In Wn + w.) + ( r e + v.) s. + u. i 0 i I I I = In WQ + r e s . + u. + w. + v .Sj . . . . .(30) Now, in the simple regress ion of In W g on s, the expected value of the / \ —e estimated slope coefficient r is g iven by E(~re) = E [ I s. In W. / | s 2 ] = E [ J s . ( r e s . + u. + w. + v.s.)/y s 2 ] . i i i I I . i i i = r e + E H usJl sh * E[l WjSj/I s f ) i i i i + E (£ v.s2/£ s 2 ) , i i assuming, just for the moment, that both In W and s have been scaled in deviations from their respect ive means. Note that although s. is a f ixed number for any g iven i, it is nevertheless stochastic in the sense that the identity of the i t n indiv idual will va ry randomly in repeated samplings. If the s imple-regression estimate is to be unbiased, the terms involv ing u., v . , and Wj must v a n i s h . In other words, u and w 2 must be uncorrelated with s, and v must be uncorrelated with s . T h e requirement perta in ing to u is, of course , a s tandard assumption of the linear regression model. T h e same requirement ex- tends natural ly to w, which contr ibutes in parallel fashion to the observable e r ror (u. + w. + v . s . ) . Here, we isolate w to expose analytical ly whatever bias may result from this one e r r o r component. In fact , some degree of bias appears h ighly probab ly , since it is d i f f i - cult to believe that s and w could be independent . Factors which promote initial earning capacity seem certain to affect schooling as well . In part icu lar , s and w may be related empirically through a mutual dependence upon abil ity and family b a c k g r o u n d . If the relat ion- -e ship is posit ive, r will have an upward b ias . S u r p r i s i n g l y , however, some theoretical arguments suggest a negative relat ionship. Since these arguments hinge on the prec ise treatment of abi l i ty and family b a c k g r o u n d , they are best reserved for the subsection devoted to this topic . O u r immediate concern is the requirement that v be indepe- 2 dent of s . A l though the human-capital l i terature does not investigate this rather special hypothes is , it does supply abundant evidence of a general association between schooling and the rate of r e t u r n . T h e American studies already c i ted document a fall in r e , and therefore in v , as s r i ses . If we may thus infer a negative correlat ion between v and s , it would appear that the simple- regression estimate of r e will contain a downward b ias . T h i s factor may help to explain the low rate-o f - re turn estimates typical ly der ived us ing the s imple-regression approach . In Canada , however, there is some evidence that rates of re turn increase with the level of school ing. A s we have seen, Podoluk encountered h igher returns among un ivers i ty than among secondary- school graduates . Calculat ions performed by Dodge for several h ighly trained occupations show increasing returns in three out of four c a s e s . 6 One must therefore be alert to the possibi l i ty of an upward bias in regression estimates computed from Canadian data . The empirical work reported in Chapter III addresses this problem. Mincer approaches the question of individual variation in the 2 rate of return by expanding the regression model to include s . The d e r i v a t i v e 7 d • In W/ds = rjj + 2 r e s then prov ides an estimate of the marginal re turn to school ing. T h i s will be dec l in ing if r e < 0 and in Mincer 's initial t r ia ls , r e is indeed both negative and s ign i f i cant . However, the signif icance d isappears when Mincer standardizes for the number of weeks worked d u r i n g the sample y e a r . ' On the strength of this empirical result , he concludes that rates of re turn computed on the basis of weekly wages are near ly constant , and that the apparent association between s and r e is due mainly to the employment effects of s c h o o l i n g . 1 0 By implication, therefore, estimates obtained us ing weekly wages will be unb iased . Yet , a problem of interpretation now ar i ses . The rate of r e t u r n , as it is normally unders tood , includes all the benefits a t t r i - butable to school ing. Relative immunity to unemployment is poss ib ly one of these. If so, holding weeks of work constant violates the standard concept . T h i s procedure may well fu rn i sh an unbiased estimate, but not of the parameter we or iginal ly set out to measure. What we obtain ins tead—the weeks-constant rate of r e t u r n — i s a limited notion, with limited usefu lness , perhaps , in assess ing ind iv id - ual investment behav iour . Blaug implicitly adopts the broad ra te-o f - re turn concept when he argues that Mincer's result is actually rather paradoxica l . It is a fact that average weeks worked per year increase with the level of school ing. Hence, if we standardize for the numbers of weeks worked per year by calculat ing rates of re turn to schooling from weekly rather than annual e a r n i n g s , the decline in rates of re turn to successively h igher levels of schooling should increase, not decrease, the more so as there is some evidence that weekly earn ings tend to be posit ively corre lated with weeks worked per y e a r . 11 T h e paradox noted here is really a matter of confusion over Mincer's fa i lure to d is t inguish between the weeks-constant and the weeks- variable rate of r e t u r n . For Blaug and others , "rate of r e tu rn" means only the latter. Empir ical ly , the two competing measures lie rather far apar t . In a pair of comparable regress ions reported by Mincer , the f i rs t stands at 12%; the second, evaluated at the mean 12 year of school ing, equals 18%. Hence, one cannot just i fy the f i rst measure as an approximation for the second . Whether one may legitimately hold constant weeks worked per year , or any other var iable l inked to school ing, is in fact a recur r ing problem in rate- o f - re turn estimation. We shall meet this dilemma again later. 2 Right now observe that when Mincer adds s to the simple- e ~ regression model, he is implicitly letting v. = r^. + v. , where v. represents another d i s turbance . Subst i tut ing this hypothesis into Equation (30) y ie lds In W. = In W + r S . + r ? s 2 + u. + w. + v .s (32) i 0 0 i 1 i i I I I e —e 13 e e with r^ replacing r . Estimates of r Q and r̂  will now be unbiased (subject to the prev ious restr ict ions on u and w) as long as v is i n - 2 3 dependent of s and s . If the express ion for v. succeeds in captur ing the true relationship between schooling and the rate of r e t u r n , there is no fu r ther reason to suspect that v might be corre lated with s, raised to any part icular power. One may as well assume u n - b iasedness . However, because s appears in the composite e r ro r terms of (30) and (32), both models will presumably suffer from hetero- skedast ic i ty . Estimates of r , or of r Q and r^ will not be eff ic ient, and the standard e r r o r s will be biased downward. T h i s problem will 68 not y i e l d , moreover, to any simple transformation, since the composite d is turbances are nonhomogeneous in s . Of course , one might postulate functional relat ionships between s and r that are more complicated than the linear hypothesis examined here . A n endless number of ad hoc models may be generated in this way. A n alternative strategy which seems more promising is to make v a funct ion of other var iables besides school ing. One then a r r i ves at some vers ion of the "interactions model ," descr ibed in Chapter I. In this context , the squared term appear ing in (32) represents the inter- action of school ing with itself . From an econometric point of view, one's goal in spec i fy ing fu r ther interactions is to explain v in such a way that the ultimate res idua l , v , emerges as a "clean" stochastic term, uncorre lated with any of the independent var iab les . Bias is thus eliminated, a l though the problem of heteroskedast ic i ty 14 remains. It is important to note, in conc luding this subsect ion, that the issue of individual variation in the rate of re turn is a cruc ia l one for human-capital theor is ts . Econometric d i f f icult ies as ide, if the rate of re turn (like the velocity of money or the marginal propens i ty to consume) is not a stable constant when viewed in the relevant d imens ion—across otherwise dissimilar g roups of i n d i v i d u a l s — t h e n , the power of human-capital theory is great ly attenuated. T h i s power lies in the notion that individual d i f ferences may be reduced to a single var iab le , the stock of "human cap i ta l . " Mult ip ly ing the value 69 of the stock b y a simple parameter, the "rate of r e t u r n , " y ie lds i n d i v i d - ual ea rn ings . However, when the stock of human capital and the rate of return both depend on (possibly nondisjoint sets of) individual a t t r ibutes , much of the initial c la r i ty , even as a descr ip t ive framework, is lost. T h e interactions model, even though it follows quite natural ly from Becker 's supp ly-and-demand framework, violates the spir i t of orthodox human-capital ana lys is . Endogeneity of Schooling A s soon as one pays expl icit heed to the market processes which underl ie the statistical relationship between schooling and e a r n - ings , it becomes apparent that schooling need not be an exogenous var iab le . On the demand side of the labour market, schooling determines earn ings ; but on the supply side, where indiv iduals make investment decis ions, earn ings determine schoo l ing .^ 5 Equation (29) may thus contain a degree of simultaneity b ias . Formally, we may think of the following static equi l ibr ium system: L d e m = L d e m ( w ^ ^ . . . ^ . . . ^ . ^ ^ _ ^ ( 3 3 ) L s u p = L s u p ( y ^ tf^ ... tff w*; s, z 2 ) . . . U34) \ 70 Ws = f(W s) s = 0, 1, n . . . .(35) - d 6 m = L S U P . . . . .(36) T h e f i rs t two equations are a demand and a supply function respec- t ive ly . A s in the preceding text, L's stand for aggregate numbers of ind iv iduals , bars over the W's indicate means, and aster isks denote 16 ex ante var iab les . Two stochastic elements, and z^, allow for maximizing e r ro rs and other , unspeci f ied inf luences. The th i rd equation l inks observed and expected w a g e s . 1 7 The last is an equi l ibr ium condi t ion . Subst i tut ing into it from (33), (34), and (35), 18 we obtain the locus M (W, s, z M ) = 0 , . . . .(37) where z,, is a function of z, and z_. Now, the schooling model imposes upon this locus of equ i l i - br ium points a part icu lar functional fo rm—that d isp layed in Equation (29). Us ing microdata instead of grouped observat ions , we must of course insert the individual d is turbance var iable u in place of z^ . However, nothing in the der ivat ion of the schooling model requires that we solve (37) for W. We could as well have written Sj = - ~ | - In W Q + In W. - u . }• , . . . .(38) 71 which also y ie lds an estimate of r e . In genera l , this estimate will 19 not agree with one obtained from Equation (29). Since (38) and (29) both implement the fundamental postulate of equal present va lues, it is not clear a pr ior i which one the researcher should employ. T h i s simple view of the endogeneity problem is re inforced when we consider expl ic i t ly the indiv idual 's optimizing behav iour . Recall that in Chapter I we der ived the optimality condition * * dW./ds = W.r. . By the chain rule , i I I dwf/ds = (dWJdW.MdW./ds) . I e Let us suppose that W. = Wge1* s i + u., where r e is the "true" rate of re turn available in the market. T h e n , e dW./ds = r e W„e r s. + u. i O i l We noted in Chapter I that the second-order condition for optimality will be satisf ied only if d 2W.*/ds 2 < 0. Assuming that dW.*/dW. = f'(W.)>0, 7 i A I I I ' we can meet this requirement by making r a decl in ing funct ion of s . Let us do so implicitly in order to keep the ensuing algebra relatively . . 20 simple. 21 The preceding resu l ts , together with Equation (35), now imply that for optimality to hold e f'(W.) • ( r e W 0 e r S j + u.) = f(W.) r. , 72 or ln[f '(W.)] + In r e + In W Q + r e s . + u. = ln[f(W.)] + In r. . . ' . s. = - i - j - In W Q + ln[f(W.)] - u + ln[r / r e ] r - ln[f '(W.)] If expectations coincide with exist ing market opportunit ies , f(W.) = W., * e f'(W.) = I, and for the marginal investor at least, r. = r. = r . In i 3 I I this case, (39) reduces to (38). If this analysis is cor rec t , (29) and (39) form a simultaneous 22 system in which s depends negatively upon u . Single-equation estimates of (29) may, therefore, y ie ld values of r that are biased downward. Results reported by Cr i l i ches suggest that the downward 23 bias may be as much as 40%. If so, we cannot dismiss the problem l ight ly . Defenders of the single-equation approach may nevertheless argue that in cross-sect ional data schooling is a predetermined var iab le . C u r r e n t levels of schooling are the product of decisions taken in the past on the basis of expectations formed in the past . These expec- tations may depend , in t u r n , upon market condit ions prevai l ing in per iods even fur ther removed from the present . In the case of some older workers , we may thus be dealing with time spans as long as 40 or 50 y e a r s . Under such c i rcumstances, a d i rect behavioural l ink between schooling and c u r r e n t wage rates is c lear ly impossible. We know, however, that wage s t ruc tures evolve rather slowly. A t the same time, indiv iduals may not be totally unsuccessfu l in fo re- cast ing the f u t u r e . We may, therefore, encounter a s ignif icant statistical relationship between schooling and c u r r e n t wages. A s Gr i l i ches expla ins , "To the extent that the 'e r rors ' (from the point of view of us as observers) in the ex-post and ex-ante earnings funct ions are corre lated, they will be 'transmitted' to the schooling equation and induce an additional correlation between schooling and these d i s t u r b a n - 24 c e s . " The result will be simultaneity b ias . In the formal model sketched here , the requi red "transmission" role is performed by (35). That this equation may depict correlation rather than causal ity is of no great importance. It might fur ther be argued that schooling is not dependent upon earnings because it is not, to any signif icant degree, the subject of optimizing behav iour . Accord ing to this view, such things as tastes, socioeconomic b a c k g r o u n d , and the decisions of parents serve as the main determinants of individual school ing. Actua l ly , parental dec is ion- making need not affect our earl ier ana lys is . If parents are altruist ic and as well informed as their c h i l d r e n , they may plan to maximize ch i ldren 's lifetime earnings in just the way we have prev ious ly 25 hypothes ized . It may be that a great many factors—tastes and socioeconomic background among them—determine school ing; but if the set of determinants excludes earn ings , a dilemma appears . With- out some l ink between schooling and earn ings , there is no mechanism for disequi l ibr ium adjustment. 74 If levels of schooling observed in c ross section are predetermined, the supply funct ions of the preced ing market model descr ibe vert ical l ines. With demand funct ions g i v e n , the result ing locus of shor t - run equi l ibr ium points may look like (29), or it may not. A t best, we have a problem of interpretat ion. The rates of return der ived using (29) are themselves short run in character . More prec ise ly , they are the rates a c u r r e n t investor in schooling might earn if the cu r ren t wage s t ruc - ture were to pers i s t . T h e y are not necessar i ly the long-run rates of return envis ioned in der iv ing the ex ante vers ion of the schooling model. The nature of the dilemma should now be ful ly apparent . If we wish to interpret our regress ion coeff ic ients as l o n g - r u n , equi l ibr ium rates of r e t u r n , we must recognize the endogeneity of school ing; but if we recognize the endogeneity of school ing, we must concede that our regression coeff icients may harbour simultaneity b ias . In upholding the schooling model as a behavioural theory , we encounter an econo- metric problem. The obvious solution is to adopt a simultaneous-equation a p p r o a c h . Whatever method one chooses, its success will ultimately depend on f inding exogenous var iables which perform well as p r e - dictors of individual school ing. Census data do not seem especially r ich in this r e g a r d . The present study will not explore the endogeneity question f u r t h e r , though it remains an important topic for fu ture research . 75 Expectat ions and Economic Growth One might gather correct ly from the br ie f and somewhat tentative remarks of the preceding subsection that the human-capital l i terature has v e r y little to say on how expectations are formed. Freeman, who has written most on the topic, d ist inguishes three general inf luences: c u r r e n t wages, their rates of change, and nonwage fac tors . However, in his empirical investigations, he takes only cu r ren t wages 27 as his p roxy for expected lifetime earn ings . He thus assumes what might be called "myopic" expectat ions. The standard ra te-o f - re turn 28 studies ignore expectations almost completely, leaning implicitly toward an ex-post interpretation of resu l ts . From an econometric standpoint, the most important general question we have to consider is whether the practice of ignor ing expectations leads to a misspecification of the earnings function through 29 the omission of signif icant explanatory var iab les . It might be argued that if "conditions" and recent economic t r e n d s — i n a part icular reg ion, at a part icular time—seem to favour a part icular level of schooling as an investment goal , we should then observe in our cross-sect ion data a larger number of indiv iduals than would normally occupy the given age-school ing cohort . If, in addi t ion, workers belonging to the var ious cohorts are not perfect subst i tutes for one another in p ro - duct ion , we might also observe a lower than average wage for the 30 g iven cohor t . Th i s wage d ispar i ty may follow the group in quest ion throughout its life h i s t o r y . T o allow for the poss ib i l i ty , one might cons ider add ing age and region of schooling to the prev ious earn ings 76 funct ion . Accord ing to the argument just out l ined, these var iables would represent the state of expectations prevai l ing at the time and in the place educational decis ions were made. The trouble with the foregoing interpretation is that it seems, to prec lude our saying anyth ing in general about the effects of age and region of school ing. Suppose we learn, for example, that f i f ty - f ive year old h igh-school graduates from Br i t i sh Columbia enjoy an earn ings advantage over other f i f ty - f i ve year old Canadians at the same level of educat ion. If we adhere str ict ly to our s tate-of - investor- expectations hypothes is , we cannot make any predict ions whatever c o n - cern ing Br i t i sh Columbia high-school graduates who reach f i f ty - f i ve years of age at some point in the f u t u r e . Age and region merely flag once-and- for-a l l d is turbances in the pattern of educational investment. St i l l , if these var iab les , represent ing t rans i tory inf luences, are ignored, their omission may bias any attempt to measure the "normal ," "permanent ," or " long- run" rate of r e t u r n . Accord ing to the familiar e r ro rs - in -var iab les argument, the bias will be toward ze ro—in the present case , negat ive . Age and region combat it by se rv ing as proxies for the swings in expectations which produce "er rors" (from our point of view) in the schooling var iab le . These e r r o r s , if we may refer to them as such , ar ise not from statistical measurement, but from the "mistakes" individuals make because they cannot foresee market developments. Whether or not individuals foresee and act upon detailed changes in the educational wage s t ruc ture , they may still take into account general wage advances due to economic g rowth . T h i s factor g ives r ise 77 to another problem in estimating both the ex-ante and the ex-post rate of re turn to school ing. Recall that in der iv ing the basic schooling model, we assumed that annual wage rates would remain constant throughout the individual 's working l ife. T h e more realistic assumption- that real wages will grow exogenously over t ime—requi res some modif i- cation of the prev ious resul t . Let us suppose that wages are expected to r ise accord ing to the growth formulae g*t W(s, t) = W(s, 0) • e s s = 0, 1, n , where W(s, t) measures the reward to s years of schooling at time t, and the g* stand for expected rates of growth, allowed for the moment to d i f fer by level of school ing. If we again enforce the equalization of d iscounted lifetime earn ings , it is a simple matter to show that r e - g * ( r e - g * ) s W(s , 0) = — — . w ( 0 , 0) • e s . . . . ( 4 0 ) r - g 31 replaces (3) as the equi l ibr ium condition at t = 0 . Equation (40) indicates how the equi l ibr ium wage s t ruc ture may become distorted when expected growth rates d i f f e r . In genera l , individuals trade present earn ings for future g a i n . When expected growth rates are all equal or cannot be d ist inguished on account of great uncerta inty , (40) reduces to In W(s, 0) = In W (0 , 0) + (r - g*) s . . . . ( 4 1 ) after letting g* = g* = ••• = g* = g* and taking logarithms. If we how attempt to estimate (41) us ing a regression equation like (30), we encounter an elementary sort of identification problem. The slope coeff icient we obtain measures ( r e - g * ) rather than r e . If we recognize depreciation (in effect , negative growth) , it measures e 32 e (r ' + d - g * ) . To " identi fy" r' , we must have some independent estimate of ( d - g * ) . Even if we are interested only in the net rate of return (r ) , forgett ing about growth may lead us to underestimate i the value of this parameter. Miller appears to have been the f i rs t to call attention to the 33 problem of underest imation. He observed that economic growth causes the lifetime earnings profi les of successive age cohorts to shift upwards . A t any g iven time, the lowest of these prof i les will therefore belong to the oldest members of the populat ion. A s a resul t , when we draw a cross-section age-earn ings prof i le , we obtain a c u r v e that is f latter than any of the lifetime earnings trajectories we are in fact t r y i n g to represent . T h i s f lattened cross-sect ion profi le y ie lds an underestimate of the return to school ing. In Human Capital, Becker recognized the problem and computed separate rates of return for each of several 34 assumed rates of economic g rowth . Whether one computes the rate of re turn d i rect ly from age-earn ings profi les or adopts the regression approach favoured by Mincer , a reasonable assumption concern ing g* (or its ex-post realization g) seems the only possible recourse in most 35 cases . The situation is d i f ferent when the researcher has at his disposal a series of repeated c ross sect ions. T h e n it is possible to estimate g by following the respect ive cohorts over some period of actual calendar time. In this manner, Johnson and Hebein a r r i v e at 36 exogenous growth rates in the 3-5% range . Haley's estimates are 37 a little lower, fal l ing roughly in the 2-4% interva l . These f igures , imprecise as they are , g ive some idea of the correct ion one must think of app ly ing to s ingle-cross-sect ion estimates based on Equation (41) . Omission of Ab i l i ty and Family Background Without quest ion, the most pers istent challenge to the schooling model has come from the broad stream of empirical research which seeks to measure the effect on earnings of abil ity and family b a c k g r o u n d . Embedded in the result ing cont roversy are at least three major issues . One concerns the relative importance of school ing, v e r s u s background 38 and abi l i ty , in expla ining the level and d istr ibut ion of earn ings . Another concerns the problem of "screen ing" and the extent to which 39 education t ru ly enhances worker p roduc t i v i t y . The last has to do with estimating, in an unbiased manner, the absolute importance of school ing—that is to say, the rate of r e t u r n . T h i s final issue is the one which has provoked the greatest argument and the one which bears most heavi ly upon the work of the present s t u d y . The core of the problem is simple and well known. From the v e r y beginn ing of the human-capital e ra , it has been conceded that 80 if background and abi l i ty exert a d irect influence on the level of e a r n - ings, neglect ing their contr ibut ion may lead one to overestimate the impact 40 of educat ion. Earn ings di f ferent ia ls due in fact to super ior abil it ies and to the high socioeconomic standing of parents will be c red i ted mis- takenly to the additional schooling which these favourable attr ibutes tend to encourage. In more prec ise terms, the omitted-variable formula of econometric analysis states (us ing the standard "dot" notation) that S\ S\ /\ /S 6 = 6 + 3 3 . . . .(42) Ws Ws-a Wa-s as - e Here, 3yy s cor responds to r , and a ' s tands for some abil i ty or back- ground variable exc luded from the simple model. The degree of bias in the zero-order coeff icient B^s depends on the direct influence of a on earn ings (Byy ) and on the strength of the association between a and schooling (6 ) . If both are posit ive, so is the result ing 3 S bias . Interestingly enough, it is not clear a pr ior i that 6 _ must be 3 S greater than zero . In the Ben-Porath model, background and abil ity may be thought to affect the parameters H Q (initial human capital or earn ing capacity) and a (personal ef f ic iency in the product ion of fu r ther human cap i ta l ) . Yet , as we noted in Chapter I, these two factors influence the period of specialization in opposite ways . If s measures, at least rough ly , the per iod of special ization, and if a is a variable which governs both H_ and a, then it follows that 3 may be negat ive . Empir ical ly , U 3 S of course , there is general agreement that s is posit ively associated with 81 41 the standard proxies for abi l i ty and family b a c k g r o u n d . Given the model, one must conclude either that a (the posit ive influence) is more important than HQ or that the standard proxies favour it on average . A t the same time, one might ask whether f inancing imperfections associated with b a c k g r o u n d , but ignored by the model, are not an important factor in the empirical resu l t . In any event . Mincer points out that if abi l i ty or background affects earn ings only by way of additional school ing, will suf fer 42 ~ ~ no b ias . A l though 3 may be posit ive, (L. = 0. In this case, 3 S W 3 *S schooling is an essential input used for conver t ing latent advantages into marketable sk i l l s . Hause, on the other h a n d , has a rgued that 43 abi l i ty and schooling are really complements. A s s u c h , they enter the earn ings funct ion interact ive ly . Under these c i rcumstances , not only is 8 ^ a nonzero, but its value depends also on the part icu lar level at which s is held constant . The consensus among American studies has been that where a measures IQ or some other test score, 3yya , s ' s small but statistical ly d i f ferent from zero . Though results v a r y , the typical estimate of 44 45 bias in 3,., is rather small as wel l . Gr i l i ches and Mason, for Ws example, f ind it to be on the o rder of 11-15%. Dodge reaches a similar conclusion with respect to a sample group of Canadian profess ionals , 46 although his results are b y no means unambiguous. In the extreme, 47 Behrman, Taubman, and Wales obtain a bias estimate as high as 62% us ing a sample of male twins. 82 Elsewhere, Taubman and Wales come to the rather d is t ress ing 48 inference that the percentage bias var ies across age cohorts . If so, we cannot think of app ly ing any overal l "ability correct ion" to the zero-order coeff icient BWs. Gr i l i ches has re inforced this view with the general observat ion that a s tandard percentage adjustment must have - e 49 - e 8 w s = r as its denominator. Yet , r is bound to v a r y , perhaps widely, depending on the g roup of indiv iduals in question and on the precise specification of the estimating equat ion. There is no reason to believe that the absolute bias (the numerator) will va ry in order to keep the percentage bias constant . F inal ly , to compound the u n - cer ta inty , Welch has argued that if s and a , our proxies for "education" and "abi l i ty ," harbour a s ignif icant degree of measurement e r r o r , even ~ 50 the direction of bias in $•», is indeterminate. Ws Because the census data employed in the present s tudy offer no reasonable proxies for abi l i ty or socioeconomic b a c k g r o u n d , we shall not inquire fu r ther into the preceding d i f f icu l t ies . A l though the results d isp layed in Chapters III and V remain v e r y use fu l , they cannot , on this account , fu l ly escape qual i f icat ion. Omission of Other Var iab les It was noted in Chapter I that Mincer's "reduced-form equat ion"—the schooling model—contains no exogenous var iables from the demand side of the labour market. It is now appropr iate to inquire whether the omission of such var iables might not also bias the estimated return to school ing, just as in the case of abi l i ty and family b a c k g r o u n d . Over the years , in ter industry s t u d i e s 5 1 have isolated a number of factors which seem to be important in determining wage levels . These include working condit ions, unionizat ion, capital intensity, concentrat ion, prof i tabi l i ty , the growth rate, and plant s ize. If the schooling of the typical worker in an indust ry happens to be corre lated systematically with any of the preceding var iables , bias should theoretical ly ensue . Whether an empirical bias does in fact ar ise through the omission of indust ry var iables remains to be d i scovered . The inter- indust ry studies do provide some evidence of an interaction among wages, school ing, and other var iab les . Weiss detects a re lat ionship, f i r s t , between schooling and indust ry concentrat ion, and second, 52 between schooling and the level of unionizat ion. Haworth and Rasmussen f ind that median labour-force school ing, adjusted for qual i ty , adds s ignif icant ly to the explanatory power of their in ter- 53 indust ry wage regress ions . However, because they focus upon the coeff ic ients of the industry variables and not upon the one assoc ia- ted with school ing, their results offer little help in answering the question posed here . Most authors of the human-capital school have simply ignored the problem, but Hanoch has taken expl ic it pains to deny its re levance. He argues that 84 . . . a high degree of mobility exists among occupations and among industr ies , and this mobility depends strongly on schooling and age . . . . In other words, an ind iv id - ual who completes more years in school would expect to move upward in the occupational scale and perhaps to work in a bet ter -pay ing i n d u s t r y . T h i s is in fact the main channel by which he can realize re turns on his additional investment in educat ion. . . . A s a result , it was decided to exclude occupation and industry variables from the equations and thus avoid serious biases in the estimated coeff ic ients of schooling which, after a l l , are the target estimates of this a n a l y s i s . 5 4 T h e r e are two related points to consider he re . One has to do with mobil ity; the other, with decid ing which variables are to be held constant and which are to be left free in estimating the return to school ing. Let us deal with each of these issues in t u r n . Leaving aside for a moment the specif ic problem of occupat ion, one must concur that if mobility enforces long-run equi l ibr ium (as seen by investors in human cap i ta l ) , then indust ry var iables require no separate cons iderat ion . T h e schooling model represents the only possible wage s t ruc ture , and any long-run adjustment of factor p r o - portions needed to maintain it will arise without fa i l . A s we observed in Chapter I, human-capital theorists rely completely on this assumption. Whether labour mobility in the real world is actually suff ic ient to keep the wage s t ructure near long-run equi l ibrium at whatever point one might happen to choose for cross-section study is nevertheless an open quest ion . "Temporary" disequi l ibr ium present at the time a c r o s s - section is gathered may g ive a false p icture of the equi l ibrium wage s t r u c t u r e . Sustained market imperfection may do the same. However, if industry var iables capture both k inds of d istort ion, inc luding them 85 in the earnings function may eliminate these two potential sources of b ias . We now come to the second issue. It is Hanoch's contention that inc luding industry var iables (perhaps as a set of dummy regressors) will cause a bias in the schooling coeff ic ient . He argues that one cannot legitimately measure the rate of return to schooling with indust ry of employment held constant . T h e two var iab les , industry and school ing, are related, he says , h ierarch ica l ly , with the latter being the pr imary determinant of wages. One may infer that the use of both in the earn ings function will g ive r ise to a problem of redundancy somewhat akin to mult icol l inearity. The schooling coeff ic ient, or rate of r e t u r n , will be underestimated as a resu l t . It is noteworthy that in a similar situation involv ing weeks worked. Mincer chose to include the additional v a r i a b l e . 5 5 Hanoch, in comparison, allows schooling "the benefit of the doubt . " He ass igns to it all the earn ings covariance mutually explained by school- ing and i n d u s t r y . In the absence of a proper ly specif ied multi- equation model to predict the worker 's indust ry of employment, there is unfortunately no clear test with which to refute this p rocedure . Yet , in the face of Hanoch's rather extreme assumption, it seems only prudent to investigate the alternative case . It may turn out that inc luding indust ry of employment adds little to the explanatory power of the earn ings function and leaves the schooling coeff icient substantial ly unaf fected . From the latter outcome, if it should t ransp i re , one might conclude that industr ia l mobility is not an im- 86 portant factor in real izing the re turns to educat ion. We shall come back to this point in assess ing the empirical results of Chapter III. Meanwhile, let us concede that Hanoch's argument gains c o n - siderable force when appl ied in the case of occupat ion. Without quest ion , occupation and schooling are intimately connected . Empir ic - a l ly , however, the strength of any statistical association will depend on how occupations are de f ined . A classif ication scheme grounded pr inc ipal ly in education will obviously lead to a h igher correlat ion than one based upon industr ia l func t ion . Disequi l ibr ium and "permanent" imperfection in the occupational wage s t ruc tu re are also poss ib le . T h u s schooling and occupation will not be completely interchangeable in account ing for the var iance of ea rn ings . A s in the case of i n d u s t r y , it appears worthwhile to include the questionable factor , occupat ion, in the earn ings funct ion , at least on a provis ional bas is , to establ ish the degree of statistical over lap with schooling and to limit thereby the range of doubt concern ing the independent impact of each var iab le . It is , f ina l ly , somewhat s u r p r i s i n g in view of Hanoch's treatment of indust ry and occupation that he does not recognize geographic mobility as a proximate source of the return to educat ion . B y computing separate rates of re turn for Americans in the North and 56 South , he in effect holds place of residence constant . Yet , one could presumably a r g u e , in the manner of the prev ious quotat ion, that h ighly schooled indiv iduals obtain part of the return on their investment through migration to (or residence in) high-wage areas . 87 Schooling and migration (residence) may be related h ierarchica l ly in the same way as schooling and i n d u s t r y . On the other hand , place of residence may exert its own i n - f luence on earn ings . Geographic immobility may prevent the equal iz- ation of wages in the long and in the short r u n . In some resource- r ich areas labour may succeed in bargain ing economic rents away from rival factors . Whatever the precise c ircumstances, it is unl ikely that all of the re turn to l iv ing in a part icular place will be attr ibutable in the end to school ing. Part will be due to the residence decis ion, just as part of the return to industry and occupation will be due to investment in job search and career p lann ing . Hanoch seems justi f ied therefore, despite the apparent inconsistency of his approach , in ho ld - ing place of residence constant . We shall likewise insert this var iable , along with indust ry and occupat ion, in the expanded earn ings funct ions of Chapter III. In each case, the rationale for inclusion is, f i rs t of a l l , to capture any fundamental disequi l ibr ium present in the earnings s t r u c - ture , as seen from the perspect ive of the schooling model. Forming part of any apparent d isequi l ibr ium may be the equal izing dif ferentials thought to compensate for var ious nonpecuniary items in the employ- ment set t ing . These dif ferentials are the resul t , not of market imperfection, but of markets funct ioning in a smoothly competitive manner. Even so, the three var iables in question may assist in measuring the pecuniary rate of re turn to schooling by impounding statistically the wage dif ferentials associated with nonpecuniary fac tors . Indust ry , occupat ion, and place of residence would appear to be reasonable proxies for many of the factors one could name. The use of these var iables seems especial ly warranted in view of ex ist ing evidence which reveals a s ignif icant correlat ion between nonpecuniary gains or losses and s c h o o l i n g . 5 7 Bias in the schooling coeff ic ient is otherwise a strong poss ib i l i ty . Besides i ndus t ry , occupat ion, and place of res idence, there are a number of census var iables one might think of add ing to the earnings function on an experimental bas is . The list inc ludes: marital status, family membership, family size, rural or urban res idence, per iod of immigration, official language, ethnic g r o u p , re l ig ion, place of highest grade in school, major source of income. In the case of each var iable , it is a simple task to formulate one or more reasonable hypotheses which define some link with earn ings . We shall leave details of such hypotheses to Chapter III. Here , it is suff ic ient to note that if any of the preceding var iables are corre lated with school- ing , their inclusion or omission is bound to affect the schooling coeff ic ient . F ind ing out how the latter responds each time a new variable is added to the earn ings funct ion would appear to be a wor th- while under tak ing . The information der ived from this empirical exercise should place us in an improved position to judge the compact specif ication favoured by most human-capital theor is ts . Normally part of this speci f icat ion, though an "omitted var iable" from the standpoint of the schooling model, is time worked . Since Chapters IV and V deal at length with the issues sur round ing time worked, we need not d iscuss them here , except to mention a few br ie f points which will short ly become s igni f icant . F i rs t of a l l , as soon as we consider variat ion in time worked , it is necessary to d is t inguish between the wage rate and earn ings . So far we have used these concepts interchangeably . Now let us make W stand only for the periodic wage, Y for annual earn ings , and h for the number of per iods worked per y e a r . If W and h are unrelated, we might specify Y. = W.h.u'., or In Y. = In W. + In h. + u., where u. = In u'.. i I I I i I I I i i Accord ing to this simple argument, the elasticity of earn ings with respect to time worked should equal un i t y . If we look upon the schooling model as explaining W, s u b - stitution from (29) implies In Y . = In W Q + r e s. + (1 + 6) - I n h . +u. , . . . .(43) with 6 = 0. In Mincer's research , 6 is nowhere constra ined and always 58 tu rns out to be s igni f icant ly greater than zero . Hence, either the estimation procedure is biased in some way, or wage rates in fact depend upon time worked . These are the questions we shall explore in Chapters IV and V . For now, we may general ly observe that if the wage rate and time worked both depend on personal attr ibutes (other than schooling) for which time worked is an effective p r o x y , then it is reasonable that § should be nonzero . T h e introduction of var iables more closely por t ray ing the at tr ibutes in question should cause 59 its value to decl ine. St i l l , under certain condit ions, 8 may cont inue to exceed zero if an overtime premium f igures heavi ly in the typical r e - muneration formula. In Mincer's regression estimates, time worked is essential ly an ad-hoc insert ion . Appended to the human-capital earn ings funct ions , it great ly increases their explanatory p o w e r . 