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Economic behaviour of self-employed farm producers Lopez, Eugenio 1981

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ECONOMIC BEHAVIOUR OF SELF-EMPLOYED FARM PRODUCERS by RAMON EUGENIO LOPEZ M.Sc, The University of B r i t i s h Columbia, 1977 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES Department of Economics We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October 1980 (c)Ramon Eugenio Lopez, 1980 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head of my department or by h i s o r her r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of ECONOMICS  The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date March 10th. 1981 ABSTRACT This d i s s e r t a t i o n proposes a model oriented towards integrating farm households' production and consumption decisions into a u n i f i e d t h e o r e t i c a l and econometric framework. It i s argued that some commod-i t i e s such as household's labour and, in some circumstances, outputs produced by the farm are traded within the household-farm unit. The implication of this i s that, i n contrast with other forms of economic organization, farm households' u t i l i t y and p r o f i t maximization decisions are not l i k e l y to be independent. Thus, the general objectives of the thesis are to develop a model appropriate to estimate farm households' supply and demand responses which e x p l i c i t l y considers the interdepen-dence of u t i l i t y and p r o f i t maximization decisions as well as to formally test the hypothesis of independence using Canadian farm census data. A model which considers two labour supply equations, i . e . , on-farm and off-farm labour supply, and f i v e net output supply equations i n c l u d -ing one aggregated output and four inputs (land and structures, hired labour, animal inputs, and farm c a p i t a l ) has been j o i n t l y estimated using Canadian farm data. The main hypotheses tested are independence of u t i l i t y and p r o f i t maximizing decisions and homotheticity of house-holds' preferences. This i n v e s t i g a t i o n suggests that u t i l i t y and p r o f i t maximizing decisions are not independent and, moreover, that there are s i g n i f i c a n t gains i n explanatory power and e f f i c i e n c y by estimating the consumption i i ( i . e . , the .labour supply equations) and the production equations j o i n t l y . Another f i n d i n g of the study i s that farm households' preferences are not homothetic. Estimates regarding the quantitative e f f e c t s of changes i n cost of l i v i n g index, output p r i c e , wage rates, and other farm input prices on households' on-farm labour supply, off-farm labour supply, and net output supply are provided. A d d i t i o n a l l y , the e f f e c t s of farm operators' educational l e v e l on t h e i r labour supply, output supply, and input demand decisions are also measured. i i i TABLE OF CONTENTS ABSTRACT i i LIST OF TABLES v i LIST OF FIGURES v i ACKNOWLEDGEMENTS v i i Chapter I INTRODUCTION 1 II THE MODEL . 10 2.1 Conditions for Independence of U t i l i t y and P r o f i t Maximization Decisions 18 2.2 Seasonality of the Self-Employment A c t i v i t i e s 25 2.3 Some Comparative S t a t i c Results 30 2.4 Review of the L i t e r a t u r e 34 2.5 Summary 40 III ESTIMATION OF HOUSEHOLD'S SUPPLY RESPONSES WITH FIXED FACTORS OF PRODUCTION 45 3.1 Properties of the Indirect U t i l i t y Function . . 49 3.2 Derivation of the Demand and Supply Equations 52 3.3 Further Implications of the U t i l i t y Maximization Hypothesis 54 3.4 A Stochastic S p e c i f i c a t i o n 59 3.5 Non-traded Outputs 63 IV THE ESTIMATING MODEL 68 4.1 Functional Forms for the Indirect U t i l i t y Function and for the Conditional P r o f i t Function 68 4.2 The Econometric Model 82 4.3 Testing for Independence of U t i l i t y and P r o f i t Maximization Decisions 88 i v 4.4 The Data 96 4.4.1 Off-farm Work and Off-farm Wages 97 4.4.2 On-farm Work, Returns to Farm Work and Non-Labour Income 98 4.4.3 Output and Input Prices 99 4.4.4 Other Variables Used and Taxes 100 4.4.5 Data Problems 101 4.4.6 Was 1970 a "Normal" Year? 102 V EMPIRICAL RESULTS 110 5.1 Hypothesis Testing 110 5.2 Supply and Demand Responses 114 5.3 Further Implications of the Results 129 VI SUMMARY AND CONCLUSIONS 141 REFERENCES . 149, APPENDIX 1: Proof of Propositions 154 APPENDIX 2: Data Transformations 160 APPENDIX 3: The Data Used 168 APPENDIX 4: Dispersion of the Estimated E l a s t i c i t i e s Across the Sample Points 173 v LIST OF TABLES T a b l e 1 C h i - s q u a r e S t a t i s t i c s f o r the V a r i o u s H y p o t h e s i s T e s t s 113 2 Parameter E s t i m a t e s of the Consumption and P r o d u c t i o n E q u a t i o n s ( E q u a t i o n s 55 and 60) 115 3 Labour S u p p l y E l a s t i c i t i e s 118 4 Compensated Demand E l a s t i c i t i e s 121 5 Labour Su p p l y E l a s t i c i t i e s w i t h Respect t o Net Output P r i c e s C a l c u l a t e d u s i n g E q u a t i o n (20) 122 6 C o n d i t i o n a l Net Output Supply E l a s t i c i t i e s 124 7 U n c o n d i t i o n a l Net Output Supply E l a s t i c i t i e s C a l c u -l a t e d u s i n g E q u a t i o n (22) 124 A . l Mean, S t a n d a r d D e v i a t i o n and Extreme V a l u e s of Some Impo r t a n t V a r i a b l e s C o n s i d e r e d 165 A.2 D i f f e r e n t E x p e n d i t u r e s as a P r o p o r t i o n of F a m i l y Labour Farm R e t u r n s 167 A.3 Cost Shares of the D i f f e r e n t F a c t o r s 167 A. 4 P r i c e Indexes of Net Outputs and A f t e r Tax Labour R e t u r n s , E x p e n d i t u r e s and Farm O p e r a t o r ' s S c h o o l i n g Y e a r s 169 A.5 Q u a n t i t i e s of Net Outputs by Census D i v i s i o n 171 A.6 Mean E l a s t i c i t i e s , S t a n d a r d D e v i a t i o n s , Minimum and Maximum V a l u e s 174 LIST OF FIGURES F i g u r e 1 20 2 29 3 30 4 73 v i ACKNOWLEDGEMENTS I would l i k e to thank Alan Woodland for h i s constructive c r i t i c i s m s during the various stages of elaboration of th i s research. His comments have l e d to very s u b s t a n t i a l improvements i n the form and content of th i s t h e s i s . I am also very g r a t e f u l to Erwin Diewert for providing important comments and encouragement at an early stage of t h i s t h e s i s . In addition I wish to thank Rick B a r i c h e l l o for his constructive comments and for providing part of the data used. Other people have also contributed to the improvement of th i s research with t h e i r comments and c r i t i c i s m s . Robert Evans, Jonathan Kesselman, James MacMillan, and William Schworm, among others, should be mentioned. J e f f Tyson provided research assistance i n data c o l l e c t i o n and computer work. F i n a l l y , I wish to acknowledge my largest debt to my wife, Ivonne. Her encouragement, patience, and support have been e s s e n t i a l i n the improvement and completion of th i s t h e s i s . v i i CHAPTER I INTRODUCTION The focus of attention i n th i s study i s the analysis of the econ-omic behaviour of self-employed farmers. S p e c i f i c a l l y , i t i s concerned with the supply and demand responses of households which also own and operate a firm ( i . e . , a farm). D i s t i n c t i v e features of these economic units are: (a) a s i g n i f i c a n t proportion or usually the t o t a l i t y of the labour input used by the household's firm i s supplied by i t s proprietors, i . e . , the household's members; (b) the returns from the family farm's operation may constitute an important proportion of the household's income a v a i l a b l e for consumption and other purposes; (c) i n many cases a substantial part of the family farm output i s oriented to s a t i s f y d i r e c t l y the household's own consumption n e c e s s i t i e s . This l a t t e r feature may be important f o r farm households i n developing countries. Thus, recognizing these rather close linkages between households viewed as consumers and producers one may conceptualize them as household-firm u n i t s . An important im p l i c a t i o n of feature (a) i s that family labour i s traded within i n d i v i d u a l households. Family labour i s "produced" by the household and used by the family firm. Hence, there exists a household's supply schedule and a firm's demand schedule f o r family labour, and the i n t e r n a l equilibrium shadow pri c e of family labour i s given by the 1 2 i n t e r s e c t i o n of these two schedules. The supply schedule of family labour i s , therefore, dependent on household's preferences, household's income, and on the vector of consumption good prices faced by the house-hold. Thus, a change i n any of these variables w i l l a f f e c t not only consumption demand responses (including leisure) but also p r o f i t maximizing decisions, i . e . , output supply and input demand responses. This i s so, because changes on the consumption side w i l l s h i f t the family labour supply schedule faced by the f i r m and hence the l e v e l of family labour used by the firm i s a l t e r e d . S h i f t s i n the supply schedule of family labour (which i s usually the most important factor of production i n t h i s type of firm) w i l l lead to changes i n output supply and input demand responses. Thus, the consumption side exerts an impact on the production side v i a the supply of family labour. S i m i l a r l y , the family labour demand curve i s dependent on the firm's production technology and output and input prices faced by the household's firm. Hence, changes i n these variables w i l l s h i f t the labour demand schedule faced by the household and, hence, the shadow pri c e of l e i s u r e w i l l be affected. This w i l l imply that a change on the production side w i l l exert an influence on u t i l i t y maximizing decisions not only through the e f f e c t of the firm's p r o f i t on family income (effect associated with feature [b]) but also by a l t e r i n g the relevant shadow pri c e of l e i s u r e . Hence, con-sumption decisions w i l l also be dependent on the production side. The implication of th i s i s , therefore, that u t i l i t y maximizing and p r o f i t maximizing decisions are not i n general independent as occurs i n the 1 conventional case. An i m p l i c a t i o n of feature ( c ) , on the other hand, i s that i f those outputs produced f o r s a t i s f y i n g the household's own consumption 3 ne c e s s i t i e s are not traded outside the household then the prices of these outputs are endogenous, to the household-firm un i t . The i n t e r n a l trade equilibrium takes place i n an i d e n t i c a l manner as i t occurs with family labour. Hence, non-traded outputs w i l l play the same r o l e as family labour i n e s t a b l i s h i n g a linkage between the consumption and the produc-t i o n sides. Thus, interdependence of u t i l i t y and p r o f i t maximizing decisions are i n t h i s case related to the endogeneity of the shadow pri c e of non-traded outputs. The previous discussion suggests that household-firm unit's u t i l i t y and p r o f i t maximizing decisions are, i n general, prone to be interdepen-dent. However, th i s interdependence may not e x i s t i f a l l goods produced 3 by the household's firm are at least p a r t i a l l y traded and i f the follow-4 ing assumptions are made: Assumption 1: Households u t i l i t y depends on t o t a l labour supply, not on the a l l o c a t i o n of that supply among d i f f e r e n t working a c t i v i t i e s . In p a r t i c u l a r , on-farm work and off-farm work as wage earners are perfect substitutes i n consumption. Assumption 2: Household's members working i n the family f i r m and hired labour used by the family firm are perfect substitutes i n production. It w i l l be shown, however, that although either of these assumptions i s necessary for independence of p r o f i t and u t i l i t y maximizing decisions, they are not s u f f i c i e n t . Assumption 1 implies that households can provide only one type of labour ser v i c e s , i . e . , that the s a l a r i e d working a c t i v i t i e s performed by households outside the family farm are the same as the self-employment a c t i v i t i e s . Indeed, i f t h i s were not the case, Assumption 1 could not be j u s t i f i e d . It has long been recognized that the d i s u t i l i t y assoc-iated with diverse working a c t i v i t i e s i s generally d i f f e r e n t (see, for example, Benewitz & Zucker; Diewert, 1971; F i e l d s & Hosek; and Rottenburg). The one type of labour services assumption i s quite r e s t r i c t i v e and appears to contradict even casual observation. For example, i n the case of the a g r i c u l t u r a l sector i n Canada (where produc-t i o n i s organized e s s e n t i a l l y as household-firm units) more than 75% of farm operators doing off-farm work i n 1976 were reported to be i n non-a g r i c u l t u r a l a c t i v i t i e s (Bollman). U t i l i t y differences associated with d i f f e r e n t working a c t i v i t i e s may be even more remarkable when one of the a c t i v i t i e s i s a self-employ-ment one. To the differences a t t r i b u t e d to diverse jobs now one may add the greater time f l e x i b i l i t y , the "pride" of working for your own business and not being dependent on boss decisions, etc., implied by self-employ-ment a c t i v i t i e s i n contrast with wage earnings occupations. Thus, i t appears that the labour choice problem has an even greater importance i n the case of self-employed workers. Assumtion 2 i s also dubious i f one considers differences i n super-v i s i o n costs and differences i n educational le v e l s between firm operators and t h e i r f a m i l i e s and hired labour. A d d i t i o n a l l y , the absence of perfect s u b s t i t u t a b i l i t y between hired and non-hired labour has been em p i r i c a l l y established i n studies applied to a g r i c u l t u r e (see, for example, B a r i c h e l l o ) . The lack of independence between p r o f i t and u t i l i t y maximizing decisions has important implications from the point of view of estimating household's preferences and firm's production technologies. Consider, 5 for example, two households which face the same prices for consumption goods and l e i s u r e and that have equal incomes. If they select d i f f e r -ent commodity bundles then, using an approach based on independence of the production and consumption sides, one would conclude that the house-holds have d i f f e r e n t preferences. However, i f the hypothesis of inde-pendence does not hold then the differences on the commodity bundles selected may be due to differences i n the production technology of the household's firm (or differences i n prices faced by the firm) rather than to differences i n household's preferences or tastes. S i m i l a r l y , two firms facing same output and input prices which behave d i f f e r e n t l y i n the output and input markets would be considered to have d i f f e r e n t technol-ogies i f one uses the conventional dichotomized models based on the independence hypothesis. However, t h e i r d i f f e r e n t production responses may be due to differences i n the household's tastes or wealth, for example, rather than to differences i n production technologies. Thus, the main problem of the conventional dichotomized models of the household and of the firm i s that they cannot discriminate between changes i n production technologies and changes i n household's preferences and may wrongly i d e n t i f y changes i n production technology as changes i n consumer's preferences and v i c e versa. From the previous discussion the following conclusions emerge: (a) in contrast with other forms of economic organization, u t i l i t y maximization and p r o f i t maximization are not l i k e l y to be independent; (b) i f the problem of no independence prevailed then i t w i l l have impor-tant implications i n modelling farm household's supply and demand responses. Consequently, general objectives of t h i s thesis are to provide an empirical framework for measuring farm household's supply and demand responses which e x p l i c i t l y considers the interdependence of u t i l -i t y and p r o f i t maximizing decisions and to formally test the hypothesis of independence using Canadian farm data. The s p e c i f i c objectives are the following: 1. To discuss a general model of the economic behaviour of the household-firm u n i t . It i s intended to use t h i s model i n analyzing i t s supply and demand responses as consumers and as producers. The emphasis i s placed on studying the i n t e r r e l a t i o n s among consumption goods' demand, labour supply, output supply, and input demand of the household-firm u n i t . 2. To e m p i r i c a l l y estimate supply and demand responses of Canadian farm households. S p e c i f i c a l l y , i t i s intended to estimate the equations of consumption behaviour, labour choice, output supply and input demand for Canadian farm households e x p l i c i t l y derived from the t h e o r e t i c a l model. The use of a household-firm model i n Canadian a g r i c u l t u r e i s j u s t i f i e d considering that i n th i s sector the basic unit of production i s the family farm.'' Thus, a model which e x p l i c i t l y considers the interdependence of consumption and produc-t i o n a c t i v i t i e s i s used to answer some important a g r i c u l t u r a l p o l i c y related questions such as: (a) E f f e c t s of a g r i c u l t u r a l commodity pr i c e changes on output supply, farmer's labour supply responses and farmer's labour choice between farm and off-farm work. (b) E f f e c t s of input p r i c e changes on output supply, input demand, farm families labour supply, and labour choice. (e) E f f e c t s of changes i n non-agricultural wages on farm household's supply and demand responses. In p a r t i c u l a r , i t i s intended to 7 quantify the e f f e c t s of such changes on the supply of a g r i c u l -t u r a l commodities as well as on the derived demand f o r other inputs. (d) E f f e c t s of changes i n farm household's wealth on farmer's labour supply. Changes i n farmer's labour supply associated with changes i n wealth may have an e f f e c t i n input demand and supply of a g r i c u l t u r a l commodities which has not been estimated so f a r . 3. The model used i n analyzing Canadian farm households i s based on the assumptions that there are no fixed factors of production and that a l l outputs produced by the farm are at least p a r t i a l l y traded out-side the household. Moreover, some r e s t r i c t i o n s on the production technology were also imposed. Hence, a t h i r d objective of the thesis i s to provide a framework appropriate to econometrically e s t i -mate farm household's supply responses with fixed factors of produc-t i o n and non-traded outputs, based on general assumptions regarding the production technology. This model i s expected to have a p p l i c a -tions p a r t i c u l a r l y for studying supply responses of farm households i n under-developed countries. The remainder of t h i s thesis i s organized as follows. In Chapter II a general s t a t i c model for the household-firm i s introduced. Some s i m p l i f i c a t i o n s of the model using assumptions regarding production technologies and existence of r e n t a l markets for durable factors of production are then considered. These s i m p l i f i c a t i o n s have the advantage of s u b s t a n t i a l l y reducing the d i f f i c u l t i e s involved i n the empirical estimation of the model and also f a c i l i t a t e the comparative s t a t i c anal-y s i s . This chapter also includes an analysis regarding the precise con-d i t i o n s under which u t i l i t y maximization and p r o f i t maximization decisions 8 are independent. Chapter I I I considers t h e o r e t i c a l and empirical implications of the general model when no r e s t r i c t i o n s on the production technology are imposed and/or when the assumption of no fixed factors i s relaxed. The emphasis i s placed on analyzing the existence of d u a l i t y r e l a t i o n s between d i r e c t and i n d i r e c t u t i l i t y functions^ and on the d e r i v a t i o n of the household's supply and demand schedules using d u a l i t y , given endogen-e i t y of the family labour's p r i c e and/or of outputs produced by the family firm. Furthermore, the implications derived from the u t i l i t y maximiza-t i o n hypothesis are also discussed. Some comparative s t a t i c expressions are derived using those r e s t r i c t i o n s . In;Chapter IV the issues of the empirical implementation of the r s i m p l i f i e d version of the model developed i n Chapter II are discussed. Functional forms for the functions representing the consumption and production sides of the model (the i n d i r e c t u t i l i t y function and the p r o f i t function, respectively) are s p e c i f i e d and used to derive the estimating equations. The next section considers a stochastic framework for the estimating equations and discuss the econometric method used and some econometric problems. In the l a s t section a discussion of the data requirement v i s - a - v i s the data a v a i l a b l e i s presented. The data a v a i l a b l e and the transformations performed on the data, s t r e s s i n g the data l i m i t a t i o n s and hence the necessity of i n t e r p r e t i n g the- r e s u l t s cautiously, are discussed. Chapter V reports the major empirical findings of the thesis and i n Chapter VI a summary and conclusions are provided. 9 Footnotes ''"The problem of interdependence of u t i l i t y maximization and p r o f i t maximization decisions i s analogous to the household production function analyzed by Lancaster, Becker and Pollack, and Watcher. In t h i s con-text, u t i l i t y i s a function of commodities which are produced by the household using goods and time. Hence, u t i l i t y maximizing decisions are dependent not only on goods' prices and household's preferences but also on the household's technology or production function. 2 The importance of considering that some outputs and family labour are "traded" within households i n modelling farm households supply responses i n developing countries has been recently emphasized by Nerlove: In most developing economies the a g r i c u l t u r a l sector i s so large and so c e n t r a l to the whole process of economic growth and demo-graphic change, that supply response cannot be treated as an i s o -lated phenomenon. Moreover, i n these economies, markets, at least as we know them in.developed economies, may be poorly organ-ized or may not e x i s t at a l l ; i t follows that the relevant 'prices' motivating producer behaviour may be d i f f i c u l t or impos-s i b l e to observe d i r e c t l y . Many of the trade-offs i n the a l l o c a -t i o n of resources may take place within i n d i v i d u a l households or between those households and r e l a t i v e l y i s o l a t e d labor or product markets. (p. 3) 3 . . If a l l outputs are at least p a r t i a l l y traded outside the house-hold-firm unit then the shadow prices w i l l always be i d e n t i c a l to the external market prices and hence the relevant prices are the market p r i c e s . 4 Barnum and Squire, Bollman, and Lau, L i n and Yotopolous have i m p l i c i t l y used e i t h e r of these assumptions. ^According to A g r i c u l t u r e Canada, 53% of the t o t a l labour force i n a g r i c u l t u r e were self-employed operators, 19.5% were family workers, and only 27.5% were hi r e d workers. "Canadian farming i s characterized by a large number of small family u n i t s " (p. 69). ^The i n d i r e c t u t i l i t y function i s defined as the maximum u t i l i t y attainable given a budget constraint (Roy). CHAPTER II THE MODEL In t h i s chapter a general model of the farm household i s presented and, using some assumptions regarding the production technology and existence of renta l markets f o r factors of production, a s i m p l i f i e d , e m p i r i c a l l y a p plicable, version of i t i s derived. Some comparative s t a t i c r e s u l t s with emphasis on the interactions between the production and consumption sides of the model are also presented. The basic idea underlying the model i s that households maximize t h e i r u t i l i t y which i s a function of the goods consumed, the time spent on household a c t i v i t i e s , the number of hours of on-farm work and the number of hours of off-farm work supplied by the household members (which are re l a t e d to the l e i s u r e time a v a i l a b l e ) . I t i s assumed that households maximize such a u t i l i t y function subject to a budget con-s t r a i n t and subject to a time constraint. The budget constraint i n d i -cates that the t o t a l expenditures on consumer goods cannot be greater than the t o t a l income obtained by the household. This income consists of three parts: (a) the net income obtained from the family farm's operation, represented by a dual p r o f i t function conditional on the number of hours of work which the household's members supply to t h e i r own farm. Thus, the p r o f i t function represents the production side of the model whose basic linkage with the consumption side (represented by 10 11 the u t i l i t y function) i s the household's labour supplied to the family farm. The net income obtained from the family farm's operation natur-a l l y w i l l depend on the production technology, output and input prices faced by the family farm, and on the amount of work which the house-hold's members provide to t h e i r farm. (b) A second major component of the household's budget i s the income which household's members earn while working for other firms as wage earners. It i s assumed that the wage rate obtained while working outside the family farm i s exogenous., (c) A t h i r d source of income i s the non-labour income which includes the returns obtained from f i n a n c i a l and r e a l assets owned by the household. These returns are also assumed to be exogenous.'': More formally, the u t i l i t y maximization problem of the household firm i s : max f ( L l 5 L 2 ; ... X ; T x, ... T ) (1) L,X,T N s.t. ( i ) £ p X N i -n(q;l.i) + w 2L 2 + y n=l 2 M ( i i ) E L + I T = H i i k . m k=l m=l ( i i i ) X.. s 0, T ;> 0, L > 0 N m k where f = household's u t i l i t y function X = (X x, X^) i s the N dimensional vector of consumption goods T = ( T l 5 .... T w) i s the M- dimensional vector of time which house-M hold members spend, on the household a c t i v i t i e s excluding pro-ductive work i n the household's firm L x = number of hours of work supplied to the family firm by household's members L 2 = number of hours of work supplied to other firms by household's members p^ = r e n t a l p r i c e of commodity n consumed by household . members y = non-labour income q = price vector of the s net outputs the family firm can produce (using the convention of representing outputs as p o s i t i v e quantities and inputs as negative quantities) ^(qjLj^) = a family firm's conditional p r o f i t function as a function of q and L x H = t o t a l number of hours that household's members have av a i l a b l e for a l l a c t i v i t i e s w2 = wage rate received by household members when they work for other firms. It i s assumed that the u t i l i t y function f ( L 1 , L 2 ; X 1 , ... X^;T l 5 T ) s a t i s f i e s the following r e g u l a r i t y conditions (Diewert, 1974): A . l defined and continuous from above for X,T,L 1,L 2 S 0 A.2 quasi-concave i n i t s arguments A.3 non-decreasing i n X A.4 non-increasing i n L j and L 2 . Assumption A . l i s necessary for existence of a s o l u t i o n to problem (1). Condition A.2 i s a standard assumption which i s used i n e s t a b l i s h -ing d u a l i t y r e l a t i o n s h i p s with an i n d i r e c t u t i l i t y function associated with problem (1). Condition A.3 i s somewhat r e s t r i c t i v e , considering that i t rules out the case of " t h i c k " i n d i f f e r e n c e curves (see Debreu, 1959), but i t i s a necessary condition for the budget constraint to be binding. 13 The conditional p r o f i t function •n(q;~L1) i s defined as. follows: irCq^i) = max {q TQ : (Q;L X) e T} Q where q = [ q i » q 2 > ••• <lg] i - s a column vector of net output prices Q = [Qi» Q 2 > ••• Qg] i s a column vector of net outputs (outputs and inputs) and T i s the production p o s s i b i l i t i e s set, i . e . , the set of a l l out-put and input combinations which the firm can produce given the state of knowledge. It i s assumed that the set T s a t i s f i e s the following conditions (Debreu): B.l closed, bounded from above, non-empty subset of the S dimen-sio n a l space, B.2 i s a convex set (non-increasing marginal rates of transforma-tion) , and B. 3 i f Q 1 eT, Q 1 -> Q 1 1 then Q 1 1 eT (free disposal condition). Using Diewert's (1972) proof regarding the r e l a t i o n s between the conditions on T and on the variable p r o f i t function, i t can be e a s i l y v e r i f i e d that i f T s a t i s f i e s B then - n i q i L ^ w i l l s a t i s f y the following conditions: C. l non-negative C.2 l i n e a r l y homogeneous i n q C.3 convex and continuous i n q C.4 non-decreasing i n L x and C.5 concave and continuous i n L j for f i x e d q. 14 U t i l i t y maximization model (1) allows household members to have d i f f e r e n t preferences for the d i f f e r e n t types of labour services that 2 they can provide (to t h e i r own firm or to other f i r m s ) . Changes i n the parameters of the model w i l l induce changes i n the d i s t r i b u t i o n of labour services supplied to on-farm and off-farm a c t i v i t i e s . Diewert (1972) has discussed a problem s i m i l a r to (1) and has shown that i t can be decomposed into two maximization problems: I M L L L ^ X ! , . . . ^ ) = max {f ( L j . , L 2 ; X X , . . .X^;T1,. . . y : M 2 • E T = H - E L. } (2) , m T k m=l k=l and a second stage u t i l i t y maximization which i s max U * ( L 1 } L 2 , X ) (3) L 1 } L 2 » X s.t. ( i ) pX g -Hq;!^) + w 2 L 2 + y ( i i ) L1 + L 2 S H ( i i i ) X g: 0, L L 5 L 2 > 0 Diewert (1972) has shown that i f f ( L - , , L 2 ;X, ,. . .X ;T, ,.. . ,T„,) N M s a t i s f i e s conditions A then U*(L 1 }L 2,X) w i l l also s a t i s f y A. Thus a sol u t i o n for problem (3) w i l l e x i s t (provided the constraint sets are compact i n X,'L1 and L 2 ) and, moreover, constraint (i) w i l l be binding at the s o l u t i o n point. Now, model (3) can be represented i n a more convenient form by a further transformation i n the variables of U*: U(H - L 1,H - L 2;X 1,...X N) E U * ( L X , L 2 ; X L 9 . . . X ^ ) (4) The advantage of U(H-L]_ ,H - L2 ;X]_, . .vX^) i s that i t i s defined over the non-negative orthant and that the corresponding budget constraint may be defined using non-negative prices and p o s i t i v e income. I t i s easy to v e r i f y that i f U* s a t i s f i e s conditions A then U w i l l s a t i s f y A . l , A.2, and A.3. Condition A.4 1 for U w i l l read "nondecreasing i n H-Li,H-L2.',' Using (4) i t i s now possible to reformulate model (1): max U(H-Li,H-L 2;X) (5) H-Li,H-L 2,X s.t. (i) pX + w 2(H-L 2) S Hw2 + y + Tr{q;H-(H-Li) } ( i i ) (H-Li) S 0, (H-L 2) H , X ^ 0 ( i i i ) (H-Li) + (H-L 2) ^ H (iv) (H-Li) S H, (H-L 2) ^ H. From now on i t i s assumed that constraint ( i i i ) i s not binding. Constraint ( i i i ) implies that the t o t a l labour the household members desire to supply cannot be greater than the t o t a l time a v a i l a b l e . Thus, th i s assumption implies that at a l l wage rates and commodity prices i n d i v i d u a l s w i l l want to consume some l e i s u r e . Model (5) can be s i g n i f i c a n t l y s i m p l i f i e d i f the production technol ogy exhibits constant returns to scale. In representing the production technology by iT(q;Li) i t has been i m p l i c i t l y assumed that there are no 3 fixed factors of production. Therefore, i f the production technology exhibits constant returns to scale and i f there are no fixed factors of production then the p r o f i t function i s homogeneous of degree one i n L i and can be decomposed as follows (see proof i n Appendix 1): ir(q;H-(H-Li)) = [H-(H-Li)] • ft(q) =Li» ff(q) (6) where f f(q)"is non-negative, convex j,. , continuous, and l i n e a r homogeneous i n q. Existence of fixed or quasi-fixed factors i n the short run i s assoc iated with any of the following two causes: (1) adjustment costs are important; (2) i n d i v i s i b i l i t i e s of durable factors or imperfections i n 16 the c a p i t a l markets (credit markets) i f well developed second-hand rent a l markets for durable factors do not e x i s t . If there are i n d i v i s i b i l i t i e s the firm may not be able to purchase a new unit of a factor even i f a f r a c t i o n of i t would be optimal to incorporate i n the production process. S i m i l a r l y , i f the f i r m does not have access to appropriate c r e d i t sources, the firm w i l l have to adjust upwards a f a c t o r using i t s own funds. Thus, cash constraints may imply that a firm can only p a r t i a l l y adjust i t s durable factor stocks. How-ever, i f second-hand r e n t a l markets for durable factor services work appropriately, then adjustments of durable factors become e s s e n t i a l l y a flow problem s i m i l a r to the adjustment of any non-durable, d i v i s i b l e input. Hence, the firm can rent or l e t the flow services of the f a c t o r s , thus overcoming i n d i v i s i b i l i t y or cash constraint problems. Hence, i n order to j u s t i f y the use of (6) i t i s necessary to assume that adjustment costs are n e g l i g i b l e and the existence of perfect r e n t a l 4 markets for durable factors of production. An a l t e r n a t i v e way of j u s t i f y i n g (6) i s simply to postulate long-run equilibrium where a l l c a p i t a l stocks have been adjusted to optimal l e v e l s . ^ Thus, i f a l l factors are v a r i a b l e then, using (6), the u t i l i t y maximization problem (5) may now be written as: max U(H-L l 5H-L 2,X) (7)i s.t. (i) pX + it(q) (H-Li) + w 2(H-L 2) < H(ff + w2) + y = Z ( i i ) (H-Li.) ^ 0; (H-L 2) S O ; X 5 0 ( i i i ) (H-Li) 5 H; (H-L 2) S H. The advantage of using (7) rather than (5), i s that (7) i s a standard maximization problem with a l i n e a r constraint provided that ff(q) is known and that constraint ( i i i ) is not binding." Thus, standard dual-ity theory (see, for example, Diewert, 1 9 7 4 ) can now be applied in order to derive equations for household's commodity demand, labour supply to the household's farm, and off-farm labour supply. This is so because the wage rate received by household members working on the family farm becomes independent of the household's preferences, depending only on the output and input prices which the family firm faces as well as on its production technology. Therefore, an indirect utility function G(p,ff,w2;Z) can be defined in the standard manner: G(p,ff(q) ,w2,Z) = max { U C H - L T.,H - L 2 ,X) : H-Lx,H-L2,X (i) pX + TfCqXH-Li) + w 2(H-L 2) i Z (ii) H-L X 2 0 , H -L 2 > 0 , X 2 0 } ( 8 ) where G will be continuous, quasi-convex in p, ft and w2, nonincreasing in p, nondecreasing in Z and homogeneous of degree zero in p, ff, w2 and Z (see, for example, Varian, pp. 8 9 - 9 0 ) . From ( 8 ) it is possible to derive the Marshallian demand functions for H-L,!, H -L 2 and X using Roy's identity: Ci) H-Lx (ii) H -L 2 ( i i i ) X Furthermore, the set of conditional net supply functions can be derived from the conditional profit function using Hotelling's lemma (Hotelling). Q-Cqila) = L x • ^ 1 i=l,...S ( 1 0 ) 3G/3fT(q) 3G/3Z 3G/3w 2 3G/3Z 3G/3p 3G/3Z (j) (p ,TT , W 2 , Z ) ty ( P . TT > W 2 , Z ) e(p,ff ,w2 , Z ) ( 9 ) 18 where i s the conditional net supply of commodity i . When commodity i Ls an input then z^ = -Q^ i s defined as input i . Note that due to the constant return to scale assumption L x plays only a scale r o l e i n the determination of the net supply functions. The unconditional net supply functions are obtained by s u b s t i t u t i n g 9(i) into (10): Q (q;p,w2,Z) = [H-fiCp ,TT (q) ,w2 ,Z) ] i=l,...,S (11) l dq^ Equations (9) and (11) represent the set of supply and demand responses obtained from a model which considers consumption and produc-t i o n a c t i v i t i e s of the farm household within an integrated framework. U t i l i t y maximization decisions (represented by equation set (9)) and p r o f i t maximization decisions (represented by equation set (11)) are interdependent. Changes i n the consumption side are transmitted to the net output supply functions v i a the function 4'.