6 0 Ac tua l ly , time worked proves only a little less important than schooling in the overtak ing set, 2 61 add ing about 0.27 to the value of R . Wherever Mincer achieves his most impressive statistical r e s u l t s — i n those equations for which the 2 R exceeds 0.50—he does so through the insertion of the t ime-worked var iab le . We shall test its performance, us ing Canadian data and the same, s ingle-equation techniques, in Chapter III. T H E P O S T S C H O O L I N V E S T M E N T MODEL Somewhat i ronical ly . Mincer bases his own objection to the schooling model on an omitted-variable argument . He points out in Schooling, Experience, and Earnings that when indiv iduals spend their time acqu i r ing formal educat ion, they ineluctably sacr i f ice , along with income, the opportun i ty to engage in alternative methods of human- 62 capital accumulat ion. Time devoted to schooling obviously limits the time available for such th ings as on-the-job tra in ing and learning by d o i n g . Among indiv iduals of a g i v e n ' a g e , one would consequent ly predict an inverse correlat ion between years of school attendance and the quant i ty of postschool investment. There fo re , in omitting post- 91 school investment from the earn ings funct ion , we bias downward the estimated return to school ing. In this fash ion. Mincer accounts for the small coeff icient thrown up by the simple regression model. Cor rec t ing its probable bias means f ind ing a way to measure postschool investment. Though individuals may sometimes use post- school leisure to augment their human capita l , we normally associate investment act iv i ty with time spent on the job. Cumulative work time or "experience" thus measures potential investment. Measuring realized investment involves two steps . T h e f i rst is to to estimate years of exper ience; the second is to specify the lifetime investment prof i le . These problems occupy the next two subsect ions . T h e th i rd and final subsection in this part s u r v e y s very br ief ly the results obtained by holding postschool investment constant , f i rs t in a parametric, and then in a nonparametric manner. Estimating Years of Exper ience Because ord inary census data provide no direct information on work h istor ies . Mincer chooses as a p roxy for experience the 68 individual 's c u r r e n t age, minus his age at school leav ing . The latter equals mean years of school attendance for those in the ind iv id - ual's schooling category , p lus f ive years , the presumed age at school e n t r y . In effect , Mincer assumes that, between the end of formal schooling and retirement at age s i x ty - f i ve , indiv iduals never take a holiday from the labour force or become unemployed. 92 In the case of pr ime-age males, whose commitment to the labour force is seldom in ter rupted , this assumption is perhaps admissible as a f i rst approximation; but in the case of women, whose labour- force partic ipation tends to be i r regular and d iscont inuous, it is h igh ly inappropr iate . For this reason. Mincer exc ludes women from his data 64 set . The present study adopts the same expedient . Problems in app ly ing Mincer's p roxy to a sample consist ing entirely of males nevertheless remain to be overcome. A l though pr ime- age males seldom desert the labour force, they c learly d i f fer with respect to lifetime unemployment. Such di f ferences are an obvious source of measurement e r r o r . Hence, if we use the suggested p roxy in a linear regression and make the simplest assumpt ion—that its e r r o r s are uncorrelated with any of the accompanying var iables or stochastic t e rms—standard econometric reasoning asserts that the coefficient of "experience" will have a downward b ias . B l inder makes the additional claim that if schooling is the only other independent 65 variable in the regress ion , its coeff icient will have an upward b ias . In fact, this contention is false. It is shown in Appendix MB that as long as schooling and experience are negatively corre la ted , the coeff ic ients of both variables will be underest imated. Actua l ly , as B l inder points out, the s tandard econometric proof does not quite fit the case under d i scuss ion . Owing to the way in which the lifetime investment profi le is usual ly specif ied (see below), the exper ience proxy does not enter the earnings function as a s ingle, l inear regressor . Fur thermore , its measurement e r ro r does 93 not have an expectation equal to ze ro . Because actual exper ience may fall short of but never exceed exper ience as def ined by the p r o x y , the embedded e r ro rs should all be nonnegat ive . In making this comment, however. B l inder fails to notice that the second of two terms used in computing the p r o x y — t h a t is, age at school leaving—may itself be measured with e r r o r . Hence, the d iscrepancy between actual and imputed experience need not always be pos i t ive . In any event , a posit ive expectation does no more than alter the constant term in the regression 66 equat ion. Apar t from the two dif f icult ies mentioned by B l inder , there are other considerat ions which may render the standard econometric proof inappl icable. One is the possibi l i ty that e r r o r s in the exper ience p roxy may be corre lated with the level of school ing. If the latter affects cumulative lifetime unemployment—the most obvious source of measurement e r r o r — i n the antic ipated direction (negat ive ly , in other words) , we must presume an negative correlation of some unknown magnitude. A fu r ther possibi l i ty is that e r ro rs in the proxy may be corre lated with the true level of exper ience, being thus heteroskedast ic . It is only reasonable to suppose that cumulative unemployment will increase along with experience over the individual 's lifetime. Th i s problem, however, will not upset any qualitative conc lus ions . A final consideration is that schooling may be measured with e r r o r . We must concede this poss ib i l i ty , if only because the data are often reported in c lass intervals rather than by specif ic y e a r . 94 Under the foregoing c i rcumstances, we cannot predict the direction of bias in either the schooling or the experience coeff ic ient a 6 7 p r i o r i . Empir ical ly , Malkiel and Malkiel (also c i ted by Bl inder) f ind that the schooling coefficient is biased upward (as B l inder guessed) by about 12% of its "true" (estimated) va lue. On the basis of the argument given in Append ix I IB, one would have to infer that this u p - ward bias is a result of the suspected inverse correlation between schooling and the e r r o r in the experience p r o x y . A s initially forecast , the experience coefficient is biased downward—by about 19%. Spec i fy ing the Investment Profile A f te r settl ing on a proxy for cumulative work time. Mincer proceeds to the second obstacle in estimating postschool investment— that of determining the proport ion of work time devoted in each period to the acquisit ion of human capi ta l . In terms of the theoretical d i s cus - sion presented in Chapter I, the problem is to specify the form of k ' ( p ) , where as before, p is the year of exper ience. Mincer advances two hypotheses : k'(p) = k{j - k'Q • p/T' 0 ^ p ^ T ' . . . .(44a) k'(p) = k;> • e 6 p .(44b) Here, 8 is a posit ive constant; and though T ' may be the date of retirement, it is more general ly the date at which g ross investment falls to zero . The f i rs t equation is a l inear relationship in which the propensi ty to invest falls from k'Q ^ 1 at p = 0 to zero at p = T ' . T h e second equation is a decl ining exponential which originates at k Q but remains positive at p = T ' . Both specif ications seem to have been chosen for their t rac- tabil ity in estimation, since neither of them closely resembles the theoretical investment profi le y ie lded by the income-maximization model. 68 Haley's ve rs ion , for example, implies a functional form with the general propert ies of a th i rd -o rder polynomial in p. Whether (44a) or (44b) might succeed in approximating such an investment profi le is d i f f icult to say . Both satisfy the minimum a pr ior i requirement that k'(p) decline over the life cyc le , but in all other respects , the two equations are ad hoc . The exponential hypothes is , (44b), is fur ther suspect insofar as it does not constra in k'(p) to zero at any point . To der ive estimating equations, one may substitute (44a) and (44b) alternately into the continuous time vers ion of (16'), namely: In W = In W. + ( r ' e - d) s p 0 + In [1 - k ' (p)] . . . . .(45) [ r X k ' ( t ) - d]dt Performing the integration and expanding the last term in a Tay lo r series up to the quadrat ic y ie lds 96 In W p = a + b l S + b 2 p - b 3 p . . . . ( 4 6 a ) w h e r e a = l n W Q - k| j (1 +kJ,/2) b 2 = r x k | j + k ^ l + k'Q) / T ' - d b , = r , e - d b = - r x k ' / 2 T ' - k ' 2 / T ' 2 in t h e f i r s t c a s e , a n d In W = a + b ^ + b 2 e H + b 3 e 2 p - d • p (46b) w h e r e a = In W + r x k j j / B b 2 = - r x k ^ / 6 - k'Q - d *>! = r'e - d b 3 = - ( k J j ) 2 / 2 , in t h e s e c o n d . T h e l i n e a r h y p o t h e s i s t h u s l e a d s to a q u a d r a t i c e s t i - m a t i n g e q u a t i o n , a n d t h e e x p o n e n t i a l h y p o t h e s i s , to a f o r m k n o w n a s t h e " C o m p e r t z c u r v e . " T h e n a t u r e o f t h e q u a d r a t i c s p e c i f i c a t i o n is b e s t a p p r e c i a t e d b y i n s e r t i n g t h e v a r i a b l e s w h i c h u n d e r l i e t h e e x p e r i e n c e p r o x y . If we let A s t a n d f o r a g e , t h e n a c c o r d i n g to t h e d e f i n i t i o n in t h e las t s u b - s e c t i o n , p = A - s - 5. In a t t e m p t i n g to e s t i m a t e ( 4 6 a ) , we a r e t h u s d e a l i n g w i t h In W = a + b 1 s b 2 ( A - 5) + b 3 ( A - 5) = (a - 5b 2 + 25b 3 ) + ( b 1 - b 2 - 10b 3 )s + b 3 s 2 + ( b 2 - 10b 3 )A + b 3 A 2 - 2 b 3 A s . . . . .(47) T h i s result d i f fers somewhat from the traditional earn ings prof i le , an 2 equation in s. A , and A . Mincer argues that the traditional form prov ides an underestimate of the return to school ing, inasmuch as b 2 > 0 . 6 9 Ac tua l l y , as the preceding algebra demonstrates, the rele- vant condition is that b 2 + 10b 3 > 0 — a requirement that nevertheless appears equally true in pract ice . • Secondly , the traditional form ignores a potentially important interaction between schooling and age . One can see from Equation (47) that Mincer's quadrat ic estima- ting function really contains two novelt ies: the use of the interaction term and a restr ict ion on its coeff ic ient . The latter is constra ined to equal - 2 b 3 . In view of the concealed restr ic t ion, it cannot be assumed 2 that adding the interaction var iab le—through the use of p and p 2 rather than A and A —wi l l improve the fit of the equat ion. However, if one were to estimate the second line of (47) expl ic i t ly , it would be possible to test the val idity of the restr ict ion a n d , indeed, the s ign i f i - cance of the interaction term when its coefficient is unconstra ined . Mincer does not examine these two minor statistical quest ions . A s for the problem of bias in the estimated return to school ing, it is l ikely true that subst i tut ing a quadrat ic in p for the traditional quadrat ic in A will increase the schooling coeff ic ient; but this effect 98 is pure ly mechanical . It must occur , g iven the way in which p has been de f ined . A n independent measure of exper ience might not lead to the same resu l t . In any event , one should be careful not to c o n - fuse the downward bias flowing from the alleged misspecification of the earn ings function (the use of A instead of p) with that ar i s ing from the outr ight omission of exper ience . Since A and p are bound to be h ighly corre lated , the second form of bias is potentially the more severe . The exponential hypothes is , implemented through (46b), does not lend itself so easily as the quadrat ic specification to comparison with the traditional earn ings funct ion . We shall be content, therefore, merely to review its performance in estimation. T h e Empirical Outcome Before we examine Mincer's quantitat ive resul ts , note that while (46a) y ie lds to the standard l inear-regress ion approach , (46b) is more demanding. Because of the v e r y large sample Mincer employs, h ighly sophistocated nonlinear techniques are no doubt impractical . Unders tandab ly , he resorts to d irect trial and e r r o r . Ass ign ing d i f fe r - ent values to 8, he computes a series of linear regress ions and chooses 2 the one (or the pa i r , as it turns out) with "the highest R and the most plausible coeff icient [ s ] . " 7 0 Unfor tunate ly , we cannot look at Mincer 's reported regress ions (see Append ix 11 A) and compare precisely the empirical performance of his two competing hypotheses . No two equations d i f fer only in this one 99 aspect . It does appear that the exponential form holds an advantage, a lthough the d i f f e rence—perhaps 2 or 3 percentage points in the value 2 71 of R — i s rather s l ight . Both models explain roughly 30% of the var iance in annual ea rn ings . The real advantage of the exponential form lies in its abi l i ty to identify the parameters r x , kg, and d . Once the latter has been estimated from the coefficient of the linear term in p—cal l the estimate d—the definit ions of b 2 and b 3 g ive us two equations in two unknowns, r and k Q . From the estimates b 2 and b 3 we may thus compute r and 72 kg . Mincer's results imply values of 12.1% and 0.54, respect ive ly , with d equal to 1.2%. A l though the estimate of kg is well below unity (the value implied in Haley's theoretical model), Mincer cons iders it "rather h i g h . " He accepts without comment the estimate of d , though a lack of inter- pretation here may be somewhat misleading. If it is t rue , as a rgued in the prev ious sect ion, that depreciation and growth are ind is t ingu ish- able except in algebraic s ign , then the coeff ic ient labelled d must really measure (d - g) rather than d alone. Growth at an assumed rate of 2 . 5 - 3.0% would therefore mean depreciation at the rate of 73 3 .7 -4 .2%. Johnson and Hebein, with data able to d is t inguish growth and deprec iat ion, encounter values of d in the range 1 . 0 - 3.4%. 74 Haley's estimates reach 4.3%. T h u s Mincer's f ind ing remains c red ib le , even though cons iderably inflated by the suggested re- in terpretat ion . Fortunate ly , we do not require a d ist inct estimate of d , but only the exist ing composite d , in order to compute values for the other p a r a - meters, r x and k^. 100 T u r n i n g to the quadrat ic specif ication (46a), we see that, unl ike the exponentia l , it does not allow us to identify any of the pa ra - meters. The definit ions of b 2 and b>3 represent two equations in four unknowns, r x , k'Q, T 1 , and d . Mincer purpor t s to eliminate one u n - known (d) by express ing the model in net terms; that is, he ignores d and subst itutes k(p) and T in place of k'(p) and I' in Equation ( 4 5 ) . 7 5 T h i s procedure raises no d i f f icu l ty in the case of the integral , but in the case of the f ina l , logarithmic term it appears inva l id . The logarithmic term, it will be recal led, por t rays the gap opened between measured and potential earn ings on account of cu r ren t investment in human- cap i ta l . T h i s gap must sure ly depend upon gross rather than net investment. Mincer's procedure seems legitimate only for the special case in which depreciation equals zero . T h e n , gross and net investment are the same t h i n g . If one were prepared to assume zero deprec iat ion, it would at f i rst seem possible to identify the remaining parameters; for in this s ituation, measured and potential earn ings reach an identical maximum where p = T = T ' . One may locate maximum measured earn ings by d i f ferent iat ing Equation (44a) with respect to p and setting the result equal to zero . The solution y ie lds p = T = - b 2 / 2 b 3 . In this way. Mincer's publ ished regression coeff ic ients imply that when weeks worked are free to v a r y , earn ings peak at 33.8 years of exper ience, and that when weeks worked are held constant , earn ings peak at 37.8 y e a r s . Insert ing these values for T in the equations def in ing b 2 and b_ leads, however, to an inadmissible solution for r and kn. 101 T h i s outcome is by no means inexpl icable. In the f i rst place, it seems unl ikely that depreciation is in fact equal to zero . Yet , if it were, one would have to recognize that under such c i rcumstances, measured and potential earn ings attain not so much a peak as a plateau, since in the absence of depreciation there is no reason for earn ings (wage rates) to dec l ine . It follows that unless T is actually v e r y near retirement, the quadrat ic functional form may be inappropr iate . In pract ice . Mincer dec ides—arb i t ra r i l y it seems—to let T equal twenty years with weeks var iable and th i r ty years with weeks held constant . Myster ious ly , however, his publ ished est imates— r x = 6.31 and k Q = 0.58 in the f i rst case, r x = 11.9% and k Q = 0.42 in the second—seem in arithmetic accord only if T were to equal 20.6 years and 33.1 years r e s p e c t i v e l y . 7 6 In view of the theoretical problems just d i scussed , one cannot in any event put great store in the preceding resu l ts . A s pred ic ted , the insertion of exper ience has a dramatic effect on the schooling coeff ic ient . With postschool investment held constant in this parametric fashion, the estimated return to schooling increases from 7% in Mincer's Equation ( S i ) to about 11% in Equations ( P 1 ) - ( G 4 ) . The exact specif ication of the investment profi le has little bear ing on the resu l t . There is, however, an even simpler method of holding post- school investment constant , and that is to cons ider only those indiv iduals at a g iven stage of the life c y c l e . Mincer argues that the appropr iate stage occurs at the point of o v e r t a k i n g . 7 7 A t the 102 overtak ing year of exper ience ( p ) , the individual earns , by def in i t ion, prec isely the amount he would have received had he not engaged in any postschool investment. Hence, the earn ings di f ferentials observed within the overtak ing set or cross-sect ion are due entirely to d i f f e r - ences in school ing. Rates of re turn computed from these earn ings di f ferentials will thus be free of b ias . We can obtain the "unbiased" estimates from the schooling model, prov ided we know the approximate period of over tak ing . A s explained in Chapter I, p i 1/r . Mincer assumes: (a) that the preceding relationship holds with equality and (b) that r x = r e . T h u s if r e were equal to 12.5% p would equal 8 y e a r s . A f te r some experimentation Mincer settles on a c ross section of individuals with 7 - 9 years of exper ience, produc ing Equations ( V I ) - ( V U ) . Cons istency demands that the rates of re turn estimated from these regress ions equal approximately 12.5%. £ In Equation ( V I ) , with weeks worked free to v a r y , r = 16.5%; in £ Equation ( V 2 ) , with weeks worked held constant , r =12.1%. T h e latter estimate is consequent ly the more pleasing of the two. However, both yield the hoped-for increase in the rate of r e t u r n . £ The weeks-constant estimate of r satisfies a fu r ther cons is - tency requirement in that it comes close to Mincer's estimates of r . Had there been a large d i sc repancy , the definition of the overtak ing set would have been suspect . A t the same time, theory demands that e x r = r at the margin; otherwise, the individual would not choose the level of schooling actual ly o b s e r v e d . Since Mincer assumes that r e and r x are constant , we must have equal ity as well in the estimated averages . We shall look for this cons istency proper ty in the results of Chapter III. 103 T H E G E N E R A L MODEL Though the empirical work of this study pertains solely to the special models of human-capita l accumulation which we have already cons idered , it will be helpful in assess ing and categoriz ing the present effort to examine, ve ry br ie f l y , the implementation of the "general model ." A t the level of theory , the general (Ben-Porath) model promises an integrated treatment of schooling and on-the-job t ra in ing . However, when we come to implementation, this potential remains substantial ly unfu l f i l l ed . So far , researchers have been forced to apply the concepts of the model to homogeneous educational g roup ings , estimating dist inct sets of parameter values in each case . What surv ives of general i ty must be found in the relatively wide c lass of postschool earn ings p r o - files which the model can suppor t . T h e pr inc ipal studies in the f ield are those of Ben-Pora th , 78 Heckman, B rown, Haley, and Moreh . We have already noted in the preced ing pages some of their quantitat ive resu l ts . Instead of merely assuming a convenient trajectory for postschool investment, this line of inqu i ry rests upon the deeper microeconomic foundation of a product ion function for human-capita l . Not unexpectedly , therefore, the estimating equations turn out to be inherently nonl inear . T h e studies named utilize a var iety of nonl inear methods. These d i f fer chief ly in the parameters which the respect ive authors choose to specify rather than estimate. T h u s Heckman f ixes the discount rate; B rown , the rate of deprec iat ion; Moreh, the product ion parameter ( y ) and the age of ret irement. Haley frees all the parameters but cannot Identify the ent ire set. His 79 estimating equation is by far the most complex of those s u r v e y e d . T h e values which it can d ist inguish are general ly plausible, and on this g r o u n d the Ben-Porath model der ives suppor t . T h e other studies turn up contradict ions . A notable feature of the preced ing work is the small number of variables which it employs. As ide from the personal attr ibutes (school ing and sex) which help in def in ing the var ious subsamples, only earnings (or their rate of change) and some var iant of calendar time (either age or experience) take part in the calculat ions. T h e authors listed above all t ry to advance the basic model, not by cap tur ing and insert ing new information through the use of additional var iab les , but by estimating increasingly complex functional r e p r e - sentations of the earn ings prof i le . Despite the theoretical basis for this research , one is tempted to label it "curve f i t t i n g . " T h e problem res ides, no doubt , in the pract ical limitations which beset nonl inear estimation p rocedures . These do not readily admit large data matrices. Because of the consequent need to restr ic t sample s izes, it is ve ry di f f icult to treat general populat ions, which manifest considerable d i v e r s i t y . In small samples that are r ich ly categor ized , the cell f requencies often fall too low to g ive meaningful resu l t s . Even with a restr ic ted sample, the researcher may not be able to include all the var iables of interest . T h e choices are therefore c lear . One may settle for the r igourous estimation of a few hypothetical parameters, as in the case of Haley and the rest ; o r , one may sacr i f ice some degree of r igour adopt an approximate specif ication for the human-capital investment prof i le , and pursue a broad investigation of the earn ings s t ruc ture T h i s s tudy takes the latter approach . APPENDIX MA T A B L E 1 MINCER'S REGRESSION R E S U L T S a Equations (dependent var iab le : In W) (S1) (PI) (P2) Main Sample:c 7.58 + .070s (43.8) 6.20 + .07s + .081p - .0012p' (72.3) (75.5) (55.8) 4.87 + .255s - .0029s - .0043ps + .148p - .0018p^ (2.34) (7.1) (31.8) (63.7) (66.2) (P3) f (D ) + .068p - .0009p 2 + 1.207 In h S (13.1) (10.5) (119.7) (G1a) 7.43 + .110s - 1 . 6 5 1 3 e " , 1 5 p (77.6) (102.3) ( G i b ) 7.52 + .113s - 1 . 5 2 e " ' 1 0 p (74.3) (101.4) (G2a) 7.43 + .108s - 1 . 1 7 2 e " , 1 5 p - . 3 2 e " 2 ( , 1 5 ) p + 1.183 In h (65.4) (16.8) (10.2) (105.4) (G2b) 7.50 + .111s - 1 . 2 9 e " ' l 0 p - . 1 6 2 e " 2 ( * 1 0 ) p + 1.174 In h (65.0) (3.5) (G3) f (D J + 1.142 In h (16.0) (107.3) s ,P (108.1) (C4) 7.53 + .109s - 1.192e ' 1 ° P - .146e 2 < - 1 0 ) P - .012p + 1.155 In h ( n . a . ) ( n . a . ) ( n . a . ) (2.4) ( n . a . ) Overtaking Set:** (VI) 6.30 + .165s (26.5) (V2) 1.89 + .121s + 1.29 In h (24.6) (30.6) .067 .285 .309 .525 .313 .307 .546 ,551 ,557 ,556 ,328 ,596 106 107 (Table 1 - continued) Equations (dependent var iab le : In W) (V3) 4.78 + .424s (10.0) .010s (6.1) .347 (V4) 1.60 + .183s (5.3) .002s + 1.270 In h (1.7) (29.7) .602 Source : Schooling, Experience, and Earnings, p. 92, Table 5.1, and p. 53, Tab le 3.3. F igures in parentheses are t ratios, written in absolute terms. Original notation has been changed to conform with that employed in the c u r r e n t text . T h e symbol D refers to a vector of dummy var iables for schooling and exper ience. 28,678 observat ions on white, nonfarm, out-of-school males with e x p e r i - ence not exceeding 40 y e a r s . 2,124 observat ions on similar individuals with 7-9 years of exper ience . APPENDIX MB BIASES IN T H E EARNINGS F U N C T I O N DUE T O E R R O R S IN T H E M E A S U R E M E N T OF E X P E R I E N C E Let us suppose that the true earnings function is where Y = XB + u , In W, In W n S 1 P i s p n *n 8 = (IIB.1) A s in the text , W stands for wages or earn ings , s for school ing, and p for exper ience—al l scaled here in deviations from their respect ive means. The d isturbance vector u is assumed to have the classical propert ies = 0 , • E(uu') = a 2 I , E (X 'u) = 0 . . . . (I IB. 2) Suppose now that we observe Y = Y and X = X + V where V , the matrix of measurement e r r o r s , is given by 108 109 V = 0 v . 0 v whence X = S 1 P i + V 1 s p + v n r n n Hence, p is the only variable measured with e r r o r . 1 We shall assume that E (V 'u) = 0. It follows that E(X 'u) = E ( X ' u + V'u) = 0 (IIB.4) Subst i tut ing into ( I I B . 1 ) , we obtain Y = ( X - v ) 8 + u = X B + u - V B ( M B . 5 ) Under these condit ions, an ord inary least-squares regress ion of Y on X will yield the estimator 0, for which the expectation is E[§] = EUX'XTVY] E [ ( X ' X ) 1X'(X6+ u - V8)] 8 - E t X ' X J ^ X ' V B ] . .(I IB . 6) Let us use B = [B , B ]' to represent the asymptotic bias in 8. 5 p A c c o r d i n g l y , 1, E r r o r s in Y merge with the components of u if we assume for them the same correlat ion proper t ies . There fo re , nothing essential is lost by letting Y = Y. 110 B = plim [ 3 - 3 ] plim [ -(X 'X) V V 3 ] plim - plim x 11 x 1 2 x 1 2 x 2 2 X 11 X 1 2 l X 1 2 X 2 2 J 1 n p, + v , • • • p + v r l 1 *n n Es.v./n i i (Zp.v +Zv. )/n 0 v , 0 v n . 6 P . • 6 P ' . (I IB . 7) where the x.. are elements of (X 'X) 1 In. that: On the basis of arguments g iven in the text , we may hypothesize Zs.v. plim — £ 0 ^ n 3 p > 0 plim Zp.v. n > 0 plim £ P : P : < 0 (I IB. 8) In addi t ion, it is obvious that plim Zv./n > 0. We assume that the pre- ceding asymptotic var iance and covar iances converge to finite limits. Now, from (IIB. 7 ) , B = - plim 1 | x n Zs.v. + x 1 2 ( Z p . v . + Zv 2 ) J • 6 p ; . . ( I I B . 9 ) I l l B p = - plim i j x ^ Z s . v . + x 2 2 ( Z p . v . + Iv.2) j • 6 p (IIB.10) T o sign these express ions , we must investigate the elements of (X'X) n. There fore , observe that Z ( p . + v . ) 2 / n -(ZSjPj + ZSjV.)/n 2 -( Zs.p. + Zs^.) In Zs. / n . . . . (MB.11) (X'X) ' = ' n n |x ' x| Since (X'X) is a posit ive definite var iance-covar iance matrix with a pos i - tive determinant, it follows with the help of our hypotheses that X 1 T X 1 2 > 0 a n d x 1 2 > 0 We now have all the requ i red information. From ( M B . 9) and (IIB.10) it is apparent that if Es.v. = 0 asymptotical ly, both 8 g and 3 will have a downward bias. A posit ive correlation between p and P v makes this bias more severe . On the other hand , if £s.v. < 0 , the bias in both coeff ic ients is indeterminate, assuming we do not know the magnitudes of the correlat ions invo lved . Within the framework explored 2 here , the schooling coefficient Bg may have the upward bias suggested b y B l inder only as a result of some negative correlat ion between s and v . 2 A sl ightly more general model has been put forward b y Maurice 0. L e v i , " E r r o r s in the Variables Bias in the Presence of Cor rec t ly Measured Var i ab les , " Econometrica, XLI (September, 1973), 985-986. T h i s der ivat ion admits any number of independent var iables but y ie lds essential ly the same results as encountered here . 112 It is of course well known that if more than one independent variable (in the present context , schooling) is measured with" e r r o r , then no qual itat ive conclusions are poss ib le . However, in the two- 3 variable case, Thei l has prov ided a helpful approximation formula, which in our c u r r e n t notation reads as follows: B. = ^-5 (9.6. - pe.B.) j # k = s, p, . .(MB.12) J T _ P 1 J J K K where p is the correlat ion coeff icient l inking s and p and 6. is the ratio of the e r ro r var iance in j to the var iance of the true var iab le . Ceter is par ibus , it would seem that e r ro rs in the measurement of schooling tend to lower both B and B , since 6 and B are posit ive and p is s p s p negat ive. 3 H . T h e i l , Economic Forecasts and Policy (Amsterdam: North- Holland Publ ish ing Company, 1961), p. 329. N O T E S C H A P T E R 2 ^Human Capi ta l , p. 159. 2 For convenient reference, all of Mincer's reported r e g r e s - sions have been reproduced in Append ix 11 A , which follows this chapter . See Equation ( S I ) . 3 The best known examples a re : Becker , Human Capi ta l ; Hansen, "Total and Private Rates of Return to Investment in School ing"; Hanoch, "An Economic Ana lys is of Earn ings and Schoo l ing ." 4 The exceptions occur at v e r y h igh and at v e r y low levels of educat ion. Accord ing to Hanoch, o p . c i t . , marginal re turns in the elementary grades sometimes exceed 1001, whereas, marginal re turns to graduate education are 7% or less. ^Incomes of Canadians, p. 42, Table 5.9. 6 R e t u r n s to Investment, p. 100, Table 5.14. 7 ^e ^e In the following express ion r^ and r̂  , are the estimated co- eff icients of s and s , respect ive ly . o See the Append ix H A , Equations (V3) and (P2) . "Equation ( V 4 ) . 1 0 S c h o o l i n g , Exper ience , and Earn ings , p. 54. 1 1 " H u m a n Capital T h e o r y , " p. 838, n . 16. 1 2 A p p e n d i x 11 A , Equations (V2) and ( V 3 ) . Note that these are marginal rates, the f i rst having been assumed constant and therefore equal to the average . T h e mean level of schooling is g iven by Mincer as 12.2 y e a r s . 113 114 '"In the simple case, with the rate of re turn assumed constant , r e se rved to represent the average over years of schooling and over individuals populating the var ious schooling g r o u p s . With the rate of re turn allowed to v a r y , the average, as opposed to the marginal r e t u r n , for individuals with s years of schooling is g iven by •s (reQ + 2 r e t ) dt Is = reQ + r e s , and the population mean is •oo Jfl ( r e + r e s) f (s) ds 0 0 1 where f(s) is the proport ion of individuals with less than s years of school ing. 1 4 Haessel and K u c h , "Earnings in C a n a d a , " employ a three- step, nonl inear, iterative procedure to circumvent the heteroskedast ic i ty problem. T h e y do not report the extent to which the result ing maximum-likelihood estimates d i f fer from those produced by ord inary least squares . 1 5 V i e w e d in detai l , the dependence of schooling on earn ings may ar ise in several ways . As noted, earnings act on school atta in- ment through the individual's investment response. If schooling is a normal consumption good as well as a repository of investment, individuals expect ing (and later realizing) high earnings will make large "purchases . " If the capital market is imperfect, initial earning capacity (embedded empirical ly in W.) may constra in both consumption and investment. A s pointed out in1 a s l ightly d i f ferent context by C . S . Tol ley and E. O lsen , "The Interdependence between Income and Educat ion ," Journal of Political Economy, L X X I X (May/June, 1971), 460-480, the preced ing considerat ions apply not only to indiv iduals but also to communities. Wealthy jur isdict ions will spend more on education then poor ones, re inforc ing individual tendencies. 16. ,, . . dem , , sup . . . In the express ions L ( s ) and L ( s y , the subscr ipt s in parentheses furn ishes a reminder that we are really measuring d i f ferent types or categories of labour on a single L ax i s . B y including s in the argument lists of (33) and (34), we are thus able to treat compactly what is essential ly a multimarket problem. Including mean earning or wage rates for the d iscrete labour types serves to emphasize the theoretical belief that quantit ies demanded and suppl ied depend on the ful l set of such rates^ A l ternat ive ly , we could have inserted the c o n - tinuous function W= W ( s ) . In this case, (33) and (34) become funct ionals . Note that in (33) demanders observe the true market averages . Imperfect knowledge on the part of demanders adds nothing of interest to the following ana lys is . 115 At this point there is no need to be v e r y specif ic about how expectations are formed. We need only be assured that expected wages respond to changes in actual market rates . In this static system we ignore whatever lags may be invo lved . 18 We assume the existence of the multimarket equi l ibr ium which this locus represents . 19 A well known result in regress ion theory states that the product of the estimated slope coeff icients must equal the square of the correlat ion coefficient between the two variables in quest ion . Agreement in the estimate of r will thus occur only if the correlation between s and W is per fec t . 20 e e e e Making r = VQ + r 1 s, with r i < 0, does not change the present analys is , except that (39) below no longer prov ides an expl ic it solution for s.. i 21 We assume that (35) captures individual expectations as well as the aggregate relationship or iginal ly p o r t r a y e d . 22 e T h i s conclusion is unaffected by making r depend on s in the manner proposed above. If we ignore the term l n [ r j / r e ] (either because it is small or because it vanishes when rj = r e ) , an expl ic it solution for s\ takes the form s. = -r* ± [ ( r * ) 2 - 4 r e { - - - + w. " - } ] V 2 Inspection will show the posit ive square root to be the relevant one. A c c o r d i n g l y , d s./dw. < 0 . 23 Zvi C r i l i ches , "Estimating the Returns to School ing: Some Econometric Problems," Econometrica, X L V (January , 1977), 1-22. T h e degree of bias may be inferred by comparing Tables I and IV. 24 I b i d . , p. 13. 25 See Tsuneo Ishikawa, "Family S t ruc tures and Family Values in the T h e o r y of Income D i s t r ibu t ion , " Journal of Political Economy, LXXXIII (October , 1975), 987-1008. 26 The latter include such things as unemployment and job vacancies, which may signal ensuing disequi l ibr ium adjustment of wages and incomes. See T h e Market for Co l lege-Tra ined Manpower, p p . 8-10. 116 27 I b i d . , p p . 59-60. In so do ing . Freeman concurs with Theodore W. Schu l tz , who earl ier suggested that uncerta inty about future earn ings was so great that indiv iduals could not possibly refer to anyth ing but c u r r e n t wages in determing their investments. See "The Rate of Return in Al locating Investment Resources to Educat ion ," Journal of Human Resources , II (Fa l l , 1967), 293-309, e s p . p p . 303-305. 28 See n . 3 above. 29 ( Misspecif ication through the use of an incorrect functional form is not something about which we can speculate with any assurance . 30 Str ic t ly speak ing , of course , we cannot determine how cur rent wages might appear without spec i fy ing in full the under ly ing production funct ion(s) and without ascertaining the regional and industr ia l pattern of output demand. However, the direction in which wages may appear to respond is in no way crucia l to the present argument . 31 We ignore, as usua l , the f initeness correct ion [ , - e ^ V 0 ) T ] / [ 1 . e - ( r e - g ; ) ( T - s ) ] . 32 Note the sign reversa l in comparison with (16') . Because the latter is essential ly an accounting formula, d enters there with a negative effect on earn ings . By the same logic, g would appear with a posit ive s i g n . T h e equi l ibrat ing function is performed, if at a l l , by r ' e . In (11) r 1 is assumed f i xed , and base-per iod ( i . e . , cur rent) earnings make the necessary adjustment. Since these move in compen- satory fash ion, they r ise with an increase in depreciation and fall with an increase in expected g rowth . 33 Herman P. Mil ler, "Annual and Lifetime Incomes in Relation to Educat ion ," American Economic Review, L (December, 1960), 962-986. 34 I b i d . , p. 73. 35 See, in part icu lar , Thomas Johnson , "Returns from Invest- ment in Human C a p i t a l , " American Economic Review, LX (September, 1970), 546-560; and Canada, Stat ist ics Canada , Economic Returns to Education in Canada . T h e latter assumes a growth rate of 2.5%. 117 ""Thomas Johnson and Freder ick J . Hebein , "Investment in Human Capital and Personal Income, 1956-1966," American Economic Review, LXIV (September, 1974), 604-615, Table 1. 37 "Estimation of Earn ings Prof i les ," p. 1233, Table III. These and the preced ing estimates appear to depend on how successfu l the authors are in accounting for endogenous growth through postschool investment. 38 Suppor ters of the human-capital doctr ine tend, natura l ly , to emphasize schooling and to minimize the role of all factors that are outside the individual 's cont ro l . For a su rvey of the arguments see: F. Thomas Jus te r , "Introduction and Summary ," in Educat ion, Income, and Human Behavior , edited by F. Thomas Juster (New Y o r k , McGraw- Hill Book Company, Inc . , 1975); or Sherwin Rosen "Human Cap i ta l : A S u r v e y of Empirical Research" (Discu feion Paper 76-82, Department of Economics, Un ivers i ty of Rochester , 1976). Note especial ly Zvi Gr i l iches and William M. Mason, "Educat ion, Income, and A b i l i t y , " Journal of Political Economy, L X X X (May/June, Supplement, 1972), S74-S103, and Samuel Bowles, "Schooling and Inequality from Generation to Generat ion ," Journal of Political Economy, L X X X ( May/ June , Supplement, 1972), S219-S251. 39 See the fol lowing: Herbert G int is , "Educat ion, Techno logy , and the Character is t ics of Worker P roduc t i v i t y , " American Economic Review, LXI (May, 1971), 266-279; Paul J . Taubman and Terence J . Wales, "Higher Educat ion, Mental Ab i l i t y , and S c r e e n i n g s , " Journal of Political Economy, LXXXI ( January/February , 1973), 28-55; Kenneth J. A r r o w , "Higher Education as a F i l te r , " Journal of Public Economics, II (Ju ly , 1973), 193-216; R ichard L a y a r d and George Psacharopoulos, "The Screen ing Hypothesis and the Returns to Educat ion ," Journal of Political Economy, LXXXI I (September/October, 1974), 985-998; J. E. St ig l i tz , "The T h e o r y of Sc reen ing , Educat ion, and the D i s t r i - bution of Income; American Economic Review, L X V (June, 1975), 283-300; John G . Ri ley, "Information, Sc reen ing , and Human Cap i ta l , " American Economic Review, LXVI (May, 1976), 254-260. 40 On this account Becker deflated the ra te -o f - re turn estimated in Human Capital by 20%. Following Edward F. Denison, T h e Sources of Economic Growth and the A l ternat ives Before Us (New Y o r k : Committee for Economic Development, 1962), Bert ram, T h e Contr ibut ion of Education to Economic Growth , appl ied a deflator of 40% to the Canadian data . See Gr i l i ches and Mason, op . c i t . 118 42 School ing, Exper ience , and Earn ings , p. 139. 43 John C . Hause, "Earn ings Prof i le: Ab i l i ty and Schoo l ing , " Journal of Political Economy, L X X X (May/June, Supplement, 1972), S108-S138. 44 See G int is , op . c i t . , and Finis Welch, "Human Capital T h e o r y : Educat ion, Discr iminat ion, and Life C y c l e s , " American Economic Review, L X V (May, 1975), 63-73. 45„ O p . c i t . 4 c Returns to Investment in Un ivers i ty T r a i n i n g , p p . 70-75. H / J . Behrman, Paul J . Taubman, and Terence J . Wales, " C o n - trol l ing for and Measur ing the Effects of Genetics and Family E n v i r o n - --" ment in Equations for Schooling and Labor Market S u c c e s s , " in Kinometr ics: T h e Determinants of Socioeconomic Success Within and Between Families, edited by Paul J . Taubman (Amsterdam: N o r t h - Holland Publ ish ing Company, 1977). 48 Paul J . Taubman and Terence J . Wales, "The Inadequacy of Cross-Sect ion A g e - E a r n i n g s Profi les When Abi l i ty is Not Held Cons tant , " Annals of Economic and Social Measurement, I ( Ju ly , 1972), 363-370. , 49 "Estimating the Returns to Schoo l ing , " p p . 4-6. 50-. .. „ O p . c i t . , p. 67. 3 ' A m o n g the most prominent a re : John T . Dunlop, "Produc t i v - ity and Wage S t r u c t u r e , " in Income Employment a n d Public Policy (New Y o r k : W.W. Norton 5 C o . , I n c . , 1948); Sumner , H . S l i chter , "Notes on the S t ruc ture of Wages," Review of Economics and Stat ist ics , XXXII ( F e b - r u a r y , 1950), 80-91; Joseph Garbar ino , "A T h e o r y of In ter indust ry Wage S t r u c t u r e , " Quar ter ly Journal of Economics, LXIV (May, 1950), 283- 305; Leonard E. Weiss, "Concentrat ion and Labor E a r n i n g s , " American Economic Review, LVI (March , 1966), 96-117; Stanly H . Masters , "Wages and Plant S ize : A n Inter industry A n a l y s i s , " Review of Economics and Stat ist ics , LI ( A u g u s t , 1969), 341-345; Michael L. Wachter, "Relative Wage Equations for United States Manufac tur ing , 1947-1967; Review of Economics and Stat ist ics , LN (November, 1970), 405-410; W. Hood and R . O . Rees, " Inter - Indust ry Wage Levels in United Kingdom Manufactur- i n g , " Manchester School of Economics and Social S tud ies , XLII (June , 1974), 171-183. 119 O p . c i t . 53 C . T . Haworth and D.W. Rasmussen, "Human Capital and Inter-Industry Wages in Manufac tur ing , " Review of Economics and Stat ist ics , LMI (November, 1971), 376-380" " p. 312. "An Econometric Ana lys is of Earn ings and Schoo l ing , " 5 5 S e e p. 66 above. O p . c i t . 5 7 S e e Greg J . Duncan , "Earnings Funct ions and Nonpecuniary Benef i t s , " Journal of Human Resources, XI (Fa l l , 1976), 462-483; and Robert E. B . Lucas , "Hedonic Wage Equations and Psychic Wages in the Returns to Schoo l ing , " American Economic Review, L X V M (September, 1977), 549-558. 5 8 S e e Append ix 11A. 59 T h i s explanation does not easily apply in the case of Mincer, who uses weeks rather than hours as the empirical counterpart of h . 6 °A fur ther motive for inc lus ion, as we have seen, is to cancel variation in the rate of re turn to school ing. 6 1 S c h o o l i n g alone explains about 33% of the earn ings var iance . See Equations (VI) and ( V 2 ) . 6 2 S e e p p . 45-47. 6 3 School ing, Exper ience , and Earn ings , p. 84. 64 Haessel and K u c h , "Earnings in C a n a d a , " include women but subtract from exper ience a constant number of years for each ch i ld b o r n . For other approaches see Jacob Mincer and Solomon W. Polachek, "Family Investment in Human Cap i ta : Earn ings of Women," Journal of Political Economy, LXXXI I (March/Apr i l , Supplement) , S76-S108; and Solomon W, Polachek, "Dif ferences in Expected Post- School Investment as a Determinant of Market Wage D i f ferent ia ls ," International Economic Review, XVI (June, 1975), 451-470. 120 65 "On Dogmatism in Human Capital T h e o r y , " p. 14. 66 Observe_that indiv idual measurement e r r o r s can always be written in the form v + v. , where v represents the mean. If the latter exceeds zePo, th'e3 mean leve f̂ of experience will be inflated by a corresponding amount; but this distort ion will not affect the value of any slope coeff ic ients . 6 7 "Male-Female Pay D i f ferent ia ls ," Tables 1 and 2. 68 "Human Cap i ta l : T h e Choice between Investment and Income," p. 937, F igure 5. 6 9 S c h o o l i n g , Exper ience , and Earn ings , p. 84. 7 0 1 b i d . , p. 93. T h e favoured values of 8 are 0.10 and 0.15. Mincer reports that: "While R 2 changes little in a wider internal , the partial repression coeff ic ients are sensit ive to the specification of 8." 7 1 T h e nearest comparison is probably between (PI) and ( C l a ) or ( C 1 b ) , or between (P3) and (C2a) or ( C 2 b ) . 72 School ing, Exper ience , and Earn ings , p. 94. 73 "Investment in Human Cap i ta l , " p. 610, Table 1. 7 Z | "Est imation of the Earn ings Prof i le ," p. 1233, Table III. 7 5 S e e School ing, Exper ience , and Earn ings , Chapter 4. 76 I b i d . , p. 94. Except for the last f igure , these numerical results appear incorrect . T h e reader may wish to v e r i f y , us ing k Q = b 2 T + 2 b 3 T 2 and r x = b 2 / k Q - ( 1 + k Q ) / T 1 that the reported parameter estimates, together with the assumed values of T imply the following r x = 4.0% and kg = 0.66 in the f i rst case; r x = 11.5% and kg = 0.42 in the second. Only in the second case is the d iscrepancy small enough to be attr ibuted to rounding e r r o r . 7 7 l b i d . , p p . 47-49. 121 78 Ben-Pora th , "The Production of Human Capital Over T ime"; Heckman, "Estimates of a Human Capital Production Funct ion"; B rown, "A Model of Optimal Human Capital Accumulat ion"; Haley, "Estimation of the Earn ings Prof i le"; Moreh, "Investment in Human Capital over T i m e . " In fact, Haley's specification must surely be one of the most complex ever to appear in the econometric l i terature. See o p . c i t . , p p . 1228-1229, Equations (9) and (13). C H A P T E R III T H E EARNINGS F U N C T I O N : S I N G L E - E Q U A T I O N E S T I M A T E S FOR C A N A D A T h i s chapter has two main object ives. T h e f i rs t is to present a series of estimates which reproduce with Canadian data the study of earn ings funct ions carr ied out for the United States by Jacob Mincer . T h o u g h it is not everywhere prudent , g iven the multiple aims of the cur rent s tudy , or possible, g iven the data, to imitate Mincer's methods exact ly , the procedures employed here yield results that are reasonably comparable. Some of the resu l ts , as we shall see, are v i r tua l ly . identical to Mincer 's ; others are s t r ik ing ly d i f ferent . The second objective pursued in this chapter is to extend Mincer's investigation by adding to the earn ings function var iables which do not arise within a str ict human-capital framework. Obv ious ly , there are a number of factors besides school ing, exper ience, and weeks worked which influence the level of earn ings . It is useful to isolate these factors statistically and to measure their relative importance, even though the associated hypotheses remain ad hoc . Omitting them cou ld , if nothing else, bias the estimated coeff ic ients of the human-capital var iab les . Whether or not any potential for bias actually ex is ts , the expanded earnings funct ions appear to 122 123 offer the best empirical s tandard against which to judge the performance of Mincer's undi luted human-capital speci f icat ion. In the same way, these single-equation estimates serve as a basis of comparison for the system estimates reported in Chapter V . The rest of the cu r ren t chapter is d iv ided into three sect ions. T h e second and th i rd d i scuss , respect ive ly , a Mincer- l ike set of human- capital regressions and a contrast ing group of earnings funct ion- estimates, expanded in the ways suggested ear l ier . Before we look at these empirical resu l ts , however, it is necessary to review the data and the methods which underl ie them. Accord ing ly , the f i rst section below descr ibes in detail the pr inc ipal data source used in compiling * this s tudy , the choices made in drawing the requ i red sample, and the procedures followed in def in ing the many var iab les . Throughout this prel iminary d iscuss ion , we shall take special note wherever an adopted procedure confl icts with one employed by Mincer . T H E D A T A , T H E S A M P L E , A N D T H E V A R I A B L E S T h e Principal Data Source Al l the basic information used in this study originates with the 1971 Census of Canada . Except for one special tabulation, all of it comes, speci f ica l ly , from the Public Use Sample, a vast set of indiv idual records drawn from the Census Master F i le . T h e Public Use Sample (PUS) prov ides microdata on (1) indiv iduals , (2) house- 12a holds, and (3) families res ident in (a) the prov inces or (b) the metro- politan areas of Toronto and Montreal . T h e r e are consequently six separate f i les, each furn ished on magnetic t a p e . 1 T h i s s tudy employs the file on individuals resident in the p rov inces . The Individual Fi le, in common with the rest , is a one- in-one- hundred sample of the Canadian populat ion. It is based on the Census long-form quest ionnaire, which was administered randomly to one- th i rd of all households. A strat i f ied random selection of one in every t h i r t y - three and one- th i rd such records prov ides the eventual one- in-one- hundred sample. The strat i fy ing var iables consist of age (three categor ies) , sex (two categor ies) , mother tongue (three categor ies) , relation to head of household (three categor ies) , and community type (three categor ies) . T h e sample is thus representat ive of one hundred s ixty-two dist inct s t ra ta . Each sample record suppl ies coded information on f i f ty-e ight var iab les . T h e character ist ics por t rayed include among other things age and sex, place of residence, community type, the level of schooling and its geographic o r ig in , the quant i ty , v intage, and type of voca- tional t ra in ing , var ious aspects of family membership, the individual 's language, c i t i zenship , migration h is tory , ethnic and religious back- g r o u n d , labour- force status, indust ry and occupat ion, weeks and hours worked, total income, family income, income from wages and salaries, and income from self-employment—in. short , a large ar ray of economic and personal a t t r ibutes . Needless to say, the PUS data do not supp ly any direct information on individual abilities or job 125 exper ience. Of the f i f ty-e ight character ist ics available for s tudy , twenty-nine contr ibute to the present research . To preserve individual anonymity, the PUS tapes record much less detail for some character ist ics than do the publ ished Census repor ts . Indust ry , occupat ion, and place of residence are the variables chief ly a f fected. In the case of indust ry and occupat ion, the f iner levels of d isaggregat ion have merely been s u p p r e s s e d . There are twelve separate codes for indust ry and eighteen for occupat ion. In the case of res idence, it was decided not to identify geographic areas with populations of less than 250,000. A s a result , indiv iduals l iv ing in Prince Edward Island, the Y u k o n , and the Northwest Ter r i to r ies were dropped from the sample. T h i s omission, while u n - fortunate from the standpoint of completeness, could scarcely have had much effect on the overal l regress ion estimates. T h e r e is in general much similarity between the PUS data and the one- in-one-thousand sample Mincer obtains from the American c e n s u s . However, in one important respect , the two bodies of information are quite incomparable. Mincer's sample pertains to 1959; the PUS data, to 1970. Hence, if we f ind some d ispar i ty in the regression estimates, it may be that Canada and the United States d i f fer s t ruc tura l l y ; or it may be that the s t ructures are identical but chang ing , and that we are simply measuring them at d i f ferent points in time. For the purpose of evaulat ing theoretical ly based arguments, it would be des irable , no doubt , to examine only contemporaneous comparisons. On other g r o u n d s , the problem of d i f fer ing time per iods does not seem especial ly s igni f icant . If Mincer's general izations are "wrong" for Canada , it does not always matter whether they are wrong because they are outdated or because they fail to descr ibe some unique features of the Canadian economy. It is chief ly important that such general izations may prove misleading. Nevertheless , if the conclusions reached here contradict some of Mincer 's , the theoretical appeal of the human-capital model is indeed diminished, since it is seen not to place b inding restr ict ions on the data . Most researchers in the field would probably argue that the s t ructures under considerat ion change rather slowly and that the greater part of any d iscrepancies uncovered must be the result of d i f ferences between the two count r ies . For this reason and for the others mentioned, the analysis presented below will not shr ink from drawing the obvious comparisons, despite the incongruence in time per iods . T h e Sample T h e PUS file selected as the pr inc ipal data source contains information on just over 214,000 ind iv idua ls . The f i rst step in the research was to draw from this pool of records a working sample of manageable size and appropr iate composit ion. With regard to sample size, the goal was to obtain 20,000- 30,000 observat ions . T h i s number is of the same order as that employed by Mincer and is well within the gross data-handl ing capabil it ies of the available computer software. It is also large enough to prov ide adequate representation within all the designated population s t rata . With regard to sample composition, the problem was to exclude those indiv iduals to whom the earn ings model does not a p p l y . Since the model, as it s tands , does not incorporate a theory of labour-force participation or unemployment, it cannot apply to individuals who report no work a n d , hence, zero earnings for the census 2 year . Negative earn ings , which may ar ise through self-employment, 3 are likewise inadmissible. Individuals who d id not work or suf fered nonposit ive earn ings d u r i n g 1970 were therefore exc luded from the sample. For essential ly the same reason — inattention to time off w o r k — the standard empirical model fails in attempting to explain the earn ings of women. A s we observed in Chapter II, the proxy designed to measure experience through the use of a single census cross section performs reasonably well only in the case of males. Females thus had to be eliminated from the sample. Three other groups were also exc luded : these in full-time attendance at a school or un ivers i ty , those employed in the publ ic service ( including the armed fo rces) , and those whose indust ry of employment was "unspeci f ied or unde f ined . " T h e in-school population was exc luded , f i r s t , because it is obvious that in this g roup i n d i v i d - uals have not yet achieved the des i red levels of education and , second, because any earn ings they might report would likely be most atypical of what they could receive as full-time members of the 128 labour force . Public servants were eliminated in order to focus as much as possible on indiv iduals whose employers could be assumed to behave H as prof i t maximizers. Workers in unspeci f ied or undef ined industr ies were too few and too poorly character ized to warrant separate ana lys is ; yet , they could not be combined sat isfactori ly with any other g r o u p . The best solution was therefore to ignore them. A precise summary and technical statement of the sampling cr i ter ia may be found in Table 2. In l ight of the test for nonppsit ive employment incomes, the ones for zero weeks and zero hours are logically redundant but were nevertheless imposed to guard against incons istency. A l l the listed cr i ter ia were appl ied in the g iven se- quence to records from the PUS Individual F i le . T A B L E 2 SAMPLING C R I T E R I A Individual At t r ibute PUS V a r i a b l e 3 Codes Rejected 3 Remarks 1. Sex Sex 1 Exc ludes females. 2. Weeks worked in 1970 NUMWEEKS 0, 1 Exc ludes nonworkers , persons under 15 y e a r s . 3. Hours usual ly worked per week U S U A L H R S 0 Exc ludes "not app l i cab le ." U. Employment income INCWAGES + I N C S E L sum ^ 0 Exc ludes those with zero or negative earn ings . 5. School attendance A T T E N D 1 Exc ludes full-time attenders (Part-t ime accepted) . 6. Industry of employment INDUST 00, 11, 12 Excludes nonworkers and persons under 15 years , workers in publ ic adminis- tration and defence, workers in industr ies u n d e f i n e d . a S e e Canada , Stat ist ics Canada , Public Use Sample T a p e s : User Documentation. 129 T h e following procedure was used to obtain the des ired sample s ize. T h e beginning record—e i ther the f i rst or the second—was chosen at r a n d o m , 5 and the indicated tests were appl ied to every second observat ion in the source f i le . In a l l , 107,010 records were scanned to create a working sample of 22,682 ind iv idua ls . These numbers suggest the fraction of the total population (21.2%) to which the conc lus - ions of this s tudy a p p l y . Since the PUS file records are ar ranged initially in random order within provinc ia l b locks, and since the proport ion tested is v e r y large, there is little reason to fear a biased or unrepresentat ive sampling, despite the lack of any expl icit stratif ication in the selection p rocedure . Some feeling for the character and composition of the sample may be gained by looking at Tables 16-25, which form Append ix IIIA. These tables report the distr ibut ion of employment income, total income, and family income by size category , and the d istr ibut ion of age, res idence, and indust ry by level of educat ion, showing in the last two cases both the number of individuals in each cell and their average earn ings . Also included are d istr ibut ions cover ing occupat ion, period of immigration to Canada , ethnic i ty , and religious aff i l iat ion. Mean earnings for the 22,682 indiv iduals in the sample were $7,233, about 10% higher than the pub l i shed statistic for all males 15 and over who worked in 1970. 6 Mean age was 39.8 y e a r s , and the mean level of school ing, 10.0 y e a r s . T h e sample descr ibed in Append ix IIIA is "large" in the style of Mincer, statistical ly speak ing , but d i f fers somewhat in composit ion. 130 A s we have prev ious ly noted. Mincer studies "white, nonfarm, n o n - student m e n . " 7 T h e present research excludes women and full-t ime students but does not reject farm residents or nonwhites. Because of the desire to su rvey the Canadian population as fu l ly as possible, and because of the data-process ing overhead requ i red to draw a second sample solely for comparative purposes , it was decided not to implement Mincer's f i rs t two c r i t e r i a . S ince it is a relat ively simple matter to hold ethnic g roup and association with farming constant in the regress ion analys is , little is lost by adopting this p rocedure . In genera l , it is not clear why the human-capital model should not apply to farmers and nonwhites. It may be that whites and nonwhites d i f fer in ways that affect the model parameters, and it may be that farmers receive substantial nonmarket earnings or that they report as labour income part of the re turn on phys ica l capita l ; however, it seems best to provide for such complexities through appropr iate statistical techniques. T h e present research does eliminate publ ic servants and military personne l , whom Mincer apparent ly inc ludes. If governments merely follow the lead of prof it-maximizing f irms in sett ing the wage s t ruc ture (and if pub l i c -serv i ce unions s t r ive to imitate pr ivate-sector b a r g a i n s ) , one might a rgue that the human-capital model—or more prec ise ly , these aspects of it which depend on prof it-maximizing b e h a v i o u r — could still app ly . To have assumed such a "competitive" outcome would, though, have violated the spir i t of the cu r ren t s tudy , which is to 131 investigate the interplay of human-capital processes and market imper- fect ion. It would seem a pr ior i that this interplay is best observed in the pr ivate sector . In addition to the four cr i ter ia already d i scussed . Mincer imposes two alternate restr ic t ions , thus def in ing a pair of samples. One excludes indiv iduals 65 and over ; the other , indiv iduals with more than 40 years of work exper ience . In fact . Mincer publ ishes results 9 only for the latter. He does not provide any expl ic it justif ication for the exc lus ions, but one might reason that the hypothesized e x p e r - ience profi les are unl ikely to fit well at the upper end of the age s c a l e . 1 0 In any event , it was decided not to implement either of Mincer's restr ict ions here . Owing to the inclusion of farm residents and older workers , the cu r ren t sample is probably somewhat more heterogeneous than the one Mincer chooses. The level of inequality is certa in ly g reater . Tak ing the logarithm of earn ings , we f ind that here its variance is 0.767. In the case of Mincer, it is 0.694 in the group aged under 65 and 0.668 in the g roup with 40 or fewer years of e x p e r i e n c e . 1 1 How much of the evident d ispar i ty is the result of d i f ferences in sample composition and how much, the result of intercountry compar- ison, is impossible to determine. 132 The Var iables T h i s subsection defines all the regress ion var iables used in the present s t u d y . For quick reference. Table 3 (below) introduces the symbolic name af f ixed to each, lists the PUS source var iable , and offers a br ief descr ip t ion . T h e ensuing text explains the construct ion of the most important var iables in some detai l , analys ing the var ious choices which presented themselves. T A B L E 3 SUMMARY O F T H E V A R I A B L E S Regression Variable PUS Source Var iable(s) Descr ipt ion A C E A S Q DF E T H 1 - E T H 7 FAMSIZ G E 0 1 - C E 0 6 A C E A C E U S F A M I N C , INCWACES, I N C S E L F I N C T O T A L , INCWAGES, I N C S E L F U S E T H N I C FAM SI ZE C E O - C O D E A g e , Age squared . Dummy: = 0 when INCFAM = 0; when INCFAM £ 0. Dummy: = 0 when INCOTH = 0; when INCOTH Z 0. Ethnic or cu l tura l g r o u p : 1 = Br i t i sh Isles*; 2 = Western European; 3 = Eastern European; 4 = Chinese and Japanese; 5 = Jewish; 6 = Native Indian; 7 = Negro, West Indian, other . Number of persons in the indiv idual 's "census family" (= 1 in the case of a "nonfamily p e r s o n " ) . Place of res idence: 1 = At lant ic reg ion; 2 = Quebec; 3 = Ontar io*; 4 = Manitoba- Saskatchewan; 5 = A lber ta ; 6 = Br i t i sh Co lumbia . Table 3 (continued) 133 Regression Var iable PUS Source Var iab le(s) Descr ipt ion H E A D IM1-IM4 INC INCFAM I N C O T H IND1-IND10 L A N 1 - L A N 4 L E N C 1 - L E N C 4 MAJ OC1-OC12 FAM-MEMB PRDIMMIG INCWACES, INCSELF U S F A M I N C , INCWACES , I N C S E L F I N C T O T A L , INCWAGES, I N C S E L F INDUST O F F - L A N G L E N C R S MAJSINC O C C U P A T Head of a census family; 0 = nonhead or nonfamily person ; 1 = h e a d . Period of immigration to Canada: 1 = before 1946; 2 = 1946-1965; 3 = 1966- 1971; 4 = Canadian born*. Income from wages and salaries and employment (= INCWAGES + I N C S E L F ) . In logs. Family income in excess of INC (includes all p roperty income and the earn ings of other family members). In logs. Nonemployment income of the individual (= I N C T O T A L - INCWAGES - I N C S E L F ) . In l ogs . Industry of employment: 1 = a g r i c u - l ture; 2 = fo res t ry ; 3 = f ish ing and t rapp ing ; 4 = mining and oil wells; 5 = manufactur ing*; 6 = const ruct ion ; 7 = t ransport , communications, u t i l - it ies; 8 = trade; 9 = f inance, insurance, real estate; 10 = community, bus iness , and personal s e r v i c e . Off icial language: 1 = Engl ish only*; 2 = French on ly ; 3 = both; 4 = neither. Length of vocational t ra in ing; 1 = no tra in ing*; 2 = 3-5 months; 3 = 6 months- 3 y e a r s ; 4 = more than 3 y e a r s . Major source of income: 0 = sources other than self-employment; 1 = self- employment (farm or nonfarm). Occupat ion: 1 = managerial; 2 = natural and social sciences; 3 = teaching; 4 = medicine and health; 5 = c ler ica l ; 6 = sales; 7 = serv ices*; 8 = farming and other pr imary ; 9 = process ing , fabr icat ion, assembly, and repa i r ; 10 = construct ion; 11 = t ransport operat ion; 12 = other ( includes religion and the a r t s ) . 134 Table 3 (continued) Regression Var iable PUS Source Var iab le(s) Descript ion PSQ PX P2X R E L 1 - R E L 4 S S C O S T S P H C 1 - S P H C 7 SSQ T M A R G T Y P E USMAR WEEKS WTIME X I N C F A M D F X I N C O T H D I A G E , E D U C A T A G E , E D U C A T A G E , E D U C A T A G E , E D U C A T U S - R E L I G E D U C A T , A G E , G E O - C O D E E D U C A T , A G E , G E O - C O D E S C H O O L , P L C B I R T H E D U C A T , A G E G E O - C O D E (See text) TYPE-71 U S M A R S T NUMWEEKS NUWEEKS Exper ience (= A G E - B', where B*= S + 5.67 when B > 15, and B 1 = 15 otherwise) . Experience s q u a r e d . E x p ( B P ) , B = 0.05, 0.10, . " , 0 . 3 0 . E x p ( 2 B P ) , B = 0.05, 0.10, •••,0.30 . Rel ig ion: 1 = Protestant*; 2 = Roman Catholic and Orthodox; 3 = Jewish and other ; 4 = none. Years of schooling (estimated). See t e x t . Years of schooling with posit ive opportunity cost (= S - 9 if S < 9; = 0 o therwise) . Place of highest grade in school (up to secondary leve l) : 1 = Atlantic reg ion: 2 = Quebec; 3 = Ontar io*; 4 = Manitoba-Saskatchewan; 5 = A l b e r t a ; 6 = Br i t i sh Columbia; 7 = the Yukon and Northwest Ter r i to r ies or outside Canada . Defaults to place of b i r th for those with no schooling . Years of schooling s q u a r e d . 1 - marginal tax rate (est imated). In logs , Community type : 1 = u r b a n , population 30,000 and o v e r ; 0 = u r b a n , p o p u - lation under 30,000, p lus r u r a l , farm and nonfarm , Marital s tatus: 0 = s ingle, widowed, d i vo rced , separated; 1 = mar r i ed . Weeks worked d u r i n g 1970, d iv ided by 50. In l ogs . Weeks in 1970 times usual hours per week, d iv ided by 50-40 = 2000. In l o g s . Interact ion: INCFAM*DF Interaction: INCOTH"DI 135 Table 3 (continued) Regression PUS Source Variable Var iab le(s) Descr ipt ion X P C E 0 1 - X P C E 0 6 _ Interaction : P X E O XPIND1-XPIND10 - Interaction : P'IND X P O C 1 - X P O C 1 2 - Interaction : P"OC X P S Q G E 0 1 - X P S Q C E 0 6 - Interaction : PSQ"GEO X PSQ1N D1-X PSQ1N D10 - Interaction : PSQ ' IND X P S Q O C 1 - X P S Q O C 1 2 - Interaction : P S Q ' O C X S G E 0 1 - X S C E 0 6 - Interaction : S ' G E O XSINS1-XSIND10 - Interaction : S ' IND X S O C 1 - X S O C 1 2 - Interaction : S*OC X S P - Interaction : S*P ZINC (see text) T M A R C + 1 NC * Denotes reference group of a dummy set. T h e var iables appear ing in Table 3 may be sorted for fur ther d iscussion into the following six categories: 1. Income var iab les : INC, M A J , I N C O T H , INCFAM, DI , D F , X I N C O T H D I , X I N C F A M D F , T M A R G , Z I N C ; 2. T ime-worked var iab les : WEEKS, WTIME; 3. Human-capital and l i fe-cycle var iab les : S, S S Q , S P H G , P, P S Q , PX , P2X, X S P , A G E , A S Q , L E N C ; 4. Var iables thought to represent immobilities and other market fac tors : G E O , T Y P E , IND, O C , all interactions involv ing these at t r ibutes ; 5. Family-status var iab les : H E A D , U S M A R , FAMSIZ ; 6. Persona l -background var iab les : L A N , E T H , R E L , IM. 136 We shall consider each group in t u r n . 1. Income variables. T h e pr inc ipa l dependent variable used in this study is INC, the sum of wages, salaries, and self-employment earn ings , expressed in logarithms. Two problems arose in its c o n - s t ruct ion . The f i rst is one f requent ly encountered in working with income data: the highest incomes are grouped together in a s ingle, open-ended c lass . A l though the PUS source variables INCWACES and I N C S E L F communicate actual dollar amounts rather than dollar ranges for most indiv iduals , those report ing an income of $75,000 or more are shown as receiving exactly $75, 000. Th i s d i f f iculty was met by assuming a Pareto d istr ibut ion for the upper tail and comput ing, on 12 that basis , the mean in the open-ended c lass . Individuals were then ass igned this level of income. In fact, however, the problem turned out to be insignif icant, as INC—much less INCWACES or I N C S E L F separate ly—exceeded $74,999 for only 18 observat ions, or 0.08% of the ent ire sample. T h e other , more serious problem had to do with the compo- sition of self-employment earn ings . It is likely that amounts reported under this heading are a mixture of the re turns to both human and nonhuman capi ta l . Ideally, one would like to estimate the proport ion attr ibutable to nonhuman sources and subtract it in computing I N C . Unfortunate ly , the available data {on unincorporated business) do not appear to warrant such an attempt. An alternative would have been simply to exclude indiv iduals with posit ive (or large) self-employment 137 earn ings . T h i s tactic would obviously have injected its own bias into the resu l ts , el iminating, for example, most indiv iduals in the p r o - fess ions. A s a compromise, it was decided to include self-employment earn ings in the var iable INC but to def ine, in addi t ion, the independent dummy variable M A J , which equals 1 when self-employment earn ings are the major source of total income, and 0 otherwise. For indiv iduals rece iv ing only self-employment earn ings , the use of MAJ is equivalent to assuming that the proport ion of such earn ings attr ibutable to n o n - human capital is constant (though estimable and not specif ied in a d v a n c e ) . However, since self-employment may affect equi l ibr ium earn ings in 1 3 var ious ways, we cannot impose any narrow theoretical interpretation on the coeff icient of M A J . Apar t from the descr ipt ive information to be gathered from this var iab le , its main purpose will be to counteract biases threatening other regress ion coeff ic ients on account of the problem just d i s c u s s e d . T h e income var iables remaining in the list after INC and MAJ all contr ibute , in Chapte rs IV and V , to the empirical analysis of time worked . For completeness we shall nevertheless review their def init ions here . I N C O T H is a theoretical const ruct best understood as depict ing the proper ty or nonempldyment income of the indiv idual after personal income taxes. It was computed by subtract ing from total income (PUS var iable I N C T O T A L ) the sum of (a) estimated tax payments a n d (b) employment earn ings multiplied by one minus the marginal tax rate (see below). As explained in Chapter IV, the result is used in mapping the indiv idual 's budget const ra int . A l te rnat ive ly , INCFAM 138 measures all income of the family in excess of what the individual in question earns from employment. It was found by subtract ing the two prev ious ly stated quantit ies (a) and (b) from total family income, as g iven by the PUS variable U S F A M I N C . Since the latter is in g rouped 14 form, class midpoints were used in this calculat ion. Observe that (the antilog of) INCFAM equals (the antilog of) I N C O T H plus both the property and employment incomes of other family members. However, INCFAM does not take into account other members' tax payments. These definit ions raise one complication: when, as sometimes happens, "other" incomes and own taxes equal zero, we cannot t rans - form into logarithms. T h e solution in such instances was to let INCOTH or INCFAM equal some arb i t ra ry value and define the interaction terms XINCOTHDI and X I N C F A M D F . A s explained in Table 3, the dummy variables DI and DF equal zero when the associated income variables equal zero; hence, so do XINCOTHDI and X I N C F A M D F . Otherwise, XINCOTHDI = I N C O T H , and X I N C F A M D F = I N C F A M . In pract ice, then, a dummy-interaction pair does the work of INCOTH or I N C F A M , which never actually appear in any regress ion . The last two variables related to income—ones also needed in the analysis of time worked—are T M A R C and Z I N C . A s stated in the table, T M A R C equals one minus the indiv idual 's estimated marginal tax rate (in logar i thms). ZINC is simply T M A R C + INC. Since the latter are both in logarithms, we have—once again in logar i thms—the quant i ty (1 - marginal tax rate) x (employment e a r n i n g s ) . T h i s some- what unorthodox construct ion stems from the analysis reviewed in Chapter IV. 139 Estimating T M A R C meant, of course , simulating as care fu l ly as possible the indiv idual 's personal income tax r e t u r n . T h i s task re - qu i red certain assumptions and approximations. In the case of married family heads, it was assumed that the income of other family members ( INCFAM - INCOTH) belonged solely to the spouse (here , necessar i ly , the wife) and that family size minus two measured the number of wholly dependent c h i l d r e n . 1 5 In the case of nonmarried family heads, the number of potential ly dependent ch i ldren was assumed to equal family size minus one, with other income d iv ided evenly among the subordinate ind iv idua ls . Those who were not family heads were assumed to claim no dependents . A l though the preced ing f ive assumptions doubtless fail in many instances, they probably represent the great majority of family situations occur r ing in the present sample. These assumptions, together with information on the 1970 tax 1 6 s t ruc tu re , would have been suff ic ient to determine total personal exemptions, except for one deta i l . T h e allowance for a dependent ch i ld var ied in 1970, as it does c u r r e n t l y , with the ch i ld 's age. T h e present data source does not prov ide this information. A c c o r d i n g l y , an average claim ($341) was c o m p u t e d 1 7 and employed in all cases . T o a r r i v e at taxable income, the simulation routine added to personal exemptions an average f igure represent ing var ious common deduct ions which indiv iduals are al lowed. These involve reg is tered pension fund and retirement savings plan contr ibut ions , medical expenses , char i table donations, and union or professional d u e s . Separate averages (of all such items combined) were computed in each of fourteen income c l a s s e s . ' u T h e appropriate f igure was then added to personal e x - emptions, as stated, and the result subtracted from total income (PUS variable I N C T O T A L ) to estimate taxable income. T h e final step in the routine was to search through a table of effect ive marginal rates to f ind the one app ly ing to the individual in quest io . Since the combined federal and provincia l rates vary across the c o u n t r y , it was necessary to take into account the indiv idual 's 1 9 prov ince of residence (PUS variable C E O - C O D E ) . A federal tax reduction prevai l ing in 1970 and special provis ions relating to Quebec were also cons idered . T h e result ing estimate, labelled T M A R C , is probably the best that can be inferred us ing census data . T h o u g h undoubtedly subject to e r r o r , it does not appear misleading in any systematic way. 2. Time-worked variables. WEEKS and WTIME are the two alternative measures of employment constructed here . T h e y serve as independent var iables in the earn ings- funct ion estimates reported in this chapter and as endogenous var iables in the simultaneous-equation estimates to be presented later. Let us f i rs t consider the definit ion of WEEKS. T h i s variable is based on the number of weeks dur ing which the individual worked, for however short a time, in 1970. The Census a n d , consequent ly , the PUS variable NUMWEEKS do not furn i sh much precision in this a rea , break ing down the f i fty-two-week year into just f ive intervals (1-13, 14-26, 27- 39, 40-48, 49-52). WEEKS was obtained by taking the f ive class midpoints, * u d iv id ing each by 50, and transforming into logar- ithms. Roughly speak ing , therefore , WEEKS is measured in terms of years ; more prec ise ly , it is scaled so that the employment of " fu l l - time" workers (49-52 weeks) equals un i t y . In view of the logarithmic transformation, this normalization affects only the constant term in the forthcoming regress ions T h e alternative employment var iable WTIME takes into account both weeks and hours . It is the product (in logarithms) of weeks worked in 1970 and hours usual ly worked each week. T h i s measure, or ones similar to it, have been used widely by economists and statistic 21 ians to estimate annual hours , notwithstanding the acknowledged imprecis ion. T h e main problem aff l ict ing WTIME stems from the hours component. In the Canadian census , hours are reported either for the job held in the week preceding enumeration day (July 1, 1971) or , in the case of persons then unemployed, for the job of longest durat ion held since January 1, 1970. One would obviously prefer an average of hours worked per week in 1970, if such a thing were pract i ca l . T h e Canadian def in i t ion, which stresses usual hours , is probably less objectionable than the American counterpart , which tradit ional ly asks for hours worked "last week"; but both are c lear ly subject to t rans i tory , shor t - run d i s turbances . Fortunate ly , it is not essential for purposes of this s tudy to use WTIME in computing the hour ly wage rate . T h i s common procedure is one which places the most strain on the cred ib i l i ty of the var iab le . 142 Since the PUS source variable U S U A L H R S is again d i s c o n t i n u o u s — there a re , to be exact , seven in terva ls—i t was necessary to employ the class-midpoint approximation, as in the case of NUMWEEKS. In this instance, however, there was a f ina l , open-ended class (50 or more hours/ week) to deal with. Unhapp i ly , there does not also exist a well-establ ished theoretical d istr ibut ion which one may apply in order to estimate the mean in this open-ended c lass . A n a rb i t ra ry value of 54 hours/week was there- fore ass igned . The chosen f igures were d iv ided by 40 and transformed into logarithms, and the result for each individual was added to WEEKS in order to a r r i ve at WTIME. The latter is consequently scaled in terms of a work year f ixed at 2000 hours . 3. Human-capital and life-cycle variables T h e f i rst human- capital measure we have to define is , of course , school ing. T h e PUS variable E D U C A T d ist inguishes twelve di f ferent levels: no school ing, less than grade 5, grades 5-8, grades 9-10, grade 11, grade 12, grade 13, 1-2 years un ive rs i t y , 3-4 year (without degree) , 3-4 years (with degree) , 5 or more years (without degree) , 5 or more years (with d e g r e e ) . T o define the continuous regression variable S, we must translate each given level of education into an appropr iate number of y e a r s . "No school ing" prov ides an obvious zero point for the scale, and it is natural to let grades 11, 12, and 13 equal 11, 12, and 13 years 22 of instruct ion respect ive ly . T h e other eight levels demand a keener ana lys is . 143 In view of the emphasis accorded schooling by the present s t u d y , it was thought essential to measure this var iable with as much accuracy as the Census itself would allow. There fo re , it was decided not to resort to the standard c lass-midpoint assumption in translat ing the PUS variable E D U C A T . Instead, special tabulations were obtained from Statistics Canada g iv ing the number of out-of-school males at each single grade of publ ic school or year of un ive rs i t y , by age group and 23 place of highest g r a d e . It was then possible to compute, for each schooling interval (except the last two), a mean value conditional upon age group and place of highest g r a d e . These conditional means were used to estimate the schooling attainment of the individuals included in the sample. For most of these fal l ing into the last two, open-ended 24 c lasses , values of 17.5 and 18.5 years respect ively were ass igned . T h e exception was for those schooled in Ontar io , which maintains a th i r teen-year system of publ ic educat ion. Here , the assumed f igures were 18.5 and 19.5 years . It is di f f icult to say how much the preceding refinements affect the subsequent regression estimates. Within the lower school- ing interva ls , which contain a large proport ion of ind iv iduals , v a r i a - tion among the computed means was not insubstant ia l . In the second schooling interval (grades 1-4), the range was 2 .72- 3.66 years ; in the th i rd (grades 5-8), it was 6 .47- 7.67 y e a r s ; and in the fourth (grades 9-10), 9 .35-9 .91 y e a r s . Variat ion within the narrower, postsecondary intervals was rather s l ight , but most of the computed values fell 144 uniformly 0 .10-0 .20 years above or below the class midpoint, depending on the interval in quest ion . T h o u g h it is d i f f icul t to assess the effect of subst i tut ing c o n - ditional means for class midpoints, one may at least be confident that the 25 schooling var iable S will not suf fer any contamination from age or region 26 as a result of the presentation of the data in grouped form. Hence, the t rue impact of S will not be attr ibuted to either of these other factors . It is worth not ing , f ina l ly , in connection with S that the source variable E D U C A T furn ishes somewhat more detail than Mincer had at his d i sposa l . Instead of the twelve schooling categories available here , he could consult only e ight . It is not clear how Mincer dealt with the g roup ing problem. Though later d iscussion will concentrate upon S, an a l ternat ive measure S C O S T was def ined in an ef fort to por t ray the number of school years with a positive (market) opportuni ty cost . On the assumption that Individuals cannot work in the market pr ior to age f i f teen, S C O S T was set equal to S-9 if S < 9 and equal to zero otherwise. Besides stat ing the individual 's level of educat ion, the PUS data tell where the subject completed his last year of publ ic school . T h i s information permits the construct ion of a r o u g h , though perhaps useful set of proxies for the quality of school ing. T h e dummy str ing S P H C — p l a c e of highest g rade—was accord ing ly def ined in the manner set out in Table 3. Note that the Yukon and Northwest Ter r i to r ies and "outside Canada" have been merged into one g roup—ca l l it "outside Southern Canada"—and that place of highest grade defaults in the case of those with no schooling to place of b i r t h , identically categor ized. S P H C , together with A C E , fix unambiguously the indiv idual 's educational milieu at a part icular stage of schooling and thus jointly stand in place of a qual i ty index. Str ic t ly speak ing , however, we obtain a means of holding qual i ty constant only for one year of s t u d y . A s an overal l measure of schooling qual i ty , SPHC (plus A C E ) will be inaccurate to the extent that individuals migrate interregional ly d u r i n g their years .of publ ic school . Moreover, SPHC has nothing to say about post- secondary educat ion. In view of how S was cons t ruc ted , us ing SPHC, redundancy may also be a problem. Let us now consider exper ience . The basic variable P was computed in the manner descr ibed ear l ier—that is, by subtract ing from age the sum of years schooling (S) plus age at school e n t r y . However, no individual was credi ted with experience ostensibly gained before age 27 28 15. Age at school entry was assumed to equal 5.67 y e a r s . T h i s value, an average, spr ings from two pr ior assumptions: (a) that b i r thday are spread uniformly over the calendar year , and (b) that ch i ldren begin school in September of the year d u r i n g which they achieve age 6. Notice that the special tabulations which assist in the c o n - struct ion of S also contr ibute to the estimation of P. With the mean level of schooling and age inverse ly correlated within schooling intervals , the standard procedure would have led to a modest overestimate of P for young individuals and to a similar underestimate for older ones . In the same way, P would have been underestimated for those schooled, and possibly stil l res ident i n , educationally depr ived reg ions . T h e variables der ived from P — P S Q , PX , P2X, and X S P — r e q u i r e only br ie f comment. P and its square , PSQ , implement the quadrat ic functional form d iscussed in Chapter II. PX and P2X do the same for the exponent ia l . The latter take on d i f ferent values as the parameter 6 29 is iterated in steps of 0.05 from 0.05 to 0.30. XSP is of course the exper ience-school ing interaction which appears in Mincer's work. T h e last human-capital factor to note is L E N C , a str ing of dummy variables represent ing the durat ion of any vocational course or apprent iceship undertaken by the individual (or if more than one, that of longest d u r a t i o n ) . Unfortunate ly , owing to a lack of detail in the PUS source data, it was impractical to attempt any decoding into time e q u i - va lents . Vocational tra ining in the formal sense is not a factor g iven separate treatment by Mincer . Investigating its impact on earnings is therefore a matter of special interest , even though the data permit only the roughest sort of empirical ana lys is . 