(p,Tf (q) ,w2 ,Z) i n (11). For example, changes i n consumer preferences w i l l have an impact on L x which i n turn w i l l a f f e c t optimal net output supply responses (because the l e v e l of /<|^ (0 i s changed). S i m i l a r l y , changes i n the production side w i l l a f f e c t u t i l i t y maximization decisions not only v i a Z but also by changing the shadow pri c e of L i , i . e . , by changing ff(q) i n (9). Thus, i f output prices increase, for example, then there w i l l be changes i n net output supply responses and also the household w i l l reconsider i t s con-sumption and labour supply a l l o c a t i o n s because the increased output prices w i l l imply a higher l e v e l fair the shadow pri c e of on-farm work (ff(q)). 2.1 Conditions for Independence of U t i l i t y and P r o f i t  Maximization Decisions In t h i s section the conditions for independence of u t i l i t y and p r o f i t maximization decisions are analyzed using model (5). In other 19 words, the aim i s to determine under what conditions the linkages between equation system (9) and the net supply equation (11), discussed above, are disrupted. This amounts to asking under what conditions the shadow price of on-farm labour w i l l be unrelated to the (conditional) p r o f i t function and the unconditional net output supply functions can be defined indepen-dently of the TeveLof the' household's labour supply. The propositions presented i n t h i s section are concerned with the conditions f o r independence of p r o f i t and u t i l i t y maximizing decisions under the maintained assumption that a l l outputs produced by the family firm are at least p a r t i a l l y traded. Thus, some propositions concerning the conditions required for the independence hypothesis are presented (see proofs i n Appendix 1). The propositions make reference to Assumptions 1 and 2 which are defined i n Chapter I. Assumption 1 r e f e r s to the case where households derive the same u t i l i t y (or d i s u t i l i t y ) by working for t h e i r own firms or elsewhere as wage earners, even i f the outside a c t i v i t y i s e n t i r e l y d i f f e r e n t from t h e i r own firm's work. Assumption 2 indicates that the family labour and the hi r e d labour used by the family firm are perfect substitutes i n production. In proving the following propositions,model (5), modified to consider the s p e c i f i c assumptions underlying the d i f f e r e n t assumptions associated with Propositions (1) to (3), i s used. Proposition 1 If Assumption 1 i s true then p r o f i t maximization decisions are independent of u t i l i t y maximizing decisions i f the household members also work as wage earners f o r other f i r m s . 7 In th i s case the imputed p r i c e to household members' work i n the family firm i s the wage rate (assumed parametric to households) received by the household's members working as wage earners outside the 20 family firm. Hence the p r o f i t function w i l l be a function of th i s wage rate as well as of the output and other input p r i c e s . S i m i l a r l y , the i n d i r e c t u t i l i t y function w i l l depend on the same wage rate which i s the unique pr i c e of l e i s u r e . Notice that i f family members do not do outside work, then i n general the supply shadow p r i c e of family labour w i l l be dependent on preferences and prices of consumption goods and hence p r o f i t maximizing and u t i l i t y maximizing decisions are not independent. The s i t u a t i o n described by Proposition 1 can be represented by using Figure 1: Family Labour Figure 1. In Figure 1, the curve S represents the family labour supply schedule, that i s 9U(H-Li-L 2,X) S = 8(H-Li-L 2) X which i s the marginal valuation of family labour where X and the Lagrangean m u l t i p l i e r , X are evaluated at t h e i r u t i l i t y maximizing l e v e l s for any given H-Li~L 2 value. The family labour supply schedule i s unique due to Assumption 1 ( i f t h i s assumption does not hold then there are two d i f f e r e n t supply schedules: on-farm and off-farm labour supply 21 schedules). Curve D i s the household's firm demand schedule for family aTrCq;]^) labour ( i . e . , the shadow pri c e function — — ) which i s downward sloping under non-constant returns to scale. Line w2 i s the off-farm wage rate which i s assumed exogenously. given. Under these conditions the equilibrium l e v e l of ( t o t a l ) labour supply w i l l be OA of which OB w i l l be on-farm work (L x) and BA w i l l be off-farm work ( L 2 ) . The important thing to notice i s that any change on the consumption side (cost of living•changes or changes i n household's wealth, f o r example) w i l l have no e f f e c t on on-farm work provided that household's members s t i l l work off-farm a f t e r such a change. Thus, changes i n the consump-t i o n side w i l l s h i f t the labour supply schedule, f o r example, from S to S 1 i n Figure 1. This w i l l imply that t o t a l labour supply i s reduced from OA to OA1. However, the l e v e l of on-farm work w i l l remain at the same l e v e l (OB) a f t e r the s h i f t i n S has taken place. Hence, the production sector i s not affected ( i . e . , net output optimal supply l e v e l s remain the same) by changes on the consumption side. S i m i l a r l y , a change on the production side (say, changes i n q or the production technology) w i l l s h i f t the D'. schedule but,, i f off-farm work s t i l l occurs a f t e r such s h i f t , t o t a l labour supply w i l l not be affected. A s h i f t i n D w i l l only imply changes i n the d i s t r i b u t i o n of work between on-farm and off-farm a c t i v i t i e s , but i t w i l l not a f f e c t t o t a l labour supply. Given that from the viewpoint of the household as a consumer, the r e l e -vant d e c i s i o n v a r i a b l e i s t o t a l labour supply rather than i t s d i s t r i b u -t i o n between on- and off-farm a c t i v i t i e s (due to assumption (1)), the s h i f t i n D w i l l have no e f f e c t on the u t i l i t y maximizing decisions. Thus, i n contrast with the model represented by equation (9) and 22 (11), under the conditions of proposition (1) the linkages between the production and consumption sides of the model are disrupted with the only g exception of the e f f e c t of t o t a l p r o f i t on household's income. It i s important to notice that i n 1976 only 34% of a l l farmers i n Canada and 28% of commercial farmers (those with farm sales above $2,500 per year) reported "some days" of off-farm work (Bollman, 1978). There-fore, even i f assumption 1 holds, the s u f f i c i e n t condition for proposition (1) i s s a t i s f i e d by only 34% of a l l farmers i n Canada. Proposition 2 If assumption 2 i s true then p r o f i t maximization decisions can be considered independent from the consumption parameters i f the house-hold firm uses hired labour. In th i s case the p r o f i t function w i l l be a function of the conventional parameters, i . e . , output and input p r i c e s , using the hired labour wage rate paid by the family firm as the p r i c e of a l l labour (family and hired) used i n the production process. S i m i l a r l y , demand for consumption goods and services (including l e i s u r e ) w i l l depend on consumption goods prices and on the hired labour wage rate as the unique price of l e i s u r e . It i s important to mention that according to Agriculture Canada, i n 1976 less than 35% of a l l farms used hired labour and less than 7% employed hired labour on a yearly b a s i s . Thus, even i f i t i s assumed that family labour and hired labour are p e r f e c t l y i d e n t i c a l inputs i t would not be s a t i s f a c t o r y to use the conventional model based on inde-pendent consumption-production decisions i n studying Canadian a g r i c u l -ture supply and demand responses. 23 Proposition 3 If assumptions 1 and 2 are both true, then p r o f i t maximization d e c i -sions can be considered independent of the consumption parameters i f e ither one of the following s i t u a t i o n s occur: (a) the house-hold's f i r m uses h i r e d labour; (b) the household members work as s a l a r i e d workers i n other firms. The following two c o r o l l a r i e s to proposition 3 may be useful as em p i r i c a l l y testable, predictions from a model based on the j o i n t hypoth-esis that assumptions 1 and 2 hold. Co r o l l a r y 3.-1 If assumptions 1 and 2 hold, and i f the wage rate that household's members can obtain outside the family f i r m i s greater than the wage rate the family firm pays to i t s hired labour, then house-hold's members w i l l not work i n t h e i r own firm. They w i l l a l l o -cate t h e i r t o t a l working time outside the family firm. Corollary 3.2 If assumptions 1 and 2 hold, and i f the wage rate that household's members can obtain outside the family firm i s lower than the wage rate the family firm pays to i t s hired labour then household's members may work outside the family firm only i f the family f i r m does not use hired labour. Thus, c o r o l l a r y 3.1 says that i f one observes that household's members work i n t h e i r family f i r m when the wage rate they obtain outside the family f i r m i s greater than the wage the family firm pays to i t s hired labour, then a model based on the above two assumptions can be rejected. S i m i l a r l y , when the conditions of c o r o l l a r y 3.2 are met and i f household's members work outside the family firm and such fi r m does 24 use hired labour then one can also r e j e c t a model based on assumptions 1 and 2. It i s i n t e r e s t i n g to note that the average off-farm a f t e r tax wage rate obtained by Canadian farmers i s s i g n i f i c a n t l y greater than the wage rate paid to hired labour used by farmers (see Appendix 2, Table A . l ) . Given that farmers do an important amount of on-farm work, both assump-tions (1) and (2) cannot simultaneously hold f or Canadian farmers accord-ing to c o r o l l a r y 3.1. Proposition 4 If assumptions 1 and 2 do not hold then u t i l i t y maximizing and p r o f i t maximizing decisions w i l l not be independent. This proposition i s indeed a c o r o l l a r y to propositions 1 to 3 and can be r e a d i l y proved using the f i r s t order conditions associated with a u t i l i t y maximization problem s i m i l a r to (5). The above propositions show that even the two standard assumptions together, i . e . , a unique pr i c e of l e i s u r e (assumption 1) and perfect s u b s t i t u t a b i l i t y between family labour and hired labour (assumption 2) are not i n general s u f f i c i e n t to guarantee independence of the p r o f i t function from consumption parameters. It i s important to notice that the s u f f i c i e n t conditions f or inde-pendence of p r o f i t maximization and u t i l i t y maximization decisions do not depend only on tastes and technology but also on the relevant p r i c e structure faced by the household firm u n i t . For example, whether a firm uses hired labour or not w i l l depend not only on production technology, tastes, etc., but also on the r e l a t i v e p r i c e of hired labour with respect to output, input, or other p r i c e s . An econometric model aiming to provide quantitative predictions should be v a l i d f o r a r e l a t i v e l y wide 25 range of r e l a t i v e p r i c e s . An econometric model based on the assumptions which allow propositions 1, 2, or 3 to be v a l i d may break down i f there i s a s u f f i c i e n t p r i c e change to v i o l a t e those assumptions. For instance, a labour supply model using the hired labour wage rate as the imputed p r i c e of l e i s u r e w i l l not be v a l i d i f the r e l a t i v e p r i c e of hired labour reaches a l e v e l where the family firm w i l l not use hired labour. At t h i s p r i c e l e v e l quantitative predictions w i l l be i n v a l i d because the p r i c e of l e i s u r e w i l l not be the hired labour wage rate. Thus, the shortcomings of econometric models using the assumptions underlying propositions 1, 2, or 3 are serious, because those assumptions do not depend only on r e l a t i v e l y stable c h a r a c t e r i s t i c s of the household-family firm u n i t s , such as technology, tastes, etc. Therefore, i t i s not even appropriate to estimate such models f o r households which at given prices s a t i s f y those assumptions i f the purpose i s to use them for quantitative p r e d i c t i o n s . Thus, the importance of developing a more general model which does not r e l y on propositions 1, 2, or 3 i s evident not only f or those house-holds which do not meet the s u f f i c i e n t conditions f or p r o f i t maximization-u t i l i t y maximization independence but also f o r those which do meet them at p a r t i c u l a r p r i c e l e v e l s . 2.2 Seasonality of the Self-Employment A c t i v i t i e s Many self-employment a c t i v i t i e s are characterized by rather strong seasonal patterns. This i s p a r t i c u l a r l y relevant i n a g r i c u l t u r a l produc-t i o n (mainly f o r crop farms). Thus, farm households' a c t i v i t i e s i n the slack season (say from November to A p r i l ) are very d i f f e r e n t from the a c t i v i t i e s performed during the busy season (say May to October). Normally, i n the slack season most household's time i s spent on l i v e s t o c k 26 related a c t i v i t i e s , farm improvements such as construction work, machinery r e p a i r s , etc. On the other hand, the busy season involves much more he c t i c and d i v e r s i f i e d works including crop planting and harvesting. This seasonal pattern presents a rather serious problem because the marginal p r o d u c t i v i t y schedules of labour i n the two seasons may be expected to be very d i f f e r e n t . In f a c t , empirical studies which have considered labour used i n the two seasons as d i f f e r e n t productive inputs, have shown that the two schedules d i f f e r to a large extent and that the value of the marginal product of labour (at a given l e v e l of work) i s s u b s t a n t i a l l y lower during the slack season (Nath). Given that data on farm labour are seldom a v a i l a b l e at a disaggregated seasonal l e v e l , the seasonality of self-employment a c t i v i t i e s may be a serious problem i n empirical a n a l y s i s . An important question i s : under what conditions i s i t consistent to aggregate these two labour categories by simply adding-up hours of work performed during the slack and busy seasons? In other words, under what conditions the p r o f i t functions •n(q;L^, ,L 1 ) = 7r(q;L..) where L,, and L. are hours of household's on-farm lb Is 1 lb Is work i n the busy and slack seasons, r e s p e c t i v e l y , and L, = L„, + L, . 