4. Immobilities and other market factors. If the market for skil ls were everywhere perfect ly competit ive, as human-capital theory presumes; if the adjustment to momentary disequi l ibr ium were always rap id ; and if the nonpecuniary re turns to various jobs were un important— then it would be unnecessary , in attempting to explain individual ea rn ings . 147 to look much beyond the human-capital variables already d i s c u s s e d . T h e sole aim of empirical research would be to produce ref ined estimates of the human-capital s tock. Yet , it seems hard ly p rudent , when viewing the labour market, to assume a priori that immobilities and other imperfections, "momentary" d isequi l ibr ia , and nonpecuniary factors will all be negl ig ib le . T h e acceptance or denial of such a proposit ion demands empirical i n q u i r y . We shall therefore consider a number of variables which one may interpret as standing for nonpecuniary di f ferentials or market imperfect ion. The f i rst of these is the dummy vector C E O , s ign i fy ing place of residence (on enumeration day , Ju ly 1, 1971). If indiv iduals are perfect ly mobile and have no geographic preferences , the regress ion coeff ic ients of C E O should all turn out ins igni f icant . Note that C E O departs s l ightly from the standard f ive-region segmentation of Canada , d is t ingu ish ing the relatively populous and industr ia l ly separate economy of A lber ta from those of the other two Prair ie p rov inces . A related dummy variable T Y P E denotes community s ize . In- d iv iduals in rura l areas and those in "small" towns (population under 30,000) were grouped together ( T Y P E * 0) pr imari ly in order to stress the earn ings experience of those in large cities ( T Y P E = 1). A s promised in Chapter II, s t r ings of dummy variables were also def ined to represent indust ry and occupat ion. T h e construct ion of IND was a stra ight forward decoding of the PUS variable I N D U S T . It is nevertheless important to observe that the industry associated with each indiv idual is the one which prov ided either the job held in the week 148 pr ior to enumeration, or fai l ing that , the job of longest durat ion held since January 1, 1970. T h e r e is thus no guarantee that reported 1970 earnings (INC) were der ived wholly, or even par t ly , from employment in the reported i n d u s t r y . To the extent that individuals changed industr ies d u r i n g the period under considerat ion, we must expect IND to contain some e r r o r . However, since the e r ro r is unl ikely to be in any way systematic, its only effect should be to weaken the explanatory power of the indust ry var iab les . If these remain signif icant despite the e r r o r , the case against the human-capital variables as the sole determinant of earnings is s trengthened all the more. The same remarks apply to the vector of occupational dummies, O C — t h o u g h as conceded in Chapter II, the case for inc luding occupation in an equation with schooling already present is not so strong as that for inc luding region or i n d u s t r y . With regard to the detailed spec i f i - cation of O C , it was found necessary to exercise some mild restraint in the number of variables de f ined . A s a result , eighteen PUS categories were collapsed into twelve. The need to economize on the number of variables arose pr inc ipal ly on account of the desire to investigate the interaction of IND, O C , and C E O with the human-capital measures S , P, and P S Q . Even so, the number of interaction terms in this set reached seventy - f i ve , not count ing those perta in ing to reference g r o u p s . For reasons of economy and for other reasons which will become clear when we examine the results of the next section, interactions involv ing the forms PX and P2X were not de f ined . 149 5« Family-s tat us variables. These factors were included in some of the earnings equations pr imari ly for descr ipt ive purposes . T h o u g h one may conceive hypotheses in which they exert causal effect on earnings (perhaps via "reservation wages") or in which they serve as proxies for certain "abil ity" at t r ibutes , it would be a mistake, no doubt , to consider them wholly exogenous. The f i rst of these var iab les , H E A D , d ist inguishes those who head a "census fami ly ." The latter comprises either a h u s b a n d , a wife, and any never-marr ied c h i l d r e n , or one parent and at least one never - married c h i l d , all l iv ing together . Th i s nuclear aggregation was chosen for s tudy in preference to the so-cal led "economic fami ly ," on which in - 30 formation was also p r o v i d e d . "Head" always refers in the census definit ions to the husband or parent (here , necessar i ly , the father) of any age . The second var iable , USMAR, d ist inguishes married ind iv idua ls . Those who are s ingle , d i vo rced , separated, or widowed—that is to say , those who report no current spouse—were grouped together in the reference category (USMAR = 0) . T h e last var iable , FAMSIZ , r e p r e - sents the number of persons in the census family, except that in the case of nonfamily persons , FAMSIZ equals one. Where the PUS source var iable FAM-SIZE indicated "ten or more persons" (another open-ended c l a s s ) , FAMSIZ was set a r b i t r a r i l y — at eleven if USMAR equalled zero, and at twelve if USMAR equalled one. In ef fect , the number of ch i ldren was assumed constant , on average, in the two cases . 150 6. Personal-background variables. These factors also play a descr ipt ive role in the regression equations, though it is reasonable to treat them as exogenous. A s in the case of family-status var iab les , hypotheses have been suggested l inking them to earnings and employment. We shall not stop to consider such arguments here, but rather in the appropr iate empirical sections which follow. The definit ions of L A N , E T H , R E L , and IM are all relat ively s t ra ight forward . LAN is based on official language instead of mother tongue (also available) because of the policy signif icance adher ing to the former in Canada . With regard to ethnic g roup ( E T H ) , twenty-one PUS categories were combined for purposes of this study into a more manageable seven . In the shortened descr ipt ion of Table 3, "Western European" includes F r e n c h , A u s t r i a n , F inn ish , German, Italian, Nether lands, and Scandinav ian; "Eastern European" includes C z e c h , 31 Hungar ian , Pol ish, Russ ian , S lovak, and Ukra in ian . With regard to religious group ( R E L ) , the procedure was to d ist inguish Protestants, Catholic and Orthodox , non-Chr i s t i ans , and those profess ing no re l ig ion. Th i r teen PUS categories were combined into four . F inal ly , with regard to period of immigration (IM), the rationale was to identify "early immigrants (before 1946), postwar immigrants" (1946-1965), and "recent immigrants" (1966-1971). "Canadian born" fu rn i shed the natural reference g r o u p . T h u s ten PUS categories were again col lapsed into f o u r . H U M A N - C A P I T A L EARNINGS F U N C T I O N S In this section we shall treat only a few of the one h u n d r e d s ix ty-e ight var iables just de f ined . Repl icating Mincer's orthodox human- capital approach , we shall see how his t ight ly specif ied earnings funct ions performed with the Canadian data . These equations d i f fer from one another, most fundamental ly, in the way experience is held constant . As we observed in Chapter II, Mincer attacks this problem either by restr ic t ing the sample to one experience cross section (the overtak ing set) or by postulating the form of the investment prof i le . In fact . Mincer tests two functional forms, 32 the exponential and the quadrat i c . We thus have three approaches to cons ider . T h e next three subsect ions deal with each one separately , in the order just stated. Before we proceed to the resu l ts , one or two general comments are in order concerning the mechanics of estimation. Because the decoded raw data matrix had the intimidating dimensions 22,682 by 168, it would have been h igh ly ineff ic ient, if not impossible in pract ice , to process it in the usual manner, reading each observat ion into the computer and c a r r y i n g out var ious prel iminary calculations every time a new series of regress ions was r e q u i r e d . Fortunate ly , all of the stat is- tical procedures contemplated in this s tudy ( inc luding the three-stage least squares of Chapter V) could be performed knowing only the moment matrix of raw data . Ac tua l l y , since the matrix is symmetric, only one tr iangle was needed. 152 33 A versati le and eff icient regression programme known as RLS was used to compute the moment matrix, which was then stored for easy access . In pract ice , the final matrix was itself built up in stages, by the simple process of matrix addi t ion. T h e intermediate matrices prov ided dist inct random subsamples of the large main sample. These were used for prel iminary tes t ing . Final estimates were then carr ied out for the full set of observat ions . T h i s procedure tends to minimize the statistical dangers of hypothesis testing when the data are to be extensively "mined" by comparing a number of alternative specif icat ions. A l l the estimates d i s - p layed here were obtained us ing R L S , which accepts moment matrices as input . T h e Over tak ing Set As we observed in Chapter II, Mincer tends to favour an empirical definit ion of the overtak ing set which includes individuals with 7-9 years of exper ience. In the present sample there turned out to be 1,238 individuals who met this cr i ter ion (speci f ica l ly , 7.0 J P < 9 .0) . The i r mean years of schooling were 10.85—somewhat greater than for the full sample— and the var iance of logged earnings was 0.629—as expected , somewhat less . Results for this g r o u p , correspond ing to Mincer's Equations 34 ( V 1 ) - ( V 4 ) , are d isp layed in Table 4. T h e simple regress ion of INC on 2 S implies a re turn to schooling of 10.0%. T h i s rate and the level of R fall cons iderably short of the values obtained by Mincer . The addition of WEEKS lowers the estimated return by about one q u a r t e r . T h i s fraction presumably measures the re turn component which individuals receive ind i rect ly , through increased employment rather than through h igher wages. Note that, cont rary to Mincer's f ind ings , the coefficient of WEEKS does not depart s ignif icant ly from one. Earn ings are almost exactly proport ional to weeks worked ; by implication, wage rates do not depend on the volume of employment—not even through a mutual posit ive correlat ion of both factors with worker ab i l i ty . T A B L E 4 E S T I M A T E S FOR T H E O V E R T A K I N G S E T 3 Equation Number Equations (dependent variable = INC) (CV1) (CV2) (CV3) (CV4) 0.5214 (6.88) 1.003 (15.3) 0.4609 (2.66) 1.188 (8.16) 0.1001 S (14.9) 0.0741 S (13.1) 0.1117 S (3.66) 0.0392 S (1.55) 0.9573 WEEKS (24.1) 0.0005 SSQ (0.39) 0.0015 SSQ -t (1.42) 0.9617 WEEKS (24.2) I .425 1238 observat ions on indiv iduals with 7-9 years of exper ience . F igures in parentheses are t rat ios, written in absolute terms. 154 Schooling squared (SSQ) does not achieve signif icance whether or not WEEKS is inc luded. The rate of re turn appears to be constant even when employment is allowed to v a r y . T h u s Mincer's argument on this point turns out to be i r re levant , at least for the present g r o u p . However, 35 looking at Equation ( C V 4 ) , where even S is insignif icant, one begins to suspect that the quadrat ic functional form may be inappropriate in the Canadian set t ing . A s we have noted, d irect estimates for Canada have prev ious ly shown a somewhat i r regular (nonmonotonic) pattern in the rates 36 of re turn to school ing, rather than the near ly continuous schedule of decline familiar in United States s tud ies . O n the whole, Equations ( C V 1 ) - ( C V 4 ) do not seem especial ly favourable to the use of the over tak ing concept . Except in ( C V 3 ) , the implied rates of re turn are not consistent with the assumed length of the over tak ing per iod (recall that if costs are constant , p = 1/r x ) . One must bear in mind, however, that the length assumption, which defines empir- ically the overtak ing set, vyas simply copied from the work of Mincer . If rates of re turn are lower in Canada than in the United States, a some- what longer period of overtak ing might have g iven better resu l ts . Since the search for a new empirical definit ion appears methodologically dub ious , we shall not pursue this problem here . Instead, we shall turn to the full sample of indiv iduals and to parametric methods of holding exper ience constant . 155 Exponential Exper ience Profiles Besides holding experience constant so that one may estimate the re turn to schooling in an unbiased manner, the exponential form of the exper ience profi le should allow one to estimate the initial propensi ty to undertake postschool investment (kg) , the typical net postschool rate of re turn ( r x ) , and even the rate at which human capital depreciates ( d ) . From Equation (46b) in Chapter II it follows that k Q = ( - 2 b 3 ) * and r x = 6 [ ( b 2 / k Q ) + 1 ] , where b 2 and b 3 a re , in the cur rent notation, the coefficients of PX and P2X respect ive ly . T h e coefficient of P, when that variable enters the regression along with PX and P2X, furn ishes the estimate of d . To be admissible, the implied value of kg must fall within the closed unit in terva l ; that of r x must surely be nonnegative (otherwise no one would think of i nves t ing) . The preceding requirements place certain reasonableness restr ict ions upon b 2 and b 3 , namely: b 2 < - k Q and - 1/2 < b 3 < 0 . If these condit ions are not met simultaneously, the model fa i ls . The outcome of experiments with the exponential form appears in Table 5. These resu l ts , obtained by iterating for d i f ferent values of 3 in the same way as Mincer , are not very encourag ing . A s 8 increases. b 2 decl ines and b 3 r i ses , each monotonicaily. None of the specif ic values t r ied for B produces coeff icients which meet the reasonableness requ i re - ments. Viewing Equations ( C C 2 ) , one might expect , on the basis of monotonicity, to encounter reasonable coeff ic ients when B is in the 0.15- 0.20 range . Unfortunate ly , there does not appear to be much hope of ref in ing this estimate. A s was reportedly the case with Mincer's sample, 2 the value of R does not change s igni f icant ly within the plausible range of B. It is not clear what other cr i ter ion one could possibly use . Mincer, of course , relies on the plausibi l i ty of the coefficients themselves, or x equiva lent ly , upon r and k^; but this course is not open here . One could presumably search over values of B in the 0.15-0.20 range and obtain x 0 plausible f igures for r and k , but one could not then claim to have "estimated" these parameters. In view of how sensit ive b 2 and b^ seem to be , a great many pairs of values would likely be found acceptable. One's general conclusion must be that the exponential form is not a sat isfactory device for estimating the investment parameters in the case of Canadian males. The other results presented in Table 5 reinforce this inference. In Equations (CGI) and ( C C 5 ) , the admissible values of 6 must be some- what less than 0.05. It is di f f icult to believe that an optimal plan would dictate such a low rate of decline (under 5%) in the net propensi ty to invest , g iven the length of the average working l i fespan. In (CCU) the coeff icients of P, interpreted as rates of deprec iat ion, are not alone implausible; but in light of the suspicion s u r r o u n d i n g , f i r s t , the value of B a n d , second, the functional form, they cannot be taken v e r y 157 T A B L E 5 F U L L - S A M P L E 3 E S T I M A T E S USING E X P O N E N T I A L E X P E R I E N C E PROFILES E q u a - tion Coeff ic ients of No. 8 S SSQ P PX P2X WEEKS R 2 (CG2) .05 0.0745 (52.9) - - 2.160 (27.9) -3.026 (37.7) 0.8411 (82.4) .406 .10 0.0768 (53.5) - — 0.3274 (5.36) -1.604 (20.7) 0.8531 (82.9) .397 .15 0.0782 (54.6) - - -0.6097 (9.72) -0.6539 (7.71) 0.8587 (83.4) .395 .20 0.0784 (55.2) - - -1.287 (19.2) 0.1099 (1.19) 0.8588 (83.5) .396 .25 0.0780 (55.2) - - -1.814 (25.1) 0.7270 (7.30) 0.8573 (83.4) .397 .30 0.0772 (54.8) — — -2.238 (28.9) 1.228 (11.6) 0.8561 (83.2) .396 (CG4) .05 0.0732 (51.3) - -0.0309 (13.9) -1.572 (5.61) -0.8364 (4.72) 0.8405 (82.6) .411 .10 0.0734 (51.3) -0.0204 (22.6) -2.080 (16.9) 0.2621 (2.33) 0.8398 (82.4) .410 .30 0.0730 (50.7) — -0.0057 (12.8) -2.950 (31.0) 1.885 (16.0) 0.8466 (82.4) .401 (CG5) .05 0.0082 (1.51) 0.0030 (12.3) -0.0316 (14.2) -1.565 (5.61) -0.8639 (4.90) 0.8430 (83.2) .415 < .10 0.0086 (1.58) 0.0030 (12.3) -0.0209 (23.2) -2.066 (16.9) 0.2261 (2.01) 0.8421 (82.8) .414 .30 0.0098 (1.78) 0.0029 (11.9) -0.0062 (13.9) -2.946 (31.0) 0.844 (15.7) 0.8488 (82.8) .404 22,682 observat ions T h e dependent variable is INC. F igures in parentheses are t rat ios, written in absolute terms. Constants , though present in all the regress ions , are not shown. ser ious ly . Equations (CC5) depart s l ight ly from Mincer in adding S S Q . Here , in contrast to ( C V 4 ) , the term is s igni f icant , though S itself is not. T h e positive coefficient implies that r increases with the level of schoo l ing—by about 0.6% for each additional yea r . In view of Podoluk's 37 results from the 1961 census , this f ind ing is not a complete s u r p r i s e , though again it is at var iance with United States exper ience . Here , the indicated re turn at the mean year of schooling is just under 7%. Quadratic Exper ience Profi les Estimates obtained using quadrat ic experience profi les are shown in Table 6. These results are no more helpful in attempting to evaluate r x and k Q than are the ones der ived us ing the exponential form, but they are perhaps easier to interpret from a pure ly descr ipt ive standpoint . Before we examine what little the estimates have to of fer concern - ing the investment parameters, let us look at var ious other , more t ransparent implications. Note f i rst of all the schooling regression (CS1) inserted in Table 6 for purposes of comparison. A s it tu rns out, the schooling coeff ic ient, when rounded , precisely matches that of Mincer . 2 On the basis of R , schooling may be said to explain 7.3% of (log) earning var iance—just a little more than in Mincer's sample. The addition of the experience term in (CP1) causes the schooling coeff ic ient to r ise , as expec ted—though not quite so markedly as in Mincer's ( P 1 ) . Di f ferent iat ing with respect to P (remembering that PSQ = 2 P ) and sett ing the result equal to zero show that earnings reach a peak 159 T A B L E 6 F U L L - S A M P L E 3 E S T I M A T E S USING Q U A D R A T I C E X P E R I E N C E PROFILES Equations (dependent variable = INC) (CS1) (CP1) (CP2) (CP3) (CP4) (CP5) (CP6) Constant .9906 -.0714 -.3663 .5397 .5809 .3944 -.7484 (57.0) (2.86) (5.51) (23.6) (9.85) (17.2) (17.2) S .0695 .0891 .1009 .0715 .0393 .0775 .0624 (42. 4) (54.8) (11.3) (49.9) (5.04) (53.5) (46.0) SSQ — .0009 (2.91) - .0022 (7.92) - - P - .0829 .1029 .0583 .0683 .0572 _ (63.3) (44.3) (49.7) (33.1) (47.5) PSQ - -.0014 -.0016 -.0010 -.0011 -.0010 _ (58.8) (58.0) (46.5) (45.0) (42.8) X S P — - -.0014 (10.3) - - .0007 (5.72) - - A C E — - - - - - .0983 (49.6) ASQ — — - - - - 0011 (47.2) WEEKS — — - .8629 (85.3) .8615 (85.3) - .8576 (84.8) WTIME — .6589 (78.7) — R 2 .073 .213 .220 .405 .409 .382 .407 22,682 observat ions, F igures in parentheses are t rat ios, written in absolute terms. 160 at 29.6 years of exper ience . Holding weeks constant , in ( C P 3 ) , lowers the estimated rate of re turn from 8.9% to 7.2%—that is, by about one f i f t h — b u t leaves peak earn ings , at 29.2 years , little c h a n g e d . T h e insertion of SSQ and X S P , in Equations (CP2) and (CPU) , helps to delineate fu r ther the shapes of the earn ings prof i les . A s before with the Canadian data, the coefficient of SSQ is posit ive and s igni f icant , though admittedly rather small in the f i rst case . Holding weeks constant does not eliminate the apparent rise in the rate of re turn but , in fact, seems to strengthen it . T u r n i n g to X S P , we f ind that its coefficient is s igni f icant ly negat ive. A s Mincer points out, this result implies that exper ience prof i les for the var ious levels of school ing tend to converge over the life cyc le , since earn ings rise less (or decline more) with exper ience at high levels of schooling than at low levels . The degree of convergence indicated here is nevertheless relat ively small in comparison with that observed by Mincer . When we take both SSQ and X S P into account, the implied rate of re turn to schooling for indiv iduals with mean levels of schooling and exper ience (10.03 and 23.14 years respect ive ly) tu rns out to be 8.7% with weeks variable and 6.7% with weeks held constant . For mean-schooled ind iv iduals , measured earnings peak at just under 28 years of exper ience in both cases . Di f ferent iat ing the express ion for the peak-earn ings year with respect to S shows that an additional year of schooling hastens the peak by 0 . 3 - 0 . 4 years in terms of exper ience . In terms of age, the peak is therefore postponed by 0 . 6 - 0 . 7 y e a r s . 161 Replacing the quadrat ic in exper ience with a quadrat ic in age reveals in (CP6) that (weeks-constant) earn ings peak, on average, at 44.7 years of age . A t normal ret irement, earn ings will have receded b y almost 20%, accord ing to the estimates. T h e age quadrat ic fits the Canadian data just as well as , if not better than, the experience quadrat i c ; but in the former case, the implied rate of re turn to schooling is lower a n d , perhaps , negatively b iased . Coeff ic ients of the employment var iab les , represent ing elast ic- ities, are s ignif icantly less than one throughout Table 6. Th i s f ind ing contrasts sharp ly with that of Mincer , who observed elasticities in the neighbourhood of 1.2. It is also at var iance with the outcome in the overtak ing set , for which the measured elasticities are not s igni f icant ly d i f ferent from one. The indication is that low wages and high levels of employment go together . T h i s seems especial ly to be the case when we consider hours (WTIME) rather than weeks in ( C P 5 ) . T h e implied elasticity drops from 0.86. to 0.66. T h e fit is s l ight ly weaker than in ( C P 3 ) , ref lect ing perhaps the e r ro rs to which WTIME is subjec. "E r ro rs in var iables" may indeed have some part in depress ing the coeff icients of both WEEKS and WTIME. However, it should be noted that Mincer's employment var iab le , with which we are making comparison, suf fers the same shortcoming. One may of course rationalize in var ious ways the apparent i n - elasticity of earn ings with respect to employment. A backward-bend ing supply c u r v e of labour would explain this resul t , especial ly if one assumes perfect competit ion. Workers conf ined to low-wage jobs may v e r y well seek long hours or "moonlight" in order to reach equi l ibr ium. In an environment of d iscrete choices, some workers may have such a s t rong taste for income that they eschew high-wage jobs with s tandard , inflex- ible weeks and hours in favour of low-wage jobs with weeks and hours unconst ra ined . T h e latter may occur even though indiv idual supp ly c u r v e s are posit ively s loped . T h e trouble with both these arguments is that they require us to postulate radical ly d i f ferent pre ferences , or d istr ibut ions of pre ferences , among the Canadian and American work forces . A super ior explanation may therefore lie in the pronounced seasonality of economic act iv i ty in Canada . If seasonal workers are involuntar i ly unemployed d u r i n g part of the year (or if they are simply earn ings maximizers), they will demand, and in competitive equi l ibr ium rece ive, h igh wages as a compensation for low h o u r s . Despite the plausibi l i ty of this argument, it is probably unwise to speculate v e r y far on the basis of the present single-equation estimates, which may be b iased, and which doubtlessly entangle l abour -supp ly , labour-demand, and investment responses . We shall take up the elasticity question again in l ight of the simultaneous-equation estimates reported in Chapter V . It remains in this section to explore br ief ly what the present estimates imply concern ing the investment parameters. From Equation (46a) and the accompanying def init ions it follows that 163 k b = b 2 T ' + 2 b 3 T ' 2 + d * T ' and r x = b 2/k|j - (1 + k>Q) /T ' + d / b 2 , where b 2 and b 3 are the coeff ic ients of P and PSQ respect ive ly . Since it is not possible to identify the four unknowns ( r x , k'Q, T ' , and d) us ing only the preceding pair of express ions , we must be content to examine a range of numerical combinations in order to see where the most plausible values l ie. Table 7 shows, in the weeks-var iable case, the values of r x a n d kjj which ar ise in connection with certain specif ied values of T ' and d . Because one may wish to interpret the latter as the d i f ference between depreciation and expected growth , some nonposit ive values have been included for t r i a l . A s much as a n y t h i n g , Table 7 seems to emphasize the-inadequacy of the present technique for measuring the rate of re turn to postschool investment. If one is prepared to assume the val idity of the model, then it is possible to rule out "large" values of T ' and d ; but there is little else that one may say . Over the six admissible cases—those in which , say , 0% < r x < 30% and 0 < k'Q < 1 — r x ranges from 3.9% to 20.2%. T h e r x - k | j pair cor respond ing to T ' = 20 and d = 0 is perhaps worthy of special note, since it is the combination implied by Mincer's assumptions. The values obtained here are similar to the ones Mincer repor ts ; but as the table demonstrates, they are too sensit ive to the assumptions concern ing T 1 and d to warrant much conf idence. T A B L E 7 V A L U E S OF r x A N D k'Q C O N S I S T E N T WITH SPECIFIED V A L U E S OF T ' A N D d ( W E E K S - V A R I A B L E C A S E ) 3 d T 1 (years) 20 25 30 -.01 X r = 14.8% 103.4% -27.1% k 1 - K o .34 .08 - . 3 3 0.00 x r = 7.7% 20.2% -273.2% K o .54 .33 - .03 .01 x r = 3.9% 9.9% 30.2% K 0 .74 .58 .27 .02 x r = 4.3% 10.6% 30.0% k' = K 0 1.14 .83 .57 .03 x r = 4.8% 10.9% 30.0% k' K 0 1.34 1.08 .87 .04 x r = 5.1% 11.2% 29.9% k 0 = 1.54 1.33 1.17 a S e e Table 6, Equation ( C P 1 ) , in which b_ = 0.083 and b 3 = -0.0014. 165 E X P A N D E D E A R N I N G S - F U N C T I O N E S T I M A T E S Having considered the str ict human-capital speci f icat ion, we may now view the results obtained by expanding the earn ings funct ions to include variables typical ly ignored by human-capital theor is ts . We shall pay part icular attention to any changes which occur in the schooling coefficient as new variables are a d d e d . More genera l ly , we shall be able to assess the relative importance of human-capital and other factors in determining the employment incomes of Canad ians . T o begin the analys is , we must choose one of the human-capital earnings funct ions as a standard of comparison. T h e quadrat ic Equation (CP5) seems best suited for this purpose . It is simple to estimate and to interpret , and its functional form is by far the most ' widespread in the l i terature. Though ( C P 3 ) , containing WEEKS, fits s l ightly better , statistical concerns ar is ing later in connection with the system estimates of Chapter V favour the use of WTIME as the employ- ment var iab le . Hence, (CP5) is to be p r e f e r r e d . We shall not ignore, however, the variables SSQ and X S P , which are missing from it. These terms will ultimately be included in the expanded regress ions . The latter are d isp layed and d iscussed in the f i rst subsection below. T h e second deals with a part icu lar vers ion of the so-cal led "interactions model. " 166 T h e Impact of Previously Omitted Var iables Ear l ier in this chapter , variables which might be thought to influence employment earn ings were grouped under several head ings . Restated here for convenience, they a re : (1) human-capital and l i fe- cycle var iab les , (2) variables thought to represent immobilities and other market imperfections, (3) family-status var iables , (4) persona l - background var iab les . The text and the tables which follow review each set of factors in t u r n . Fur ther div is ions examine an alternative to the initial specif icat ion, analyse the occupational dimension of employment earn ings , and present a br ie f summary. It must be noted, to beg in , that the order in which variables enter succeeding regressions may have an effect on the interpretation of resu l t s . Because here , and in genera l , the independent var iables of concern are correlated with one another , there will always be some area of indeterminacy in the assignment of explanatory s igni f icance. 2 The amount by which a part icular variable increases the level of R is one estimate of its importance, but only a conditional estimate for the set of regressors included by pr ior select ion. The order of selection establ ished here follows pr inc ipal ly from the emphasis g iven by this s tudy to the var iables in g roups (1) and (2) above, we shall devote special attention to the indeterminacy or var iance-attr ibut ion problem as it affects the preced ing factors . 167 1. Human-capital and life-cycle variables. T h e main factors in this group which do not appear in the orthodox specif ications are L E N C and S P H C . With regard to the former, Table 8 shows that br ief voca- tional courses (LENC1) have no signif icant effect upon earn ings . Programmes of intermediate length (LENC2) have a modest effect at best (see also Table 9) . However, long vocational programmes, which one might guess consist mainly of classical apprent icesh ips , add as much as 18% to the level of earn ings (see the coefficient of L E N C 3 in Equation 38 ( C P 7 ) ) . Holding additional variables constant nevertheless reduces this apparent premium cons iderab ly . Vocational preparat ion is ev ident ly correlated to a signif icant degree with both place of residence (CEO) and indust ry ( I N D ) , especial ly the latter . A t a minimum (in Equation (CP13) , Table 10), the apparent earnings premium associated with LENC4 falls to 8.0%. A s d iscussed ear l ier , SPHC (place of highest grade) may be considered a proxy variable for schooling qua l i ty . Not s u r p r i s i n g l y , however, SPHC and C E O (place of cur rent residence) turn out to be closely corre la ted . When both are entered in the same regress ion , some coeff ic ients of C E O s u r v i v e the ensu ing mult icol l inearity; but those of 39 SPHC become uniformly ins igni f icant . SPHC on its own does not match the performance of C E O under identical c i rcumstances . P re - l in inary tests support ing these observat ions may be found along with other , miscellaneous regress ions in Append ix 1MB. Fur ther work ut i l iz ing SPHC was not attempted. 168 T A B L E 8 REGRESSION E S T I M A T E S 3 OF T H E E X P A N D E D EARNINGS F U N C T I O N , I E q u a t i o n s 0 (Dependent variable = INC) m a . Var iable (CP7) (CP8) (CP9) Constant .3977 (17.4) .6323 (25.8) .6616 (27.0) S .0763 (52.5) .0688 (47.0) .0705 (47.5) P .0564 (46.8) .0548 (46.3) .0525 (45.7) PSQ -.0009 (42.0) -.0009 (41.5) -.0008 (39.4) WTIME .6577 (78.7) .6569 (80.0) .6804 (84.7) LENC2 -.0014 (0.05) -.0011 (0.04) -.0086 (0.31) L E N C 3 .0363 (2.39) .0260 (1.75) .0115 (0.80) LENC4 .1782 (9.42) .1460 (7.85) .0998 (5.53) GEOI - - -.1770 (9.87) -.2085 (11.9) GE 0 2 - - -.0604 (5.33) -.0604 (5.51) C E 0 4 - - -.2601 (15.2) -.1422 (8.43) G E 0 5 - - -.1058 (5.91) -.0480 (2.75) G E 0 6 - - .0491 (3.11) .0306 (2.00) T Y P E - - .1987 (21.0) .1151 (11.9) IND1 - - - - - .7047 (31.1) IND2 - - - - -.0111 (0.32) IND3 - - - - -.4090 (6.89) IND4 - - - .1943 (6.99) IND6 - - - - .0557 (3.52) IND7 - - - - .0365 (2.46) IND8 - - - - -.1573 (11.5) IND9 - - - - .0099 (0.42) IND10 - - - - -.1282 (9.42) MAJ - - - - -.0477 (2.86) R 2 .385 .409 .452 Main sample, 22,682 observat ions The f i r s t f igure in each set is a regression coeff ic ient; the second, in parenthes is , is the corresponding t ratio, written in absolute terms Also relegated to the appendix is an i l lustrat ive equation employ- ing S C O S T in place of S . Recall that S C O S T counts only those years of schooling registered after about age f i f teen. It does so on the specu- lation that early school attendance may entail no opportuni ty cost and thus should not be presumed costly in der iv ing the model. A s one might expect , especial ly in view of the results concerning S S Q , the rates of re turn implied for S C O S T exceed those for S, the addition being about 1.5 percentage po ints . A s one might also expect , S C O S T does not 2 yield as high an R as S. Though di f ferences in schooling at the low end of the scale may not reflect investment decis ions, such d i f ferences are evident ly recognized and rewarded in the market, either because schooling in the range under d iscussion enhances product iv i ty or be- cause it serves as a p roxy for abil ity and background character ist ics which we are otherwise unable to measure. A c c o r d i n g l y , S would appear to be the variable of choice in the analysis of earn ings determination and d is t r ibut ion , even though its t runcated var iant S C O S T might possibly g ive better ra te-o f - re turn estimates. Since replacing S with S C O S T had little effect on the coeff ic ients of other var iab les , we shall not pursue fur ther experiments with the latter but will instead c o n - centrate on S in order to present results of maximum comparative interest . 2. Variables thought to represent immobilities and other market imperfections. Prime candidates under this heading are C E O and IND. These are added sequentia l ly , along with T Y P E and M A J , in 170 Equations (CP8) and ( C P 9 ) , Table 8. A s explained prev ious ly , the co- eff ic ients measure percentage d i f ferences in earnings relative to the chosen reference g r o u p . In the case of ( C P 9 ) , the reference group c o n - sists of, nonmetropolitan Ontar io residents without formal vocational t ra in ing employed as wage-earners in manufactur ing. It turns out that all the coeff ic ients of C E O and T Y P E are s ign i f i - cant at the 0.05 level or better ; indeed, all but one are signif icant at the 0.01 leve l . The regional ranking implied by (CP8) is perhaps a little s u r p r i s i n g , inasmuch as Manitoba-Saskatchewan rather than the At lant ic Provinces falls at the bottom of the earnings l ist . Holding the industrial mix constant , in ( C P 9 ) , y ie lds the ranking one would have predicted for the time ( 1970): Br i t i sh Columbia, Ontar io , A lber ta , Quebec, Manitoba- Saskatchewan, the At lant ic Prov inces . That this pattern should pers ist in the face of considerable standardization says much about the p ro - foundness of regional d ispar i ty in Canada . A s for T Y P E , the 11.5% earnings advantage of metropolitan-area residents in (CP9) appears general ly consistent with expectat ions. If geographic mobility, the supply of information, and competition for employment were both perfect and cost less, one would expect the coeff icients of GEO and T Y P E to be ins igni f icant . It may be, of course , that the observed geographic and metropo l i tan-versus-rura l -and-smal l - town di f ferent ia ls are really of an equal iz ing na ture—the competitive outcome of v a r y i n g tangible and intangible benefits and costs . Equation (CP9) then implies that the At lant ic Provinces supply the largest, and Br i t i sh Columbia the smallest, real amenity total . It would sure ly be presumptuous to attempt an objective assessment of this propos i t ion . One may say, comparing (CP8) and ( C P 9 ) , that the net effect of equal iz ing dif ferentials and market imperfection is to lower the es t i - mated return to schooling by 0.75 percentage po ints . Together , C E O and T Y P E explain an additional 2.4% of the earnings var iance , or about one- th i rd of the amount ascr ibed to schooling in ( C S 1 ) . 2 40 IND adds a fur ther 4.3% to the value of R . Seven of its nine coefficients are s igni f icant . Hence, C E O , T Y P E , and IND, at a minimum, contr ibute almost as much (6.7%) as S at its maximum (7.3%). When S is d ropped from ( C P 9 ) , 4 1 R 2 falls by 5.5 percentage 42 points , indicating the minimum effect of the var iab le . Of course , schooling does not pretend to measure the individual 's total stock of human capi ta l . If the latter is g iven by S, P, and PSQ, we may estimate its contr ibut ion from (CP1) at 21.3%. T h e market- imperfect ion variables have about one- th i rd the explanatory power. T h e y lower the implied rate of re turn to schooling by almost 2 percentage points . T h e negative coefficient obtained for MAJ suggests that, on average, indiv iduals pay a premium for being sel f -employed. T h e size of the premium may actually be somewhat larger than is indicated here , since one would expect the present coefficient to be biased u p - wards through the inclusion in earn ings of some returns to non- human capi ta l . On the other hand , because the self-employed category is extremely heterogeneous, the average f igure may not be especial ly use fu l . 172 Family-s tat us variables. T h e results of adding H E A D , FAMSIZ , and USMAR are d isp layed in Equation (CP10) , Table 9. These var iables are included here pr imari ly for descr ipt ive purposes , since we have not su rveyed any r igourous theoretical arguments for their inser t ion . One might speculate that family and marital responsibi l i t ies could have some effect upon the individual 's "reservation wage" d u r i n g periods of job search . Those who have held out for a high wage at some time in the past , either because of perce ived high subsistence requirements or because of available support from secondary earners , will tend to record h igh cu r ren t incomes as a resu l t . Discrimination in favour of married family heads may also be a factor . One should nevertheless be on g u a r d against the s t rong l ikelihood that the var iables in quest ion are endogenous. Earn ings may v e r y well predetermine family s tatus . A t the very least, earn ings and family status may be related solely th rough a common dependence upon some unmeasured qual ity of the ind iv idua l . A t any rate, H E A D is uniformly signif icant with a large coeff ic- ient. USMAR is s ignif icant at the 0.05 level or better in all but Equation ( C P 9 ) . FAMSIZ is nowhere signif icant in Table 9, but it becomes so in (CP14) and (CP15) , Table 10, where WTIME has been deleted. Hours of work apparent ly interact with size of family to create a link between the latter variable and earn ings , though size of family bears no relationship to the implicit wage. H E A D , U S M A R , and FAMSIZ together account for a modest 1.5% of total earnings var iance . 