1 lb Is If the relevant p r i c e of household's labour i s an exogenous, wage rate then the Hicks:' composite commodity theorem applies, and permits one to aggregate a n ( l ^is> anc^ ^he f a c t that they have the same pri c e means that the relevant composite commodity i s simply . However, i f the p r i c e of family labour faced by the firm i s not given exogenously from outside the house-hold f i r m (as i t occurs when the conditions for propo-s i t i o n s (1) and (2) do not hold) then a necessary condition (although not s u f f i c i e n t ) f or the composite commodity theorem to apply i s that house-holds be i n d i f f e r e n t of working i n the busy or slack seasons. To see t h i s , consider a u t i l i t y maximization problem where busy season house-hold's labour (L., ) and slack season labour (L, ) are considered d i f f e r -lb Is ent inputs i n production. Thus, assuming that households are i n d i f f e r e n t between working i n the two seasons, the u t i l i t y maximization problem would be: max UCH-L,, - L n ,X) (12) (i) pX < Tr(q;L l b,L l s) + y ( i i ) H - L, - L.. > 0, x > 0 Is lb ( i i i ) L l f l * 0, L l b i 0 (iv) L l b 5 H b , L l s , H s , H s + H b =H where IL^  and H g are the maximum number of hours that households have a v a i l ^ able for work and l e i s u r e during the busy and slack seasons, respectively and a l l other variables have previously been defined. From the f i r s t order conditions of the above problem the following r e l a t i o n may be obtained: a ^ i b ' V , _ ^ i b ' V , M . I ( 1 3 ) 3 L l b + M b 3 L l s + M s 3(H-L l b-L l s) X Q 3 ) where 1^, Mg and \ are the Lagrangean m u l t i p l i e r s associated with con-s t r a i n t s L^ b S R^, L^ g S H g and the budget constraint, r e s p e c t i v e l y . Now, i f =M g = 0, i . e . , i f there exists an i n t e r i o r s o l u t i o n for both L,, and L. then = and, moreover, one can consider the ib is 91*,, 9L.. lb Is the shadow prices of L „ and L, faced by the family firm as i d e n t i c a l , lb Is J J 9U 1 equal to , =r- • — . Thus, i n th i s case Hicks' aggregation con-3 (.n-L -L ) A lb Is d i t i o n i s s a t i s f i e d and therefore hours of work used i n the slack and busy seasons can be co n s i s t e n t l y aggregated. Hence, the necessary and 28 s u f f i c i e n t conditions for the composite commodity theorem to apply are that households be i n d i f f e r e n t of working i n the busy or slack seasons and that an i n t e r i o r s o l u t i o n for L , and L_ e x i s t s . Furthermore, lb Is given that the equilibrium shadow prices of a n d are i d e n t i c a l , a consistent aggregate would be 5 + L- s^> i - e . , the simple addition of hours of work of the two seasons. Hence, i f households are i n d i f f e r -ent between working i n any of the two seasons and i f t h e i r u t i l i t y i s maximized by working s t r i c t l y less than the maximum possible hours of work i n each season, ( i . e . , i f l e i s u r e time i s not zero i n e i t h e r season) then there are no aggregation biases by using data which does not d i s t i n -guish between hours of work i n the slack and busy seasons. In t h i s case the co n d i t i o n a l p r o f i t function can be written as Tr(q;L x). Note that for s i m p l i c i t y of notation, off-farm work has not been considered. However, one can v e r i f y that zero off-farm work during the busy season i s not a necessary nor a s u f f i c i e n t condition for obtaining a corner s o l u t i o n i n L,. . Thus, the fact that farmers do not do off-farm work lb i n the busy season (as i s l i k e l y to be the case), does not imply that L., = H, . Indeed the condition for an i n t e r i o r s o l u t i o n for L_. and lb b lb L i s rather weak; i t only requires that farmers have some l e i s u r e time and/or work off-farm during the busy season and that they do some on-farm work i n the slack season. Therefore, an i n t e r i o r s o l u t i o n i n L,, and L„ i s consistent with the r e a l i s t i c s i t u a t i o n that farmers work lb Is only on-farm during the busy season and work off-farm and on-farm i n the slack season. Graphically, the argument may be expressed i n Figure 2 8U 1 where S = , r- • — i s the marginal valuation of family labour lb Is schedule along the u t i l i t y maximizing path as a function of + ^ j _ s * quantity of family labour/year Curve D i s the t o t a l demand for labour schedule, which indicates the quantity of L , and demanded at each labour p r i c e . The i n t e r s e c t i o n lb Is of curve D and the supply schedule S w i l l give the equilibrium shadow price of family labour. In the above f i g u r e , t h i s equilibrium shadow pri c e i s R and at t h i s p r i c e L* of slack season work and L* units of busy Is lb J season w i l l be performed. Thus, the equilibrium shadow price of L n i and lb L^ s i s i d e n t i c a l equal to R. The c r u c i a l element i s that under the assump-t i o n of indif f e r e n c e between working at d i f f e r e n t seasons of the year, there i s only one supply schedule of family labour and hence that house-holds w i l l f r e e l y switch hours of work between the two seasons u n t i l the value of the marginal labour i n the two seasons becomes i d e n t i c a l . This i s true under the assumption that e q u a l i z a t i o n of the shadow prices i s achieved before the maximum possible number of hours of work (H^ or H ) i s reached i n any of the seasons. The necessity of an i n t e r i o r s o l u t i o n for L _ and L as a condition lb Is for having an i d e n t i c a l equilibrium shadow pri c e for L.. and \L may be Is lb i l l u s t r a t e d using Figure 3. Figure 3 depicts a s i t u a t i o n where the constraint = i s Figure 3. binding. Households would desire to a l l o c a t e L 0,, hours i n the busy lb season but, given t h e i r time constraint, they can only work H^ hours. This means that while the equilibrium shadow pri c e of labour i n the slack season w i l l be R, the shadow pri c e i n the busy season w i l l be higher equal to R*5. In t h i s case, aggregation of L n and L,, would not be con-is ib s i s t e n t and hence a p r o f i t function Tr(q,Li) where L„ = L,, + L„ w i l l not 1 lb Is e x i s t . 2.3 Some Comparative S t a t i c Results Using model (7) the interdependence between the consumption and production sides of the model i s now examined by considering the e f f e c t s of changes i n net output prices on the household's a l l o c a t i o n of i t s working time between on-farm and off-farm a c t i v i t i e s . Furthermore, the e f f e c t s of changes i n consumption parameters (such as y or p) on supply of net outputs as well as the e f f e c t s of net output prices on the net outputs supplied by the family firm are also considered. As mentioned e a r l i e r equations (9) are the s o l u t i o n of the 31 conventional s t a t i c model of the behaviour of consumer workers (Hicks, 1946) and the standard comparative s t a t i c r e s u l t s can be derived i n the usual manner. Thus, using 9 ( i ) : 3(H-L 1) 3(j>(p,ff,w2;Z) 3(J> (p ,ff ,w2 ;Z) = + . H (14) 9ff 3TT 3Z Now, i t i s a well known r e s u l t that (Samuelson, 1938) : 34> Cp, if ,w2;Z) = (15) 9ff ~™ v 3Z ' which i s the Slutsky equation, e „ i s the Hicks 1' own s u b s t i t u t i o n term, if if M o, CT11J.^w ^ ^ ^ ^ ^ c ^ ^ j - y ^ „ i s the income e f f e c t . Hence (14) can be 3Z written as: 3(H-L X) = e. + L 3if TTTT 1 3Z ' which implies that the e f f e c t of ff on the supply of family labour to the household's fi r m ( L X ) i s : 3LT_ 3LJ_ = - e _ + L, . (17) 3TT TTTT 1 3Z 9^1 If l e i s u r e i s not an i n f e r i o r good then — — < 0 and hence, although 9Z 3 L X -e S 0, the sign of i s unknown, which i s a standard r e s u l t . ffff 377 The e f f e c t of a change of a net output p r i c e , q^, on the supply of family labour to the household's fi r m can be now examined using (10): 9^1 9^1 3ff 99 i 3if 9q i 3L r - e _ + L-L — V TTTT " 3 Z Q i IT • ( 1 8 ) i 3 1 ^ Thus, i f q. i s an output price then the sign of w i l l be 9 q i 3L^ 3l"i i d e n t i c a l to and, and w i l l have opposite signs i f q. i s an 3 if 3q^ 3TT i input p r i c e ( r e c a l l i n g that i n t h i s case = -z^) . Hence, i f q^ i s an 32 z. input p r i c e then the s u b s t i t u t i o n e f f e c t e^^^~ =0- Family labour w i l l always appear to perform as a net complement with other inputs used i n the family firm. This also holds for the r e l a t i o n between family labour and hired labour. Any increase i n an input p r i c e w i l l induce a f a l l i n the demand shadow pr i c e of family labour used by the family firm. This w i l l lead the household to decrease the a l l o c a t i o n of family labour to i t s own firm and to increase labour supplied elsewhere (ignoring the income e f f e c t on l e i s u r e ) . The e f f e c t of a change i n on the a l l o c a t i o n of family labour out-side the family f i r m can be analyzed i n a s i m i l a r manner: l l L 2 -e ~ + L2 .-K-=-W2TT 3Z Q i ^ . (19) L2 where e ~ i s the Hicks cross s u b s t i t u t i o n term. W TT 2 9 L z dff . . . . Again T ; — and -~— w i l l have i d e n t i c a l signs i f q. i s an output & 3q. 9q. l l l p r i c e and opposite signs i f q^ i s an input p r i c e . Relations (18) and (19) can be represented i n terms of e l a s t i c i t i e s : q-Q-EL.q. E L . f f - ^ - i , j = 1, 2 (20) i i 1 rt, J where £ i s the e l a s t i c i t y of supply of labour service j with respect L.q. J 1 to a change i n net output p r i c e q. and ~ i s s i m i l a r l y defined as the j e l a s t i c i t y of supply of labour service j with respect to fr. The e f f e c t of a change i n net output p r i c e q^ on net output G\ can be r e a d i l y derived using (10): 33 Notice that i f i = j , the e f f e c t of output p r i c e i on output i , (21) w i l l not be unambiguously p o s i t i v e as i t occurs i n the non-family fir m case. In t h i s case, convexity of ft i s s u f f i c i e n t to sign the f i r s t term on the right-hand-side of (21) as p o s i t i v e but the second term's sign depends on whether -r— i s p o s i t i v e or negative. Thus, the p o s s i -q j b i l i t y of downward sloping output supply or upward sloping input demand functions cannot be ruled out. The reason f o r t h i s i s that an increase i n an output p r i c e may induce a net reduction i n the supply of labour to the family fi r m i f the income e f f e c t on l e i s u r e (assuming l e i s u r e i s a normal good) predominates over the s u b s t i t u t i o n e f f e c t . Hence, although the supply of output i per unit of family labour w i l l always increase, i t i s possible that the amount of family labour used by the family firm be reduced and therefore a reduction i n t o t a l output i i s pos s i b l e . Equation (21) can also be expressed i n e l a s t i c i t y terms: q i Q i e. . = S. . + e - ,. —M- , (22) 1J l j LiTT TT where e.. i s the output i e l a s t i c i t y with respect to q., S.. i s the e l a s t i c i t y of Q . / L i with respect to q. and eT ,. i s the e l a s t i c i t y of L i with respect to ft. The e f f e c t of an increase i n household's wealth w i l l imply an expan-sion of the non-labour income a v a i l a b l e . Thus, using (7) one can obtain the e f f e c t of a change i n the non-labour income, y, on the net outputs 3Q 3 L i 3Lj . _ • 1 = 9Z_ _9JL = J*2L_ or>\ 3y 3Z 3y 3q^  3y 3q. " 3 L i Hence i f l e i s u r e i s not an i n f e r i o r good, then — — < 0 and i n 3Z 34 general one can say that J < 0. Thus, the e f f e c t of an increase i n dy household's wealth (and hence non-labour income) w i l l be a reduction i n productive a c t i v i t i e s . Obviously given that L i has a scale e f f e c t i n productive a c t i v i t i e s , i t w i l l a f f e c t a l l net outputs prop o r t i o n a l l y and the net output supply e l a s t i c i t i e s with respect to y(e. ) w i l l a l l be i d e n t i c a l and equal to the e l a s t i c i t y of labour supply e with respect L i y to y: e. = e. = e T . (24) j y i y L i y S i m i l a r l y , the e f f e c t of other consumption parameters, say p or W2 on net output supply w i l l be: e. = e. = e T . (25) JW2 1W2 L]W2 Notice that, given the assumptions used, the quantitative value of the e f f e c t s of changes i n the parameters of the consumption side of the model (p,w2,y) on production decisions can be expected to be as important as the cross e f f e c t s of p, W2 and y on L i . 2.4 Review of the L i t e r a t u r e In t h i s section a b r i e f review of the l i t e r a t u r e concerning house-holds a l l o c a t i o n of time and household's production function i s presented. Walras was one of the e a r l i e r writers to analyze the a l l o c a t i o n of time between work and non-work a c t i v i t i e s using a u t i l i t y maximization frame-work. He was the f i r s t to consider the case where consumers could supply many d i f f e r e n t types of labour services which have d i f f e r e n t e f f e c t s on consumer's u t i l i t y . More recently, Becker, Mincer, and Lancaster have extended Walras's model to a household model st r e s s i n g the production of commodities by the household (which are the elements a c t u a l l y entering the u t i l i t y function) 35 using goods and household time. A t y p i c a l household production a c t i v i t y i s f o r example cooking, where raw food (goods) and household time are com-bined to produce cooked food which are the commodities which enter into the u t i l i t y function. Although Becker, Mincer, and others have extended the o r i g i n a l Walras's model within a time and income a l l o c a t i o n framework they implic-i t l y assumed that i n d i v i d u a l s are q u a l i f i e d to o f f e r only one type of labour service. Diewert (1971) generalized Becker's model by developing a model f or in d i v i d u a l s capable of o f f e r i n g many d i f f e r e n t types of labour services simultaneously. Diewert also provided a framework for the empirical estimation of the consumer-worker preferences. He defined a u t i l i t y function dependent on goods and time spent i n d i f f e r e n t a c t i v i t i e s which combined the parameters of the o r i g i n a l u t i l i t y function (defined i n the commodity space) and the household production technology. Thus, i f the major i n t e r e s t i s on estimating the combined household technology-household preferences reduced form function then one can use Diewert's approach. The concern of t h i s thesis i s the self-employed consumer-worker (or household) which, i n general, i s an analogous problem to the household production function model analyzed by Becker and others. In the house-hold production function model households use goods and household's time to produce commodities (and leis u r e ) which are consumed by the household. The self-employed consumer-worker uses production inputs and part of his time to produce goods and income f or household's consumption. Thus, l e i s u r e and non-traded goods produced by the self-employed worker c o n s t i -tute the major'linkages between household's preferences and the produc-t i o n technology. Hence, i n the:same way as i n the model analyzed by 36 Becker the demand for goods and the household's time a l l o c a t i o n r e f l e c t not only consumer preferences but also the household technology, i n the case of the self-employed worker, his demand for consumer goods, hi s time a l l o c a t i o n , his demand for factors of production and output supply responses j o i n t l y r e f l e c t h i s consumer preferences and the technology of his firm. In the household production function model the linkage between production decisions (how much goods and time to use) and consumption decisions (how much commodities to consume) takes place because commodity (shadow) prices and l e i s u r e prices are endogenous and dependent on the household's production technology. Lancaster and Becker by ( i m p l i c i t l y ) assuming a d i s j o i n t household's production technology, constant returns to scale and that households were i n d i f f e r e n t among a l t e r n a t i v e a l l o c a -tions of t h e i r time, concluded that commodity shadow prices faced by the household as consumer were exogenous to the l e v e l of commodities demanded although dependent not only on goods' prices but also the household's production technology. Also the demand for goods are dependent (up to a scale l e v e l ) on the l e v e l of consumption of commodities. Thus, the importance of the Lancaster-Becker model was to emphasize that to under-stand market demand for goods i t i s necessary to consider not only goods' prices and the structure of household's preferences but also the tech-nology of the household's production function. Pollack and Watcher have shown that i f households are not i n d i f f e r e n t among a l t e r n a t i v e a l l o c a -tions of t h e i r time (or i f either of the assumptions regarding the pro-duction technology does not hold) then commodity shadow prices and l e i s u r e p rice(s) are endogenous. to the levels of commodity and l e i s u r e demanded and hence commodity shadow prices are not only dependent on goods' prices and the household production technology but also on consumption p r e f e r -ences. In t h i s case production decisions ( i . e . , goods' demand) are dependent on commodity demand i n a more complex manner than up to a scale l e v e l . In the case of the self-employed consumer-worker, the major linkage between consumption and production decision l i e s i n the endogeneity of the p r i c e of the household's labour used by the family firm. As i n the household production model, i f some assumptions regarding the production technology are made ( i . e . , no f i x e d inputs and CRS) then the shadow pri c e of l e i s u r e i s exogenous to the l e v e l of l e i s u r e demanded although the demand f o r l e i s u r e (and for consumption goods) are dependent not only on net output prices but also on the production technology of the household's firm. S i m i l a r l y , the supply of net outputs by the firm are dependent up to a scale l e v e l on l e i s u r e consumed (or labour supplied) by the house-hold. If the assumptions regarding the production technology are relaxed then the shadow p r i c e of l e i s u r e i s endogenous to the l e v e l of household's demand and i n t h i s case net output supply w i l l be dependent on household's l e i s u r e demand i n a more complex manner than simply to a scale l e v e l . An a d d i t i o n a l source of interdependence between the production and consump-t i o n sectors occurs i f some outputs produced by the family firm are not traded and are e n t i r e l y consumed by the household (the household does not buy or s e l l such output). In t h i s case the shadow prices of those out-puts are also endogenous dependent on the family firm technology and household's preferences. . . . . I i Empirical applications of the household production model are scarce and empirical work concerning estimation of supply and demand responses of self-employed consumer-producers have ignored the interdependence 38 between the production and consumption sides of the model. Most of the studies on self-employed consumer-producers are related to a g r i c u l t u r a l producers which i s a sector where th i s type of organization of production predominates. Lau, L i n , and Yotopoulos have estimated a system of expenditure functions (including l e i s u r e , a g r i c u l t u r a l commodities, and non-agricultural commodities) for farm households i n Taiwan, using a u t i l i t y maximization approach. Using a p r o f i t function estimated by Yotopoulos et a l . , f o r the same area, they estimated the household marketed surplus. The only source of i n t e r r e l a t i o n s between the consump-t i o n and production sides of the model was the e f f e c t of p r o f i t from the farm operation on the household's income. It was assumed that the p r i c e of l e i s u r e was unique and equal to the p r e v a i l i n g (exogenous) wage rate and that a l l outputs were at least p a r t i a l l y traded. In de f i n i n g a unique exogenous p r i c e of l e i s u r e i t was i m p l i c i t l y assumed that house-holds were i n d i f f e r e n t among a l t e r n a t i v e a l l o c a t i o n s of t h e i r working time and that a l l households do off-farm s a l a r i e d work.'''"'" A s i m i l a r approach was adopted by Barnum and Squire i n analyzing consumption and production responses of farm households i n Malaysia. Bollman analyzed off-farm supply of labour using Canadian farm data under s i m i l a r assump-tions regarding l e i s u r e p r i c e . In contrast with Lau et a l . , his method i s more ad hoc i n the sense that he does not derive the estimat-ing equation from an e x p l i c i t u t i l i t y maximization framework. The functional forms used i n the Lau et a l . and Barnum and Squire studies were also quite r e s t r i c t i v e . Lau et a l . assumed a homothetic to the o r i g i n preference function and a Cobb-Douglas production technol-ogy. Barnum and Squire also used a Cobb-Douglas production function but assumed a u t i l i t y function which i s homothetic to a point i n the 39 commodity space which i s independent of commodity prices (affine homo-t h e t i c preferences). In summary, previous empirical studies of the self-employed consumer-producer have been characterized by the following features: (1) they have f a i l e d to consider the labour choice problem ( p a r t i c u l a r l y choice between d i f f e r e n t working a c t i v i t i e s which may generate d i f f e r e n t d i s u t i l i t y ) which has implied that the i n t e r r e l a t i o n s between consumption and production a c t i v i t i e s have been r e s t r i c t e d to one d i r e c t i o n , from the production side to the consumption side, only v i a the income e f f e c t . (2) Despite the fact that a number of the studies have been concerned with farm households i n underdeveloped countries where the existence of non-marketed outputs can be expected to be important, they have not considered t h i s problem i n the s p e c i f i c a t i o n of the models. (3) The studies have ei t h e r estimated t o t a l labour supply or they have concen-trated on the estimation of the off-farm labour supply equation only. None of the studies have j o i n t l y estimated consumption demand, house-hold's labour supply to the household's firm, labour supply to other firms, and the input demand and output supply equations. (4) The functional forms used for specifying preferences and production technol-ogies have been very r e s t r i c t i v e . The analysis i n t h i s thesis i s directed to improve upon the above shortcomings by: (1) considering labour choice and the i n t e r r e l a t i o n s between the production and consumption sides of the model; (2), developing a model appropriate to t h e o r e t i c a l l y analyze and econometrically e s t i -mate household's supply and demand responses under conditions of f a i l u r e s of some output markets and general assumptions regarding the family firm's production technology (see Chapter III) ; (3) estimating the consumption and production branches j o i n t l y , emphasizing i n t e r r e l a t i o n s within and between the two branches; (4) using more general fu n c t i o n a l forms f o r the production technology and households' preferences. In p a r t i c u l a r , the f u n c t i o n a l forms used f o r spe c i f y i n g preferences allow to formally t e s t whether preferences are homothetic to the o r i g i n or whether they are homothetic to a point i n the commodity space which i s independent of p r i c e s . 2.5 Summary In t h i s chapter a general model of the farm household economic behaviour has been discussed. The model does not r e l y on the indepen-dence of the production and consumption sides of the model. Next by the use of some assumptions the model i s s i m p l i f i e d and transformed into a standard optimization problem which allows one to use we l l established d u a l i t y r e s u l t s f o r comparative s t a t i c and empirical estimation of the model. A d d i t i o n a l l y , the exact conditions under which the independence hypothesis would hold have been discussed. F i n a l l y , some comparative s t a t i c r e s u l t s were derived emphasizing the i n t e r r e l a t i o n s between the production and consumption sectors of the model. 41 Footnotes Real assets include the durable factors of production owned by the family firm. The returns associated with these assets correspond to t h e i r r e n t a l p r i c e s . These r e n t a l prices w i l l be exogenous even i n the short run provided there e x i s t perfect r e n t a l markets for a l l durable factors of production. I f such markets e x i s t then households w i l l rent or l e t a proportion of t h e i r assets u n t i l the returns obtained from those assets used by the family farm are equal to t h e i r r e n t a l p r i c e s . In Chapter III the case where such markets do not exist and hence where returns to r e a l assets are endogenous, dependent on the amount of family labour used by the family firm and on the l e v e l of such assets, i s also considered. 2 Notice that i n problem (1) the variables entering the u t i l i t y function have been aggregated across household's members. A more gen-e r a l formulation of preferences would be: f ( X L X , X 1 2 , . . . ,X^;X 21,. . . ,X^;L ! L 5 L 1 2 ; L 2 1 } L 2 2 ;L 3 1 , . . . > J\J2^ where X „ is the consumption of goods j by household's member i . S i m i l a r l y , L^^,L^2 a r e hours of work supplied by the kth household's member to the family firm and to other firms, r e s p e c t i v e l y . Now, i f for any consump-t i o n goods j the prices of X „ are i d e n t i c a l for a l l i , i - 1,...,M then Hicks' aggregation theorem can be used and the u t i l i t y function may be written as: f = f(Xly.. . ,X^;L11,L12;L21,L22;L3l, . .. ,^,1^^ where M X. = z X.., j = 1,...,N. i= l J S i m i l a r l y , i f one assumed that prices of L are i d e n t i c a l for k = 1,...M and i f the prices of L ^ are equal for k = 1,...,M (indeed i t i s s u f f i c i e n t to assume that the price r a t i o s W k l are i d e n t i c a l for k = 1.....M, m = 1,...,M across the observations) w ml then the household's preferences can be represented as i n (1): where f - f(L x,L 2;X 1,X 2,...>X^) M M L i = • E L k l and L 2 = E L f c 2 k=l i = l These assumptions allow to s u b s t a n t i a l l y simplify the empirical applications of the model and are frequently used i n the household's l i t e r a t u r e (see, f o r example, Lau, L i n , and Yotopolous). An a l t e r n a t i v e procedure to aggregate would be to assume that X „ , for i = 1,...,M are consumed i n fixed proportions and that L ^ and L ^ J for k = 1,...,M are also i n fixed proportion. That i s to use a Leontief aggregation procedure. In th i s case one can define an aggregate commod-i t y X. = min(-±- X. ., .. ., X^.) for a l l j = 1,. .. ,N and labour supply J Oi £ 1J Oij^  rlj 42 aggregates Lj = min(-p L 1 1 ; ... -~- L ^ ) and L 2 = min(-3- L 1 2 , ... L ^ ) i M i M where a., p. ( i = 1.....M) are the f i x e d c o e f f i c i e n t s . In i i i e q u i l i b r i u m X.. = a,X., L. n = 3.L, and L.„ = p.L 2 f o r i = 1.....M i j i j i l 1 i 2 i M Hence, the aggregate p r i c e of X. would be P. = £ a.p. . and the aggre-J J i = l 1 1 J M gate wage r a t e f o r L j would be wl = Z S.w... and the wage f o r L 2 would be i = l 1 X M w2 = Z p.w . T h i s a g g r e g a t i o n procedure was not used because, as a 1 = 1 1 d i f f e r e n c e w i t h H i c k s ' method, i t imposes r a t h e r s t r o n g r e s t r i c t i o n s on the s t r u c t u r e of household's p r e f e r e n c e s . I t i m p l i e s t h a t t h e r e are no s u b s t i t u t i o n p o s s i b i l i t i e s among the consumption l e v e l s of the d i f f e r -ent i n d i v i d u a l s i n the household. T h i s assumption was judged more u n r e a l -i s t i c than those r e q u i r e d f o r the H i c k s ' p r o c e d u r e . 3 I f f i x e d f a c t o r s of p r o d u c t i o n e x i s t then the p r o f i t f u n c t i o n would be •n(q;L1,k) where k i s a f i x e d f a c t o r . 4 The assumption of p e r f e c t r e n t a l markets f o r d u r a b l e f a c t o r s does not seem too u n r e a l i s t i c at l e a s t i n the case of a g r i c u l t u r e i n developed c o u n t r i e s . R e n t a l markets f o r l a n d and a g r i c u l t u r a l machinery, which c o n s t i t u t e the major d u r a b l e f a c t o r s used i n t h a t s e c t o r , appear to be w e l l d e v e l o p e d . ^Given t h a t i n the e m p i r i c a l a n a l y s i s c r o s s - s e c t i o n a l data are used, a l o n g - r u n e q u i l i b r i u m i s p o s t u l a t e d i n or d e r to j u s t i f y the assump-t i o n t h a t a l l f a c t o r s of p r o d u c t i o n are v a r i a b l e (see Chapter I V ) . I f c o n s t r a i n t ( i i i ) i s b i n d i n g then f o r some households L1 = 0 or L 2 = 0. There are r a t h e r s e r i o u s econometric d i f f i c u l t i e s a s s o c i a t e d w i t h c o r n e r s o l u t i o n s ? a n d t h e y have deserved the a t t e n t i o n of a number of s t u d i e s (Amemiya, Wales and Woodland, 1980). Given t h a t the data used i n the e m p i r i c a l a n a l y s i s are aggregated, both L j and L 2 are g r e a t e r than zero at a l l sample p o i n t s and hence c o n s t r a i n t ( i i i . ) i s not b i n d i n g at any of the o b s e r v a t i o n s (see T a b l e A . 4 ) . ^Olsen has argued t h a t under the c o n d i t i o n s of assumption (1) and i f t h e r e i s f r e e e n t r y and e x i t i n the self-employment and s a l a r i e d a c t i v -i t i e s then, i n long run e q u i l i b r i u m , the shadow p r i c e of on-farm work i s equ a l t o the o f f - f a r m wage r a t e even i f household's members do not work o f f - f a r m . Hence, u t i l i t y and p r o f i t maximizing d e c i s i o n s would be i n d e -pendent even i f households do not work o f f - f a r m . However, many a g r i c u l -t u r a l a c t i v i t i e s are s u b j e c t to r a t h e r important i n s t i t u t i o n a l r e s t r i c t i o n s and f r e e e n t r y i s c e r t a i n l y f a r from r e a l i t y . Moreover, even i f the f r e e e n t r y and e x i t c o n d i t i o n s p r e v a i l , i t i s easy to v e r i f y t h a t under r e a l i s t i c c o n d i t i o n s r e g a r d i n g s t a n d a r d day hours (or week-hours) of work i n the s a l a r i e d a c t i v i t i e s , O l s e n ' s c o n c l u s i o n i s not v a l i d . 43 The argument can be better i l l u s t r a t e d using the following f i g u r e : 0 d L e i s u r e In the f i g u r e , TKqjLjJ represents the household's farm conditional p r o f i t function, MH i s the exogenous wage rate p r e v a i l i n g i n the labour market and I, II and III are i n d i f f e r e n c e curves of the self-employed people. If there are no r e s t r i c t i o n s on the number of hours of work i n the labour market then point A (where the shadow price of labour i s less than the p r e v a i l i n g wage rate) would not be a long run equilibrium. In t h i s case self-employed people would move towards the labour market becoming s a l a r i e d workers. This would imply that less resources are devoted to the self-employed a c t i v i t i e s which would change net output pr i c e s q (presumably i t may also lower MH) and thus causing an upward s h i f t of Tr(q,L 1). This process would p e r s i s t u n t i l the • H q ^ i ) becomes tangent with the MH l i n e , i . e . , u n t i l the on-farm shadow pri c e of family labour i s equal to the off-farm wage rate. Thus, the relevant wage rate for the remaining self-employed households w i l l be equal to MH regardless whether they do part-time off-farm work or not. However, th i s movement w i l l not n e c e s s a r i l y take place under the more r e a l i s t i c assump-ti o n that there e x i s t s a standard work hours per day. Suppose, for example, that an i n d i v i d u a l must work at least H-d hours per day (say, 8 hours) i f he i s to p a r t i c i p a t e i n the labour market at a l l . Then the f e a s i b l e set i s MOdC and edH (H i s the zero work option). In t h i s case in d i v i d u a l s w i l l maximize t h e i r u t i l i t y at A and, although point B would imply a larger u t i l i t y , that point i s not f e a s i b l e . Hence, i n spite that returns to on-farm labour are lower, self-employed households w i l l not move out of t h e i r a c t i v i t i e s because i f they worked as hired labour they would be at point C which y i e l d s a lower u t i l i t y . Hence point A, where the shadow pri c e of on-farm work i s lower than the off-farm wage rate, may be a long run equilibrium. g Notice, however, that the model associated with equations (9) and (11) assumes a constant return to scale production technology (CRS). Thus, CRS implies that curve D" i n Figure 1 would be a h o r i z o n t a l l i n e . Hence, CRS under the conditions of proposition (1) would imply that house-hold's members work only off-farm ( i f w2 i s greater than ff (q)) or only on-farm ( i f w2 i s less than ff(q)). 44 ^The term e _ i s defined from an expenditure function IT IT e(p ,ff,w 2;y) = i min px + f f(H-Li) + w 2(H-L 2) : U(H-Li ,H-L2 ,X) 2= y} X,H-Li,H-L 2 where y stands for u t i l i t y l e v e l . The expenditure function e(p,fr,w 2;y) i s a p o s i t i v e l i n e a r homogeneous, increasing and concave function of p, fr, and w2 and an increasing function of y (Varian, 1978). Using func-t i o n e one can e a s i l y obtain the basic comparative s t a t i c s r e s u l t s assoc-iated with the u t i l i t y maximization problem (8). The term e~~ corre-9 2e sponds to the second d e r i v a t i v e e with respect to ff, -^-i , which i s non-p o s i t i v e given that e i s concave. ^See for example, the survey on labour supply by Heckman et a l . ^The authors used aggregated data given as average values of each of f i v e classes for the eight a g r i c u l t u r a l regions of Taiwan f or the years 1967 and 1968. The authors do not say what percentage of house-holds i n each observation engaged i n off-farm s a l a r i e d work. I t i s clear that i f a sizeable proportion of the households did not work o f f -farm then the use of an off-farm wage rate as a unique p r i c e of household labour i s not appropriate even i f households were i n d i f f e r e n t among al t e r n a t i v e a l l o c a t i o n s of t h e i r working time (see proposition 1). CHAPTER III ESTIMATION OF HOUSEHOLD'S SUPPLY RESPONSES WITH FIXED FACTORS OF PRODUCTION The model of the previous chapter was j u s t i f i e d by assuming that a l l factors of production were v a r i a b l e and a constant returns to scale production technology. The f i r s t assumption appears reasonable i n a long-run equilibrium context. In short-run analysis, however, such an assumption can only be j u s t i f i e d i f perfect ren t a l markets for durable factors of production e x i s t . While i t might be reasonable to assume the existence of perfect r e n t a l markets i n the a g r i c u l t u r a l sector of a devel-oped economy, t h i s assumption may not be as p l a u s i b l e for a g r i c u l t u r e i n developing countries. Thus, i n this chapter a model appropriate to estimate the household's supply and demand re l a t i o n s i n the short-run under conditions of imperfections i n second-hand factor markets ( i . e . , when some inputs are f i x e d or quasi-fixed), which does not r e l y on the constant returns to scale assumption i s provided. Constant returns to scale and no: fixed factors of production were assumed i n Chapter II i n order to obtain a l i n e a r budget constraint for the household's u t i l i t y maximization problem. This allowed us to use standard d u a l i t y theory i n order to derive and characterize the functions which describe the optimal values of the u t i l i t y maximization problem. However, i f any of the above assumptions are relaxed then the budget 45 46 constraint i s non-linear and hence one i s faced with an optimization problem which i s non-linear i n both the objective and constraint func-t i o n s . Conventional d u a l i t y theory does not apply to t h i s class of problems and, thus, standard d u a l i t y cannot be used to specify and char-a c t e r i z e the household's optimal responses. Moreover, the shadow pr i c e of family labour faced by the household w i l l be dependent on hours of (on-farm) work. Thus, assuming for s i m p l i c i t y no off-farm employment''" and defining Ll as on-farm work the u t i l i t y maximization problem i s : max U(H-L 1 (X): H-L l 5X ( i ) pX S T r t q-K.Li) + y (26) ( i i ) X 5 0, L x g 0; R-L1 S 0 One may write the s o l u t i o n of (26) as: H-Li = L^p.q.K.y) and X = X(p,q,K,y) (27) 3 T r(q,K,L 1) One approach would be to define w E — = g 1(q,K,L 1) as the 3L j_ shadow p r i c e of labour and r = ^ ( l ' ^ ' ^ = g 2(q,K,L 1) as the shadow pr i c e of c a p i t a l and express the budget constraint as px + w(H-Lx) i wH + rK + y