173 T A B L E 9 REGRESSION E S T I M A T E S 3 OF T H E E X P A N D E D EARNINGS F U N C T I O N , II Equations (dependent variable = INC) i n a . Var iable ( C P 1 0 ) C ( C P 1 1 ) C ( C P 1 2 ) C Constant .6007 (23.9) .6422 (24.8) .6421 (23.9) S .0675 (45.9) .0653 (43.2) .0651 (42.2) P .0416 (34.2) .0414 (33.8) .0415 (33.9) PSQ -.0007 (30.9) -.0007 (30.7) -.0007 (30.8) WTIME .6498 (81.1) .6494 (81.1) .6488 (80.9) LENC2 -.0021 (0.79) -.0191 (0.71) -.0188 (0.70) L E N C 3 .0075 (0.53) .0086 (0.61) .0085 (0.60) LENC4 .0811 (4.56) .0843 (4.73) .0833 (4.66) HEAD .2404 (7.69) .2360 (7.55) .2309 (7.39) FAMSIZ -.0020 (0.81) -.0019 (0.79) -.0018 (0.75) USMAR .0590 (1.93) .0683 (2.12) .0696 (2.28) IM1 - - .0268 (1.21) .0267 (1.20) IM2 - - -.0245 (1.84) -.0190 (1.35) IM3 - - -.1050 (4.70) -.0856 (3.67) LAN 2 - - -.1091 (5.45) -.1195 (5.43) LAN 3 - - .0150 (0.72) -.0004 (0.26) LAN 4 - - -.0282 (0.65) -.0321 (0.73) ETH2 - - - - .0054 (0.44) E T H 3 - - - - -.0202 (1.02) ETH4 - - - - -.1059 (2.08) ETH5 - - - - .2301 (5.65) E T H 6 - - - - -.0460 (0.84) ETH7 - - - - -.0512 (2.22) REL2 - - - .0141 (1.13) REL3 - - - - -.0724 (3.12) REL4 — — - - .0166 (0.88) R2 .467 .469 .470 Main sample, 22,682 observat ions The f i rst f igure in each set is a regression coeff ic ient; the second, in parentheses, is the cor respond ing t ratio, written in absolute terms. i n c l u d e d but not shown are G E O , T Y P E , IND, and M A J . 174 Personal-background variables. A l though the four charac ter - istics identif ied here—that is, IM, L A N , E T H , and R E L — a p p e a r to c o n - tr ibute negl ig ibly to earnings inequality "at the marg in , " individual coeff icients supply a fair amount of useful information. A s might be expected , recent immigrants (IM3) suffer a modest earnings disadvantage (8.61 vis a vis the reference group in (CP12)), but those who have l ived in the country for some time do approximately as well as the Canadian b o r n . Uni l ingual francophones (LAN2) earn 11-12% less than uni l ingual anglophones and less, even , than individuals who have no f luency in either Engl ish or French ( L A N 4 ) . A t the same time, biligualism (LAN3) does not seem to confer any signif icant advantage. Adherence to a non- Chr is t ian religious faith (REL3) signals below-average earn ings . Of the six coefficients for ethnic g r o u p , three are signif icant at the 0.05 level or better . G iven the standardization enforced in (CP12) , we f ind that Jews in the sample (ETH5) earn an average of 23.0% more than the reference g r o u p , Chinese and Japanese ( E T H 4 ) , 10.6% less, and Negro, West Indian and "other" ( E T H 7 ) , 5.1% less . Native Indians (ETH6) also suffer a d isadvantage, but it is not statistically s igni f icant . One should not assume, however, that the preceding ethnic coeff icients measure the full extent of any discrimination which may be present . T h e r e is , f i rst of a l l , some degree of multicoll inearity between E T H and each of the other three background variables IM, L A N , and R E L . Secondly , it must be remembered that in (CP12) , as in most of the other earn ings funct ions, time worked is held constant . D iscr imin- 175 ation may well manifest itself more s igni f icant ly through h i r i n g , t u r n - o v e r , and so on than through the payment of d i f ferent iated wages. Table 10 therefore presents some fur ther ev idence . We see in (CP13) that removing IM, L A N , and R E L does not have much overal l effect , but it does lower the coefficient of E T H 5 rather markedly . T h e reason is simple: as shown in Table 24 (Appendix I M A ) , E T H 5 and REL3 are pract ical ly the same var iable , since most non-Chr i s t ians in the sample are ethnical ly Jewish. In fact, the coefficient of ETH5 in (CP13) is v i r tua l ly the algebraic sum formed by the coefficients of ETH5 and REL3 in (CP12) . Removing WTIME has a profound effect on the coeff icient of E T H 6 . T h e disadvantage borne by Native Indians does indeed appear to stem much more from employment than from wage rates . On average, native people earn 34-35% less than those in the reference g r o u p . Overal l in (CP15) , four of the six ethnic coef f ic- ients turn out to be s igni f icant . Variable returns to schooling. By including only the l inear term S in Table 8-10, we have so far dictated a constant rate of re tu rn to school ing. Table 11 relaxes this assumption by re- in t roduc ing the squared term SSQ and the exper ience interaction X S P . A s before, the coefficient of SSQ is both posit ive and h ighly s ignif icant, implying that the rate of re turn increases with the level of school ing. T h e coeff icient of S is d r i ven to ins igni f icance. That of X S P remains s igni f icant ly negat ive . T h u s even after extensive standardizat ion. T A B L E 10 176 REGRESSION E S T I M A T E S OF T H E E X P A N D E D EARNINGS F U N C T I O N , III Ind. Var iable 1̂ Equations (dependent var iable = INC) ( C P 1 3 ) C ( C P 1 4 ) C ( C P 1 5 ) C Constant .6177 (23.7) .2554) (8.51) .2243 (7.70) S .0667 (44.5) .0732 (41.9) .0749 (44.2) P .0418 (34.4) .0608 (44.5) .0614 (45.4) PSQ -.0007 (31.2) .0011 (43.4) .0011 (44.3) WTIME .6492 (80.9) - - - LENC2 -.0213 (0.79) -.0418 (1.37) -.0451 (1.48) L E N C 3 .0076 (0.54) .0093 (0.58) .0090 (0.56) LENC4 .0800 (4.50) .1020 (5.02) .1010 (5.01) H E A D .2334 (7.47) .3134 (8.84) .3164 (8.91) FAMSIZ -.0021 (0.86) -.0113 (4.09) -.0116 (4.22) USMAR .0655 (2.14) .1375 (3.96) .1341 (3.85) IM1 - - -.0261 (1.04) _ _ IM2 - - -.0003 (0.02) - _ IM3 - - -.1056 (3.99) - - LAN 2 - - -.1402 (5.62) _ _ LAN 3 - - -.0157 (0.84) - - LAN 4 - - -.0204 (0.41) - - ETH2 -.0042 (0.39) -.0102 (0.72) -.0059 (0.48) ETH3 -.0158 (0.86) -.0299 (1.33) -.0284 (1.35) ETH4 -.1362 (2.72) -.1075 (1.86) -.1396 (2.46) ETH5 .1649 (4.73) .2424 (5.25) .1778 (4.50) ETH6 -.0362 (0.66) -.3510 (5.67) -.3440 (5.56) E T H 7 -.0771 (3.65) -.0497 (1.90) -.0785 (3.28) REL2 - - .0030 (0.21) _ _ REL3 - - -.0785 (2.98) - - REL4 — - .0038 (0.18) - - R 2 .468 .317 .315 Main sample, 22,682 observat ions T h e f i rst f igure in each set is a regress ion coeff ic ient; the second, in parentheses , is the correspond ing t ratio, written in absolute terms Also included but not shown are G E O , T Y P E , IND, and MAJ 177 T A B L E 11 T H E E X P A N D E D EARNINGS F U N C T I O N WITH A V A R I A B L E R A T E O F R E T U R N ( E Q U A T I O N (CP16)) a Ind. u Ind. Var iable Coeff icient (t r a t i o ) 0 Variable Coeff icient (t r a t i o ) 0 Constant .8626 (14.5) IND9 -.0051 (0.22) IND10 -.1581 (11.6) S .0057 (0.74) SSQ .0033 (12.0) MAJ -.0677 (4.14) P .0498 (24.2) PSQ -.0008 (30.7) HEAD .2151 (6.92) FAMSIZ -.0017 (0.69) USMAR .0703 (2.31) X S P -.0005 (4.65) WTIME .6482 (81.2) IM1 .0259 (1.17) IM2 -.0312 (2.22) IM3 -.1036 (4.46) LEN 2 .0012 (0.44) LEN3 .0352 (2.48) LEN 4 .1049 (5.89) LAN 2 -.1455 (6.63) LAN 3 -.0182 (1.11) LAN 4 -.0979 (2.23) GEOI -.2342 (13.4) GE0 2 -.0246 (1.52) G E 0 4 -.1458 (8.63) ETH2 .0052 (0.42) G E 0 5 -.0499 (2.88) ETH3 -.0367 (1.86) G E 0 6 .0434 (2.85) ETH4 -.1270 (2.51) ETH5 .2081 (5.14) ETH6 -.0967 (1.77) T Y P E .1132 (11.6) E T H 7 -.0689 (3.00) IND1 -.6731 (30.3) IND2 .0300 (0.86) REL2 .0124 (1.00) IND3 -.4295 (7.23) REL3 -.0782 (3.39) IND4 .1836 (6.74) REL4 .0009 (0.05) IND6 .0474 (3.05) IND7 .0305 (2.10) •y IND8 -.1533 (11.5) R 2 .476 Estimated for the main sample, 22,682 observat ions . Absolute va lues . 178 experience profi les continue to exhibit convergence . A t mean levels of schooling and exper ience, the estimated return to schooling (dINC/dS) is 6.01. Inserting SSQ and X S P in (CP16) , Table 11, raises the R 2 by 0.6 of a percentage point. One might therefore be tempted to conclude that variation in the rate of re turn to schooling is not a very important source of earn ings inequal i ty . One cannot assume, however, that all variation in the rate of re turn expresses itself through SSQ and X S P . Much may be left in the res idua l . A l though Mincer develops a way of part it ioning the residual var iance to obtain a maximum estimate of the 43 component associated with variable re tu rns , his argument is i n - applicable here because it assumes the independence of S and r e . We have no recourse, it seems, but to account expl ic it ly for variat ion in the rate of return through the use of additional determinants. T h e interactions model reported below pursues this problem. Otherwise, the re- introduct ion of SSQ and X S P vaults three more variables into the "signif icant" category , namely: L E N C 3 (6 months - 3 years vocational t ra in ing) , IM2 (immigrated 1946- 1965), and LAN4 (neither Engl ish nor F r e n c h ) . C E 0 2 (Quebec residence) becomes ins igni f icant . Comparing (CP15) and (CP16) , one can see that the general pattern of coeff ic ients is not much af fected. 44 The occupational dimension. It has been argued that inc luding occupation in the earn ings funct ion along with schooling will necessar i ly bias downward the estimated rate of r e t u r n , since indiv iduals 179 appear to reap the benefit of their schooling investment by moving upward through the occupational h i e r a r c h y . Holding occupation constant thus imposes an unnatural constra int . Nevertheless , it seems useful to examine the occupational dimension of earn ings , not only for descr ipt ive purposes , but also in order to test the empirical s ignif icance of the preceding objection. Its practical val idity must depend to a great extent on how "occupation" is de f ined . A s usua l , the researcher is ve ry much at the mercy of the data . If the available categorization scheme rests on hierarchical factors such as the level of t ra in ing , the degree of status, or the span of respons ib i l i ty , then the bias problem just mentioned will be more severe than if the system is grounded in some abstract analysis of work funct ion , the nature of the indus t ry , or the type of good or service p r o d u c e d . In the latter case, occupational wage dif ferentials are again l ikely to be of the equal iz ing var ie ty , or else they are the result of noncompetitive forces . T h e part icular categorization scheme embodied in the PUS data is not easy to character ize in the preceding terms. Status, funct ion , and industry all seem to play a role. T h e headings are broad (since there are only twelve used h e r e ) , and all would appear to admit individuals with widely va ry ing levels of school ing. Schooling and occupat ion, as cu r ren t l y def ined, are nonetheless correlated to a degree . It seems prudent therefore merely to let the results speak for themselves. The effects of adding occupation to the earn ings function are d isp layed in Table 12. 180 T A B L E 12 T H E E F F E C T S OF O C C U P A T I O N 3 Ind. Var iable E q u a t i o n s 0 (dependent var iable = NC) (CP17) (CP18) (CP19) c (CP20) C O N S T A N T .4382 (15. 7) .6850 (12.8) .7868 (10 .4) .2681 (4. 42) S .0534 (32. 5) .0255 (4.57) .0236 (2. 48) .0281 (4. 42) SSQ - - - .0006 (1 . 81) - - P .0531 (45. 8) .0529 (45.7) .0474 (23 .3) .0783 (61 .6) PSQ -.0009 (40. 4) -.0009 (40.5) -.0007 (29 .7) -.0013 (56 .8) X S P - - - -.0005 (4. 42) - - WTIME .6596 (81. 8) .6587 (81.9) .6420 (81 .6) - - OC1 .6743 (25. 9) . 5229 (5.61) .4608 (4. 84) .5869 (5. 53) OC2 .5103 (18. 3) -.0305 (0.30) -.1042 (1 . 02) -.0492 (0. 43) OC3 .4628 (14. 1) -.3414 (2.19) - .1727 (1 . 11) - .5693 (3. 20) OC4 .5151 (13. 7) -.8489 (6.66) -.7206 (5 . 73) -.8714 (6. 00) OC5 .2340 (10. 1) .0814 (0.94) .0468 (0. 56) .0646 (0. 66) OC6 .2298 (10. 7) -.1384 (1.90) -.0031 (0. 04) -.0178 (0. 21) OC8 -.2842 (13. 2) -.4812 (7.10) -.1111 (1 . 60) -.5195 (6. 73) OC9 .2618 (13. 7) .0604 (0.99) .0525 (0. 88) .0784 (1. 12) OC10 .3303 (15. 6) .1173 (1.76) .0698 (1 . 06) .0656 (0. 86) OC11 .2117 (9.03) -.0285 (0.35) -.0318 (0. 41) -.0634 (0. 67) OC12 .2456 (11. 4) .1783 (2.69) .0919 (1. 42) .1096 (1 . 45) XSOC1 - - .0210 (2.63) .0154 (1 . 88) .0237 (2. 61) X S O C 2 - - .0490 (6.00) .0431 (5. 09) .0566 (6. 08) X S O C 3 - - .0635 (5.88) .0495 (4. 50) .0819 (6 . 68) X S O C 4 - - .1049 (11.0) .0962 (9 . 98) .1140 (IC .5) X S O C 5 - - .0196 (2.33) .0132 (1 . 62) .0269 (2. 81) X S O C 6 - - .0397 (5.45) .0242 (3 . 41) .0392 (4. 72) XSOC8 - - .0226 (2.97) .0197 (2. 71) .0318 (3 . 67) X S O C 9 - - .0234 (3.53) .0135 (2. 13) .0289 (3 . 82) XSOC10 - - .0246 (3.38) .0144 (2. 08) .0388 (4. 68) XSOC11 - - .0277 (3.13) .0156 (1 . 84) .0364 (3. 61) XSOC12 — — .0098 (1.42) .0098 (1 . 45) .0181 (2. 28) R 2 .436 .440 . .495 .274 a Est imated for the main sample, 22,682 observat ions . T h e f i rst f igure in each set is a regress ion coeff ic ient; the second , in parenthes is , is the cor respond ing t rat io, written in absolute terms. c A l s o included but not shown are L E N C , C E O , T Y P E , IND, M A J , H E A D , FAMSIZ , USMAR, IM, L A N , E T H , and R E L . 181 2 The eleven intercept dummies in (CP17) raise the level of R by 5.4 percentage points, compared with ( C P 5 ) , and lower the implied rate of return to schooling from 7.8% to 5.3%. The latter change represents the maximum extent of the possible b ias . If it were the true extent, it could also be interpreted as measuring that component of the re turn to schooling which must be realized through occupational mobil ity. Doubt less ly , however, there exists some return to occupational mobility which is merely correlated with but not dependent upon the level of school ing. A s one might easily have forecast , managerial personnel (0C1) rank at the top of the earnings scale, followed by workers in health care ( O C 4 ) . Farm and other pr imary workers (OC8) rank lowest, preceded by service workers (the reference g r o u p , OC7) . Equations (CP18)-(CP20) add the vector of interaction terms X S O C . (CP19) includes the collection of variables treated earl ier in Table 11; (CP20) is identical to (CP18) except for the deletion of WTIME. B y add ing the respect ive coeff icients of X S O C to the coeff icient of S, one may compute the set of intra-occupational rates of r e t u r n . These are not, of course , the rates of re turn that individuals secure , having chosen to enter a part icu lar occupat ion. T h e y measure instead the rewards to educational upgrad ing within a part icular category . Hence the large f igure implied for workers in health care ( X S 0 C 4 : 0.0255 + 0.1049 = 0. 1304) must simply reflect unusual steepness in the earn ings gradient across schooling levels in this f ie ld . Teach ing (XSOC3) stands out in a similar fash ion. 182 Occupation does appear to capture some variation in the rate of r e t u r n , for in (CP19) the coeff ic ient of SSQ becomes ins ig i fn icant . A l though the interaction terms add v e r y little to the R , they are jointly s ignif icant in an F test at the 0.01 level . Permitting hours of work to v a r y , in (CP20) , does not change the general pattern of these coeff ic ients; but it does increase their va lues, as the employment factor becomes incorporated in the estimated rates of r e t u r n . Most of the intercept coeff ic ients fall a lgebraica l ly , since the earn ings-school ing gradients pivot upward to accommodate the rearranged scatter of observat ions . Summary. Now that we have looked in detail at all the variable groups cons idered for inclusion in the earn ings funct ion , it is necessary to conduct a broad comparison of their quantitat ive inf luence. For this purpose Table 13 presents a decomposition of the explained earn ings var iance ( inequal ity) and a set of F statistics perta in ing to the variable g r o u p s . These F statistics are more useful in the cur rent context than the standard t ratios g iven ear l ier , since the latter, being in part dependent upon the choice of a reference g r o u p , are bound to be somewhat a r b i t r a r y . A s noted p rev ious ly , we cannot avoid a certain degree of a r b i t r a r i n - ess involv ing the order in which var iables enter the regress ion equat ions. Since the order shown in Table 13 tends to favour (gives the "benefit of the doubt" to) the orthodox human-capital var iables by introducing them f i r s t , we must pay some attent ion. 183 T A B L E 13 T H E E X P L A N A T O R Y POWER A N D S I G N I F I C A N C E OF V A R I A B L E S IN T H E E X P A N D E D E A R N I N G S F U N C T I O N S Var iance Increment 3 F Sta tistic Var iable Group Upon Addit ion Percent of E x p . V a r . b Upon Deletion Upon Addi t ion Upon Deletion S .07332 14.82 .00014 11341 .00* 6.40 P PSQ .14011 28.33 .01999 f 17443.69 457.06 WTIME .16873 34.12 .14872 10012.27 6800.76 L E N C .00248 0.50 .00089 48.73 13.57 GEO T Y P E .02.476 5.01 .01093 228.49 83.30 IND MAJ .04210 8.51 .02213 234.76 101.20 H E A D FAMSIZ USMAR .01558 3.15 .01198 251.91 182.61 IM .00064 0.13 .00042 10.33 6.40 LAN .00122 0.25 .00128 19.64 19.51 E T H .00081 0.16 .00066 6.51 5.03 R E L .00033 0.07 .00031 5.49 5.30 SSQ X S P .00632 1.28 .00075 150.16 17.15 OC X S O C .01817 3.67 .01817 37.77 37.77 Total .49457 100.00 - - a C h a n g e in R . Var iable groups were added to the regress ion in the order shown and then deleted s ing ly . Change in R upon addi t ion , d iv ided by maximum R with all variables included (x 100) 184 as we did ear l ier , to the a l ternat ives . T h e table thus reports the 2 change in R observed upon the deletion of each variable or variable group from the full model. It is clear from Table 13, if not from all the prev ious resu l ts , that WTIME is by far the most important explanatory var iab le . T h e decision to explore this variable fur ther in Chapters IV and V thus appears well f ounded . Exper ience (or more agnostical ly, the "l ife- cycle factor") was included early and is important upon addition but very much less so upon delet ion. The linear term for schooling behaves similarly. One should note, however, that the presence of S S Q , X S P , and X S O C in the full model predisposes this resu l t . When all the human-capital var iables and their interactions are deleted, 2 the R falls by 0.042; the F statistic for their joint s ignif icance is 99.69. Converse l y , when the "unorthodox" variables C E O through OC 2 are deleted, the R falls by 0.105; and the correspond ing F statistic is 83.16. Broadly speak ing , geographic and industr ial factors seem to play an important role in earn ings and inequality determinat ion— v e r y nearly as important, perhaps , as that of school ing. Family status is associated with earn ings , although one cannot be conf ident about the direction of causa l i ty . T h e persona l -background variables identif ied here account for a v e r y small proport ion of total i n - equal i ty , at least insofar as wage rates are concerned . Neverthe less , the signif icance of individual coeff ic ients shows that some small g roups may have strongly d ivergent earn ings exper iences . 185 T A B L E 14 R A T E S OF R E T U R N T O S C H O O L I N G IMPLIED B Y V A R I O U S S P E C I F I C A T I O N S OF T H E E A R N I N G S F U N C T I O N Equation Number Estimated Return (%) Details of Specif ication a (CS1) 6.95 Includes S only (CP1) 8.91 A d d s P, PSQ (CP2 ) 8.66° A d d s S S Q , X S P (CP5) 7.75 Includes WTIME; excludes S S Q , X S P (CP17) 5.34 Includes OC (CP7) 7.63 A d d s L E N C , excludes OC (CP8) 6.88 A d d s C E O , T Y P E (CP9) 7.05 A d d s IND, MAJ (CP10) 6.75 A d d s H E A D , FAMSIZ, USMAR (CP12) 6.51 A d d s IM, L A N , E T H , R E L (CP14) 7.32 Excludes WTIME (CP16) 6 . 0 3 b Re- inserts WTIME, S S Q , X S P Changes noted are cumulative 'Calculated at mean levels of schooling and experience A s a final matter, it seems useful to compare, all at once, the schooling coeff ic ients obtained from var ious specif ications of the earn ings funct ion . These are col lected in Tab le 14. T h e largest implied rates of re turn occur with hours of work free to v a r y ; the smallest, when occupation is held constant . With hours f i xed , the 186 range is from 6.03% to 7.75%; with hours var iable , it is from 6.95% to 8.91%. In neither case does the degree of uncerta inty seem especially serious from a policy point of v iew. 46 If one were to add a correct ion for economic growth — s a y , 47 2.5%—as does the prev ious ly c ited Statistics Canada s tudy , the preceding f igures would increase accord ing ly . In the comparable (time-variable) case, they tend to exceed the Statist ics Canada estimate of approximately 8%. However, the latter takes into account the direct pr ivate and social costs of educat ion, which are ignored by the cur rent p rocedures . The present estimates imply re turns lower than found by Podoluk for Canada a decade earl ier and lower than reported by Mincer for the United States. A n Interactions Model A t several points in preced ing chapters we have cons idered the interactions specif ication put forward by Haessel and K u c h . It will be recalled that these authors attempt to explain possible d i s - parit ies in the rate of re turn to human capital by making them a function of certain independent var iab les . Since earnings are assumed to equal (at least in part) the product of human capital and its rate of r e t u r n , the resul t , upon substitution for the latter, is an estimating equation d isp lay ing a number of interaction terms. In selecting var iables to explain the rate of r e t u r n , Haessel and Kuch emphasize personal background and occupat ion. Using the former, they investigate the problem of d iscr iminat ion. T h e present s tudy is more concerned , however, with the sort of market imperfection which may be captured by the var iables " indus t ry" and "place of res idence . " Hence, the following r e g r e s - sion model is postulated: INC. = cn + r. • H. + b„ • WTIME. + u. i 0 i i 4 I I r. = a Q + alj • C E O . + • IND(. H. = h n + h,S. + h-P. + h ,PSQ . , I 0 1 i 2 i 3 I where a'̂  and a^ are row vectors of coeff icients mult iplying the columr vectors IND. and C E O . , which descr ibe individual i. A s in prev ious i i ^ notation, r. stands for the average rate of re tu rn on units of human capi ta l , the total accumulation of which is g iven by H.; and u. is an e r ro r term with classical p roper t ies . The remaining lower-case symbols are scalar coeff ic ients . Upon substitut ion into the f i rst equation we obta in : , N C i = ( c 0 + a 0 n 0 ) + a 0 h 1 S i + a o h 2 P i + a 0 h 3 P S Q , • B I | WTIME I + a ' ^ G E O j + a 2 h 0 I N D . + a ' ^ X S G E O j + a j ^ X S I N D . ' + a L h 0 X P C E O . + a ' h . X P S Q G E O . + a ' h - X P I N D . 1 2 i 1 3 i 2 2 i + a 2 h 3 X P S Q I N D . + u. 188 where the interaction terms are as def ined in Table 3. The reg res - sion coeff icients may be def ined implicitly by writ ing INCj = b Q + b ^ . + b 2 P . + b 3 P S Q j + b^WTIME. b ' G E O . + b ' IND. + b ' X S G E O . + b ' X S I N D . 5 i 6 i 7 i 8 i b ' X P G E O . + b' X P S Q G E O . + b' XPIND 9 I 10 I 12 + b' . -XPSQIND. + u. 13 I i Here , b Q through b^ are sca lars; b'5 through b ' 1 3 are row vectors . The preced ing equation is amenable to o rd inary least squares estimation by v i r tue of the fact that the express ions for r. and H. are assumed nonstochast ic . Haessel and Kuch show that if random components other than u are present , the model will be subject to heteroskedast ic i ty . T h e y consequently develop an asymptotical ly • 48 eff icient (maximum-likelihood) estimation p rocedure . Owing to the computational burden involved in treating the present sample, this refinement is not pursued here . We must therefore be somewhat cautious in accept ing the der ived standard e r r o r s , although the estimated coeff icients are presumably unb iased . From the coeff ic ients it is possible to obtain estimates of the re turn to schooling within a g iven region or i n d u s t r y . One need only compute 189 d INC /dS. = b, + b !_(dXSGEO./dS . ) +b' ( d X S I N D . / d S . ) . I I 1 / I l o I I Note, however, that this rate of re turn is not quite the same thing as r., the analytical device used above. T h e latter is the rate of return to a unit of human capi ta l ; the former is the rate of re turn to a (time) unit of school ing. Results appear in Table 15. T h e schooling interactions 2 shown in ( C M ) contr ibute only 0.004 to the value of R , though as a 49 g roup they are highly s igni f icant . T h e vectors X S G E O and X S I N D , taken in that o rder , are s ignif icant individual ly as wel l . Over regions, as shown by the former, the implied rate of re turn var ies from 7.5% in Atlantic Canada to 4.3% in Br i t i sh Columbia (for workers in the reference i n d u s t r y , manufacturing) . Since these regions are general ly regarded as being at or near opposite ends of the scale with respect to levels of education and human-capital scarc i ty , this outcome seems consistent with o rd inary demand-and-supp ly in fer- ences . Over industr ies , the range is a little larger than over reg ions—about 4.7 percentage po ints . A s in the case of occupat ion, however, it may be deemed somewhat improper to hold indust ry constant in estimating r e t u r n s . T h e relevant opportuni ty wage need not be found in the indust ry within which the individual is cu r ren t l y employed. T h i s objection is perhaps less serious with respect to the experience interact ions. Because workers tend to give up mobility 190 T A B L E 15 T H E I N T E R A C T I O N O F S C H O O L I N G A N D E X P E R I E N C E WITH I N D U S T R Y A N D P L A C E O F R E S I D E N C E a Ind. Variable Equations (dependent variable = INC) ( C M ) (CI2) Constant .8142 (22.1) .7399 (15.1) S .0571 (18.2) .0593 (18.0) P .0519 (45.5) .0559 (22.1) PSQ -.0008 (38.3) -.0009 (18.4) WTIME .6807 (85.0) .6807 (84.9) GE01 -.3860 (7. 65) -.2770 (3. 50) G E0 2 -.1266 (3. 87) -.2433 (4. 67) GE04 -.1568 (2. 87) .0444 (0. 52) G E 0 5 -.0531 (0. 89 -.1005 (1 . 13) G E 0 6 .1804 (3. 39) .3375 (4. 32) IND1 -.7041 (10.8) -.3311 (3. 10) IND2 -.0832 (0. 79) -.0779 (0. 48) IND3 -.6889 (4. 04) -.8619 (3 . 02) IND4 .1581 (1 . 75) .1828 (1 . 29) IND6 .1626. (3 . 11) .2040 (2. 44) IND7 .0312 (0. 62) .1266 (1 . 60) IND8 -.1965 (*. 05) -.2183 (3 . 17) IND9 -.0731 (0. 77) -.0199 CO. 15) IND10 -.5111 (12 .7) -.3602 (5 . 86) XSGEOI .0179 (3. 60) .0180 (3. 36) X S G E 0 2 .0059 (1 . 88) .0099 (2. 94) X S G E 0 4 .0008 (0. 16) -.0054 (0. 94) S C G E 0 5 .0008 (0. 15) .0043 (0. 73) X S C E 0 6 -.0134 (2 . 82) -.0182 (3. 61) XSIND1 -.0054 (0. 76) -.0157 (2. 03) XSIND2 .0091 (0. 76) .0040 (0. 32) XSIND3 .0346 (1 . 65) .0486 (2. 12) XSIND4 .0038 (0. 42) -.0006 (0. 07) XSIND6 -.0129 (2. 35) -.0173 (2. 89) XSIND7 .0006 (0. 12) -.0022 (0. 41) Table 15 (continued) 191 Ind . Var iable Equations (dependent variable = INC) ( C M ) (CI2) XSIND8 XSIND9 XSIND10 X P C E 0 1 X P C E 0 2 X P G E 0 4 X P C E 0 5 X P G E 0 6 XPIND1 XPIND2 XPIND3 XPIND4 XPIND6 XPIND7 XPIND8 XPIND9 XPIND10 X P S Q G E 0 1 X P S Q G E 0 2 X P S Q G E 0 4 X P S Q G E 0 5 X P S Q G E 0 6 XPSQIND1 XPSQIND2 XPSQIND3 XPSQIND4 XPSQIND6 XPSQIND7 XPSQIND8 XPSQIND9 XPSQIND10 .0038 .0087 .0337 (0.81) (1.10) (9.49) .0029 .0113 .0298 -.0151 .0052 -.0111 -.0017 -.0063 -.0193 .0114 -.0001 .0074 .0050 -.0052 .0050 -.0119 -.0093 .0003 .0001 .0002 .0001 .0001 .0003 -.0003 .0001 -.0002 -.0002 .0001 -.0001 .0002 .0001 0.58) 1.32) 7.77) 3.46) 1.86) 2.78) 0.40) 1.63) 3.98) 1.19) 0.00) 0.91) 1.18) 1.23) 1.49) 2.17) 2.80) 3.93) 1.02) 2.33) 0.92) 0.79) 3.06) 1.68) 0.37) 1.38) 2.12) 0.85) 1.84) 2.44) 2.16) ,455 ,458 Estimated for the main sample, 22,682 observat ions The f i rs t f igure in each set is a regression coeff ic ient; the second, in parentheses, is the correspond ing t ratio, written in absolute terms. 192 as they gain exper ience, rates of re turn to the latter form of human capital within part icular regions and industr ies may be of definite practical re levance. Like the schooling interactions in ( C M ) , those 2 involv ing experience in (CI2) add v e r y little to the R , but enough to be judged signif icant in an F test at the 0.01 l e v e l . 5 0 T h e " re tu rn" to an additional year of experience is lowest at the (national) mean in Manitoba-Saskatchewan (1.24%) and highest in A lberta (1.72%). It is lowest in agr icu l ture (0.88%) and highest in f ish ing (1 .88%) . 5 1 Although rates of re turn to schooling and experience do appear to vary across regions and industr ies , it cannot be claimed that such variat ion contr ibutes v e r y strongly to the preva i l ing level of earnings inequal i ty . Whereas, region and indust ry are important 52 in themselves, they do not have much effect on the earn ings potency of d iscret ionary human-capital investment. If such variation in the rate of re turn is indeed an important source of inequal ity, better data, with groups more narrowly defined than at present , will obviously be needed to establ ish the fact . APPENDIX IIIA T H E WORKING S A M P L E : D I S T R I B U T I O N S O F S E L E C T E D C H A R A C T E R I S T I C S T A B L E 16 INDIVIDUAL INCOMES BY SIZE C A T E G O R Y Size Category ($'s) Numbers of 1 ndiv iduals Employment Income Total Income 0- 999 1136 748 1,000- 1,999 1254 1108 2,000- 2,999 1462 1343 3,000- 3,999 1740 1668 4,000- 4,999 2004 2010 5,000- 5,999 2368 2362 6,000- 6,999 2519 2465 7,000- 7,999 2440 2505 8,000- 9,999 3344 3495 10,000-11,999 1838 2076 12,000-14,999 1254 1380 15,000-17,999 536 606 18,000-24,999 428 493 25,000-34,999 199 220 50,000-74,999 42 44 75,000 or more 18 25 Total 22,682 22,682 193 T A B L E 17 FAMILY INCOMES O F INDIVIDUALS BY SIZE C A T E G O R Y Size Category ( $ ' s ) Number of Individuals Loss 9 0 0 1- 999 123 1,000- 1,999 262 2,000- 2,999 529 3,000- 3,999 752 4,000- 4,999 1,020 5,000- 5,999 1,263 6,000- 6,999 1,512 7,000- 7,999 1,674 8,000- 9,999 3, 374 10,000-11,999 2,889 12,000-14,999 2,891 15,000-19,999 2,080 20,000-24,999 745 25,000-34,999 410 35,000-49,999 194 50,000 or more 98 Nonfamily Individuals 2,857 Total 22,682 195 T A B L E 18 S C H O O L I N G B Y A G E G R O U P Level of Schoo l ing 3 Number of Individuals A g e d 15-24 25-34 35-44 45-54 55+ Total 1 7 9 42 39 70 167 2 32 67 159 234 378 870 3 500 1261 1738 1647 1614 6760 4 939 1411 1219 983 752 5304 5 579 680 474 386 267 2386 6 922 923 599 472 311 3227 7 100 247 200 178 184 909 8 188 323 188 123 101 923 9 43 130 79 55 38 345 10 92 308 169 136 78 783 11 5 31 25 14 9 84 12 30 329 267 180 118 924 Total 3437 5719 5159 4447 3920 22,682 1 = no school ing; 2 = grades 1-4; 3 = grades 5-8; 4 = grades 9-10; 5 = grade 11; 6 =. grade 12; 7 = grade 13; 8 = 1-2 years u n i v e r s i t y ; 9 = 3-4 years un ive rs i t y , without degree; 10 = 3-4 years un ive rs i t y , with degree; 11 = 5 or more years un ive rs i t y , without degree; 12 = 5 or more years un ivers i ty , with degree . 196 T A B L E 19 S C H O O L I N G BY REGION Numbers of Individuals Schoo l ing 8 Atlantic Quebec Ontario Manitoba- Sask . A lberta B . C . 1 36 26 61 15 15 14 2 113 346 163 74 31 43 3 625 2319 2297 638 394 487 4 443 1446 2033 456 407 519 5 232 704 753 229 206 262 6 157 537 1291 293 391 558 7 8 73 682 15 20 111 8 63 242 373 70 60 115 9 29 86 110 29 34 57 10 36 250 317 50 61 69 11 64 237 361 66 82 114 Total 1810 6302 8572 1937 1706 2355 See footnote to Table 18. 197 T A B L E 20 MEAN EARNINGS BY REGION A N D L E V E L O F S C H O O L I N G Mean Earn ings ($ 's) Level of Schoo l ing 3 Canada A t l . Quebec Ontario Manitob; Sask . T i" A l t a . B . C . 1 4090 2892 3690 5417 3076 3190 4177 2 4740 3367 4522 5709 3862 4932 5534 3 5889 4228 5696 6464 4924 6581 6931 4 6576 6576 6333 6927 5974 6497 7298 5 7227 6720 6753 7905 6443 7022 7840 6 7371 5757 7349 7785 5775 7282 7789 7 9157 7411 10403 9235 6992 7701 8537 8 8379 8130 8310 8633 9345 7107 7914 9 8356 5944 9153 8732 6915 11184 6698 10 11190 7982 10743 12501 9397 10422 10434 11 8470 8117 9425 8110 2635 5406 9541 12 16365 12015 14808 18804 14215 13524 17612 Al l Levels 7233 5472 6793 7963 6060 7306 8019 See footnote to Table 18 198 T A B L E 21 S C H O O L I N G BY I N D U S T R Y Level of Schoo l ing 3 Numbers of Individuals Employed A g r i c u l t . Fores t ry F ish ing M i n i n g Petroleum Manufac. 1 22 8 4 7 38 2 81 34 18 23 249 3 719 173 60 184 2007 a 320 81 29 180 1598 5 92 31 11 68 679 6 109 28 5 78 944 7 16 6 0 19 281 8 27 4 0 16 220 9 4 3 1 5 72 , 10 12 2 0 19 170 11 1 0 0 1 27 12 10 2 0 14 94 Total 1413 372 128 614 6379 T r a n s p . , C o n s t r . Commun. , U t . T rade Finance Serv ices 1 22 11 17 1 37 2 120 113 92 12 128 3 961 824 944 87 801 4 616 744 962 123 651 5 226 356 493 106 324 6 286 442 653 209 473 7 56 86 157 119 169 8 66 106 154 97 233 9 19 43 65 22 111 10 16 56 76 53 379 11 5 8 7 6 29 12 12 49 29 39 675 Total 2405 2838 3649 874 4010 3 See footnote to Table 18 199 T A B L E 22 MEAN EARNINGS BY I N D U S T R Y AND L E V E L OF S C H O O L I N G Level of Mean Earn ings ($ 's) a Mining Schooling A g r i c u l . Forestry F ish ing Petroleum Manufac. 1 2353 3907 875 6173 4140 2 2981 3672 2239 5097 5528 3 3984 5175 3557 7180 6337 4 4709 6001 3244 7860 6800 5 5052 7096 3786 8504 7172 6 4531 10970 4328 7713 7753 7 4403 8373 - 9226 9831 8 3905 9342 - 9585 8942 9 5057 2770 1000 12154 8919 10 4844 14320 - 13875 11813 11 1010 - - 7900 8141 12 21463 8630 - 14150 13484 Al l Levels 4312 5931 3247 8038 7239 T r a n s p . , C o n s t r . Commun. , U t . T rade Finance Serv ices 1 5700 4950 3779 5010 3970 2 5603 5352 4038 5678 3958 3 6454 6599 5880 6395 5056 4 6695 7440 6448 7839 5658 5 7231 8551 7148 8761 5980 6 7575 7917 • 7039 8569 6323 7 8599 10307 8271 10083 8277 8 6849 8324 8162 10414 8023 9 7795 9285 9221 7791 7498 10 12716 12201 13681 15202 9685 11 8892 10602 8750 9335 8144 12 12003 12580 12912 14656 17359 Al l Levels 6819 7656 6732 9300 8451 a S e e Footnote to Table 18 T A B L E 23 O C C U P A T I O N Occupational Category Number of Individuals 1 Managerial and administrative 1241 2 Natural and social sciences, engineer ing 988 3 Teach ing 635 4 Medicine and health care 407 5 Cler ical 1702 6 Sales 2525 7 Serv ice 1592 8 Farming and other pr imary 2236 9 Process ing , fabr icat ing , repair ing 4868 10 Construct ion trades 2556 11 T ranspor t equipment operation 1597 12 A r t s , re l ig ion, other , and not stated 2335 Total 22,682 201 T A B L E 24 E T H N I C A N D RELIGIOUS C R O U P Religion Ethnic Group Protestant Catholic and O r t h . Jewish and Other No Religion Total 1. Br i t i sh Is. 6949 1610 394 824 9777 2. W. European 1861 7302 240 329 9732 3. E. European 312 978 54 103 1437 4. Chinese and Japanese 52 18 32 66 168 5. Jewish 4 1 346 11 362 6. Nat. Indian 55 73 9 6 143 7. Other 191 685 114 63 1053 Total 9424 10667 1189 1402 22682 T A B L E 25 PERIOD O F IMMIGRATION T O C A N A D A Period of Immigration Number of Individuals Before 1946 1025 1946- 1965 3073 1966 or later 953 Canadian born 17631 Total 22682 202 A P P E N D I X I I B MISCELLANEOUS REGRESSIONS (CS2) INC = .6077 (13.0) + .0688 S + -(25.0) .0588 P (26.5) .0010 PSQ (24.2) + .8776 WEEKS (45.9) - .2731 CEOI (1.16) .0569 CE02 - (1.38) .2905 CE04 - (5.91) .0669 CE05 (1.35) + .0468 CE06 (1.11) + .0725 SPHC1 (1.25) • .0492 SPHC2 + (1.09) .0486 SPHC4 + (1.02) .0174 SPHC5 (0.30) + .0516 SPHC6 (0.92) + .0192 SPHC7 (0.67) R 2 .464 number of observations = 5670 (CS3) INC .5840 (12.4) + - .0696 S + (25.2) .0599 P (26.9) .0010 PSQ (24.6) + .8777 WEEKS (46.1) t - .1446 SPHC1 (4.48) - .0011 SPHC2 - (0.46) .1425 SPHC4 - (4.67) .0311 SPHC5 (0.79) + .0971 SPHC6 (2.38) + .0043 SPHC7 (0.16) R 2 = .458 number of observations = 5670 (CS4) INC .6237 (13.7) + .0683 S • (25.0) .0587 P (26.7) .0010 PSQ (24.4) + .8775 WEEKS (46.2) .2138 CEOI (6.49) .02193 CE02 - (1.03) .2562 CE04 - (8.03) .0527 CE05 (1.57) + .0755 CE06 (2.56) R 2 = .464 number of observations = 5670 (CS5) INC 1.041 (34.8) + .0873 SCOST + (26.7) .0603 P (27.1) .0010 PSQ (25.1) + .8827 WEEKS (46.5) R 2 .454 number of observations = 5670 (CS6) INC = 1.550 (93.7) + .0426 P (36.1) .0008 PSQ + (34.4) .7143 WTIME (85.3) + .0044 LENC2 (0.15) + .0481 LENC3 (3.20) + .1614 LENC4 - (8.56) .2576 CEOI (14.0) .1343 CE02 . (11.8) - .1586 CE04 (8.98) - .0340 CEOS (1.86) .0661 GE06 + (4.11) .1632 TYPE - (16.2) .7530 IN01 (34.5) - .0607 IND2 (1.64) - .5053 IND3 (8.16) + .2000 IND4 + (6.86) .0024 IND6 + (0.15) .0547 IND7 (3.53) - .1424 IND8 (10.0) + .1556 IND9 (6.32) + .0202 IND10 (1.46) R 2 = .397 number of observations = 22682 'Figures in parentheses are t ratios, written in absolute terms. N O T E S C H A P T E R III 'For a complete descr ipt ion see Canada , Statist ics Canada , Public Use Sample T a p e s : User Documentation. 2 One might think of us ing a "Tobit" procedure in this s ituat ion; however, such an approach will not be explored here . Zero earn ings are not per se inconsistent with the model if k = 1. Yet , indiv iduals are not general ly observed to specialize in on- the- job t ra in ing . 3 There is, of course , the pure ly mechanical problem of express ing nonposit ive earn ings in logarithmic form. In any case, negative earn ings are l ikely to be a t rans i tory phenomenon for the ind iv idua l , better ascr ibed to ownership of physical capital and to ent repreneursh ip than to human capi ta l . 4 T h i s is not to say, unfortunate ly , that the sample c o n - sists only of workers in the pr ivate sector . Only those in "public administration and defence" ( S . I . C . Division II) could be exc luded . 5 A coin fl ip in fact chose the second. 6 T h e latter was $6574. See Canada , Statistics Canada , 1971 Census of Canada , V o l . I l l , p t . 6, Income of Individuals, Catalogue no. 94-768 (Ottawa: Statistics Canada, May 1975), p. 1, Table 19. S c h o o l i n g , Exper ience , and Earn ings , p. 90. g Isolating these factors completely of course demands both slope and intercept dummies. Slope dummies are not prov ided here except in the form of one interaction between agr icu l ture and years of school ing. In prel iminary testing the insertion of this latter variable and the intercept dummy for agr icu l ture lowered the schooling 203 204 by about 0.5 percentage po ints . T h i s result implies that omitting farmers might cause an even larger d ivergence between the present f ind ings and those of Mincer than is observed below. 9 He repor t s : "The regress ion coeff ic ients in the age c r o s s - section were v e r y close to those in the exper ience c ross -sec t ion , but the multiple coeff ic ients of determination were . 0 2 - .03 lower in the age set . . . . " I b i d . , p. 91, no . 7. 1 ^Presumably, such indiv iduals are no longer making posit ive gross investments. T o represent their exper ience prof i les may str ict ly require a nonsmooth func t ion . T h e exponential form may be especial ly inappropriate since as we have seen, it never falls to zero. 11 I b i d . , p. 90. 1 2 T h e Pareto d istr ibut ion is g iven by f (Y) = A y " a , where A and a are constants (a > 2) and f (y) is the proport ion of indiv iduals with income greater than Y . If V represents the largest income in the population and U, the boundary of the open-ended c lass , the mean income in this interval is g iven by V U A Y -a Y d Y [ A / ( 2 - a ) ] Y 2-a U A Y a dY [A/(1 - a ) ] Y 1-a U a as long as V is large . F i t t ing a Pareto c u r v e to the d istr ibut ion if INC within the sample y ie lded a value of 2.657 for a . T h i s implies a mean of $189,200. 13 . For example, if self-employment is like a lottery, with a few large gains and many small losses (relat ive to other oppor tun i t i es) , indiv iduals who choose to enter may wil l ingly pay a premium in the form of inferior r e t u r n s . Those with a taste for se l f -d irect ion may do the same. ed above . 1 4 T h e open-ended class was dealt with in the manner explain- 205 1 5 T h e variables U S M A R , H E A D , and FAMSIZ were used in making the requ i red determination. 16 Canada, Dominion Bureau of Stat ist ics , Pr incipal Taxes and Rates: Federa l , Provincial and Selected Municipal Governments , 1970 (Ottawa: Queen's Pr in ter , 1970). 1 7 T h e necessary f igures were obtained from Canada, Depart - ment of National Revenue, 1972 Taxation Statist ics [1970 taxation year] (Ottawa Information Canada , 1972), p. 152, Table 16. 1 8 The source was i b i d . , p p . 150-151, Table 15. 19 A problem here is that G E O - C O D E gives the individual's residence on July 1, 1971, not his residence for tax purposes in 1970. Some e r ro r may thus attach to recent interprovincia l migrants . 20 In the f i fth c lass , 50 was used rather than 50.5. 21 See, for example, Canada , Health and Welfare Canada , Character is t i cs of Low-Wage Earners in Canada , Social Secur i ty Research Report No. 01 (Ottawa: Information Serv ice Department of National Health and Welfare, September, 1976); or United States, Bureau of the C e n s u s , Statistical Abst rac t of the United States, 1970 (Washington: U . S . Government Pr int ing Of f ice , 1970). 