E Z. This approach would suggest to define an i n d i r e c t u t i l i t y func-t i o n G(p,w,Z) and obtain the demand equations i n the usual manner. The estimation technique should take care, however, of the endogeneity of w and Z (since Z i s dependent on r and w) by estimating H-Lx(p,w,Z), X(p,w,Z),. g 1(q,K,L 1) and g 2(q,K,L x) simultaneously. There are a number of inconveniences i n following t h i s approach: i n the f i r s t place, the variables w and r are unobservable and at most one could estimate them i n d i r e c t l y by estimating the function TrCqjKjLj.). Secondly, i t i s clear that the approach i s exact only under a constant returns to scale production technology. Only under t h i s assumption -niq^yhj^) = wL x + rK, r e l a t i o n which i s implied by the above approach. Otherwise the " f u l l income," Z, would not be calculated c o r r e c t l y . F i n a l l y , even i t i t were possible to observe or ca l c u l a t e w and r t h e i r values would not necessar-i l y correspond to the "true" values which the household a c t u a l l y con-sidered i n i t s u t i l i t y maximizing decisions. In estimating household's responses one e x p l i c i t l y assumes that they make errors i n t h e i r u t i l i t y maximizing decisions. In p a r t i c u l a r , the le v e l s of L x a c t u a l l y chosen by households also r e f l e c t possible optimization errors and, hence, given that the variables w and r are dependent on the lev e l s of L l 5 they w i l l also be observed or measured with errors. Thus, errors made by house-holds i n choosing L x w i l l a f f e c t w and r and, given that the actual L1 used by households i n c a l c u l a t i n g t h e i r w and r are unknown, the relevant shadow prices considered by households are also unknown. The method of estimating the reduced form l e i s u r e demand and con-sumption demand equations (27) also presents problems i f these equations need to be e x p l i c i t l y solved from problem (26). This i s v i r t u a l l y impossible to do due to the complexity of the budget constraint. Thus these functions could only be a r b i t r a r i l y s p e c i f i e d and hence the connec-t i o n between the theory and the estimating equations would be l o s t . There e x i s t s , however, an a l t e r n a t i v e which allows to estimate the reduced form equations for H-L x and X without e x p l i c i t l y solving f o r (27) , but that preserves the linkages of the estimating equations and the theory. If one can define and characterize an i n d i r e c t u t i l i t y function a s s o c i -ated with (26) and i f the household's supply and demand equations are derived from such i n d i r e c t u t i l i t y function (say, by a generalized Roy's i d e n t i t y ) then the r e s t r i c t i o n s implied by the u t i l i t y maximization I 48 hypothesis can be f u l l y considered i n the estimating reduced form equa-t i o n . The problem of e s t a b l i s h i n g d u a l i t y r e l a t i o n s h i p s for the class of optimization problems such as (26), where both the objective function and the budget constraint are nonlinear, i s n o n t r i v i a l and has recently been analyzed by Epstein (1978) . Epstein considered a general nonlinear optimization problem and showed the existence of a dual representation (an i n d i r e c t u t i l i t y f u n ction), determined general properties of the i n d i r e c t u t i l i t y function and derived a generalized Roy's i d e n t i t y . Given that Epstein's assumptions were quite general he obtained a general character-i z a t i o n of the i n d i r e c t u t i l i t y function associated with his nonlinear problem. Moreover, he did not consider the econometric problems involved i n estimating the behavioural equations derived from the model. Hence, a purpose of t h i s chapter i s to show that some more s p e c i f i c properties of the i n d i r e c t u t i l i t y function can be derived and, hence, that a d d i t i o n a l comparative s t a t i c r e s u l t s can be obtained, from a model where the non-l i n e a r budget constraint involves a conditional p r o f i t function. A second objective i s to i n d i c a t e how the optimal household-firm unit's supply and demand responses associated with (26) can be derived v i a the i n d i r e c t u t i l i t y function. F i n a l l y , the problems raised when a stochas-t i c framework for the behavioural equations i s s p e c i f i e d are discussed and an estimation procedure i s proposed. If u t i l i t y maximization problem (26) i s defined l o c a l l y f or the compact subset M, then the i n d i r e c t u t i l i t y function associated with that problem i s : G(p,q,K,y) = max {U(H-Li,X) : (i) px - TT(q;K,Li) = y H _ L 1 > X r fT\ ( i i ) (H-L l 5X) (p,q,K,y) e'r} (28) where the attention i s r e s t r i c t e d to the set of u t i l i t y l e v e l s 49 M = {u:u<;ugu, u<u} which implies that the corresponding commodity space and parameter space are compact, non-empty subsets of R 3 and R1*, r e s p e c t i v e l y . Epstein (1978) showed the existence of a d u a l i t y r e l a t i o n s h i p i n the context of a general non-linear model of which (28) i s a s p e c i a l case Hence, i t i s not necessary to show here that such a d u a l i t y r e l a t i o n also exists between the functions G ( 0 and U(») for a given function T T ( * ) This implies that an i n d i r e c t u t i l i t y function G(«) exists and, moreover, that a function U * ( H - L X , X ) , whose behavioural implications (for a given T T ( 0 ) are the same as I K H - L - L J X ) , can be retrieved from the following problem: Thus, using Epstein's r e s u l t s a d u a l i t y or one-to-one correspon-dence between U(.) and G(.) for any given function T T ( 0 can be estab-l i s h e d . 3.1 Properties of the Indirect U t i l i t y Function In the l i n e a r budget constraint case an important property of the i n d i r e c t u t i l i t y function i s that i t i s quasiconvex i n the p r i c e space. However, i f the budget constraint i s nonlinear then the i n d i r e c t u t i l i t y function i s not n e c e s s a r i l y quasiconvex, as i s shown i n the following proposition (see i t s proof i n Appendix 1): Proposition 5 The function G defined by (28) w i l l be quasiconvex i n i t s arguments i f and only i f the constraint function i s a concave function of the p r i c e (p,q) and K v a r i a b l e s . U*(H-LlfX)-.= min {G(p,q,K,y) : (i) px - ^ ( q ^ L j = y p,q,K,y (29) 50 The constraint function i n (28) i s concave i n p and q (using the properties of the p r o f i t function) but i t i s not concave i n K since -TKqjK,!^) i s convex i n K. Therefore, function G i s not quasiconvex. Is i t necessary to assume global quasiconcavity f o r U i n problem (28)? In the l i n e a r constraint case the assumption of quasiconcavity of U i s useful because i t implies that a l l points i n the commodity space ( in the p o s i t i v e orthant) w i l l be optimal f o r some set of prices and income. This insures that minimization of the i n d i r e c t u t i l i t y function subject to the budget constraint r e t r i e v e s the o r i g i n a l U. In the nonlinear budget constraint case the assumption of quasiconcavity of U i s no longer useful i n t h i s respect unless the budget constraint i s concave i n i t s arguments. In (28) the constraint function does not s a t i s f y t h i s condi-t i o n and hence the u t i l i t y function r e t r i e v e d by (29) w i l l not necessa r i l y be i d e n t i c a l to the o r i g i n a l U. Hence, the assumption of quasiconcavity of U i s no longer useful and instead the following assumption, as pro-posed by Epstein (1978), i s used: Assumption A 1.2: for a l l X°, H-L0-,^  e J" there exists, a s e t of parameters p , q , K , such that X°, H-L° 1 are optimal, solving u t i l i t y maximization problem (28). Thus, t h i s assumption i s used instead of A.2 together with assumptions A . l , A.3, and A.4 (see Chapter I I ) . Hence, i n t h i s chapter i t i s assumed that U s a t i s f i e s the follow-ing properties. Conditions A 1 on U A 1.! Defined and continuous from above. A 1.2 For a l l X°, H-L 0! there e x i s t a set of parameters p°, q°, k°, y° /P such that X°, H-L^, are optimal solving (28). A.3 L o c a l l y increasing i n X. A.4 Increasing i n . 51 Epstein (1978) has shown that i f U s a t i s f i e s A 1.2 then the ret r i e v e d u t i l i t y function obtained by minimizing the i n d i r e c t u t i l i t y function subject to the budget constraint w i l l be i d e n t i c a l to the o r i g -i n a l U. The problem with assumption A 1.2 i s that i t cannot be empiric-a l l y v e r i f i e d . Epstein j u s t i f i e s the use of A 1.2 by the fact that i f X°, H-L0]^ i s not optimal f o r any parameter vector then X°, H-L 0! w i l l never be observed and hence one can define a function U** U whose behav-i o u r a l implications are exactly the.same as U. If th i s i s the case then i t i s i r r e l e v a n t whether U* or U are re t r i e v e d from (29). Thus, a l l that i s needed i s to test whether the second order condi-tions f o r a minimum of (29) are l o c a l l y met at the so l u t i o n points. These conditions w i l l be met i f the Hessian matrix of the function G(p*,q*,K*;px- Tr(q*;K*,L 1)) i s p o s i t i v e semidefinite at p*, q*, K*. Hence, a l o c a l U(») can be re t r i e v e d from G i f th i s condition; i s s a t i s -f i e d . The following proposition summarizes the properties of G(«) as defined by (28). Proposition 6 If U s a t i s f i e s conditions A 1 and i f the p r o f i t function TT s a t i s f i e s conditions C (see Chapter II) then G(p,q,K,y) defined by (28) s a t i s -f i e s conditions D below. Conditions D on G D.l Defined and continuous from below. D.2 Nonincreasing i n p and increasing i n q, K, y. D.3 Homogeneous of degree zero i n p, q, y. D.4 V: vector that (p 0,q 0,K°,y 0) i s a so l u t i o n to (28). 52 Proof D.l Its proof i s i n Epstein (1978), page 31. D.2 Let B E {H-L l 5X : pX - i K q ; ^ - ^ 2 y} and B 1 E {H-L l 5X : p 1X-rr(q;K,L 1) S y}. Assume that p 1 S p, then B 1 <=! B because TT i s decreasing i n H-L]_.^ Hence given A 1.3 and A 1.4 the maximum U attainable cannot be increasing i n p. The argument for showing that G i s increasing i n q, K, y i s s i m i l a r . Note that TT i s increasing i n q and K. D.3 If p, q and y are m u l t i p l i e d by t > 0 then the budget constraint w i l l not change at a l l because TT i s homogeneous of degree one in q. Hence, the maximum U attainable w i l l not be alt e r e d by increasing p, q and y by the same proportion. D.4 Proof i s i n Epstein (1978), page 31. 3.2 Derivation of the Demand and Supply Equations The demand and supply equations associated with problem (28) can be derived from the f i r s t order conditions of minimization problem (29). In der i v i n g the household's supply and demand s p e c i f i c a t i o n s one i s i n t e r -ested i n obtaining the consumption functions X(p,q,K,y), H-Lx(p,q,K,y) as well as the unconditional net output supply vector Q(p,q,K,y). Hence, i t i s necessary to check whether these functions can be derived from the f i r s t order conditions of problem (29). The f i r s t order conditions of (29) are: (a) ! ^ + ^ X = 0 /, \ 8G 9G 9TT N ( b ) 9^ " 37 9K " ° (c) - r ^ - ^ r 1 ^ = 0 i = 1 S 3q i 8y 9q i ' ' 53 (d) pX - 7r(q;K,LL)-' y = 0 (30) F i r s t notice that a l l equations i n (30) are not independent.^ Hence equation (d) can be dropped and the S + 2 remaining equations can be used i n spe c i f y i n g the same number of supply and demand equations (X, H-Lx and S net output Q^). The demand equation f o r X i s d i r e c t l y obtained from 30(a). If the conditions of the i m p l i c i t function theorem are met by the function T T(0 then one can also obtain the equation H-L* x(p,q,K,y) by solving 30(b) for L x . Using Hotelling's lemma i t can be e a s i l y seen that the unconditional net output supply functions can be obtained from 30(c): 3G/3q Q. = j ~ - = i = 1,...,S (31) x i dqi 3G/9y • ' ' Hence, underlying the S equations i n 30(c) there are S net output supply equations Q^(p,q,K,y). Thus, 30(a), 30(b) and 30(c) provide the u t i l i t y maximizing equations f o r consumption goods' demand, the shadow pr i c e of fi x e d factors and the net output supply functions, r e s p e c t i v e l y . This approach has the i n t e r e s t i n g feature of integrating the d e r i v a t i o n of the production and consumption household's behavioural equations and thus emphasizing the interdependence of production and consumption d e c i -sions, which i s evident from the fac t that both consumption and produc-t i o n responses are derived from the same i n d i r e c t u t i l i t y function. Some properties of the behavioural functions can be d i r e c t l y derived from the conditions D on the i n d i r e c t u t i l i t y function G(«). In the f i r s t place, they w i l l be defined and continuous. Moreover, condition 3TT D.3 implies that the behavioural equations f o r X, —— and Q are homogen-3K eous of degree zero i n p, q, and y. Note that the net output supply functions w i l l not be homogeneous of degree zero i n q as occurs i n the 54 conventional model. However, the net output supply conditional on L X ( i . e . , assuming that changes i n q do not a f f e c t L X ) are homogeneous of degree zero i n q. The fa c t that the unconditional net output supply functions are not homogeneous o f d e g r e e z e r o i n q m a k e s i n t u i t i v e s e n s e ; i f a l l q^ are m u l t i p l i e d by t then the optimal L X w i l l change because, although the shadow p r i c e of L X w i l l be unaffected, the budget constraint i n (28) changes since the conditional p r o f i t l e v e l i s expanded by t, i . e . , i n (28.i) Tr(tq ;K JL-L) = tir (q jK,!^) . The u t i l i t y maximization hypoth-esis implies some add i t i o n a l r e s t r i c t i o n s which are discussed i n the following section. 3.3 Further Implications of the U t i l i t y Maximization Hypothesis It i s possible to define an expenditure function e(p,q,K;u) associated with u t i l i t y maximization problem (28): e(p,q,K;u) E min {px - T r C q ^ . L x ) : UCH -L^X) ^ u} (32) X . H - L - L The s o l u t i o n of (32) gives the compensated or u t i l i t y constant demand functions X*(p,q,K;u) and H - L* X(p,q,K;u). Given that a d u a l i t y r e l a t i o n between U and G exists then an expenditure function which i s continuous twice d i f f e r e n t i a b l e , increasing i n p and decreasing i n q and K w i l l also e x i s t . Moreover, a maximum for the following problem w i l l e x i s t : max E H e(p,q,K;u) - [pX - -niqiK,!,^] (33) P . q . K A d d i t i o n a l l y , E = 0 when evaluated at i t s maximum. That i s , the minimum expenditures e are never larger than actual expenditures and they w i l l be i d e n t i c a l when actual expenditures are evaluated at the s o l u t i o n of problem (33). Thus, the f i r s t order conditions of (33) w i l l define the expenditure minimizer or compensated values for 55 consumption goods, X*, and they w i l l also i m p l i c i t l y define the expendi-ture minimizer values for on-farm work, L * i as well as the net output supply values as a function of L * l 9 which i s defined f o r a constant u t i l i t y l e v e l . Thus, the f i r s t order conditions of maximization problem (33) are: ( i ) 3e(p,q,K;u) _ ^ = Q 3p ( i i ) 9e(p,q,K,uJ + = 3q 3q ( i i i ) 3e(p,q,K,u) + = 0 (34) Equation ( i ) i s Shephard's lemma i n i t s usual representation (except f o r the f a c t that q and K appear i n the.function e ( . ) ) , and d i r e c t l y provides the compensated demand function for consumption goods, X*(p,q,K;u). Equation ( i i i ) i m p l i c i t l y defines the expenditure minimizer or compensated function for on-farm work, i . e . , i f the conditions of the 3Tr.(q,K,L*!) i m p l i c i t function theorem hold for the function — then one can 3K solve ( i i i ) f o r L*i(p,q,K;u). Using Hotelling's lemma 3 Tr(q;K,L* 1) = Q*(q;K,L*!(p,q,K;u)) where Q* can be interpreted as compen-sated net output supply functions. That i s , Q* i s the net output supply f o r a given l e v e l of u. Hence, using ( i i ) one can obtain the compensated net output supply functions i n terms of the expenditure function, i . e . , 0* = - Se(p,q,K;u) ^ 3q ' The second order conditions f o r a maximum require that the Hessian matrix H f o r E be negative semidefinite i n p, q and K. Thus, using the 3 e 3 2 e convention of defining -— = e , -—*- = e and using s i m i l a r notation for 3p p' 3p 2 pp the other d e r i v a t i v e s , the following comparative s t a t i c matrix can b e obtained: PP qp "KP pq -pK (e +77 ) (e +TT _ J qq qq qK qK ( eKq + T rKq ) ( e K K + \ K ) - 3X*_ 3p 3Q* 3L* 3X* 3q 3Q* 3X* 3K 3L*i 3Q* 3L*i 3L-i 3p 3L*i 3q 3L*i 3K 3L- 3L*i 3L*] "KL*i 3p "KL*i 3q 3K (35) Negative semidefiniteness of matrix H implies that i t s diagonal elements are non-positive. That i s , as i n the conventional case, the expenditure function i s concave i n commodity p r i c e s , p. An a d d i t i o n a l implication of negative semidefiniteness of H i s that the expenditure function e(0 i s also concave i n q since e + TT < 0 implies that qq qq ~ e i 0 by convexity of TT(«) i n q. Furthermore, a testable p r e d i c t i o n of the model i s that the absolute value of TT cannot be greater than the qq absolute value of e . The expenditure function i s not necessarily qq y concave in K because TT(0 i s concave i n K. So, for example, a p o s i t i v e e may be consistent with negative semidefiniteness of H because Using the i d e n t i t i e s X*(p,q,K;u°) E X(p,q,K;e(p,q,K;u 0)), L*i(p.q,K;w°) = LjCp.q.K^Cp.q.K;!!0)-) and Q*(p,q,K;L*1) = Q(p,q,K;L 1) where X, L x and Q without stars denote uncompensated demand and net output supply functions, one can v e r i f y that the 3 x 3 matrix D D = 3X 3X 3p + 3y X 3X _ 3X 3q 3y Q 3X 3X 3k 3y \ 3Li 3 L i ( + 3Lj 3p 3y X) 3Lx 3L] 9Q " " i 3L X 3Lx 3Li 3q 3y 3Li 3K 3y K L ^KLx (3p 3Li 3Li 3Li + 3y X) T ( KL1 v3q 3Lx W 3Li 3L] Q ) 'UKL1 (3K 3y K) (36) w i l l be equal to the Hessian matrix H. Hence, matrix D w i l l be negative semidefinite and symmetric. This allows one to obtain some comparative s t a t i c r e s u l t s ; using the diagonal terms the comparative s t a t i c e f f e c t s on L-L and Q of a change i n q can be obtained v i a a modified Slutsky equation: 9L, e + TT 9L-, ( a ) _ i = _ J 3 — s a + Q ^ (b) -P = -e + TT T Q - — (37) 3q qq qL1 x 9y where the f i r s t right-hand-side term i n each equation represents the compensated ( i . e . , u t i l i t y constant) net output p r i c e e f f e c t on labour supply and net output supply. As can be expected t h i s e f f e c t i s p o s i -t i v e given that matrix H i s negative semidefinite. The second r i g h t -hand-side term i n each equation i s the income e f f e c t which i s negative 3 L l 30 i f l e i s u r e i s not an i n f e r i o r good. Hence, the signs of and — 6 ' 6 3q 3q w i l l i n general be unknown. The e f f e c t of K on ~L1 can also be considered using matrices H and D: 3 L * £ K K + ^KK , 9 L i w~= ^ — " + ^KW ( 3 8 ) If -n > 0 then the f i r s t term on the right-hand-side i s p o s i t i v e K L x and the sign of the second term w i l l be negative i f l e i s u r e i s a normal good. An increase i n c a p i t a l endowments w i l l lead to an increase i n the opportunity cost of l e i s u r e which would imply a lower demand for l e i s u r e . However, the i n c r e a s e i n K w i l l also have an income e f f e c t which w i l l be manifested i n an expansion i n the demand f o r l e i s u r e . The r e s u l t s which are not t o t a l l y obvious are those associated with 3 e the symmetry of matrix H. Thus, r e c a l l i n g that — = -Q*(q:k,L*x) and 3q 58 using elements (1,2) and (2,1) of (35) one may obtain the following testable symmetry r e l a t i o n s h i p : 3Q*(q;K,L* x) 3X_ = _ _ (39) 9q 9p Thus, the compensated e f f e c t of a change i n a net output p r i c e on the demand for consumption goods w i l l be equal to minus the e f f e c t of a change i n the consumer good pr i c e on compensated or u t i l i t y constant net output supply. S i m i l a r l y , using elements (1,3) and (3,1) of (35) an a d d i t i o n a l symmetry r e l a t i o n becomes evident: 9X* 9TT/9K , / n s 3 K " = " ~W '  (40) which implies that the e f f e c t of changes i n fixed factors on X* w i l l be i d e n t i c a l to minus the e f f e c t of a change i n the cost of l i v i n g index (p) on the shadow p r i c e of c a p i t a l , 9TT/9K. F i n a l l y , (2,3) and (3,2) of matrix D provide the following well known symmetry r e l a t i o n : 9Q = 9TT/9K , . 9K 9q K * l J Obviously^ r e l a t i o n s (39) to (41) can be expressed i n terms of uncompensated functions using the equality of matrices D and H. It i s easy to v e r i f y that i f off-farm labour supply i s considered (L 2) then 9(H-L* 2) 9Q(q,K,L* 1) 9(H-L* 2) = and — = where w-> i s the off-farm 9q 9w2 9K 3w2 z wage rate. In summary, using the r e s u l t s from sections (3.1) and (3.2) the following implications of the u t i l i t y maximization hypothesis are obtained. 9TT 9K Q = Q(ps9,K.,y) are defined, continuous and homogeneous degree zero i n p, 1. The behavioural equations X = X(p,q,K,y), ^  = ii(p,q,K,y) and 59 q and y. In p a r t i c u l a r the net output supply equations are not homo-geneous of degree zero in q. 2. The Hessian matrix of the function E as defined by (33) i s negative semidefinite which implies the following empirically testable p r e d i c t i o n s : (i ) ~r ~ Q - — S O 3q 3y ( i i ) IQ - TT T Q — > 0 K J 3q qLi 3y -3L X 3L X . . 3X* i n addition to the standard r e s u l t - — i 0. 3. Symmetry of the Hessian matrix of the function E implies that 9Q(q;K>L*i) 9 X * 3 T r/9K , 3Q 3TT/3K . ^ . .. _ -— = , —Tr- = and ~ = — . Note that the f i r s t 3p 3p ' 3K 3p 3K 3q two symmetry properties relatev production and consumption decisions , thus emphasizing t h e i r interdependence. 3.4 A Stochastic S p e c i f i c a t i o n Using (30) and (31) one can obtain the household's demand for con-sumption goods, X as well as the net output supply response s p e c i f i c a t i o n s by postulating appropriate functional forms for G(«) and T T ( 0 . With respect to L x i t i s necessary to indicate that one can only obtain an i m p l i c i t representation of i t using equation (30.b) and that L x needs to be interpreted as the equilibrium l e v e l of use of labour rather than as a labour supply schedule. This i s so because labour i s supplied and demanded within the i n d i v i d u a l household-firm unit and the i n d i r e c t u t i l i t y function G(«) defined by (28) i s obtained when the supply and 60 demand equilibrium l e v e l of L1 i s substituted into UCH-Lj^X). This can be seen more c l e a r l y by considering the f i r s t order conditions of u t i l i t y maximization problem (28) : one of these conditions i s that gtUH-L^X) x gTrCq.K,!^) — — = — , where the left-hand-side can be interpreted 3^1 A 3^1 as the household's supply schedule of family labour and the right-hand-side i s the farm's demand schedule for family labour. Thus, the optimal L1 used i n d e f i n i n g G ( 0 i s the L x which solves the above equation, that i s , the l e v e l of L x which leads to an equilibrium of supply and demand for L j . The stochastic structure of the household's equations can be s p e c i f i e d by assuming additive disturbances with zero means and p o s i t i v e semidefinite variance-covariance matrix: U ; 3G/3y + 6 1 3G/3q. 3 , ( q j K ! L ^ _ ^ M 3K 3G/3y ( i i i ) = ^ r S + e 3 (42) where e. (i=l,...,3) are the disturbance terms, i It i s evident that equation ( 4 2 . i i i ) cannot be estimated unless the shadow pri c e of the fixed factor of production K i s observed. 7 Unfor-tunately, the shadow p r i c e of K i s r a r e l y observed i f a ren t a l market for f a c t o r K does not e x i s t . Although the shadow p r i c e of factor K, ^— , 9K cannot be observed the v a r i a b l e TT ("profit") can at least be calculated; i t i s simply the net farm returns a f t e r payments for a l l v a r i a b l e inputs (except L i ) are deducted from the gross sa l e s . Hence, given that q, K and ~L1 are also observed one could i n p r i n c i p l e estimate the vector of parameters, a, which characterizes the p r o f i t function by estimating TT = T r(q,K,L 1; a) + u (43) where u i s the disturbance•term. The problem of estimating (43) i s that the v a r i a b l e L x may be correlated with the disturbance term and hence the estimates of a would not be consistent. However, i f TKO i s l i n e a r i n the parameters (as i s usually the case when f l e x i b l e f unctional forms are used) then one can use an instrumental v a r i a b l e technique (see, for example, Goldfeld and Quandt) and thus to obtain consistent estimates of a . Therefore, i f an appropriate instrumental v a r i a b l e for L x exists then one can estimate equation (43) obtaining consistent estimates (a) for the parameters of the conditional p r o f i t function. Using the estimated vector <3 one can eyal 3n-(q;K,![,.,_;£) . uate the function - which i s the "true" shadow pri c e of 9K 9 7 T(q;K,L 1; a) c a p i t a l , , measured with an error. Thus, the "true" shadow 9K p r i c e of c a p i t a l i s equal to the estimated shadow pri c e plus an error term assumed to be stochastic. That i s : 9 Tr(q;K,L 1; a) 3TT (q ;K,LX; a) 3K = 3K + ( 4 4 ) Notice that equation (44) i s not estimated, the a parameters are obtained by estimating equation (43) and substituted into . Equation 9K (44) cannot be estimated because the shadow p r i c e of c a p i t a l i s not observed, i . e . , the left-hand-side of (44) i s unobservable. In other words, by estimating a i n (43) one obtains a measure of the "true" shadow pric e of c a p i t a l subject to an error, y . If (44) i s used i n (4 2 . i i i ) K then one may i n t e r p r e t equation ( 4 2 . i i i ) as an error of measurement 62 dependent v a r i a b l e s i t u a t i o n , which o f f e r s no e s t i m a t i o n problems: airCqsK.L^a) 3G(p,q,K,y)/3K 3K 3G(p,q,K,y)/3K " 3 + e 3 (45) where e, H e, - u„. K Hence, i f e 3 and u ^ a r e n o r m a l l y d i s t r i b u t e d and independent from p, q, K, y i n (42) and (44) t h e n e 3 w i l l p o s s e s s the same p r o p e r t i e s . Thus, i n o r d e r t o e s t i m a t e (45) t h e r e i s no i n c o n v e n i e n c e i n u s i n g t h e p r e d i c t e d r a t h e r than the a c t u a l shadow p r i c e of K. A l t h o u g h an e x p l i c i t l a b o u r e q u a t i o n cannot be e s t i m a t e d , i f the parameters of (44) and (45) a r e e s t i m a t e d then one o b t a i n s an i m p l i c i t r e p r e s e n t a t i o n o f Lx on the l e f t - h a n d - s i d e of (45) and hence a l l the r e l e v a n t economic i n f o r m a t i o n r e g a r d i n g household's l a b o u r use can be 3 T r(q;K,L x,a) d e r i v e d . F o r example, d e f i n i n g ijj(q ;K,L l s a) . = r= and 3 K < K p,q,K,y;6)•= 3Q/ d v where 6 a r e t h e e s t i m a t e d parameters c h a r a c t e r i z i n g < K ' ) , the l a b o u r e l a s t i c i t y w i t h r e s p e c t to household's income ( e ) i s : £ L y = 3*/3Li I T ' ( 4 0 ) I n a s i m i l a r manner one can c a l c u l a t e the e q u i l i b r i u m l a b o u r s u p p l y e l a s t i c i t i e s ' w i t h r e s p e c t to any of the o t h e r independent v a r i a b l e s . Thus, the p r o c e d u r e i n v o l v e s the f o l l o w i n g s t e p s : f i r s t l y , t o p o s t u l a t e a f u n c t i o n form f o r G ( 0 which i s at l e a s t l o c a l l y c o n s i s t e n t w i t h c o n d i t i o n s D as s p e c i f i e d i n s e c t i o n 3.2 and a l s o t o p o s t u l a t e a f u n c t i o n a l form f o r t h e c o n d i t i o n a l p r o f i t f u n c t i o n c o n s i s t e n t w i t h c o n d i t i o n s C as d i s c u s s e d i n Chapter I I . Secondly, to e s t i m a t e the f u n c t i o n TT(«) and next to j o i n t l y e s t i m a t e the i n d i r e c t u t i l i t y f u n c t i o n u s i n g e q u a t i o n s ( 4 2 . i ) , ( 4 2 . i i ) , and ( 4 5 ) . The u t i l i t y m a x i m i z a t i o n r e s t r i c t i o n s may be checked (or t e s t e d ) by v e r i f y i n g whether the e s t i m a t e d 63 G(-) function has a positive semidefinite Hessian matrix at the observed values of p, q, K and y, whether i t satisfies conditions D.2 and D.3 and if the symmetry restrictions implied by (36) are satisfied. 3.5 Non-traded Outputs Hymer and Resnick have stressed the fact that farm households in agrarian economies devote a substantial part of their time in non-agricul-tural, non-leisure activities. A significant part of the household's time is spent on small manufacturing, construction, and other non-agricul-tural activities which are oriented to produce goods to be consumed by the same household. Thus, given that these goods are not traded, their prices are essentially endogenous to the household's decisions and con-stitute an additional source of interdependence between utility maximiza-tion and profit maximization decisions. The importance of these non-agricultural activities is that they compete with the agricultural activities in the use of resources, mainly the operator and family manpower. This implies that they may have an important effect on farmers' supply responses. Model (26) can be readily adapted to consider non-traded outputs. Assuming for simplicity no off-farm employment, (26) becomes: max U(H-L1}X,z) H"~L ^  jX j z s.t. pX S 7r(q;H-(H-L1),z,K1,K2) + y (47) x, H-Li, z > 0 where z = non-agricultural goods produced and consumed by the household KX,K2 = two fixed inputs, say land and livestock. Now, one can obtain an indirect utility function G(p,q,y,KX,K2) 64 from (47) i n the same manner as before. Minimizing G subject to the budget constraint the following f i r s t order conditions are obtained: (a) |G + | £ x = o 9p 3y ( b ) _ i £ _ i £ 1ZL_ = o i = i , . . . , s ^ 3y 3q i (c) 3G 3G 3TT 3KI- 3y 3Ki ' «> t^f-ffffr0 <48) The consumption demand equations are obtained from (a), and (b) provides the supply functions of traded outputs. The equations f o r labour used and production of non-traded outputs are i m p l i c i t i n equa-tions (c) and (d). Conditions (a)-(d) can be written as: X = - 3 G / 3 p  U X 3G/3y 3G/3q. (b 1) Q, = -TFT^- i = 1,...,S ( c 1 ) 3TT i 3G/3y 3G/3K! 3KX 3G/3y 3G/3K2 3K7 ~ 3G/3y Thus, by estimating the function T T ( 0 one can obtain the predicted (d 1) l f - = w ^ - ^ values for and and use them i n ( c 1 ) and (d 1) . The econometric 3Kx 3K2 framework would be s i m i l a r to the one described by equations (42.i), ( 4 2 . i i ) and (45) with the only d i f f e r e n c e that now two rather than one shadow pr i c e equation needs to be estimated. As before, once the parameters of G( O-and TT (•) are estimated one can proceed to obtain the relevant e l a s t i c i t i e s of L x and z with respect to the exogenous variables, Thus, assuming appropriate functional forms f o r G(») and for T T ( 0 one can e x p l i c i t l y derive a set of equations for the consumption demand functions, production responses of non-traded outputs, and supply r e s -ponses f o r traded outputs. The advantage of the approach i s that the estimating equations are e x p l i c i t l y derived from a u t i l i t y maximization scheme and the interactions between the consumption, a g r i c u l t u r a l , and non-agricultural a c t i v i t i e s can be understood and measured. The disadvantage of the approach i s that the data requirements are high. In p a r t i c u l a r , i t may be d i f f i c u l t to obtain data regarding production of non-traded outputs. However, a number of surveys have been ca r r i e d out i n underdeveloped countries which do ask questions regarding non-traded outputs and i n general non-agricultural a c t i v i t i e s on the farm. 66 F o o t n o t e s ''"This a s s u m p t i o n i s made o n l y f o r the purpose of k e e p i n g the n o t a -t i o n s i m p l e . One may c o n s i d e r o f f - f a r m employment u s i n g H-L 2 as one element of the v e c t o r X w i t h o u t c h a n g i n g the subsequent a n a l y s i s . T h i s can be done p r o v i d e d t h a t the term H-W2 be added t o the r i g h t - h a n d - s i d e of ( 2 6 . i ) . A n o t h e r a s s u m p t i o n used i s t h a t a l l o u t p u t s a r e t r a d e d . T h i s a s s u m p t i o n i s . r e l a x e d i n ' s e c t i o n 3.4. 2 Note t h a t the s o l u t i o n o f ( 2 6 ) c o n s i s t s not o n l y on o p t i m a l v a l u e s f o r X and H - L j , but a l s o the u n c o n d i t i o n a l net ou t p u t s u p p l y e q u a t i o n s are i m p l i c i t l y o b t a i n e d . The net o u t p u t e q u a t i o n s c o n d i t i o n a l t o a g i v e n l e v e l of L x , i . e . , Q(q;K,L!) can be d e r i v e d i n d e p e n d e n t l y o f the s o l u t i o n o f (26) by s i m p l y d i f f e r e n t i a t i n g T v(q;K,Li) w i t h r e s p e c t t o q. However, the u n c o n d i t i o n a l net o u t p u t s u p p l y e q u a t i o n s , i . e . , Q(q,p,K,y) = Q ( q ; K , L j ( q , p , K , y ) ) a re j o i n t l y d e t e r m i n e d w i t h the o p t i m a l s o l u t i o n s H-L.Cp.q.K.y) and X(p,q,K,y) o f ( 2 6 ) . 3 The p r o b l e m a n a l y z e d by E p s t e i n i s max U(X) : C(X,a) < B; x X £^ C", (a,B) e1^ where a and B a r e par a m e t e r s and C(X,a) i s j o i n t l y c o n -t i n u o u s i n (X,a) and U(X) i s assumed t o s a t i s f y the f o l l o w i n g p r o p e r t i e s : ( 1 ) D e f i n e d and c o n t i n u o u s f o r X (2) L o c a l n o n s a t i a t i o n i n X, and (3) V X° e S a(a°,B°) such t h a t X° i s o p t i m a l . The use of a l o c a l approach i s e s s e n t i a l i n E p s t e i n ' s p r o o f o f e x i s t e n c e o f d u a l i t y . A s s u m p t i o n (3) i s a l s o n e c e s s a r y and hence i t i s assumed i n prob l e m (28) t h a t U(') a l s o s a t i s f i e s t he a s s u m p t i o n . 4 N o t i c e t h a t the f u n c t i o n TT can be w r i t t e n as ^ ( q j K j H - C H - L j ) ) w h i c h i s c l e a r l y d e c r e a s i n g i n H-L t. ^ T h i s can be seen by m u l t i p l y i n g e q u a t i o n s (a) and ( c ) by p and q, 3 G r e s p e c t i v e l y , u s i n g Xy = - — y (where X i s the Lagrangean m u l t i p l i e r ay a s s o c i a t e d w i t h c o n s t r a i n t ( i ) i n ( 2 9 ) ) and u s i n g E u l e r ' s theorem a p p l i e d t o G (w h i c h i s homogeneous degree z e r o i n p, q, ^ and y) and t o T T'(which i s l i n e a r homogeneous i n q ) . ^ T h i s r e s u l t s from a d i r e c t a p p l i c a t i o n of E p s t e i n ' s r e s u l t s . 7 3 TT One c o u l d a l s o s o l v e the f u n c t i o n — f o r L, ( i f a f u n c t i o n a l form 3 K-3 TT f o r — i s p o s t u l a t e d ) and then p r o c e e d w i t h the e s t i m a t i o n o f h1; i f i l i S i l k i i l L . ^ ( q . K . L j ) t h e n ( 3 2 . i i i ) w i l l become: L x = ty~l [|§4^- + e,]. 