22 It may happen, of course , that grades and years do not co r respond , as students skip grades or fail to win promotion. Whereas, years of schooling measure investment costs , one may speculate that grades relate more closely the mastery of certain ski l ls a n d , hence, to p roduc t i v i t y . T h e adopted procedure thus leans, if at a l l , toward the latter interpretat ion. 23 Canada , Statist ics Canada , Data Processing D iv is ion , "Special Tabulat ions 12295A and 12295B" (unpub l i shed , September, 1976). Place of highest grade was selected a pr ior i instead of place of c u r r e n t residence because the former, being less distant in time and more intimately connected with the environmental factors determining educat ion, seemed more l ikely to be a good predictor of school ing. 2H T h i s assumption and the one below match those of Haessel and K u c h , "Earnings in C a n a d a . " 25 A s one would expect , place of residence is correlated with the schooling pred ic tor , place of highest g r a d e . In the sample, correlation coeff icients between correspond ing elements of C E O and SPHC (see below in the text, or Table 3) average about 0.8. 26 T o be more prec ise , under the standard procedure S contains a measurement e r ror which is l ikely to be correlated with the variables named. The analysis is similar to that presented in Append ix I IB. 27 Mincer apparent ly uses age 14. See School ing, Exper ience , and Earn ings , p. 48, notes to Table 3.1. 28 Mincer assumes age 5; o thers , age 6. T h i s scaling affects not only the regression constant but also the coeff icients adher ing to the var ious nonlinear transformations of P. 29 T h i s is Mincer's p rocedure . 30 For precise definit ions see Canada, Statist ics Canada, Dict ionary of the 1971 Census Terms (Ottawa: Statist ics Canada , 1972). 31 Here and below, c f . Haessel and K u c h , "Earnings in Canada . 32 These descr ipt ions apply to the earn ings funct ion . Recall that the quadrat ic stems from a linear investment prof i le . 33 T h i s programme was written by Keith Wales formerly of the Un ivers i ty of B r i t i sh Columbia Computer C e n t r e . 34 See Append ix U A . 35 Note that is the system used here to number regress ion equat ions, " C " stands for "Canada ," and other alphanumeric characters for the estimation procedure or speci f icat ion. T h u s (CV4) corresponds to Mincer's ( V 4 ) , a n d so on 207 36 See Podoluk, Incomes of Canadians. 3 7 . , . . Ibid 38„. . Since vocational tra ining was not deducted in computing exper ience , it might be argued that some "double count ing" of human capital takes place when LENC and P appear in the same regress ion . T o avoid confus ion , one must carefu l ly interpret L E N C as s ign i fy ing only the intensity of investment in relation to the average subsumed under P. 39 Haessel and Kuch "Earnings in C a n a d a , " use a dummy vector similar to S P H C , but they do not encounter the multicoll inearity problem inasmuch as their sample consists entirely of indiv iduals resident in Toronto or Montreal . 40 The contr ibut ion of MAJ is negligible in comparison. 11 See Append ix 1MB, Equation (C56) . 42 See, however. Table 14. "Minimum" relates only to the present subset of var iab les . 43 School ing, Exper ience, and Earn ings , p. 56. 44 See Chapter II. 45 See Table 3, and for a detailed explanat ion, Canada , Stat ist ics Canada , Occupational Classif ication Manual, Census of Canada , 1971 (Ottawa: Information Canada, 1972). **6See the discussion in Chapter II. 47 Economic Returns to Education . 48 The authors unfortunately do not report the extent to which their eff icient estimates di f fer from those prov ided by o rd inary least squares . 49 4 3 F = 15.36. 208 5 0 F = 5.63. ^ O n e suspects that the v a r y i n g payoff to exper ience may have something to do with the pace of technological change in the two indust r ies . Exper ience counts least where change is r a p i d . Investigation of this hypothesis is nevertheless beyond the scope of the present s t u d y . 52 Observe that, within the context of the interactive model, the intercept terms for region and indus t ry explain di f ferences in the rate of return on the individual 's initial endowment of human capi ta l . C H A P T E R IV T H E S I M U L T A N E O U S D E T E R M I N A T I O N O F H U M A N - C A P I T A L I N V E S T M E N T A N D L A B O U R S U P P L Y T h e investment models we have so far cons idered treat labour supply as an exogenous factor in earn ings determination. T h e sole problem for the individual is to choose an investment prof i le which maximizes net discounted lifetime earn ings , or "wealth." Since there is in effect only one good, wealth and uti l ity maximization amount to the same t h i n g . In pursu ing this simple objective, the individual is fu r ther assumed to ignore all systematic variation in planned or in realized hours of w o r k . 1 Hence, the work profi le is not only exogenous but also constant over the life c y c l e . Both assumptions appear untenable. Empir ical ly , the work 2 profi le is somewhat peaked, rather than hor izontal . Though it would not be v e r y di f f icult to incorporate this or any other exogenous shape into an amended wealth-maximization model, it remains to be shown whether the s tandard predict ion of monotonicaily decl in ing investment in human capital would continue to ho ld . Theoret ica l ly , it is d i f f icu l t to ignore the repercuss ions of the labour- le isure choice . That choice presumably depends upon a uti l ity funct ion which includes time in the form of leisure as an argument. Yet time is also the lone or 209 210 pr inc ipal input in the product ion of human cap i ta l . T h e rational indiv idual will no doubt wish to allocate his fixed endowment of time optimally among work, le isure, and investment. Decision-making will be simultaneous rather than sequential , cont rary to our prev ious assumption. To u n d e r - stand such behaviour , we must apparent ly d iscard the f i rm-based notion of independence between consumption and investment and extend the analysis from the maximization of lifetime earnings to the maximization of u t i l i t y . 4 At the same time, it is especial ly important to keep in mind a point raised ear l ier , in Chapter 11—namely, that the rate of re turn to any form of human capital is not well def ined unless some reference is made to hours of work. Moreover, if work and investment are planned simultaneously, rates of re turn are "doubly endogenous" in the sense that they depend not only upon total investment, as in the Becker model, but also upon the profi le of h o u r s . Though it is always possible to compute the rate of re turn to schooling ex post for a g iven cross section of ind iv iduals , such an estimate will not co r respond , even in equilibrium, to the rate apprehended by these indiv iduals if we assume the wrong hours prof i le . T h e f i rs t section of this chapter s u r v e y s a small g roup of theoretical studies which explore the simultaneous determination of human-capital investment and labour s u p p l y . From the standpoint of later empirical appl icat ion, it is chief ly important in reviewing this work to f ind the answers to a pair of broad quest ions. T h e f i r s t , a l - ready mentioned, is whether the endogeneity of individual labour supply might upset the proof that investment decl ines monotonically over the life c y c l e . If the optimal propensi ty to invest is ever r i s i n g , the human-capital interpretat ion of concave earn ings prof i les is thereby weakened; and the empirical specif ication adopted earl ier is cast in doubt . We must therefore look at the robustness of the pred ic t ion . The second question we must examine is that of the general shape descr ibed by the optimal work prof i le . Investment in human capital is thought to determine the lifetime profi le of wage rates . The two are then presumed to combine multipl icatively to fashion the profi le of ea rn ings . Disentangl ing them again statist ical ly, so that we may trace the influence paths and assess the importance of human capital and other factors , is a useful research task . To beg in , we must t ry to glean from the theoretical arguments some testable hypotheses concern ing how the wage and work profi les relate to one another—whether they are indeed concave funct ions, whether they have peaks within the relevant range, and if so, whether these peaks must occur in a g iven o r d e r . The second section of this chapter draws in an informal way upon results of the uti l ity-maximization approach . A simultaneous l inear model of work and earnings is specif ied for estimation with the c u r r e n t data set . Results are reported and d iscussed in Chapter V . 212 T H E O R E T I C A L A N A L Y S I S To date, there have been four major theoretical studies in which human-capital investment and labour supply appear simultaneously as endogenous var iab les . The earl iest pub l i shed , by Chez and B e c k e r , 5 uses traditional static uti l i ty maximization with d iscrete ly dated commodities to obtain the f i r s t -o rder condit ions which character ize the solution to the indiv idual 's p lanning problem. T h i s mode of analys is turns out to be suff ic ient to answer the two broad quest ions just posed; however, it does not prov ide a v e r y r ich unders tand ing of the dynamic, processes invo lved . T h e other studies, by B l inder and W e i s s , 6 by 7 8 Heckman, and by R y d e r , S ta f fo rd , and S tephan , employ control theoretic techniques to de r i ve , within certain qual itat ive limits, the optimal profi les for investment, wages, and work. T h i s su rvey will therefore emphasize the latter a p p r o a c h . Since all four studies reach similar conc lus ions , it is not n e c e s s a r y — a n d it would in fact be r e d u n d a n t — t o trace the mathematical details of each argument . Of greater interest are the part icu lar assumptions which the var ious authors subst i tute for one another in der i v ing their resu l t s . The interchangeabi l i ty of certain assumptions and the consistent necessity for others are the points to note in the following ana lys i s . It is hoped that reduc ing the rather complex c o n - trol theoretic studies to a s ingle, ^uniform notation will also prove enl ightening in itself . 213 Components of the Model Al l the exist ing studies begin with an individual uti l i ty function such as U = U ( C , I ) . . . .(45) def ined over C , a composite Hicksian consumer good, and I , the quant i ty 9 of le isure, measured as a proport ion of the total time avai lable. B l inder and Weiss (B-W) assume strong separabi l i ty , as do R y d e r , S ta f fo rd , and Stephan ( R - S - S ) , who specialize fur ther by letting 0 fl U ( C , I ) = ln(aC H ) . Heckman ingeniously avoids separabi l i ty by writ ing U ( C , I ) = U ( C , I *H), where as before, H is the stock of human cap i ta l . T h e latter thus serves as an augmenting factor in the consum- ption of le isure . T h i s specif ication is suff ic ient to produce determinate resu l ts , though it is not clear that it is a weaker postulate than separ- ab i l i ty . Heckman's i l lustrat ive f ind ings and most of his comparative dynamic results stem from the C E S case. Apar t from ut i l i t y -p roduc ing le isure , the competing uses of time consist of work, denoted by m, and t ra in ing , denoted by j . The time budget is simply I + m + j = 1 . . . . .(46) T o connect this with the earl ier ana lys is , ' let us define "market time" as h = m+j . Then k' = j / h . R - S - S , along with Heckman, choose I and j as control variables for the optimization problem; B-W select h and k'. Since all are determined simultaneously, and since m is made dependent by (46), the choice is pure ly one of convenience. That of B-W meshes best with the previous d i scuss ion . In addition to the time budget , the individual faces a lifetime expendi ture constra int , which at any instant takes the form A = mwH + rA - C = (1 - k') hE - r A - C , . . . . (47) where A represents nonhuman wealth, and A, its time der ivat ive . Recall that w and r s ign i fy the re turns to human and nonhuman wealth respec- t ive ly , and that E = wH is earning capac i ty . T h e pr ice of consumption goods (the numeraire) has been set to un i t y . B-W amend (47) in a subtle but important manner. In place of k' they write the negatively s loped, concave function g ( k ' ) . Whereas, Mincer util izes W/E = (I - k ' ) , they employ W/E = g ( k ' ) , with g '(k ') < 0 and g"(k') < 0. B-W alert ly point out that if the "earnings- investment f ront ier" g(k ' ) were actually l inear, as Mincer postulates, there would be no advantage to combining tra in ing and work . T h e individual could achieve any point on the front ier by d iv id ing his time appropr iate ly between pure training (k 1 = 1) and pure work (k 1 = 0) . Since g(k ' ) > (1 - k') for 0 < k' < 1, concavity makes on-the-job tra in ing uniquely p r o f i t a b l e . 1 0 It is worth noting in connection with (47) that there is no general restr ict ion forc ing A to assume nonnegative va lues . Individuals 215 are free to borrow and to lend in a perfect ly competitive financial market at the g iven rate r. Instead, one might think of implementing Becker ' s prev ious ly su rveyed demand-and-supp ly model of human-capital investment by letting r = r ( A ) , with r '(A) < 0 for A < 0 . 1 1 We shall observe short ly how this specif ication would complicate the ana lys is . The final component of the present model is an equation descr ib ing the growth (and decay) of human assets . A s in Chapter I, we may write: Q H = a ( k ' h H ) y ; • * * * ( 1 9 ) H = Q H - dH = a ( k ' h H ) y - dH , . . . .(20) except that, here, k' alone gives way to k'h in recognition of the presumed var iabi l i ty in hours of potential investment time. R-S-S use precisely the foregoing speci f icat ion. A s we have seen, their assumption that 0 < y < 1 ensures , with w constant, that the marginal cost of produc ing human capital is increas ing . Heckman, on the other hand , manages with a general functional form, restr ic ted only as to f i rst and second partial 12 der ivat ives and containing both time and purchased educational inputs . B-W employ the special assumption that y = 1 ; accord ing ly , H = (cxk'h - d) H . . . . .(20') T h e y are able to proceed in this manner on account of g ( k ' ) . Concav i ty of the latter implies increasing marginal cost even though re turns in product ion are constant . Since equi l ibr ium and the time path of investment 216 depend only on the shape of the marginal cost c u r v e (given the shadow pr ice of human cap i ta l ) , it does not appear that exchanging u < 1 for g"(k ' ) < 0 has any effect on the general i ty of the resu l t s . A formal statement of the control problem is now poss ib le . It . . 13 is to maximize T e " p t U(C,I ) d t + B [ A ( T ) ] , 0 where p is the rate of time preference and B [ A ( T ) ] is the (separable) uti l i ty of terminal assets , subject to (46), (47), and (20) and t o 1 4 h = m + j > 0 and 0 ^ k' ^ 1 . . . .(48) g iven the initial condit ions H(0) = H Q > 0 and A(0) = A Q ^ 0 . . . . .(49) The control var iables are C , m, and j (or equiva lent ly , C , h, and k ' ) ; the state var iables are H and A . Ana lys i s The Hamiltonian, based on the assumptions of B-W, may be written as follows: J = e p t - jU tC , ! - h) + X A [ g ( k ' ) h w H + r A - C ] + A H [ ( o k ' h - d) H] } . . . . .(50) 217 A s usua l , A A and A H are shadow p r i c e s . T h e necessary condit ions for an interior maximum take the following form: 3J/8C = 0 : U c = A A . . . .(51) 3J/3h = 0 : U, = A A g ( k ' ) w H + A^^ak'H . . . .(52) 8J/3k ' = 0 : 0 = A A g ! ( k ' ) w H + A R a H . . . .(53) 3J/9A = - (d/dt ) ( A A e ~ p t ) : A A / A A = p - r . . . .(54) 3J/3H = - ( d / d t ) ( A H e ~ p t ) : A^/A^ = p + d - g ( k ' ) h w A A / A H - ak'h . . . .(55) (transversal i ty) : A H ( T ) H ( T ) = 0 . . . .(56) ( t ransversa l i ty) : A A ( T ) = B ' [ A ( T ) ] (57) These condit ions hold as a set wherever h > 0 and 0 < k' < 1. However, as we found in the case of the (Ben-Porath) income- maximization model, boundary solutions occur v e r y readi ly , por t ray ing familiar stages in the typical life c y c l e . Making leisure endogenous increases the possible number of such stages from two to four , namely: (I) "school ing" (h > 0, k' = 1) ; (II) " t ra in ing" (h >0, 0 < k'< 0 ) ; (III) "work" ( h > 0 , k' = 0) ; (IV) "retirement" (h = 0, k 1 a r b i t r a r y ) . 218 Since the data set uti l ized by the present s tudy samples only from the population of individuals in stages II and III, this review will ignore the other phases of the optimal p l a n . 1 5 Before we examine the prof i les of work and investment implied by (51)-(57) , it is worth pausing br ief ly to confirm the economic inter- pretation of these condit ions. Equation (51) merely demands that the marginal uti l i ty of goods be set equal to their shadow price at each instant; (57) imposes the same requirement on the terminal stock. Equations (51) and (54) together imply the well-known l i fe-cycle result that consumption fal ls, remains constant , or r ises accord ing to whether > 16 p — r . Equation (52) states that the marginal cost of nonleisure act iv i ty (Uj ) equals , f i r s t , the benefit in the form of real earn ings (A^g(k')wH) a n d , second, the benefit in the form of increased human capita l , or future earn ings ( X^ a k ' H ) . If k 1 = 0 (stage III), the marginal rate of substitut ion between goods and le isure, Uj /U^., simply equals the real wage, wH., just as in the static ana lys is ; but otherwise, U| /U^, > w H . Equation (53) requi res that the individual allocate his market time in such a way that the marginal input cost in foregone earnings (-A^g'fk 1 )wH) equals the marginal present and future benefit of increased earning potential ( A^ a H ) . It is also convenient at this point to note the effect of making r depend on A . Only (54) is a l tered: r is replaced by r ( A ) + r ' ( A ) A . T h e change is nevertheless c ruc ia l , as it makes the evolution of the shadow price a funct ion of the state var iab le . T h i s situation great ly complicates the ensuing ana lys is , and it is not known whether all of the 219 main conclusions s tand . Based on Heckman's comparative dynamic results for changes in an exogenous rate of i n t e r e s t , 1 7 one might r isk a guess that the pr inc ipal effect would be to flatten the wage prof i le ; however, nothing more is c learly apparent . T h e other alternative assumptions—those concern ing uti l i ty and the product ion of human cap i ta l—y ie ld s igni f icant , though manageable 18 changes in the preceding set of f i r s t -order condit ions. To accommodate the d i f ferences , the three control-theoret ic papers adopt d ivergent analytical strategies, together with some fur ther restr ict ions on behav iour . The reasons in each case are most easily understood if we follow for a moment the derivat ion of B-W. These authors s tudy , among other th ings , the optimal trajec- tories in ( k ' , h ) - s p a c e . If one differentiates (52) logarithmically with respect to time, it is possible to show, us ing (53), (54), (55), and (20') , that h [ - U n /U , ] = p - ( r + d)/(1 + n) , . . . .(58) where n H -k 'g ' tk ' ) /g(k') is the elasticity of g ( k ' ) . A similar operation performed on (53) y ie lds , eventual ly , k [ g " ( k , ) / g ' ( k ' ) ] = r + d - ok'h( 1 + n ) / n . . . . .(59) These express ions define two stationary loci h = 0 and k' = 0. A t h i r d , H = 0, may be obtained d irect ly from (20') . 220 All three cu rves are shown in F igure 1, reproduced (with the 1 g appropr iate notational amendments) from B-W. It is e a s y to ve r i f y by stra ight forward manipulation of (58), (59), and (20 1) that : (a) h = 0 is a vert ical line at k ' (0 < k < 1); (b) k' = 0 r ises monotonically from [0, -g'(0) (r +d)/a] to [1, ( r + d ) / a ] ; (c) H = 0 is the rectangular hyperbola H = (d/a)(1/k'); (d) the intersection of h = 0 and k' = 0, namely (k 1 , h ) , lies above H = 0 if (but not only if) r > p of the unit square , or in other words , on the boundary of stage I, where k 1 = 1. It would appear from the indicated motions that, unl ike P, some trajectories may cyc le about the point ( k ' , h ) ; but as B-W exp la in , such paths cannot a r i se . The reason prov ides considerable insight into the problem of formulating successfu l ly a model of the present k i n d . Inspection of (58) and (59) reveals that (g iven the constants) k 1 and h depend only upon k' and h . To each point in ( k ' , h ) - s p a c e there cor responds a unique mot ion, def ined by [ k ' ( k ' , h ) , h ( k ' , h ) ] . However, to attain the vert ical axis (k 1 = 0 ) , as all trajectories eventual ly must, a cycl ical path would have to c ross itself at an angle , implying two d i f ferent motions at the 21 point of intersect ion. T h i s situation could ar ise without contradict ion if either or both k 1 and h depended on the state or costate var iab les . E n s u r i n g that they do not (and that we may consequent ly work with a two-dimensional phase diagram) is a matter for careful theor izat ion. It is clear from F igure 1 that the B-W model prov ides the hoped-for theoretical conc lus ions . t F i r s t , /the gross propens i ty to invest (k 1 ) declines monotonically throughout stage I and is therefore nonincreas ing over the whole life c y c l e . Second, the supply of market 221 F igure 1 Phase diagram in ( k 1 , h ) - s p a c e . 222 hours (h) r ises to a peak at t^ and declines thereaf ter . T h i r d , if r > p (a suff ic ient condit ion o n l y ) , the peak in hours precedes t^, the peak in human capi ta l , which as we know, precedes the peak in measured earn ings , whenever d > 0. These are the restr ict ions which, at a minimum, any empirical model must test . A s noted, the other two studies der ive similar results by alternative means. Being unable to eliminate the unwanted state variables H, Heckman eschews the phase-diagrammatic approach in favour of solving the f i r s t -o rder condit ions to obtain the demand funct ions for goods, effective leisure (IH ) , and investment ( j H ) . Despite special izing the uti l i ty and product ion funct ions to the C E S form, he cannot rule out locally increasing investment time except by means of the auxi l iary assumption that depreciation is "smal l ." Comparative dynamic invest i - gation of changes initial wealth (human and nonhuman), the rate of interest, deprec iat ion, abi l i ty , and taxes furn ishes some interest ing hypotheses, but apparent ly none which the author is able to test with the data at h a n d . R-S-S are also faced with the presence of the state variable H on account of their nonlinear product ion speci f icat ion. T h e y proceed by letting p = r = 0. It is evident from Equation (54) that in this special case X ^ , is constant . There fo re , it is possible to draw a two dimensional phase diagram in ( H , X ^ ) - s p a c e and to deduce from it the behaviour of all the control var iab les . It tu rns out that h reaches its peak at the same time as H, though again, before the peak in measured e a r n i n g s . A s in Heckman, j cannot be shown to decline monotonicaily. 223 T h i s result is not, of course , inconsistent with the B-W conc lus ion, * * * stated in terms of k'. Since j = k 'h , we have j = k'h + k ' h . The f i rst term is always negative; the second is posit ive or negative accord ing to whether h < 0. T h u s , even though the proport ion of market time devoted to investment is unambiguously fa l l ing , investment time itself may be r i s ing if total market time is increasing rapidly enough . In summary, the theoretical analysis tends to weaken the human- capital interpretation of concave wage and earnings prof i les by admitting the possibi l i ty of r is ing investment at some points in the life c y c l e . T h e analys is supports an empirical model which makes hours a peaked, concave function of age. Though certain comparative dynamic results have been adduced under s t rong assumptions, these predict ions do not y ie ld v e r y readi ly to testing with cross-sect ion data . AN EMPIRICAL MODEL Th is section introduces a simultaneous linear model of wages and hours which is simple enough to be estimated with the cu r ren t data set . Though the model is incapable of settl ing all outstanding issues and is not conventional ly r igourous in the sense of being der ived from s tandard , known uti l ity and product ion funct ions, it does appear to capture the 24 most important measurable factors af fect ing individual dec is ions. 224 St ructura l Equations T h e model consists of an identity and two behavioural re lat ionships: Y = W h or In Y = In W + In h . . . .(60) a a W g = e X ' 3 h 6 u 1 or In W g = X ' B + 6ln h + In u . . . .(61) h = e Z ' Y W m u 2 or In h = Z ' y + 6 In W m + In u 2 . . . .(62) For each individual (subscr ip t s u p p r e s s e d ) , annual employment earn ings , Y , are the product of the average hour ly wage before tax, W , and the 3 number of hours worked, h . The average wage depends, f i r s t of a l l , upon h . Converse ly , h depends upon another row vector of determinants, Z 1 , which may have elements in common with X 1 , and upon the marginal after-tax wage, Wm» Among the remaining symbols, u 1 and u 2 are stochastic terms; B, y , 6, and 6 are vector and scalar constants , as the context indicates. Observe that if we subst itute (61) into (60), the result is y = e X , 6 h ( 1 + e > U l . (63) T h e n , if T represents the marginal tax rate on earnings (assumed for the moment to be constant) , the marginal af ter- tax wage must be g iven by 225 W = (1 - T ) • 3 Y / 9 h m = (1 - T)(1 + e ) e X ' B h 6 U l = (1 - x)(1 + 6) VV = (1 - T)(1 + 6) • Y/h . Subst i tut ing into the logarithmic vers ion of (62) yields In h = Z'y + 6 In [(1 - T)(1 + 6) Y/h] + In u 2 = Z'y + 6 In (1 - T) + 6 ln(1 + 0).+6 In Y - 6 l n h + l n u 2 . Solv ing the latter for In h and taking the logarithm of (63), one finally obtains a pair of estimable equat ions: In Y = X'B + (1 + 6) In h + In u 1 . . . .(64) , N H = T4TZ'^ + i4c-,n ( 1 - T ) + r4r , n Y + r ^ ( 6 5 ) These form the basis for the work reported in Chapter V . 226 Fur ther Comment and Definition Now that the general outlines of the model are c lear , it is possible to d iscuss the specification in some deta i l . T h e preceding equations contain a number of dist inct hypotheses which require amplif ication, and it is of course essential to define the const ituents of X' and Z 1 . The f i rst thing to note is that although (61) and (62) are "s t ruc tura l" equations from the standpoint of the model, they are not the s t ructura l equations one might conventionally use to segregate supply and demand in the labour market . Here, supply and demand factors presumably mingle in forming the respect ive lists (X 1 and Z') of exogenous var iab les . T h e r e - fore, it is not immediately clear whether one should take as an endogenous var iable the pr ice firms pay for labour (W ) or the pr ice indiv iduals 3 ultimately receive for it (W ) . Equation (61) employs W , making X' ffi 3 25 and h the determinants of average gross worker p roduc t i v i t y . S ince schooling and exper ience (elements of X') are still taken to be exogenous, or at the v e r y least predetermined, the fact that individuals in a g iven cross section might once have considered W in formulating their investment m plans is not necessar i ly re levant . Equation (62) incorporates the standard labour-supp ly assumption that indiv iduals respond to the marginal net wage. A l though the insertion of W m in (62) may appear unremarkable , its use does require some justif ication in a l i fe-cycle context . When work and investment are planned simultaneously, the individual does not (except in stage III) equate his marginal rate of substitut ion between goods 227 and leisure to the net wage, as the static theory implies. Moreover , since the lifetime profi le of W m is known ex ante, the effect of this var iable upon time worked at any given moment is not of the s tandard causal va r i e ty . T h e two must harmonize in the optimal p lan ; that is a l l . A c c o r d i n g l y , one might think of replacing W m with some funct ion of age or exper ience which depicts the outcome of the initial p lanning dec is ion . The expl ic it inclusion of W is nevertheless indicated on a number r m of g r o u n d s . In the f i rst place, W m may character ize the optimal plan more accurately than a pure ly exogenous funct ion of the sort just mentioned. There is no harm in us ing the endogenous variable so long as we are not mislead into making unwarranted inferences concern ing static income and substitut ion e f fects . Secondly , though work and investment may evolve together in a planned way dur ing the period of on-the-job t ra in ing , labour supp ly may respond causal ly to that component of the net wage which is the result of predetermined schooling and the initial endowment of human cap i ta l . F ina l ly , one must concede that in the real world the wage rate will be subject to unforeseen d i s tu rbances . The individual will p r e - sumably want to adjust his work effort to these, much as the static 26 theory sugges ts . T h e use of h as a determinant of W likewise appears justif ied 3 on several counts . Moonlighting and overtime are the two which come 27 most qu ick ly to mind. Both affect the average wage by a l ter ing the remuneration earned on succeeding increments of work . If secondary employment pays less per hour than pr imary , moonlighting will inf luence 9 toward the negat ive. T h e existence of an overtime premium will deflect it toward the pos i t ive . If h acts as a p roxy for var ious motivational, ab i l i ty , and environmental factors which serve as common determinants of wages and employment, there is fu r ther reason to expect that 9 will be nonzero . Since most of the personal factors one can name would appear to operate upon wages and employment i n the same d i rect ion, it seems likely on this g round that 9 > 0. However, if the labour market actually works in an oppress ive manner, heaping long hours upon the poorly paid (and converse ly , favour ing the best paid with abundant le i sure) , then it may turn out, as in Chapter III, that 9 < 0. The same may occur , as suggested ear l ier , if seasonal workers obtain high wages to compensate for limited h o u r s . One cannot pred ic t , but it is certainly important to 28 estimate, the sign and the signif icance of this parameter. Estimation, by means of (64) and (65), is relatively s t ra ight forward once the elements of X ' and V have been de f ined . Since the approach taken here is to a certain degree experimental , it would be inappropr iate to speci fy the exact composition of these vectors in advance . However, it is useful at this point to d iscuss the most prominent candidates for inc lus ion. With regard to X 1 , only a br ie f comment is r e q u i r e d . Obv ious ly , one would wish to define this vector in terms of the variables found signif icant in the single-equation estimates of Chapter III. Though all are potentially admissible as elements of X ' , emphasis will be g iven in Chapter V to the human-capital var iables appear ing in the orthodox 229 earnings funct ion . With X' restr ic ted in this way, assessment of the latter in l ight of the simultaneous-equation estimates is great ly faci l i tated. Var iables will nevertheless be added to X' , as they were to the orthodox earnings func t ion—in the present case, to d is t inguish their separate inf luences upon wage rates and hours of work . With regard to Z 1 , more needs to be said than in the preced ing instance, since we have not elsewhere cons idered the l ikely determinants 29 of hours worked . It should be c lear , even so, that two essential components of Z 1 must be age and school ing . These var iables are key factors in the present i n q u i r y , and their use in an equation like (65) is well establ ished in the l i terature . Age will sure ly affect hours worked if the preceding l i fe-cycle theory is v a l i d . To test its predict ion of peakedness in the age-hours prof i le , we shall let Z' include both age and 30 age s q u a r e d . Schooling may affect realized hours in a number of ways: by determining the sort of job (high-unemployment or low- unemployment) that a worker may ho ld , by determining the eff ic iency of job search , by indicat ing worker qual i ty to prospect ive employers ,by 31 condit ioning the suscept ib i l i ty to layoff . It is of considerable interest to compare the effect schooling may have upon earn ings by way of hours with the effect it ev ident ly has upon earn ings by way of wage rates . Including the variable in both X ' and Z' should fu rn i sh the des i red information. Other plausible components of Z' are family status, ethnic g r o u p , indust ry and occupat ion, and place of res idence . T h e f i rs t var iab le , 230 consist ing in detail of headship and marital status, is almost universal in the l i terature, though it commonly appears not as a regressor , but as a cr i ter ion with which to define subsamples for separate estimation. Ethnic g roup may affect hours through discrimination and through var ious cu l tura l ly determined t ra i ts , as we have a lready in ferred from the s ingle-equat ion resu l ts . Industry and occupation are reasonable prox ies for the employ- ment character ist ics of the jobs thus d e s c r i b e d . Residence is another p roxy for employment condit ions, which va ry considerably across regions and no doubt influence the hours of work realized by the typical ind iv idua l . A final and v e r y important component of Z' ar ises on str icter theoretical g r o u n d s . It is routine in the static analysis of labour supply to include in the result ing empirical equations an independent var iable to por t ray the nonemployment income of the individual or family. T h e estimated coeff icient of this variable then measures the static income ef fect . Such income effects also occur in the l i fe-cycle model, though they are presumably spread over the whole p lanning per iod . In any event , they may be accounted for in the standard way. A t the same time, it is necessary to relax the assumption that the marginal tax rate x is constant . These two theoretical considerat ions combine to suggest a new income var iab le . Its definit ion is i l lustrated with the help of F igure 2. T h i s shows, in leisure-income space, the before-tax budget constraint B B ' and the af ter- tax budget constra int A A* of an indiv idual whose gross wage is F igure 2 Linearization of the budget constra int 232 constant . T h e cu rva tu re of A A ' (smoothed for purposes of i l lustration) 32 ref lects the progress iv i ty of the tax system. Following Hall's p rocedure , one may l inearize the after-tax budget constraint at the observed equi l ibr ium point E. T h e individual may then be assumed to behave as if he were facing L L ' , which (given the wage rate and the level of non- employment income B'C) is uniquely determined by the slope (1 - T) W = (1 - T) Y/h and the zero-work intercept L ' C . 3 3 T h e latter a is g iven geometrically by DG - DE - EF , where EF = h ' ( l - x J Y / h = (1 - x) Y and where DG represents total income and D E , total taxes. Knowing all these quant i t ies , one may compute L'C for each individual and obtain the des ired variable to include in Z' . Ear l ier , in Table 3, this variable was labelled I N C O T H . It must be noted that the foregoing procedure is at best appropr iate only when the individual 's gross wage is constant, as shown (or when equi l ibr ium occurs only on the r ight-most segment of a piece-wise l inear budget cons t ra in t ) . Otherwise, the slope of the budget constraint will be ( 1 - T ) «(1 + 0)W , where 6 is not known in advance . If nonzero values of 8 ar ise pure ly through the correlation of wages and hours over the cross section (that is, among di f ferent jobs) , then of course , the procedure remains ostensibly va l i d . However, if nonzero values arise for each individual (that is, within the terms of the job or jobs h e l d ) , there will be e r ro rs in the calculation of I N C O T H . It thus appears that the Hall procedure is capable of d igest ing only a certain degree of n o n - l inearity in the budget const ra int . Other dif f icult ies associated with 233 the approach—ones of an econometric nature—wi l l be reviewed in Chapter V . Meanwhile, a final point to consider in def in ing the intercept term is whether one should use merely the individual 's own proper ty earn ings or the sum of these and the total income of all other family members. 35 Notwithstanding recent analyses of family labour s u p p l y , it was found that in the present , rather heterogeneous sample "own proper ty income" performed sl ight ly better than "other family income" as a predictor of hours when (65) was subjected to prel iminary examination by o rd inary 36 least squares . Since the present purpose in estimating (65) is not to investigate labour supply as such , but rather to obtain the best instruments for use in system estimates focussing on (64), it was decided to adopt the narrower income concept—which accounts for the definit ion of I N C O T H . A l though an equation like (64) is commonly re fer red to as a labour-supp/y funct ion , this interpretation depends on a number of s t rong , usual ly implicit assumptions concern ing the nature of demand and the relative var iabi l i ty of demand and s u p p l y . Whether or not one might actual ly identify a supply funct ion in estimating (64) is d i f f icult to say 37 with complete conf idence. T h e present s tudy takes an agnost ic , empiricist approach to this quest ion . Part ly as a resul t , there were few constra ints but also little guidance in selecting a functional fo rm. T h e double-logarithmic or constant-elast ic i ty form ultimately chosen to relate hours and the wage rate is h igh ly convenient , though somewhat novel from the standpoint of the labour-supp ly l i terature, which has leaned toward the double-absolute (var iable-elast ic i ty) speci f icat ion. Regardless of whether the double- logarithmic form provides a conv inc ing a pr ior i descr ipt ion of labour s u p p l y , it appears to perform reasonably well as a predictor of h o u r s . Some ord inary- leas t -squares estimates documenting this performance, along with that of the l isted independent var iab les , are presented for inspection and comparison in the appendix which follows. 235 O R D I N A R Y - L E A S T - S Q U A R E S E S T I M A T E S O F W O R K I N G H O U R S 1 ( H I ) W T I M E - 1 . 4 5 2 2 + . 0 6 7 5 A G E .0008 A S Q ( 4 7 . 0 ) ( 4 3 . 5 ) ( 4 3 . 4 ) R 2 . 077 n u m b e r o f o b s e r v a t i o n s = 2 2 , 6 8 2 ( H 2 ) W T I M E - 1 . 3 5 8 5 + . 0 2 3 6 A G E . 6 0 0 3 A S Q .4018 Z I N C ( 4 4 . 0 ) ( 1 6 . 2 ) ( 1 5 . 6 ) ( 7 0 . 3 ) - . 1 0 6 9 X I N C O T H D I + . 1 6 4 0 DI ( 2 2 . 4 ) ( 7 . 9 8 ) R 2 . 3 0 9 n u m b e r o f o b s e r v a t i o n s = 2 2 , 6 8 2 ( H 3 ) W T I M E - 1 . 2 8 0 1 + . 0 2 2 5 A G E . 0 0 0 3 A S Q + . 4 0 6 8 Z I N C ( 3 7 . 7 ) ( 1 5 . 3 ) ( 1 5 . 0 ) ( 7 0 . 4 ) - . 1 0 2 4 X I N C O T H D I + . 1 6 2 9 DI 0055 S ( 2 1 . 2 ) ( 7 . 9 3 ) ( 5 . 5 2 ) R 2 . 3 1 0 n u m b e r o f o b s e r v a t i o n s 2 2 , 6 8 2 ( H 4 ) W T I M E - 1 . 2 4 5 4 + .0202 A G E .0002 A S Q + . 4 3 7 3 Z I N C ( 3 4 . 4 ) ( 1 4 . 1 ) ( 1 4 . 9 ) ( 7 5 . 5 ) - . 0 9 1 9 X I N C O T H D I + .1734 DI . 0 0 3 3 S + .0292 C E O I ( 1 8 . 9 ) ( 8 . 7 1 ) ( 2 . 8 9 ) ( 2 . 3 5 ) - . 0 3 1 8 C E 0 2 + .0341 G E 0 4 .0081 G E 0 5 - . 0 8 9 4 G E 0 6 ( 4 . 0 3 ) ( 2 . 8 7 ) 1 ( 0 . 6 5 ) ( 8 . 3 1 ) . 0 2 5 6 T Y P E + .3731 I N D 1 .0946 I N D 2 - . 0 7 4 6 I N D 3 ( 3 . 7 3 ) ( 1 5 . 1 ) ( 3 . 3 4 ) ( 1 . 6 5 ) - . 0 0 2 4 I N D 4 .1072 I N D 6 . 0 1 5 2 I N D 7 + . 0 7 3 9 I N D 8 ( 0 . 1 1 ) ( 7 . 7 2 ) ( 1 . 3 1 ) ( 7 . 0 9 ) - . 0 0 5 8 I N D 9 .0076 I N D 1 0 + . 0 8 1 7 M A J - . 1 0 7 9 O C 1 ( 0 . 3 3 ) ( 0 . 6 4 ) ( 6 . 9 2 ) ( 5 . 6 4 ) - . 1 0 3 3 O C 2 . 1 6 5 7 O C 3 .0952 O C 4 - . 0436 O C 5 ( 5 . 1 4 ) ( 7 . 2 0 ) ( 3 . 6 1 ) ( 2 . 4 8 ) - . 0 0 1 5 O C 6 .1092 O C 8 .0351 0 C 9 - . 1 1 7 9 O C 1 0 ( 0 . 8 4 ) ( 4 . 6 2 ) ( 2 . 2 6 ) ( 6 . 5 1 ) - . 0 3 1 2 O C 1 1 . 0 7 9 0 O C 1 2 ( 1 . 7 1 ) ( 4 . 9 0 ) R 2 . 3 5 9 n u m b e r o f o b s e r v a t i o n s 2 2 , 6 8 2 ( H 5 ) W T I M E - 1 . 1 7 6 6 + .0161 A G E .0002 A S Q + . 4980 Z I N C ( 3 1 . 9 ) ( 1 0 . 6 ) ( 1 1 . 8 ) ( 5 7 . 9 ) - . 0 8 8 7 X I N C O T H D P + . 0 8 4 6 DI .0028 S ( 1 7 . 4 ) ( 4 . 0 1 ) ( 2 . 4 3 ) + • • • ( G E O . T Y P E , I N D , M A J , O O • • • + . 1 5 2 3 H E A D .0010 F A M S I Z + . 0 4 6 0 U S M A R ( 6 . 0 0 ) ( 0 . 5 6 ) ( 2 . 1 2 ) - . 0 1 0 6 E T H 2 .0182 E T H 3 . 0 0 6 3 E T H 4 ( 1 . 3 7 ) ( 1 . 3 7 ) ( 0 . 1 8 ) - . 0 0 5 9 E T H 5 .3511 E T H 6 ( 0 . 2 3 ) ( 9 . 0 7 ) R 2 . 3 6 7 n u m b e r o f o b s e r v a t i o n s = 2 2 , 6 8 2 F i g u r e s i n p a r e n t h e s e s a r e t r a t i o s , w r i t t e n i n a b s o l u t e t e r m s . N O T E S C H A P T E R IV In the perfect ly competitive labour market implicitly assumed, the two are of course ident ical . See, for example, C a r y S . Becker , "The Allocation of Time over the Life C y c l e , " in Gi lbert R. Chez and C a r y S . Becker , The Allocation of Time and Goods over the Life Cyc le (New Y o r k : Columbia Un ivers i ty Press for the National Bureau of Economic Research , 1975). ""If time were not an inelastically suppl ied resource, independence might still be maintained, since the quantity used in consumption would then not affect the pr ice or the quant i ty available for use in investment. F ix i ty of the time endowment, rather than multiple use, is therefore the key element of the problem. It is possible, of course , to restr ict the under ly ing uti l ity funct ion in such a way that the simpler model will yet suf f i ce . Suppose that the individual is initially in equi l ibr ium, equating the marginal rate of subst i tut ion between goods and leisure to the net wage. If he then decides to allocate some nonleisure time to investment, the net wage will fall in the cur rent per iod and rise thereafter . If equi l ibr ium is to be restored without upsett ing the investment calculat ion, labour supply must not change . The uti l i ty function must render the demand for leisure . per fect ly inelastic. Needless to say, this is a v e r y strong requirement. 5 O p . c i t . g A lan S . B l inder and Yoram Weiss, "Human Capital and Labor S u p p l y : A S y n t h e s i s , " Journal of Political Economy, L X X X I V (June , 1976), HH9-H72. James J . Heckman, "A L i fe -Cyc le Model of Earn ings , Learn ing , and Consumpt ion ," Journal of Political Economy, L X X X I V (August 1976), S11-S44. 236 237 Harl E. R y d e r , Frank P. S ta f ford , and Paula E. Stephan Labor , Le isure , and Tra in ing over the Life C y c l e , " International Economic Review, XVII (October , 1976), 651-674. ' Though functional notation has been suppressed , all var iables implicitly depend on time. Note that, by hypothes is , g(1) =0 and g(0) = 1. If the holders of large positive asset portfolios obtain the highest net r e t u r n s , r ( A ) might in fact by U-shaped, with r '(A) > 0 for A > 0. A discont inuity at A = 0 is certainly to be expected . His specification is Q = F ( b k ' h H , D ) , where b is a constant qu ick ly set to equal un i t y . The presence of b avoids the part icular neutral i ty assumption implicit in making H the augmenting factor in both the util ity and the product ion funct ion . Note that, here, T designates the termination of the optimal p lan , not the point of zero net investment, as in the discussion of Mincer . '"'One may either add I > 0 and A ( T ) ^ 0 or restr ic t the uti l i ty function so that the respect ive marginal utilities become arb i t rar i l y great at zero. T h i s ensures nonnegat iv i ty in any optimum. 1 5 F o r a complete statement of the f i r s t -order condit ions see B-W, o p . c i t . , p. 457. C f . Chez and Becker , o p . c i t . , who f ind that the profi le of consumption imitates the profi le of wage rates. Th is conclusion stems from the authors ' adherence to Becker 's theory of time allocation, which suggests that indiv iduals subst itute market goods for leisure in house- hold product ion as the wage rate r i ses . O p . c i t . , p p . 526-527. '"Heckman's uti l i ty funct ion adds the factor H to the left-hand side of (52), making it possible to concel H completely, but contr ibutes the term -(1 - h)U„/X H to (55). R-S-S replace U . with the special form 62/£. T h e final term in (52) becomes A H y Q n / h, amd (53) becomes 0 = X A g ' ( k ' ) h w H + A ^ Q ^ k 1 . In (55), y Q H /H replaces ak 'h . Al l authors except B-W assume g(k ' ) = (I - k 1 ) , whence g '(k ') = - 1 . 238 1 9 0 p . c i t . , p. 464, F igure 4. 20 ' A s the reader may v e r i f y , additional propert ies of k' = 0 depend on the th i rd and higher der ivat ives of g ( k ' ) , which are unspec i f i ed . B-W choose tacitly to depict the locus as a straight l ine. B-W do not mention the apparent possibi l i ty that (k ' , h ) might be a stable focus . T h i s is ruled out by the t ransversa l i ty condit ion (56) . O p . c i t . , p . 465. 22 It continues to decline in stage III (pure work) if and only if r + d - p > 0, which B-W take to be the "leading c a s e . " O p . c i t . , p . 463. 23 Heckman, o p . c i t , p. 518. 24 Blinder uses a similar model for purposes of argument but does not pursue its implementation. See "On Dogmatism in Human Capital T h e o r y , " p p . 16-17. 25 T h i s is not to suggest that indiv idual firms ignore marginal calculat ions, only that there is an empirical market relationship between W and the var iables named . a 26 A n essential ly analogous argument relat ing consumption and earn ings appears in Keizo Nagatani , " L i f e - C y c l e - S a v i n g : T h e o r y and Fac t , " American Economic Review, LXII (June , 1972), 344-353. 27 On these topics see: Robert Sh ishko and B e r n a r d Rostker , "The Economics of Multiple Job H o l d i n g , " American Economic Review, LXVI (June , 1976), 298-308; Yoram Barze l , "The Determination of Daily Hours and Wages," Quar ter ly Journal of Economics, L X X X V I I (May, 1973), 220-238. 28 For additional d iscussion and empirical analysis based on a sample of female workers , see Harvey S . Rosen, "Taxes in a Labor Supp ly Model with Joint Wage-Hours Determinat ion," Econometrica, X L I V (May, 1976), 485-508. 29 Empirical studies of labour supp ly investigate a number of factors , general ly viewed as represent ing tastes or external const ra in ts . See, for example, Marvin Kosters , "Effects of an Income Tax on Labor 238a S u p p l y , " in T h e Taxat ion of Income from Cap i ta l , edited by A r n o l d C . Harberger and Martin J . Bailey (Washington: T h e Brookings Inst itut ion, 1969); Sherwin Rosen and Finis Welch, "Labor Supp ly and Income Red is t r ibu t ion , " Review of Economics and Stat ist ics , LMI ( A u g u s t , 1971), 278-282; the collection of art ic les appear ing in Income Maintenance and Labor S u p p l y , edited by Glen C . Cain and Harold W. Watts (Ch icago: Rand McNally College Publ ish ing Company, 1973); Julie Da Vanzo , Dennis D e T r a y , and David H. G r e e n b e r g , "The Sensit iv i ty of Male Labor Supp ly Estimates to Choice of Assumpt ions , " Review of Economics and Stat ist ics , LVIII ( A u g u s t , 1976), 313-325. 30 C f . Or ley Ashenfe l ter and James Heckman, "Estimating Labor - Supp ly Funct ions" in Cain and Watts, o p . c i t . 31 For more discussion see Farrel l E. Bloch and Sharon P. Smith, "Human Capital and Labor Market Employment," Journal of Human Resources , XII (Fa l l , 1977), 550-560. 32 Robert E . Hal l , "Wages, Income, and Hours of Work in the U . S . Labor F o r c e , " in Cain and Watts, o p . cit ., p p . 118-121. For some additional d iscuss ion see W. Erwin Diewert, "Choice on Labor Markets and the Theory of Allocation of T ime ." (Unpubl ished discussion paper , Canada , Department of Manpower and Immigration, 1971). 33 Hall actually uses the zero- le isure intercept L O . 34 The present data do not allow a fur ther subdiv is ion of other family members' income into employment and nonemployment components. A t best , one might apply the indiv idual-ut i l i ty- fami ly-constra int model of Jane H . Leutho ld , "An Empirical S tudy of Formula become T r a n s f e r e d and the Work Decision of the Poor ," Journal of Human Resources , III (Summer, 1968), 312-323. 35 * See, for example: Reuben Gronau , "The Intrafamily Allocation of T ime: T h e Value of the Housewives' T ime ," American Economic Review, LXIII (September, 1973), 634-651; Or ley Ashenfe l ter and James Heckman, "The Estimation of Income and Subst i tut ion Effects in a Model of Family Labor S u p p l y , " Econometrica, XLII ( January , 1974), 73-85. 36 Greater measurement e r ro r in the latter (original ly prov ided in c lass in te rva l s ) , the inclusion of heads and nonheads of families, and the fai lure to d is t inguish between the property and nonproperty income of family members may have contr ibuted to this resul t . T h i s neglected problem has been d iscussed by A . C . Raynor "On the Identification of the Supp ly C u r v e of Working Hours " Ox fo rd Economic Papers , XXI ( Ju ly , 1969), 293-298. ' ""An exception is the t ime-series expendi ture study of Michael Abbot and Or ley Ashenfe l ter , "Labor S u p p l y , Commodity Demand and the Allocation of T ime ," Review of Economic Stud ies , XLIII (October , 1976), 389-412. Cross-sect ion precedents inc lude: Lee L i l l a rd , "Estimation of Permanent and T rans i to ry Responses in Panel Data: A Dynamic Labor Supp ly Mode l . " (Unpubl i shed report , Santa Monica: RAND Corporat ion , 1977); Gary Burt less and J e r r y A . Hausman, "The Effect of Taxation on Labor S u p p l y : Evaluat ing the Gary Negative Income Tax Exper iment ," Journal of Political Economy, L X X X V I (December, 1978), 1103-1130. C H A P T E R V EARNINGS A N D H O U R S : S I M U L T A N E O U S - E Q U A T I O N E S T I M A T E S FOR C A N A D A T h e preceding chapter develops a simplif ied, linear version of the earn ings -and-hours model. Though we have dealt at some length with the economic content of the proposed speci f icat ion, nothing has yet been said regard ing the econometric assumptions and procedures needed to implement it. A c c o r d i n g l y , the f i rst section of this chapter d iscusses estimation. The second reports results and offers an ana lys is . ESTIMATION P R O C E D U R E Before we may consider the choice of a part icu lar econometric technique for estimating the two-equation model, it is necessary to define the stochastic framework. So fa r , no restr ict ions have been placed upon the d is turbances appear ing in (64) and (65). For convenience, these equations are restated here as a system in "stacked" matrix form: 1 In Y In h X 0 0 0 Z l n ( 1 - x ) 6 Y-1/(l +6) 6/(1 + 6) 240 211 In h • (1 + 6) In Y • 6/(1 + 6) . . . .(66) Since 6 is a constant, there is no harm in treating u 2 = ( l n u 2 ) / ( 1 +6) as an ord inary random e r r o r , like u 1 E In u 1 . It is reasonable to assume the fol lowing: E ( U l j ) = E ( u 2 j ) = 0 E ( u 1 i " l i } = a Y Y E ( G 2 i D 2 i ) = a h h J = 2 ' E ( u 1 i U 2 i } = °Yh E ( u 1 i U 1 j ) = E ( u 2 i C , 2 j ) ~- E ( u 1 i u 2 j ) = 0 1 * j .(67) Within each structura l equation individual e r rors are homoskedastic; in genera l , however, the common variances are not the same across equations ( c f y y ^ a h h ^ " ^ o r e a c n individual the covariances across equations are also uniform (equall ing O y n ) , but their common value need not be zero . Since omitted var iab les—factors special to the individual or to his part icu lar environment—may affect both.earnings (via the wage rate) and hours , one cannot assume that u^. and u2> will be uncorre la ted . One can safely assume that between all g iven pairs of indiv iduals the covar iances within and across equation will be zero . If we let 242 U' = [ulj u l ] , the var iance-covar iance matrix of s tructura l d is turbances consistent with (67) may be written as follows: i (UU') = I ® l N , where £ = °YY °Yh °Yh a h h (68) That is, E(UU') consists of four N x N submatrices, each with the c o r r e s - ponding element of £ down the main diagonal and zeros elsewhere. For purposes of hypothesis testing we shall want to assume that u^ and are normally d i s t r i bu ted . If one could ignore Oyh' it would be possible to obtain cons i s - tent, asymptotically eff icient estimates of (66) using an instrumental- variable or two-stage least-squares regress ion procedure , equation by equat ion. However, the strong probabi l i ty of a signif icant cross-equat ion covariance means that such methods are unl ikely to be asymptotically eff icient in the present case . Three-s tage least squares (3SLS) would 2 therefore seem to be a logical choice. Th i s estimator is both consistent and asymptotically efficient under g iven stochastic assumptions. Though it may di f fer numerically in finite samples from the ful l- information maximum- 3 l ikelihood estimator, the two have the same asymptotic d i s t r ibut ion . In c a r r y i n g out the 3SLS procedure , one uses , in effect (though not computat ional ly) , the residuals from the second-stage ( instrumental- variable) regress ion to form a consistent estimate of "Stage three" then amounts to performing general least squares (CLS) on the stage-two var iab les . Since the result , in genera l , is a new set of consistent ly estimated res iduals , it is possible to repeat the C L S procedure until the regress ion coeff icients cease chang ing . T h i s technique, known as iterative 3SLS, cannot be shown to increase asymptotic eff ic iency but may appear to some less a rb i t ra ry than stopping after one r o u n d . The i tera- tive vers ion of 3SLS is not adopted here, essential ly on pragmatic g r o u n d s : estimates obtained by this means appear unreal ist ic in compari- son with those obtained by ord inary 3SLS. A s ev idence, some iterative estimates are d isp layed in Appendix V . Though it might seem that we are now in a position to examine resu l ts , the fact is that several important econometric issues remain to be d i s c u s s e d . These have to do with (1) the endogeneity of the tax rate, (2) the nature of the time-worked var iable , and (3) identi f icat ion. Let us consider each problem in t u r n . F i rs t of a l l , because the marginal tax rate (T) depends d i rect ly upon earn ings , and therefore indirect ly upon time worked, it is c lear ly an endogenous var iab le . The Hall p rocedure , descr ibed in Chapter IV, requires that we use the marginal tax rate in forming a slope and an intercept term, both of which are to appear on the r ight -hand side of any time-worked equat ion. In the notation of (66) the slope var iable , obtained by combining terms, is I n ( l - i ) Y ; the intercept variable is a constituent of Z . Empir ical ly , ZINC has been defined to represent the former; I N C O T H , the latter. Furthermore, as explained in Chapter IV, INCOTH is replaced in practice by the dummy-interact ion pair DI and X I N C O T H D I . Since all three v a r i a b l e s — Z I N C , DI, and X I N C O T H D I — a r e endogenous, their use in the time-worked equation of (66) will presumably result in biased estimates unless fu r ther steps are taken . In short , though the Hall procedure achieves the mapping of indiv idual equi l ibr ia , it is not unblemished econometr ica l ly . 5 One way round the prob lem—an approach used here and elsewhere is to form instrumental-var iable estimates of the endogenous income terms. T h i s technique should yield consistent final estimates of the s t ructura l coeff ic ients, but it is di f f icult to apply in the present c i r cum- stances on account of the nonl inearity in the tax s c h e d u l e , 7 the v e r y problem which leads to endogeneity in the f i rst place. Nevertheless , ZINC and X I N C O T H D I were subjected to the instrumental-variable treat- ment, the instruments being those exogenous variables needed to o simulate the tax rate and those found important in expla ining INC . Among the instruments were, in part icu lar , the quadrat ic terms SSQ and P S Q . One would hope that these terms might go some way towards approximating the expected nonl inearity of the predict ing equat ions. Since dummy-variable s t r ings comprise the remaining instruments, functional forms were not in any event acutely const ra ined . T h e use of ZINC serves to inforce the hypothesized equal ity restr ict ion on the coeff ic ients of ln(1 - x) and I n Y . Where this was undes i rab le , it was necessary to form separate instrumental-var iable estimates of the two terms, represented empirical ly by T M A R C and INC. T h e same exogenous variables were employed in each case. 245 T h e endogenous dummy variable DI was left " u n p u r g e d , " owing to the computational expense involved and to its dichotomous nature, which prevents eff icient estimation by linear least squares . T h i s omission does not seem v e r y ser ious, since DI is not equal to one only for those g indiv iduals who fall in the zero-tax bracket and have no property income. Because the zero-tax bracket is relatively wide, DI is furthermore u n - likely to change v e r y often in response to the d isturbances in the earnings equat ion; in other words, DI and these d isturbances will not be highly c o r r e l a t e d . 1 0 T h e endogeneity problem is therefore l ikely to be minimal. r We come now to the second econometric issue, that of the time- worked var iab le . The PUS data available for measuring time worked are, on the whole, rather d isappo int ing . It was decided that WTIME, as opposed to WEEKS, should stand for the theoretical variable h, even though the latter produced sl ightly better fits in the ord inary least- squares (OLS) regress ions . Whereas WTIME may take on th i r ty - f i ve d i f ferent va lues, WEEKS is limited to only f i v e . 1 1 The former thus resembles more closely than the latter the continuous variable we have in mind. Estimation using WEEKS would appear more suited to one of the proba- bil ity models, such as the multinomial logit . Both WEEKS and WTIME constitute "limited dependent va r iab les , " but the problem with regard to WEEKS is undoubtedly the more severe . By def in i t ion, WEEKS must fall in the half-closed i n t e r v a l 1 2 (0 ,52] , with many observat ions ly ing on the upper b o u n d . WTIME must exceed zero; but apart from the limit imposed in practice by g r o u p i n g , there is no firm / 246 upper bound within the normal range of exper ience . Though observat ions are l ikely to be relat ively dense in the v ic in i ty of 2,000 hours , some indiv iduals will report working a much larger accumulat ion. Hence, the d istr ibut ion of hours , and of the d is turbance in any WTIME equation one might estimate, need not be t runcated on the upper side to any noticeable 13 degree . T h e problem of the zero bound will be ignored here . C o n - clusions regard ing hours worked will thus be of the "condit ional" va r i e ty . T h e final problem we have to cons ider is that of identi f icat ion. There is no gain in app ly ing 3SLS to a g iven equation of the system unless the 1 H other, is over ident i f ied . That the earn ings equat ion, expressed in the human-capital form, is over ident i f ied should be obv ious , since many variables to be used in expla in ing hours are exc luded from it. That the hours equation will also be over ident i f ied may not be so c lear . T h e matter rests on the empirical use of age and exper ience . On the basis of the l i fe-cyc le analysis presented in Chapter IV, and in the absence of arguments to the c o n t r a r y , A C E and ASQ were used in the hours equat ion . T h e exper ience var iables P and PSQ, which do not appear in the latter, continue on the r igh t -hand side of the earn ings equat ion . T h e i r exclusion from the hours equation would appear to settle the issue of over ident i f icat ion, but one must remember that in 2 pract ice P = A G E - S - 5.67. A c c o r d i n g l y , PSQ = P = ASQ + SSQ - 2 • A C E • S - 11.34 age - 11.34 S +32.15 . There fo re , to the extent that the hours equation is over ident i f ied , it will be through the exclusion of 247 the var iables SSQ and A C E «S. Since these terms enter the human-capital earn ings equation ( through PSQ) with an equality restr ict ion on their coeff ic ients, identification will not be so s t rong , however, as in the usua l , unrestr ic ted case. R E S U L T S Tables 26 and 27 report estimates of the structura l equations per ta in ing , respect ive ly , to earnings and to hours . Equations with the same numeric digit in their reference codes were estimated simultaneously. S ince the earn ings equation was of pr imary interest , the specification of the hours equation was held constant—the one exception being in (MH2), where the equality restr ict ion on the coeff ic ients of (1 - T) and Y ( T M A R G and INC) was br ief ly re laxed . Experiments with the earnings equation involved the addition of S S Q , X S P , C E O , T Y P E , IND, M A J , IM, and E T H to the basic human-capital formulation. Initial F ind ings The basic formulation appears in (ME1) and (ME2) . The most s t r ik ing feature of these equat ions—or for that matter, of the entire s e t — i s the dramatic r ise in the coefficient of WTIME. The values d i s - played here are more than double the one obtained by O L S . 1 5 Qual i tat ively, this outcome tends to reverse the f inding in Chapter III that earn ings respond inelastically to a change in h o u r s . Quant i tat ive ly , T A B L E 26 S I M U L T A N E O U S E S T I M A T E S : a EARNINGS 248 Equations (dependent variable = INC) n d i i u Variable (ME1) (ME2) (ME3) (ME4) Constant S SSQ 1.0021 (34.1) .0629 (36.6) .9427 (34.0) .0640 (37.1) 1.1652 (18.2) .0009 (1.26) .0025 (9.22) 1.3676 (35.6) .0525 (24.1) P PSQ X S P .0221 (15.5) -.0003 (11.8) .0265 (16.9) -.0004 (13.5) .0295 (12.2) -.0005 (14.4) .0001 (1.12) .0093 (5.64) -.0000 (1.07) WTIME 1.4567 (60.0) 1.4079 (54.8) 1.3473 (53.8) 1.8198 (57.2) C E 0 1 C E 0 2 C E 0 4 C E 0 5 C E 0 6 - - - -.1340 (5.31) -.0449 (2.57) -.1132 (4.65) -.0244 (0.98) -.1520 (6.92) T Y P E - - - .1047 (7.49) IND1 IND2 IND3 IND4 IND6 IND7 IND8 IND9 IND10 - - - -.7910 (24.6) .2261 (4.48) .0521 (0.61) .1505 (3.81) .2871 (12.4) .0504 (2.40) -.2129 (10.9) .0165 (0.49) -.0255 (1.31) MAJ - - - -.1612 (6.76) IM1 IM2 IM3 - - - .0070 (0.23) -.0282 (1.50) .0260 (0.83) ETH2 E T H 3 ETH4 E T H 5 E T H 6 E T H 7 - - - .0186 (1.24) .0221 (0.86) -.0264 (0.38) .0472 (0.99) .6521 (8.57) -.0279 (0.91) Main sample, 22,682 observat ions b T h e f i rs t f igure in each set is a regression coeff ic ient; the second, in parentheses, is the correspond ing asymptotic t ratio, written in absolute terms. the present estimates bear some resemblance to the OLS results of Mincer , though they exceed even the latter by a s ignif icant marg in . Comparing the O L S and 3SLS estimates of the hours coeff icient suggests that there is indeed a substantial endogeneity bias in the former and that the direct ion of this bias is negat ive . Unfortunate ly , there is no genera l , a priori econometric predict ion against which to test the preceding resu l t . T h e re turn to schooling implied by (ME1) is about 1.5 percentage points lower than the cor respond ing OLS estimate. Proport ionately , the exper ience coeff ic ients shr ink by an even greater amount. T h e one attached to the squared term, which measures the concavi ty of the e x p e r i - ence prof i le , turns out to be v e r y small indeed. Both results no doubt reflect the increased importance of the hours term and the fact that it depends , in the other equat ion, upon age and school ing. T h e concavi ty of the exper ience profi le is , of course , a major, implication of the human-capital model. Yet , the degree of concavi ty registered in (ME1), or in any of the s t ructura l earn ings equat ions, does not prov ide especial ly strong support for the theory . On-the- job investment, if it is indeed the key factor in shaping the exper ience p r o - f i le, must not decl ine v e r y rapid ly over the life cyc le ; but in that case, it must not begin at a v e r y high level e i ther , since the model requi res that investment cease on or before ret irement. Much of the observed concavi ty in earn ings prof i les is apparent ly due to the behaviour of h o u r s . 250 T A B L E 2 7 S I M U L T A N E O U S E S T I M A T E S : 8 H O U R S R i g h t - E q u a t i o n s 6 ( d e p e n d e n t v a r i a b l e = W T I M E ) H a n d V a r i a b l e s ( M H 1 ) ( M H 2 ) ( M H 3 ) ( M H 4 ) C o n s t a n t - 1 . 1 0 5 2 ( 2 7 . 3 ) - . 7 4 4 4 ( 6 . 4 3 ) - . 6 3 5 7 ( 6 . 4 8 ) - . 8 3 3 3 ( 1 4 . 0 ) s - . 0 1 5 8 ( 1 3 . 4 ) - . 0 2 4 7 ( 1 4 . 0 ) - . 0 1 8 9 ( 9 . 9 7 ) - . 0 1 8 0 ( 1 2 . 6 ) A C E . 0 0 7 7 ( 4 . 4 5 ) - . 0 1 1 0 ( 2 . 5 3 ) - . 0 1 7 9 ( 4 . 2 3 ) . 0 0 1 7 ( 0 . 6 7 ) A S Q - . 0 0 0 1 ( 5 . 1 5 ) .0001 ( 2 . 0 5 ) .0002 ( 4 . 0 4 ) - . 0 0 0 0 ( 2 . 5 8 ) Z I N C . 4 2 0 9 ( 1 1 . 8 ) - . 8 4 6 5 ( 1 0 . 8 ) . 4 3 5 4 ( 1 0 . 2 ) I N C - . 6 3 7 9 ( 9 . 1 1 ) - - T M A R C - . 2 4 9 4 ( 0 . 6 9 ) - - — — X I N C O T H D I .0340 ( 1 . 0 5 ) - . 1 0 4 0 ( 1 . 3 1 ) - . 2 6 1 6 ( 4 . 7 9 ) .1441 ( 5 . 3 7 ) DI .4561 ( 6 . 8 8 ) .0438 ( 0 . 3 2 ) - . 2 6 8 1 ( 1 . 9 5 ) . 3 4 5 0 ( 4 . 8 0 ) C E O I - . 0 2 6 0 ( 3 . 1 4 ) - . 0 3 4 4 ( 0 . 2 2 ) - . 0 0 4 7 ( 0 . 3 9 ) . 0 7 4 7 ( 5 . 2 8 ) C E 0 2 - . 0 0 6 1 ( 0 . 5 2 ) - . 0 3 6 8 ( 1 . 7 0 ) - . 0 9 9 0 ( 5 . 1 4 ) .0664 ( 5 . 2 1 ) C E 0 4 - . 0 1 3 8 ( 1 . 7 3 ) . 0 0 8 4 ( 0 . 6 3 ) . 0 3 5 3 ( 3 . 0 2 ) .0576 ( 4 . 2 5 ) C E 0 5 - . 0 0 6 2 ( 0 . 8 3 ) . 0 0 4 5 ( 0 . 5 2 ) .0236 ( 2 . 7 1 ) . 0110 ( 0 . 8 5 ) C E 0 6 - . 0 3 3 6 ( 5 . 0 2 ) - . 0 2 3 5 ( 3 . 1 5 ) - . 0 1 3 5 ( 1 . 7 9 ) - . 0 9 5 3 ( 8 . 4 7 ) T Y P E .0158 ( 3 . 6 3 ) . 0 0 1 5 ( 0 . 2 7 ) . 0 0 4 6 ( 0 . 8 0 ) - . 0 5 1 1 ( 6 . 7 4 ) IND1 . 0 0 9 9 ( 0 . 8 0 ) . 0 7 3 0 ( 3 . 3 6 ) . 1 0 0 3 ( 4 . 5 1 ) .3811 ( 1 8 . 7 ) I N D 2 - . 0 4 0 5 ( 2 . 6 6 ) - . 0 3 6 0 ( 2 . 1 5 ) - . 0 1 7 8 ( 1 . 0 6 ) - . 1 4 9 5 ( 5 . 2 8 ) I N D 3 - . 1 4 0 4 ( 4 . 8 4 ) - . 0 4 1 7 ( 1 . 0 7 ) . 0 2 2 9 ( 0 . 5 4 ) - . 1 2 0 3 ( 2 . 5 6 ) I N D 4 . 0 3 1 9 ( 2 . 6 4 ) .0001 ( 0 . 0 0 ) .0042 ( 0 . 2 8 ) - . 0 7 6 0 ( 3 . 6 1 ) I N D 6 - . 0 5 6 8 ( 7 . 2 1 ) - . 0 4 7 6 ( 4 . 8 2 ) - . 0 2 1 5 ( 1 . 9 6 ) - . 1 7 6 6 ( 1 4 . 3 ) I N D 7 - . 0 0 0 3 ( 0 . 0 5 ) - . 0 0 7 3 ( 1 . 1 2 ) - . 0 0 7 1 ( 1 . 0 2 ) - . 0 2 8 7 ( 2 . 6 3 ) I N D 8 . 0 0 9 9 ( 1 . 6 3 ) .0288 ( 3 . 5 4 ) . 0 3 7 4 ( 4 . 5 0 ) . 1150 ( 1 0 . 9 ) I N D 9 .0101 ( 1 . 0 2 ) . 0150 ( 1 . 4 3 ) . 0 3 3 7 ( 3 . 0 1 ) .0016 ( 0 . 8 9 ) I N D 1 0 - . 0 3 6 8 ( 5 . 1 3 ) - . 0 0 7 0 ( 0 . 6 3 ) . 0 0 1 4 ( 0 . 1 2 ) - . 0 0 4 5 ( 0 . 3 8 ) M A J .0391 ( 4 . 4 9 ) . 0 1 7 3 ( 1 . 3 7 ) - . 0 1 2 1 ( 1 . 0 0 ) . 1130 ( 8 . 6 8 ) H E A D . 0 2 5 6 ( 1 . 7 7 ) . 0 2 9 0 ( 1 . 3 4 ) . 0 6 1 9 ( 3 . 8 1 ) - . 0 1 7 6 ( 1 . 7 3 ) F A M S I Z - . 0 0 5 8 ( 2 . 6 9 ) . 0 0 2 9 ( 0 . 5 6 ) . 0 1 1 3 ( 3 . 3 8 ) - . 0 1 0 0 ( 5 . 8 9 ) U S M A R . 0 2 9 7 ( 2 . 2 8 ) . 0 2 1 3 ( 1 . 0 5 ) . 0 0 2 9 ( 0 . 1 9 ) - . 0 0 1 8 ( 0 . 1 8 ) E T H 2 . 0 0 1 5 ( 1 . 3 2 ) - . 0 0 2 5 ( 0 . 8 4 ) - . 0 0 5 7 ( 1 . 7 3 ) - . 0 1 1 3 ( 1 . 4 2 ) E T H 3 . 0 0 0 4 ( 0 . 1 8 ) .0038 ( 0 . 7 8 ) .0008 ( 0 . 1 4 ) - . 0 0 8 9 ( 0 . 6 5 ) E T H 4 . 0 2 4 9 ( 3 . 9 4 ) .0121 ( 0 . 8 2 ) .0040 ( 0 . 2 5 ) . 0 4 6 4 ( 1 . 2 6 ) E T H 5 - . 0 1 8 1 ( 3 . 7 1 ) .0006 ( 0 . 0 6 ) . 0 0 8 7 ( 0 . 7 4 ) - . 0 5 1 2 ( 1 . 9 9 ) E T H 6 - . 0 6 2 1 ( 6 . 2 1 ) . 0 0 6 9 ( 0 . 3 3 ) - . 0 3 8 2 ( 1 . 7 6 ) - . 3 3 3 2 ( 8 . 2 1 ) E T H 7 . 0 1 3 0 ( 4 . 6 5 ) .0061 ( 0 . 9 9 ) . 0 0 5 3 ( 0 . 7 6 ) . 0 2 0 5 ( 1 . 2 5 ) oci - . 0 4 2 0 ( 4 . 2 2 ) - . 0 2 6 2 ( 1 . 3 6 ) - . 0 6 9 7 ( 3 . 2 7 ) - . 0 9 6 3 ( 7 . 4 5 ) O C 2 - . 0 3 0 4 ( 4 . 5 8 ) - . 0 2 0 5 ( 1 . 3 6 ) - . 0 5 9 9 ( 3 . 9 6 ) - . 0 7 0 0 ( 7 . 7 7 ) O C 3 - . 0 2 0 5 ( 3 . 0 0 ) - . 0 1 9 9 ( 1 . 0 1 ) - . 0 9 1 5 ( 5 . 2 2 ) - . 0 6 2 2 ( 5 . 9 8 ) O C 4 - . 0 1 2 6 ( 1 . 3 9 ) - . 0 1 8 6 ( 0 . 9 2 ) - . 0 6 8 7 ( 3 . 0 3 ) - . 0 5 8 0 ( 4 . 2 7 ) O C 5 - . 0 0 4 2 ( 1 . 2 6 ) - . 0 0 3 9 ( 0 . 4 2 ) - . 0 2 7 9 ( 3 . 0 6 ) - . 0 1 9 3 ( 3 . 7 3 ) O C 6 - . 0 2 5 5 ( 4 . 5 0 ) - . 0 2 0 1 ( 1 . 6 7 ) - . 0 2 4 2 ( 2 . 1 6 ) - . 0 5 8 2 ( 8 . 8 5 ) O C 8 - . 0 2 9 4 ( 6 . 9 8 ) - . 0 0 3 4 ( 0 . 3 4 ) - . 0 1 6 3 ( 1 . 5 4 ) - . 0 2 0 3 ( 3 . 4 3 ) O C 9 - . 0 0 6 5 ( 1 . 9 7 ) - . 0 1 0 7 ( 1 . 1 5 ) - . 0 3 1 2 ( 3 . 5 6 ) - . 0 2 7 1 ( 5 . 3 6 ) O C 1 0 - . 0 1 7 6 ( 5 . 1 2 ) - . 0 0 4 1 ( 0 . 4 3 ) - . 0 4 0 7 ( 4 . 4 8 ) - . 0 2 5 1 ( 4 . 7 8 ) O C 1 1 - . 0 1 0 6 ( 3 . 4 8 ) - . 0 0 9 3 ( 1 . 1 3 ) - . 0 2 1 8 ( 2 . 6 4 ) - . 0 2 0 3 ( 4 . 3 6 ) O C 1 2 - . 0 1 6 4 ( 5 . 0 7 ) - . 0 0 9 1 ( 1 . 0 1 ) - . 0 3 4 5 ( 4 . 1 3 ) - . 0 2 9 6 ( 6 . 1 6 ) a M a i n s a m p l e , 2 2 , 6 8 2 o b s e r v a t i o n s b T h e f i r s t f i g u r e in e a c h s e t Is a r e g r e s s i o n c o e f f i c i e n t ; t h e s e c o n d . In p a r e n t h e s e s , i s t h e c o r r e s p o n d i n g a s y m p t o t i c t r a t i o , w r i t t e n in a b s o l u t e t e r m s 251 In this connect ion, it must be understood that the predict ions of the s t ructura l equations do not relate to the experience prof i les one might casual ly observe and plot . T o obtain the counterpar ts to observat ion , we must compute the earn ings reduced-form equation by subst i tut ing (MH1) into (ME1), bear ing in mind that A C E = P + S + 5.67, that ASQ = A C E 2 , and that ZINC = INC + T M A R C . T h e implied reduced- form coeff icients of P and PSQ are 0.0471 and -0.0009 r e s p e c t i v e l y . 1 6 Those values are only a little smaller than these encountered in the correspond ing OLS equat ion, ( C P 5 ) — a fact which indicates rough cons is - tency on the part of the simultaneous estimates. T h e reduced-form coeff ic ients suggest that, on average , earn ings peak at 27.8 years of exper ience, or ve ry near the OLS estimate. The s t ructura l coefficients place the earnings peak at 35.8 y e a r s . For mean- schooled ind iv iduals , this point corresponds to 52 years of age . In comparison, hours reach their peak in (MH1) at 30 years of age. T h i s f ind ing is obviously consistent with the predict ion of the l i fe-cycle model that the peak in hours comes before the peak in the wage rate. Since hours are decl in ing when earnings peak (that is, at age 52), it follows that the wage rate must still be r i s ing and that it will attain its own peak, if at a l l , somewhat later. Actua l ly , since d *ln W /dp = d • In Y/dp - d • In h/dp, one can easily calculate the peak-wage year of exper ience using the same s t r u c - tural coeff icients just employed. Subst i tut ing for the two der ivat ives on the r i gh t -hand s ide, setting the d i f ference equal to zero, and solving for p (the theoretical counterpart of P ) , one a r r i ves at a f igure of 51 y e a r s . T h i s point corresponds to age 67 for individuals with mean school ing. In other words, according to the s t ructura l estimates of (MEI)-(MH 1), wage i, rates do not reach a peak or decline at all pr ior to the normal age of ret irement. Th i s result agrees , more or less, with Mincer's observat ion concern ing the "weekly earn ings" of U . S . m a l e s . 1 7 However, it does not offer much comfort to the human-capital theor ist . Accord ing to the model, self- investment should not be propel l ing wages upward when the individual is close to retirement, part icular ly if depreciation is signif icant On the other hand , since the slope of the wage profi le is rather s l i g h t — one might almost call it f l a t—in the years approaching retirement, one could still argue on behalf of the theory that investment and depreciation both simply approximate zero d u r i n g this stage of the life c y c l e . Such an interpretat ion, though logically admissible, serves mainly to i l lustrate how dif f icult it is to submit the human-capital model to the legitimate jeopardy of scientif ic fa ls i f icat ion. Focuss ing on (MH1) alone, we f ind that the coefficient of ZINC is posit ive and rather large in absolute terms. On the basis of (66) the implied estimate of 6 is 0.73. Such a high value for the elasticity of hours with respect to wages is certa in ly surpr i s ing when one cons iders the typical results reported in the labour-supp ly l i terature . T h e most common f ind ing for males appears to be that the wage elasticity is negat ive . The present result therefore raises some susp ic ion . It must be emphasized, however, that (MH1) makes no pretense at being an identif ied labour-supp ly func t ion . 253 One may think of several reasons to account for the seemingly large value of <S,though none is altogether p leas ing . At some level of intuit ion, it is not su rp r i s ing that the coefficient of ZINC (and hence 6) is large, since ignor ing taxes, we are actually regress ing In h on the variable (InW + In h ) . There would appear to exist a strong tendency for this sum and In h to be posit ively corre la ted . For many of the labour 'supp ly studies, which use wage rates rather than earn ings , there is the opposite tendency: h is regressed on Y / h . In both cases, the econometric problem is essential ly one of endogeneity . Since the exist ing studies rely mainly on OLS estimates, bias and inconsistency are to be expected . Here, however, endogeneity receives explicit treatment; thus if the present approach has been success fu l , incons is tency—and perhaps bias, g iven the large sample—will have been avo ided . On a more r igourous level , it turns out that in the general case, with several exogenous variables and correlated e r ro rs in the structura l equat ions, nothing can be proven about the direction of bias in the coefficient of Z I N C . In at least one simplified case, it appears that the direct ion of 18 bias is indeed pos i t ive . A comparison of the OLS estimates in Append ix IV and the present 3SLS results tends to confirm this suggest ion . T h e 3SLS procedure yie lds a fall in the ZINC , coefficient, though not one of suff ic ient magnitude to turn 6 negat ive. Another factor in the present outcome may be the imposition of the constant-elast ic i ty functional form, which has been little used in the ex ist ing r e s e a r c h . Di f ferences in functional form can obviously have a profound effect upon resu l t s . It is not dif f icult to imagine a labour- 254 supply c u r v e which reverses slopes part way through its range, y ie ld ing a posit ive elasticity estimate with the log-l inear specif ication and a negative elasticity estimate with some other form. A supply c u r v e of this sort , which seems theoretically p lausible, may also g ive contrad ic tory results for d i f ferent samples or data sets if these are drawn for some reason from di f ferent parts of the range . F inal ly , it is worth repeating that the hours equation may not be "strongly ident i f i ed ," in the sense that its s t ruc ture is unquest ionably revealed by var iables which produce broad and prec ise shifts in the earn ings equat ion. T h e possibi l i ty ex ists that in computing the hours regress ion , we are to a great extent merely runn ing the earn ings regress ion in r e v e r s e . A strong positive relationship between wages and hours in the earn ings regression would then c a r r y over into the hours estimates. A l though this consideration tends to limit interest in the latter, it does not affect the val idity of results y ielded by the earn ings equat ions. With regard to the remaining coeff icients in (MH1), (66) implies that all must be multiplied by (1 + 5) to obtain estimates of the s t ructura l parameters compris ing y . Even if (I + 6) is as large as prev ious ly i n - dicated (that is, 1.73), only three of the corrected estimates surpass 20 0.1 in absolute va lue . Since the raw coefficients change a good deal in any event as one moves across the table, fu r ther calculations are left at this stage to the interested reader . Before we turn to the other equations, some additional features of (MH1) deserve comment. Note f i rst of all that the raw coefficient of 255 the income-intercept term ( X I N C O T H D I ) is posit ive but (asymptot ica l ly) ins ign i f i cant—not an uncommon resul t in the orthodox labour-supp ly l i terature. If one were interpret ing (MH1) as an identif ied labour-supp ly schedule, theory would of course predict a negative coefficient as long as leisure is a normal good. School ing, unexpected ly , reduces time worked, both here and in the single-equation estimates d isp layed in Appendix IV. It would appear that any advantage which the more schooled hold over the less schooled in avoid ing unemployment is negated by d i f ferences between these g roups in labour-supp ly behaviour or in the t ime-worked character is t ics of their respect ive jobs. One must be alert , however, to the possibi l i ty that school ing, being related d irect ly to earn ings , is merely acting as an earn ings p r o x y , thus counterbalancing the latter to some degree and making the functional form less const ra ined . A s one might casually have forecast, self-employment increases time worked . T h o u g h a number of other variables in (MH1) likewise d i s - play s igni f icance, their coeff ic ients p roved general ly rather sensit ive to the part icular specif ication in force and are therefore best cons idered in l ight of all the resu l ts . Fur ther Experiments Equations (ME2)-(MH2) show the effects of insert ing INC and T M A R C separately in the hours regress ion . On the earn ings side, the coeff ic ients change v e r y little a n d , hence, require no additional comment. However, in the hours regress ion itself, the modification is c r u c i a l . T h e 256 coeff ic ients of INC and T M A R C , f i rst of a l l , are s ignif icant ly d i f ferent from each other , contrary to standard theoretical reason ing . Taxes appear much less important than gross ea rn ings . Nevertheless , in view of the problems in estimating the tax rate and in purg ing T M A R C of its endogen- 21 ei ty , one cannot treat this result as more than suggest ive . Second, in response to the change, the coeff ic ients of A C E and A S Q switch s igns , indicating a convex rather than a concave structura l profi le of h o u r s . T h i r d , most of the other coefficients become asymptotically less signif icant 22 than in (MH1) . T h e use of the two income-related terms in place of ZINC tends, it seems, to overpower the other var iab les . Equations (ME3)-(MH3) restore the use of ZINC in order to investigate the effects of SSQ and X S P in the earn ings regress ion . A s before, the coefficient of SSQ is s ignif icantly posit ive, but that of X S P is ins igni f icant . For individuals with mean levels of schooling and exper ience, the implied rate of re turn to the former is 6.2%—again, somewhat lower than estimated by O L S . T h i s f igure r ises (falls) by 0.5 percentage points for each year of schooling above (below) the mean. The reduced-form earnings profi le turns out to be convex rather than 23 concave, thereby cast ing general doubt upon this vers ion of the model. A s in (MH3), the structura l profi le of hours is also convex . We come now to the expanded earn ings funct ion, (ME4) . T h e insertion here of twenty-f ive additional var iables causes some marked changes in the coeff icients upon which we have been focuss ing . T h e indicated re turn to schooling falls by approximately one fur ther p e r c e n - tage point to 5.3%. T h e increases in earn ings on account of exper ience 257 become v e r y small indeed, and the concavity of the earn ings prof i le (as registered in the structural estimates) d isappears . A s a compensation, the importance of hours worked great ly increases. The elasticity of wages with respect to hours is g iven as 1.82. Overa l l , then , the inf luence at t r ibuted to the orthodox human-capital proxies , S, P, and PSQ, when these change ceteris par ibus , is substantial ly d iminished. T h o u g h it is arguable , because of l inked mobility patterns , whether ceter i s -par ibus 24 measurements are actually legitimate, the present estimates serve to show the effect of not conceding to the human-capital var iables , as Mincer and others do, the "benefit of the doubt . " It will be observed that, among the variables added in (ME4) to the basic human-capital specif icat ion, the coeff icients of many remain very s izable. For example, residence in At lantic Canada (CEOI) is a disadvantage worth 2.6 years of school ing; residence in Br i t i sh Columbia (CE06) is an advantage worth 2.9 y e a r s . Employment in agr icu l ture (IND1) is an immense handicap (79% of re ference-group e a r n i n g s ) , whereas employment in construct ion (IND6) yei lds top earn ings (29% more than in manufac tur ing) . Period of immigration (IM) is not s ignif icant, but rura l or small-town residence ( T Y P E ) and self-employment (MAJ) c o n t i n u e , as in the OLS resu l ts , to exact substantial earnings penalt ies. The coeff ic ients of ethnic group ( E T H ) perhaps deserve special comment. The one perta in ing to individuals of Jewish descent (ETH5) remains positive but is no longer s igni f icant , as it was in the OLS regress ions . The coefficient perta in ing to Native Indians (ETH6) is the only one which is signif icant here, and it is both positive and v e r y la rge , cont rary no doubt to one's casual pred ic t ions . It must be remem- b e r e d , however, that the coefficient in question measures the effect of Native Indian or ig in with other var iables such as school ing, exper ience, h o u r s , location, and indust ry held constant—a situation we do not casually observe in the real wor ld . The calculated reduced-form co- efficient is much smaller (0.0457), since hours at least are permitted to v a r y ; st i l l , for the most par t , ceter is par ibus appl ies . Though the present result may yet seem anomalous, it receives some support from 25 the f ind ings of Haessel and K u c h . One might speculate that, as an apparent ly d isadvantaged g r o u p . Native Indians benefit par t icu lar ly from socially or institutionally standard rates of pay , which they receive when employed, despite inferior qual i f icat ions. As for the hours s t ructura l equat ion, (MH4), it will be observed that in every case but one, the s igns of the added variables are the reverse of those in the earn ings s t ructura l equat ion . Within part icu lar categories, hours worked tend to offset high earn ings . T h i s result may be a fu r ther clue to the apparent high value obtained for the coefficient of Z I N C . When hours are low and earnings h i g h , implicit or actual wage rates per hour must be h igh as well . We thus come upon some indication of a negative relationship between wage rates and h o u r s . If negative aspects of the overal l relationship are closely l inked with the added variables ( C E O , T Y P E , IND, et ce tera) , these will tend to reflect the negative s ide, leaving the coefficient of ZINC relatively large. T h i s tendency will operate to some extent even when the variables in question do not appear in the earn ings equat ion; then , since fewer attr ibutes are held constant across the ent ire system, and the need for offsett ing coeff icients is less pronounced, one would expect those which remain in the hours equation to lie closer to zero . T h i s pattern does emerge in the comparison of (MH4) and (MH1). However, the change in the coeff icient of ZINC, while in the anticipated d i rect ion, is rather small. One can say only that add ing var iables to the system—hold ing their influence constant , in other words—may be in part responsible for the f inding with respect to Z I N C . The observat ion that wages and hours are broadly offsett ing when viewed across regions and industr ies tends to redeem the speculation concerning seasonality made earlier in connection with the OLS estimates. If seasonality is indeed the ru l ing factor in the creation of of fsett ing wage di f ferent ia ls , it is by no means su rp r i s ing that we should observe the effect through regions and industr ies , which seasonality str ikes u n - 26 even ly . In the OLS equations the seasonal effects cannot manifest themselves except through the coefficient of WTIME. In the 3SLS equations the latter is free to reflect other l inks between wages and h o u r s , such as the rates earned moonlighting, the premium for overtime, and the unmeasured abi l i ty variables which influence wages and hours in common. It is worth not ing , f inal ly , that (MH4), like all the other s t r u c - tural equations, d isp lays scant concavity in the implied experience or age prof i le . There is a very flat peak in hours at 10.0 years of exper ience . T h i s result nevertheless satisfies the predict ion of the l i fe-cycle model, since earnings peak (structura l ly) well beyond the r e l - evant range—at 141 years , to be prec ise . From the standpoint of the computed reduced form, hours and earnings peak at 15.8 and 20.4 years of experience respect ive ly . These points come a little earl ier than calculated p rev ious ly . One may wonder, g iven that the change in specif ication has been to hold additional variables constant, whether individuals thus use geographic and inter industr ia l mobility to stave off earn ings and hours peaks . If such moves benefit individuals at var ious points in their l i fe, one should indeed notice a hastening of- the peaks when this recourse is disallowed statistically in c r o s s - sect ion. 261 APPENDIX V E S T I M A T E S O B T A I N E D BY I T E R A T I V E T H R E E - S T A G E L E A S T S Q U A R E S 1 (ME5) INC = 1.1920 + 0.0589 S + 0.0087 P - 0.0001 PSQ + 1.5019 WTIME (46.3) (33.5) (10.3) (5.28) (66.8) (MH5) WTIME = -0.6742 - 0.0357 S - 0.0062 A G E + 0.0000 ASQ (20.4) (29.1) (4.68) (1.10) + 0.6324 ZINC + 0.1451 X I N C O T H D I + 0.2480 DI (27.1) (10.4) (6.66) + 0.0081 GEOI + 0.0333 G E 0 2 + 0.0056 C E 0 4 + 0.0017 G E 0 5 (2.19) (7.00) (1.60) (0.72) - 0.0078 G E 0 6 + 0.0020 T Y P E + 0.0024 IND1 - 0.0076 IND2 (3.46) (1.16) (0.28) (1.61) - 0.0404 IND3 + 0.0027 IND4 - 0.0098 IND6 + 0.0025 IND7 (3.42) (0.59) (3.03) ( 1 .25) - 0.0043 IND8 - 0.0079 IND9 - 0.0071 IND10 + 0.0195 MAJ (1.62) (2.54) (1.98) (6.22) - 0.0258 H E A D - 0.0096 FAMSIZ - 0.0I98 USMAR ( 5.73) ( 11 .6) (4.56) - 0.0023 ETH2 + 0.0005 E T H 3 + 0.0027 ETH4 + 0.0200 ETH5 (2.03) (0.27) (0.50) (5. 13) - 0.0361 ETH6 - 0.0007 E T H 7 + 0.0368 OC1 + 0.0160 OC2 (4.05) (0.32) (5.26) (3.19) + 0.0141 OC3 + 0.0475 OC4 +.0.0016 OC5 + 0.0063 OC6 (2.38) (6.43) (0.53) (1.70) - 0.0086 OC8 + 0.0017 OC9 - 0.0005 OC10 - 0.0009 OC11 (2.27) (0.58) (0.16) (0.33) + 0.0001 OC12 (0.03) Number of observat ions = 22,682 Number of iterations = 11 F igures in parentheses are asymptotic t ratios, written in absolute terms. N O T E S C H A P T E R V ' The symbols Y and h now stand for vectors , both of then N x 1, N being the number of observat ions in the sample. T h e prev ious ly de- f ined vectors X{ and Z- , i = 1,2, •••,N (i formerly suppressed) make up the rows of X and Z respect ive ly . 2 A s a general reference the reader may wish to consult J . Johnston, Econometric Methods (second ed i t ion; New Y o r k : McGraw-Hi l l Book C o . , 1972), p p . 395-398. 3 For proof see Phoebus J . Dhrymes, "Small-Sample and Asymptot ic Relations between Maximum-Likelihood and Three-S tage Least -Squares Estimators; 1 Econometrica, XLI (March , 1973), p p . 357-364. See A lber t Madansky, "On the Ef f ic iency of Three-Stage Least -Squares Est imation," Econometrica, XXXII ( J a n u a r y - A p r i l , 1964), 51-56. 5 F u r t h e r d iscuss ion on this point is p rov ided b y Terence J . Wales and Alan D. Woodland, "Labour Supp ly and Progress ive T a x e s , " Review of Economic Stud ies , XLVI ( January , 1979), 83-95. Besides dealing with endogeneity, these authors investigate what they call "specification e r r o r , " which results from a stochastic d iscrepancy between the actual and des ired labour supply of the ind iv idua l . However, this problem really ar ises only within an expl icit uti l ity framework, when one is assuming the identif ication of a l abour-supp ly func t ion . See Wales, "Estimation of a L a b o u r - S u p p l y C u r v e for Self- Employed Bus iness P ropr i e to rs . " 7 S ince ZINC stands for In(1 -- x) Y = ln( 1 - x) + In Y the determin- ants of ln(1 - x) and In Y at least combine addi t ive ly in the present formulat ion. Note from the definit ions of Chapter III that ZINC is furthermore an addit ive component in the calculation of X I N C O T H D I . 8 T h e list reads as fol lows: S, S S Q , P, PSQ, X S P , L E N , G E O , T Y P E , IND, M A J , O C , H E A D , FAMSIZ , U S M A R . g T h e reader may ver i f y the point by consult ing F igure 2 along with the definit ions of Chapter III. 262 263 1 0 S e e again Wales, o p . c i t . 1 1 S e v e n hours categories and f ive weeks categories y ie lds t h i r t y - f ive possible combinations. 12 The interval is half closed because indiv iduals who worked zero hours were prev ious ly exc luded from the sample. Such exclusion gives rise to a problem now known in the l i terature as "sample select ivity b i a s . " See Reuben C r o n a u , "Wage Comparisons - A Select ivity B i a s , " Journal of Political Economy, LXXXII (November/December, 1974), 1119-1143. Estimates are "biased" in the sense that they are conditional upen the individual 's working at some time d u r i n g the measurement per iod ; hence, they may not hold in the aggregate and maybe misleading for policy purposes if not correct ly in terpre ted . See also Michael J . Bosk in , "The Economics of Labor S u p p l y , " in Income Maintenance and Labor S u p p l y , edited by Glen C . Cain and Harold W. Watts (Ch icago: Rand McNally College Publ ishing Company, 1973). 13 For a more sophist icated treatment see C iora Hanoch, A Mult i - variate Model of Labor S u p p l y : Methodology for Estimation (Santa Monica: T h e Rand Corporat ion , September, 1976)7 14 See A r n o l d Zel lner and Henr i T h e i l , "Three-Stage Least Squares : Simultaneous Estimation of Simultaneous Equat ions ," Econometrica, X X X ( January , 1962), 54-78. 1 5 C f . Equation ( C P 5 ) , Table 6. 16 Note that the entr ies in the tables have been r o u n d e d . Hence , the results stated here and below may not appear ent ire ly consistent with the reported f i g u r e s . 1 7 S c h o o l i n g , Exper ience , and Earn ings , p. 70. 1 8 Cons ider the h ighly abbrev iated model In Y = x + (1 + 6) In h + u , In h = a + D In Y + u 2 , where D = ^ + ^ , a is a constant , and x represents the sum of all factors determining In W . Assume that u , is uncorre lated with both x and u 2 and that the latter are themselves uncor re la ted . T h i s case is just \ 264 sl ightly more general than one treated by Johnston, o p . c i t . , p p . 342-344. The reduced forms a re : l n Y = T^DTT+eJ [aM+6) + x + u 1 + u 2(1+6)] l n h = 1-D(1+9) - [ a + D x + D G 1 + G 2 ] • Following Johnston, we may compute moments (denoted mjj1, and probabi l i ty l imits. In l ight of our assumptions, we f ind that the relevant moments are 1 2 m . = r i r. 11 , n\ I 6 [ + m~ ~ + m~ ~ (1+0) 1 y y [1-D(1 + 0)J 2 xx u i u i u 2 u 2 1 m . = j-z—p.I. , v , , [Dm + Dm- ~ + m~ ~ (1 + 9) ] . h [ 1 - D ( 1 + e j ] 2 xx u 1 u l U 2 U 2 T h e OLS estimate of D is D = m . /m . It follows that y h y y Dm + Dm- ~ + m - - (1 + 8) XX U.U U - ) U T plim D = — z 1 1 — m + m - - + m ~ - ( 1 + e ) 2 xx u ] u 1 U 2 U 2 if plim m.. = m.. < °° for all i, j . T h e asymptotic bias is therefore (1 - D ) ( 1 +6) m - - plim D - D - 1 2 m + m — + m ~ ~ ( 1 + 8 ) 2 xx u 1 u 1 U 2 U 2 which is posit ive as long as 9 < - 1 , since D = 6/(1 + 6) cannot exceed un i t y . T h e reader may ver i fy that if any of the noncorrelation assumptions are violated or if the hours equation contains additional nonorthoganal independent var iab les , the direct ion of bias in indeterminate. 19 See Yoram Barzel and R ichard J . McDonald, "Assets , Subs is tence , and the Supp ly C u r v e of L a b o r , " American Economic Review, LXIII (September, 1973), 621-633. 265 These are associated with DI (real ly , just an adjunct to the constant) , with IND3 (f ishing and t rapp ing , a seasonal i n d u s t r y ) , and with ETH6 (Native Ind ians) . In part icu lar , it may ar ise because of col l inearity between T M A R C and INC . Multicol l inearity is again a problem in the case of the family- status var iables H E A D , FAMSIZ, and USMAR, which contr ibute initially to the estimation of T M A R G . " ' T h e reduced-form coeff icients of P and PSQ are -0.1146 and 0.0010, respect ive ly . 24 See again the discussion in Chapter II. 25 "Size Distr ibut ion of Earn ings in C a n a d a . " In this study the largest coeff ic ients, which are v e r y nearly identical , belong to Native Indians and to Chinese and Japanese. Perhaps on account of small numbers , those pertaining to Native Indians (an intercept dummy and a school ing interaction) are ins igni f icant . T h o u g h an exact comparison is impossible because of d i f ferences in def in i t ion, the ceter is -par ibus earn ings advantage estimated by the authors for mean-schooled indiv iduals appears to be about the same as that implied by the present s t ructura l equat ions. 26 Other factors besides seasonal i ty—proneness to s t r ikes , for example—might also fit this c r i te r ion ; but the a l ternat ives, which may contr ibute something to the explanat ion, do not account as plausibly as seasonality for the d iscrepancy between Canadian and American resul ts under O L S . C H A P T E R VI SUMMARY A N D C O N C L U S I O N S T h i s chapter provides a condensation of the arguments and in fer - ences stated in the preceding text . It reviews the assorted methodological, theoretical , and econometric objections raised against the human-capital model and attempts, in light of these objections, to place the empirical exerc ises of the c u r r e n t s tudy in the proper perspect ive . Results are summarized for a large cross-sect ion of Canadian males who worked in 1970. C H A P T E R I Three models are cons idered , in ascending order of their gener- a l i ty : (1) the basic schooling model and (2) the model of postschool investment, both employed by Mincer, and (3) the earn ings maximization model, suggested by Ben Porath . The f i rst two deal with investment in human capital at part icu lar stages of the life cyc le ; the th i rd contains the others as special cases . T h e schooling model asserts that proportionate d i f ferences in earn ings accompany absolute di f ferences in years of formal school ing; that is, In W g = In W Q + r s, where the parameter govern ing the re lat ionship, 266 e r , is interpreted as the rate of re turn to educat ion . T h e assumptions needed to sustain this interpretation are , however, exceedingly powerfu l . T h e fundamental postulate is that indiv iduals receive the same capital ized sum in lifetime earnings no matter what their level of school ing. If the supposed equality of present values is merely conceptual , then r is at best an ex post internal rate of r e t u r n . One must say "at best" because the present-va lue accounting prescr ibed by the model is v e r y r o u g h . Schooling is presumed to entail no d i rect expendi tures or subs id ies , no present or future nonpecuniary benefits or costs , and no opportunit ies for part-time employment. Hours of work and the r i sks of unemployment are held constant, over the life cyc le of the individual and across schooling g r o u p s . Though estimates based on these assumptions may, even so, prov ide some useful descr ipt ion of the earn ings s t ruc ture , they cannot be regarded as fu rn i sh ing tests of any maintained hypothes is . On the other hand , if the equality of present values is presumed to be actual , then (subject to the preceding approximations) r may be thought of as an ex ante, long-run equi l ibr ium rate of r e t u r n . Mincer, and other writers of the human-capital school , are nevertheless mainly silent on how the labour market might function to b r ing about long-run equ i l ib r ium. No analysis of indiv idual choice is ever prov ided within the context of the schooling model, though it is possible to devise one if indiv iduals are assumed to ignore leisure in favour of maximizing a single object ive, d iscounted lifetime earn ings . A g a i n , however, the exerc ise fails to place any important restr ict ions on the data . 268 Whereas the supply side of the labour market gains at least a shadowy presence, the demand side suf fers complete neglect . A l though the schooling model is unquest ionably a reduced-form relationship from the labour-market standpoint, no exogenous demand variables appear in i t . Market imperfections, associated perhaps with region or i ndus t ry , are deemed unimportant, as are any quant i ty imbalances which might cause a "temporary" departure from long-run equi l ibr ium. Since one cannot tell whether long-run equi l ibr ium actually obtains at any g iven moment of observat ion , there is no conceivable way of test ing the school- ing model. That its parameter r might supply an adequate ex post empirical descr ipt ion thus remains the strongest admissible claim. Becker 's model of the individual 's market for human capital also turns out to be barren of testable implications. It is consistent £ with any sort of cross-sect ional relationship between r and the level of school ing. T h e "interactions model ," which Haessel and Kuch der ive £ from it, nevertheless holds some promise. In this analys is , r is at least made to depend upon some measurable attr ibutes of the ind iv idua l . £ T h o u g h the hypotheses l inking r and these attr ibutes remain essential ly ad hoc, they lead one, as theory should , to investigate new dimensions of the empirical earn ings s t r u c t u r e . Distr ibutional arguments flowing from the schooling model typical ly rest on the assumption of independence between schooling and its rate of r e t u r n . T h e evidence against such an assumption is, however, v e r y widespread. T h e only unambiguous implication is that earnings will follow the lognormal d istr ibut ion (or weaker, be skewed to the r ight) if schooling follows the normal (is not heavily skewed to the l e f t ) . Unfortunate ly , as Oulton points out, there is no theory to specify the d istr ibut ion of school ing. The postschool investment model elaborates upon the schooling model by allowing individuals to d iv ide their time between training and pure work in accordance with a second parameter k. Le isure is again held constant , and the model is der ived through a series of identities and approximations. It is shown that if k, the propensi ty to invest in human capita l , decl ines over the life cyc le , then the model is consistent with the pr incipal sty l ized fact concerning age-earn ings prof i les, namely, that they are concave from below. However, concavity may also be due to biological aging or to costless learning by do ing . Only an appeal to competitive equi l ibr ium will rule out the latter. Hence, all the crit ic isms d irected at the schooling model still app ly . Empirical "tests" do not discriminate among all three competing hypotheses . Us ing the concept of "over tak ing , " Mincer der ives the p r e - diction that the cross-sect ional var iance of earnings will d isplay a minimum at roughly p = l/r e years of exper ience. T h i s hypothesis is not s t rongly confirmed by Mincer's data; but since it is conditional upon, there being only a small correlat ion between earn ings at school leaving and the propensi ty to engage in postschool investment, the model proves in the end to be immune from falsification on this account . 270 T h e Ben Porath model seeks to prov ide a behavioural theory of k based on a formal analysis of the conditions for maximizing discounted lifetime earn ings . It is shown that the optimal values of k do in fact decline over the life cyc le , the reason being that the time period over which to amortize successive investments becomes increasingly shor t . There fore , present-va lue maximization is general ly consistent with the concavity of age-earnings prof i les . Ye t , in detailed test ing , the shapes of these profi les do not conform to expectat ions. It has been suggested that the fault lies in the "neutral ity hypothes i s , " which restr ic ts the form of the human-capital input in its alternative uses . However, with- out the neutral i ty hypothes is , the model is untestable. A l though the three models surveyed may be useful as an aid to thought and as a framework for empirical descr ipt ion , they fa i l , for the most par t , to generate cr it ical hypotheses by means of which to test the central notion that earnings are the result of indiv idual investment dec is ions. C H A P T E R II Even if implementation of the var ious models t u r n s out to be merely an exercise in descr ipt ion , it is still necessary to consider the problems which may hinder unbiased estimation. Descr ipt ive resu l ts , even if correctly interpreted as such , should not be misleading from a quantitat ive point of v iew. B y means of simple regress ion . Mincer estimates the return to schooling at 7%—a f igure much below the values obtained d irect ly from age-earn ings prof i les in other s tud ies . He attr ibutes the apparent down- ward bias to the omission of experience (postschool investment) , which is negatively correlated with school ing. It is argued here that any net bias may involve several fac tors : (1) individual variation in rate of re turn (schooling coef f ic ient) , (2) the endogeneity of school ing, (3) expectations and growth , (4) omission of abil ity and family back- g r o u n d , (5) omission of other var iab les . If the individual rate of re turn falls as schooling increases, the s imple-regression estimate of r will have a downward b ias . Yet , in the case of Canada, one might look for an upward bias, since exist ing research g ives some hint of r is ing r e t u r n s . Mincer a rgues , with respect to the Uni ted States, that the apparent fall in the rate of re turn is due to the variation in weeks worked . It may not be ligitimate, however, to estimate r with weeks worked held constant . A n alternative approach is to account expl ic it ly for individual d i f ferences in r , either by letting the variable "years of schooling squared" appear in the regress ion , or by resort ing to the more elaborate interactions frame- work. In genera l , the power of the human-capital model suf fers to the extent that r tu rns out not to be a stable parameter. If schooling is really an endogenous var iable , the estimated re turn will again be subject to b ias . Proponants of the model must therefore confront a dilemma: endogenous schooling leads to biased estimation, but exogenous schooling means that there is no market 272 mechanism to enforce long-run equi l ibr ium. Expectat ions, mainly with regard to the growth of wages, must also be cons ide red . If long-run equi l ibr ium is assumed, the schooling coefficient will measure only the di f ference between the net rate of e * return and the average expected rate of real growth (that i s , - r - g ) . A n underestimate of the former, caused by misinterpretat ion, may thus o c c u r . Age and place of highest grade might serve as proxies for the state of expectations at a part icular time in a part icular locale. Among all the potential sources of bias in estimating the rate of r e t u r n , the one which has received the most attention has been the omission of abil ity and family b a c k g r o u n d . It is argued that if abi l i ty and family background have an independent effect on earn ings , and if these variables are posit ively correlated with school ing, then their omission will bias the schooling coefficient upward , as the latter "picks up" earn ings variance which is not causally attr ibutable to it . The census data used here and in the comparable study by Mincer do not, of course , prov ide the abi l i ty and background variables with which to investigate this problem f u r t h e r . It is possible to investigate potential biases from the omission of other var iab les . It is a rgued here that i ndus t ry , occupat ion, and place of residence may capture components in the apparent rate of re turn which are the result of market imperfection, shor t - run d isequ i l i - br ium, and prev ious ly ignored nonpecuniary factors . Such components will not be available to every investor in school ing. Though it may be that schooling is a prior cause of industr ia l , occupational, and geo- graph ic mobility, one cannot assume that the variables mentioned have no independent ef fect . T h e human-capital model makes this assumption and thus attr ibutes all the doubtful earn ings var iance to school ing. Mincer holds weeks worked constant , but none of the suggested var iab les . When he inserts weeks worked, the implied rate of re turn to schooling fa l ls . It turns out that the elasticity of earnings with respect to weeks is greater than un i ty . Implementation of the postschool investment model requ i res , f i r s t , that one estimate the amount of time an individual has spent on the job (his "experience") a n d , second, that one specify the proport ion of time (k) devoted in each period to t ra in ing . To estimate exper ience. Mincer and others use age minus schooling minus f i ve . Th i s proxy assumes no unemployment or nonparticipation in the labour force , to- gether with constant hours . The associated e r ro rs of measurement may bias the schooling coefficient up or down in the eventual formulation; however, empirical evidence suggests an upward b ias . T o speci fy the time prof i le of k, Mincer proposes two funct ions, one of which decl ines l inear ly , and the other , exponent ia l ly . The former leads to a quadrat ic estimating equat ion; the latter, to another exponent ia l . Neither spec i f i - cation quite matches the theoretical form implied by the Ben Porath model, although both may give a tolerable approximation. The exponential form has the advantage of ident i fy ing all the parameters of the empirical model. Besides holding experience constant in the preceding parametric fashion. Mincer uses the alternative method of app ly ing the schooling model to a single experience cohort , the one estimated to be at o v e r t a k i n g . Within this cohort , earn ings di f ferent ia ls are thought to be ent irely attr ibutable to school ing. In either case, the schooling coefficient r ises considerably as p r e d i c t e d . Implementation of the Ben Porath or income-maximization model is complicated by the unavoidable nonl inearity of functional forms. T h i s problem dictates relatively small sample sizes with few var iab les . As a resul t , it has been impossible to test hypotheses of real interest—those which link the theoretical parameters to individual a t t r ibutes . Attempts at implementation have been, to a great extent , exerc ises in c u r v e f i t t ing , as earnings are regressed on age or experience transformed in d iverse ways. "Reasonable" parameter estimates are then taken as c o n - firmation of the theory . C H A P T E R III Here , the prev ious ly surveyed aspects of Mincer's empirical work on the human-capital model are reproduced using Canadian microdata. T h e standard earnings function is then expanded by means of additional var iab les , the aim be ing , on the one h a n d , to prov ide an improved descr ipt ion of the Canadian labour economy a n d , on the other , to establ ish an alternative benchmark against which to judge the orthodox speci f icat ion. The pr incipal data source for this effort was the one- in-one- h u n d r e d Public Use Sample drawn from the 1971 c e n s u s . T h e working sample comprised 22,682 out-of-school males who were employed at some time d u r i n g 1970 in any of the 10 identifiable industr ies making up the pr ivate sector . Each observat ion consisted of individual data on 168 var iab les . Results for the basic schooling model were v i r tual ly identical to those reported by Mincer . T h e schooling coefficient or "rate of r e t u r n " was measured at 7%. T h e simple regress ion explained 7% of the earn ings var iance or " inequal i ty ." A s in Mincer's work, experience was held constant in three ways: by examining the overtak ing cohort and by estimating, f i rs t , the expon- ential , and then , the quadrat ic speci f icat ion. T h e overtaking subsample consisted of 1,238 individuals with 7-9 years of exper ience . For this g roup the schooling coefficient reached 10% but fell by one-quarter when weeks worked were held constant . T h e insignif icance of schooling squared implied, accord ing to the orthodox interpretat ion, that the return to schooling d id not v a r y . However, the level of the return was not ent i re ly consistent with the definition of the overtak ing set laid down in part with the aid of Mincer's reciprocal rule of thumb. The elasticity of earn ings with respect to weeks was not s ignif icantly di f ferent from u n i t y . T h e exponential form of the exper ience profi le was invest igated by iterating a l inear equation for d i f ferent values of 8, the exponential rate at which k declines over the life cyc le . Since we must have 276 0 ^ k ^ 1 together with a positive re turn on postschool investment (r > 0) , it was possible to deduce certain reasonableness restr ict ions with which to screen the estimates. In.one var iant of the model reas- onable coeff ic ients were implied for 8 in the 0.15-0.20 range, but the results proved far too unstable to use in computing estimates of r x and k Q (the initial propensi ty to i n v e s t ) . In another var iant 8 would have had to be somewhat less than 0.05. A s for the other coeff ic ients, that of schooling squared was s ignif icantly posit ive; that of weeks was s igni f icant ly less than un i t y . When the quadrat ic functional form was used to portray e x p e r - ience, the implied rate of re turn to schooling was 8.7%. With weeks held constant, the f igure decl ined by about one-sixth to 7.2%. T h e coeff icient of schooling squared , though relatively small, was signif icantly positive whether or not the weeks var iable was inc luded . Earn ings peaked at 29-30 years of exper ience—a little earl ier than in the United States sample. Exper ience prof i les had a sl ight tendency to converge over the life cyc le , just as Mincer o b s e r v e d . Each additional year of schooling postponed the earnings peak by only 0 . 6 - 0 . 7 y e a r s . Mincer's assumptions with respect to depreciation and the length of the net invest- ment stage produced estimates of 7.7% for r and 0.54 for kjj. However, a wide range of values were obtained by va ry ing these assumptions within reasonable limits. General ly speak ing , the introduction of exper ience, by what- ever means, had considerably less effect here in raising the coefficient of schooling than it d id in Mincer's research . On average, rates of 277 re turn (if one chooses to interpret the schooling coefficients as such) appear to be lower in Canada than in the United States . A s noted, how- ever , d i f ferences in sample composition and in time period may contr ibute to this resu l t . There is nevertheless a firm contrast in the tendency of Canadian re turns to r ise with the level of schooling and in the o b s e r v a - tion that, over the full sample, earnings did not r ise in proport ion to the number of weeks worked . Over both the expanded and the orthodox earnings funct ions, implied rates of re turn var ied from 6.9% to 8.91% with hours ignored and from 6.03% to 7.75% with hours held constant . Cor rec ted for anticipated real g rowth , these values exceeded the most recent estimates of Statist ics Canada but were still well short of those computed by Podoluk a decade ear l ier . Addit ional variables in the expanded regress ions did not account for r i s ing re tu rns unti l occupation was in t roduced; then, schooling squared became ins igni f icant . Returns that rise in cross section are not, of course , inconsistent with a competitive equi l ibr ium. T h e elasticity of earn ings with respect to hours was considerably less than unity in all the single-equation estimates. Among the added var iab les , " long" vocational t ra in ing was associated with an earn ings premium of 8% to 18%, depending on the speci f icat ion. Industry and place of res idence, the var iables taken here to represent market imperfections, d isequi l ibr ia , and nonpecuniary 2 factors , were h ighly s ignif icant, contr ibut ing almost as much to R upon addition as school ing, and somewhat more upon deletion from the full model. In fact , the deletion of all human-capital variables lowered 2 R by 0.042, whereas the deletion of all "unorthodox" variables lowered it by 0.105. Th i s result leaves open to question whether the emphasis accorded human capital in the exist ing l i terature has been wholly just i f ied . A number of var iables contr ibuted in only a minor way to the 2 value of R but were nevertheless of some interest . For example, self- employment proved to be a signif icant earning handicap on balance, as d id recent immigration. However, immigrants suf fered no lasting d i s - advantage . Marr ied heads of families turned out to receive 30-31% more than the reference g r o u p . Uni l ingual francophones earned 11-12% less. Among ethnic g roups , those of Jewish origin led the r a n k i n g . Native Indians fared worst, though not on account of wages that were low (ceteris p a r i b u s ) , but on account of meagre employment. F inal ly , a vers ion of the interactions model was estimated to d iscover whether indust ry or place of residence affected the earnings potency of schooling and exper ience . The schooling interactions were not s igni f icant; the experience interact ions, moderately so. Neither set 2 added impressively to the value of R . 279 C H A P T E R IV T h e preceding analysis abstracts completely from all p lanned variation in hours of work . At best, the models deal with the maxi- mization of lifetime earnings rather than with the maximization of ut i l i ty . However, since time is presumably both an argument of the uti l i ty funct ion and an input in the production of human capital , decisions concern ing its allocation among work, le isure, and investment are best treated simultaneously. Three control-theoret ic studies of simultaneous decis ion-making were su rveyed in order to compare their assumptions and to obtain predict ions with respect to several broad inqui r ies . It was found that these ut i l i ty-based analyses tended to undermine the assertion of the simpler human-capital models that investment declines monotonicaily over the l i fe -cyc le . Cases were uncovered in which investment might increase d u r i n g a g iven per iod . The concavity of the earnings prof i le was therefore seen as depending more heavily than in the earlier models upon the concavity of the hours prof i le . The latter was forecast to be unambiguously concave. It was deduced in one study that if the market rate of interest exceeded the rate of time preference , then there would be a peak in hours pr ior to successive peaks in earn ings and in wages. Unfor tunate ly , the studies su rveyed produced no equations which were amenable to d i rect estimation, and there was again scant d iscussion of hypotheses which might associate unobservable parameters with the observable attr ibutes of ind iv idua ls . 280 A c c o r d i n g l y , a simplified empirical model of wages and hours was postulated in an attempt to deal with the gross facts involv ing simultan- e i ty . T h e practical aim of this two-equation linear model was to obtain estimates of the earn ings funct ion which would be free of the bias s u s - pected on account of an endogenous hours var iab le . In the proposed speci f icat ion, earnings were (identical ly) the product of hours and the average wage. Owing to such things as moon- l ight ing , overt ime, and seasonality, the average wage was allowed to depend (stochastical ly) not only upon school ing, exper ience, and so fo r th , but also upon hours worked. The latter was made a (stochastic) function of certain exogenous variables and the marginal wage. T h e average and marginal wage rates d i f fered both through the dependence of the former on hours and through personal income taxes. Though it could be a rgued from a l i fe-cycle perspect ive that, since wage rates are p lanned, there is no need to include them in the hours equation a long- side the age var iable , the marginal wage was introduced separately in order to represent unforeseen inf luences, initial endowments, and var ious unmeasured qualit ies of the ind iv idua l . A l though the hours equation resembled the " labour-supp ly" funct ions f requent ly estimated in the l i terature, no attempt was made to press this interpretation in view of the strong assumptions requi red to guarantee the ident i f i - cation of a pure supply re lat ionship. 281 C H A P T E R V T h e two-equation model was estimated by the method of three- stage least squares . T h i s procedure allows not only for endogenous var iables on the r i gh t -hand side but also for the possibi l i ty that the e r ro r terms may be correlated across equat ions. Under stated assumptions, the resu l t ing estimates are consistent and asymptotical ly ef f ic ient , though they may d i f fer numerical ly from those of maximum l ikel ihood. Fur ther econometric di f f icult ies involved the endogeneity of the tax rate, the l imited-dependent-var iable status of the hours term, and the identif ication of the hours equat ion. Instrumental-variable estimates were used in an attempt to r id the pr inc ipal tax-re lated terms of their endogenous components. In view of the obstacles one meets in t r y i n g to approximate the progress ive tax s t ruc ture , this effort must un for - tunately be judged somewhat speculat ive . It was pointed out that, s t r ic t ly speak ing , the hours term const i tuted a d iscrete and limited dependent var iable , but that the problem was less severe than if weeks alone had been employed. A simple but important caveat was e n t e r e d — namely, that all estimates must be regarded as conditional on par t i c i - pants working posit ive h o u r s . It was f inal ly noted tht the identif ication of the hours equation might be somewhat tenuous, since it was obtained through the omission of only two var iab les , schooling squared and an age-school ing interact ion, which were probably of minor importance. Among the initial f i nd ings , the most s t r ik ing was the r ise in the estimated elasticity of earn ings with respect to h o u r s — f r o m 0.6-0.7 in the s ingle-equation regress ions to 1.4-1.8 in the simultaneous resu l t s . 282 The implied re turn to schooling fell to 6.3% in the orthodox earn ings funct ion , and the exper ience coeff icients were also diminished in s ize . T h e structura l earn ings profi le registered very sl ight concavi ty , thus cast ing some doubt on the human-capital interpretation of wage rates . Hours were found to peak before earn ings , as the choice-theoretic model suggested ; wage rates appeared not to peak in the relevant range, as Mincer d iscovered in the United States. The hours equation proved rather sensit ive to changes in model speci f icat ion, y ie lding in some cases a concave, and in others a convex, hours prof i le . Moreover, the implied elasticity of hours with respect to the wage rate was a good deal larger than one might have forecast on the basis of conventional labour-supp ly studies . Problems of estimation b ias , d i f ferences in functional form, and di f ferences in var iables and methods may explain this apparent d i sc repancy . Cont ra ry to United States exper ience, the coefficient of school- ing squared was again signif icantly posit ive: each additional year of schooling raised the estimated re turn by 0.5 percentage po ints . T h i s f ind ing is consistent with the suggest ion, occasionally voiced in Canada, that this country has a relative scarc i ty of workers at the h igher levels of educat ion. The expanded earn ings funct ion, estimated simultaneously with hours , prov ided considerable detail on the pattern of rewards prevai l ing across the Canadian work force . Two general observations were: that high wages tended to offset low hours , perhaps because of market equalization between jobs with high and low r i sks of unemployment or . high and low seasonality; and that earn ings peaks were hastened when geographic and inter industr ia l mobility was, in ef fect , disal lowed. T h e estimated mean rate of re turn to schooling was a mere 5.3%, and the exper ience profi le of earn ings was v i r tua l ly l inear rather than str ict ly concave. F I N A L R E M A R K S It has been argued in this study that the human-capital approach to earn ings determination lacks the hard testabil ity requ i red of a scientif ic theory and that it may serve , at best , as a framework for empirical descr ip t ion . A descr ipt ive prof i le , based loosely on the human- capital paradigm, was therefore drawn to portray the Canadian earn ings s t ruc tu re , which has not been analysed extensively in this fash ion . From a pure ly empirical standpoint, it d id not turn out that the orthodox human-capital variables were of overwhelming importance. Indust ry , place of residence, and other factors were also s igni f icant; and it cannot be assumed a pr ior i that all are simply means through which individual investment plans are real ized, as some have contended. Even if the preceding assert ion is cor rec t , one would have to concede on the basis of the present results that mobility with respect to the factors just named is an important c o n c e r n . If education and mobility are both essential for the realization of a g iven earn ings increment, pol icy init iat ives, if any are needed, cannot a f ford to sl ight either one. 284 Pr ivate , pecuniary rates of re turn to school ing, estimated rough ly , with hours constant, by the method of semi-log regress ion , fell in the 5-8% range . T h i s is well below Mincer's estimates for the United States, though it is worth repeating here that the years of observat ion d i f fered by a decade. If one were to apply Becker 's eff ic iency cr i ter ion and thus compare real market rates of re turn on human and nonhuman capi ta l , one's conclusion would have to be that there is some prima facie evidence of under investment in education on the part of ind iv iduals . Yet , it is di f f icult to say what r isk premium has been attached to educational investment, and it must be emphasized, lest the reader attempt to make policy inferences, that pr ivate and social re turns may d i f f e r . Though the present study has succeeded in adduc ing a number of interest ing facts with regard to the Canadian earnings s t ruc ture , problems remain which will not y ield to the data and methods employed here . The empirical regular it ies so far uncovered merely point the way of maximum interest for future theoriz ing and research . Ideally, one would wish to g r o u n d both the demand and the supply side of the labour market on an expl icit theory of optimal choice . T h e next step, as noted above, would be to speci fy var ious hypotheses making the theoretical parameters funct ions of the observable character ist ics d isp layed by in - d iv iduals and f i rms. Such hypotheses would be no less ad hoc than the ones tested here , but they would enter the analysis on a h igher theoretical plane and would thus be more readi ly interpretable using concepts familiar to economists. T o test such a model, one would need microdata not 285 only on individuals but also on the firms which employ them. By this means, it should be possible to d ist inguish supply and demand influences with much greater certa inty than has been establ ished here . It is to be hoped that data sets of the k ind mentioned become available in Canada before too long. R E F E R E N C E S Abbott , Michael, and Ashenfe l ter , O r l e y . 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