3 L 3 G / 3 y T h i s method i s c e r t a i n l y not a p p r o p r i a t e because, g i v e n the non-l i n e a r s t r u c t u r e of the e q u a t i o n , i t i s not i n g e n e r a l f e a s i b l e t o s e p a r a t e an e r r o r term w h i c h i s independent of the e x p l a n a t o r y v a r i a b l e s . M oreover, i t i s e x t r e m e l y d i f f i c u l t t o f i n d an e x p l i c i t r e l a t i o n s h i p between the new e r r o r term and the independent v a r i a b l e s ; hence the d i s t r i b u t i o n o f t h i s e r r o r w i l l be g e n e r a l l y unknown even i f e 3 i s n o r m a l l y d i s t r i b u t e d . D e f i n i n g ^ (q;Ki,K2,Li,z;a) = "3Li comparative s t a t i c vector 3TT and ^ 2 ( - ) - 9 1 1 3K2 then the 3B 3z ' 3B , where B i s any independent v a r i -able, need to be obtained simultaneously by so l v i n g : - 5 L i -3B 3B 3z a cp2 _ 9 B _3B 3G/3K2 where <j> - and <()' Thus, i f the conditions of the 3G/3y y " 3G/3y ' i m p l i c i t function theorem are s a t i s f i e d by the function i r ( » ) then i t i s possible to obtain the comparative s t a t i c vector and hence the relevant e l a s t i c i t i e s f o r L i and z. CHAPTER IV THE ESTIMATING MODEL This chapter discusses the empirical implementation of the s i m p l i -f i e d model presented i n Chapter II as applied to Canadian cr o s s - s e c t i o n a l a g r i c u l t u r a l data.''" Four major problems are analyzed: 1. the s e l e c t i o n of f u n c t i o n a l forms for the i n d i r e c t u t i l i t y function and for the c o n d i t i o n a l p r o f i t function and the derivation of e x p l i c i t formulations f o r the supply and demand equations to be estimated; 2. the econometric model, discussing econometric problems and the proce-dures used to overcome them; 3. the discussion of an econometric procedure designed to formally test the hypothesis of independent u t i l i t y and p r o f i t maximizing decisions; 4. the data required v i s - a - v i s the data a v a i l a b l e emphasizing data l i m i t a t i o n s . 4.1 Functional Forms for the Indirect U t i l i t y Function and for the  Conditional P r o f i t Function The model to be estimated i s the one described by the c o n d i t i o n a l p r o f i t function defined by equation (6) and by the i n d i r e c t u t i l i t y function defined by (8) i n Chapter I I . A major consideration i n choos-ing f u n c t i o n a l forms for these functions i s that the cross-sectional data used i n the study are aggregated by census d i v i s i o n s (see Section 4.4). This w i l l imply r e s t r i c t i o n s on the l e v e l of generality of the f u n c t i o n a l form postulated for the i n d i r e c t u t i l i t y function. 68 69 U t i l i t y maximization problem (8) defines an i n d i r e c t u t i l i t y func-t i o n G(ff,w2,p;Z) . Now, f o r estimation purposes i t i s necessary to postu-l a t e a functional form for G(TT,W2,P;Z) which i s continuous, p o s i t i v e , non-increasing, quasi-convex function of i t s arguments (Diewert, 1974). Four a l t e r n a t i v e s are a v a i l a b l e given that the data a v a i l a b l e are aggre-gated : 1. To assume a very general u t i l i t y function and that income i s d i s t r i -buted i n fi x e d proportions among household groups which have equal preferences. This a l t e r n a t i v e i s not appropriate because i t would require an assumption that there i s a high c o r r e l a t i o n between the type of households' preferences and t h e i r share i n t o t a l income. Obviously, t h i s i s u n r e a l i s t i c . Furthermore, although consistent aggregated demand functions can be obtained, they are not subject to any l o c a l r e s t r i c t i o n s except Walras's law i f the number of consumers i n the aggregate i s greater or equal to the number of commodities (see Sonnenschein or Diewert, 1977). In p a r t i c u l a r , symmetry and negative semi-definiteness r e s t r i c t i o n s do not apply to such system of demand functions. 2. To impose some r e s t r i c t i o n s on- the u t i l i t y function which may lead to consistent aggregate demand functions. S p e c i f i c a l l y , homothetic and i d e n t i c a l preferences are s u f f i c i e n t conditions f o r obtaining consistent aggregate demand functions (Chipman, 1974). Under these assumptions the system of demand functions w i l l not only be consis-tent but i t w i l l also s a t i s f y the symmetry and negative semi-def initeness r e s t r i c t i o n s . However, rather important differences i n educational, race, and other variables among households lead one to consider the equal tastes assumption u n r e a l i s t i c . Furthermore, 70 homotheticity implies unitary income demand e l a s t i c i t i e s f o r a l l goods which contradicts Engel's law. 3. An approach developed by Berndt et a l . (1977) allows one to assume a very general u t i l i t y function i d e n t i c a l f o r a l l households which y i e l d s consistent demand functions provided that information on income d i s t r i b u t i o n i s e x p l i c i t l y considered i n the market demand equations. Thus, the market demand equation (X^) f o r commodity i would be /OO 0 <j)(Y) • x ±(p/Y) dY where M i s the number of households Y i s the income which i s d i s t r i b u t e d according to a density function cb(Y) and x^(p/Y) i s the household's demand function which i s a function of p r i c e s , p, and Y^. This approach seems appealing mainly because i t does not impose any r e s t r i c t i o n s on the household's u t i l i t y function. Unfortunately, information on income d i s t r i b u t i o n at the census d i v i s i o n l e v e l i s not a v a i l a b l e . In other words, the density function <KY) cannot be determined given the data a v a i l a b l e and hence the procedure cannot be applied. 4. I t has been shown that homotheticity to the o r i g i n of preferences i s not a necessary condition for consistent aggregation (Gorman). The Gorman Polar Form (GPF) appears to be the most general r e s t r i c t i o n on preferences which allows for consistent aggregation and where the demand system s a t i s f i e s the i n t e g r a b i l i t y conditions. Considering . t h i s a GPF f o r G(tT,W2,p;Z) i s used i n t h i s study. 71 The GPF i n d i r e c t u t i l i t y function can be written as „/- ~ N Z- A(TT , W 2 ,p) , K n . G ( T r,w 2,pZ) = , ,~ ' — ^ (50) ty (TT , W 2 , P ) where A and ty are continuous, concave, nondecreasing and p o s i t i v e l y homogeneous of degree one i n f f , w2, and p (Blackorby et a l . , 1978). In order to consider some aggregation properties of the demand functions derived from (50) i t i s convenient to analyze the expenditure function associated with the Gorman-Polar Form i n d i r e c t u t i l i t y function, which i s obtained by simple inversion of G i n Z (Blackorby et a l . , 1978): e ( f f,w 2,p;y) = vty (TT > W 2 ,p) + A ( f f , w 2 , p ) , : where e i s the expenditure function and u denotes u t i l i t y l e v e l . In general, the expenditure function i s continuous, nondecreasing, homo-geneous of 'degree one and concave i n pr i c e s (see for example Varian). The Gorman-Polar expenditure function w i l l s a t i s f y these conditions pro-vided the compensated demand functions, X , s a t i s f y the following condition: X > V A ( T T , W 2 , P ) . The GPF does not necessarily define preferences over the en t i r e non-negative orthant and i f X < V A ( T T » W 2 , P ) then y w i l l be negative i n which case e(Tf,w2,p;y) w i l l not necessa r i l y be concave and increasing i n i T , w2 , p. That y < 0 i f X < V A ( T T , w2,p) can be e a s i l y v e r i f i e d using Shephard's lemma (Shephard). If there are N households, aggregation w i l l be possible i f each household has the following expenditure function: e n( f f > w2 »P;vO = (TT ,w2 ,p) + A h ( f f , w 2 , p ) h = 1,...,N (51) In t h i s case the aggregated expenditure function w i l l be: N N e ( T r,w 2,p;y) = i|;(ff,w 2,p) j ; p , + j A , ( t r » w ,p) (52) h=l h=l 72 Note that the function >Kft,w2,p) must be i d e n t i c a l f o r a l l consumers but A , may be d i f f e r e n t among households. The invariance of ty implies h that changes i n the d i s t r i b u t i o n of income (and hence changes i n y^ N keeping E y, constant) w i l l not a f f e c t aggregate demand i f t o t a l income h=l h does not change. Using Shephard's lemma one may obtain the compensated demand function: N . X = Ve^if ,w2,p;y) = ViJ;(ff ,w2,p) E y + V A ( T T , W 2 , P ) (53) h=l h where N A ( i f,w 2,p) = E A^( f f,w 2,p). h=l N Now, changes i n y keeping E y. constant w i l l have no e f f e c t on X h h=l provided the ty function i s i d e n t i c a l for a l l households. Obviously, the assumption of i d e n t i c a l ty functions can be relaxed i f i t i s assumed that income d i s t r i b u t i o n among households i s approximately constant. In any case the GPF allows f o r quite d i f f e r e n t preferences ( d i f f e r e n t A ^ func-tions) and hence i t i s possible to estimate a mean u t i l i t y function. So f a r , i t has been i m p l i c i t l y assumed that households' character-i s t i c s such as l e v e l of education and number of family dependents do not a f f e c t preferences. A number of empirical studies, however, have concluded that these c h a r a c t e r i s t i c s s u b s t a n t i a l l y a f f e c t preferences and hence the labour supply and commodity demand patterns (Huffman, 1980; Wales & Woodland, 1976). An approach commonly used has been to separate households into groups of approximately homogeneous c h a r a c t e r i s t i c s and then to proceed with the estimation of preferences for each homogeneous group separately. The approach followed here makes use of the property 73 of the GPF which allows f o r d i f f e r e n t households' preferences v i a changes i n the function A(») or i n p r i c e s (changes i n prices lead to s h i f t s i n the preference map i n the commodity space). Thus, i t appears reasonable to assume that households' educational l e v e l (E) and number of dependents (F) a f f e c t preferences e s s e n t i a l l y by changing the reference or base surface, i . e . , by a f f e c t i n g A ( * ) - Thus, instead of estimating a function l i k e G i n (50) for various homogeneous households' groups, i t i s preferred to estimate (50) using A(Tf,w2,p;E,F) considering a l l households at the same time. Changes i n households' c h a r a c t e r i s t i c s are thus assumed to s h i f t the expansion path i n a p a r a l l e l manner. If the set B = {X : X = VA(p;E,F)} i s defined as the base surface (where X i s the vector of commodities and l e i s u r e consumed, p i s the vector of p r i c e s ) , then households with d i f f e r -ent E or F w i l l have a d i f f e r e n t set B where B and B 1 correspond to two base surfaces f or households with d i f f e r e n t education, for example, and KK and K^K1 are the corresponding expansion paths. Note that i f the function 4>(p) i s independent of E and F then KK and K 1K 1 are p a r a l l e l l i n e s . The reason to assume that households' c h a r a c t e r i s t i c s only a f f e c t the function A(») i s to preserve the p a r a l l e l expansion paths for those groups (or census d i v i s i o n s ) which face same p r i c e s . If E or F vary among households within a group then the aggregation conditions 74 would be v i o l a t e d i f ^(•) was also dependent on households' character-i s t i c s . This i s so, because the expansion paths of households within a group would not be p a r a l l e l i f these c h a r a c t e r i s t i c s vary. Thus, i t i s assumed that educational l e v e l s and number of family dependents a f f e c t optimal commodity or l e i s u r e r a t i o s consumed but that the marginal propensity to consume (when income changes) i s not affected. Blackorby et al.(1978) - proposed a functional form for the GPF i n d i r e c t u t i l i t y function. This consists i n a CES form for the ^(ff,w 2,p) and a generalized Leontief for the A (if ,w2 ,p ;E ,F) function. Accordingly, the GPF i n d i r e c t u t i l i t y function w i l l be 3 3 h h 3 3 Z - T E E 6..p.p. + E £.p.E + E b.p.F] •_1 -_1 1] 1 ] ._ n 1 1 . _ i 1 1 G = 1 - 1 3 - 1 — — , (i,j=l,2,3) (54) « 1 P P 1 ] 1 / P i = l where <5 = 6 . , £ . , b . , a . and p are parameters to be estimated and 13 ] i i i i Pi = ft; p 2 = wz and p 3 = p. Using Roy's i d e n t i t y one can derive the demand equations i n expenditure form: 3 3 Is % 3 3 a,P P.[Z- E E 6 p p - E £ P E - E b p F ] 1 1 1=1 j = l 1 3 1 J i = l 1 1 i = l 1 1 A S i " —3 + E a.p p. i = l 1 1 3 p. h 3 3 P.T 2 6.'. (-4 + E A.E + E b.Fj 1 = 1 , 2 , 3 (55) I 1 - • 1 11 P. . T 1 -l 1 4 J=l J *1 1=1 1=1 where S^ = pj(H-L^) S 2 = P 2(H-L 2) 7 5 Note that i t i s possible to test f o r homotheticity to the o r i g i n , by t e s t i n g i f a l l 6.. =0. If- 6.. = 0 for i ^ i then preferences would be homothetic to a s i n g l e point i n the p o s i t i v e orthant and t h i s point would be independent of r e l a t i v e p r i c e s . Thus, the GPF ( 5 4 ) i s chosen considering a number of reasons: (a) f i r s t l y , because the component A ( « ) of ( 5 4 ) belongs to the class of f l e x i b l e f u n c t i o n a l forms, which may be interpreted as second-order approximation to any a r b i t r a r y function. (b) Secondly, the f u n c t i o n a l form ( 5 4 ) allows one to test f o r homotheticity of preferences to the o r i g i n and for homotheticity to a fixed point i n the p o s i t i v e orthant. Given that a number of studies have estimated farm household's demand equa-tions imposing homotheticity to the -origin" (Lau et al'.')and homotheticity to a f i x e d point ( i . e . , Barnum & Squire) i t i s important to test whether those assumptions are appropriate for Canadian farm households. To test f o r homotheticity to the o r i g i n i s p a r t i c u l a r l y important because t h i s assumption allows to avoid the use of nonlinear estimation techniques for the expenditure functions which i s an expensive and d i f f i c u l t compu-t a t i o n a l undertaking. Thus, i f the' hypothesis of homothetic preferences i s not rejected then further studies of Canadian farm households can be undertaken using l i n e a r expenditure systems. Given that the t o t a l expenditures cannot exceed the a f t e r tax income rather than the gross income i t i s necessary to modify model ( 5 5 ) i n order to consider taxes. The budget constraint i n (8) now considering taxes can be expressed as (Wales & Woodland, 1 9 7 6 ) : px 4- f f C H - L x ) + w 2 ( H - L 2 ) 4 H ( f f + w2) + y - T ( Y T ) ( 5 6 ) T where T(Y ) are the t o t a l taxes paid as a function of the household's T taxable income, Y E i f L j + w 2 L 2 + y - Ex., where Ex. are the tax 76 exemptions. The tax function can be approximated by: T(Y T ) = T. + 3.(Y T - Y T.) (57) 1 1 i where T Y . = smallest taxable income i n tax bracket i l T T. = taxes paid at income Y . l l 3^ = marginal tax rate i n tax bracket i . Hence, using (56) and (57) and defining i f = (1 — 3^)TT and ^2± = (1 ~ 3^)w2 the a f t e r tax budget constraint i s : px + i f.(H-L|) + w„.(H-L 2) < H(ff. + w„.) + (l-f3. )y + g . Y T . -l 2 i =- l 2 i l i i x. + 3.Ex. = Z. (58) i i l Thus, equations (35) are estimated using the a f t e r tax values T T \ , w„., and Z. as defined above. 2 i l In order to estimate the production side of the model i t i s neces-sary to estimate the con d i t i o n a l p r o f i t function of the household's firm. According to the d e f i n i t i o n provided i n Chapter II (page 13), the condi-t i o n a l p r o f i t function i s dependent on a vector of net output p r i c e s , q, on the amount of family labour used by the fir m and on the e f f i c i e n c y of production ( i . e . , the production technology broadly defined). The prices considered are one aggregate output p r i c e (q^) a n < 1 the following factor p r i c e s : r e n t a l p r i c e of land and structures (q 2) » h ired labour wage rate (q3), r e n t a l p r i c e of l i v e s t o c k c a p i t a l (qi+) , and r e n t a l p r i c e of other forms of c a p i t a l (qs). In a cros s - s e c t i o n a l framework, e f f i c i -ency differences among the observations might a r i s e because: (a) differences i n the educational l e v e l s of farm households, (b) regional differences i n climate and s o i l q u a l i t y , and (c) regional differences i n output composition (this f actor may be 77 important p a r t i c u l a r l y when output i s aggregated). Factor (a) may lead to improvements i n productive e f f i c i e n c y by a f f e c t i n g the technology which farmers s e l e c t . Education may a f f e c t farm p r o f i t s and the supply of net outputs i n a non-neutral way. Thus, the v a r i a b l e education i s considered as a factor a f f e c t i n g p r o f i t s , and hence net output supply, allowing f o r measuring d i f f e r e n t i a l e f f e c t s of education on the demand for the d i f f e r e n t inputs. Factors (b) and (c) may also a f f e c t the l e v e l of p r o f i t and the net output supply functions i n a non-neutral manner, i . e . , these factors may have d i f f e r e n t e f f e c t s on the supply of the d i f f e r e n t net outputs at a given l e v e l of p r i c e s . For example, differences i n weather conditions among regions may lead to biases 'towards more intensive use of some inputs and le s s intensive use of other resources. It was therefore decided to 2 add dummy var i a b l e s to the conditional p r o f i t f o r four regions. Consequently, assuming constant returns to scale and spec i f y i n g a generalized Leontief condi t i o n a l p r o f i t function, which i s a f l e x i b l e f u n c t i o n a l form i n the sense that i t provides second order approximations to any l o c a l function, the p r o f i t function i s : 5 5 h h 5 5 4 TrCq^i) = L x [ E E b q q + E a q E + E E C ,feD q ] (59) 1=1 j = i ^ 1 J i = i 1 1 i = l k=l where b . = b.., a. and C., are parameters and D, i s the dummy correspond-13 j i l l k k ing to region k. Given (59) the net output supply responses per unit of family labour can be obtained using Hotelling's lemma. Thus, the net output supply equations are: Q. 5 q. i 4 7^ = E b... (-Jy* + a . -E + E C..D,, i=l,...5 (60) L l j = l 1 3 q i 1 k=l l k k 78 where Ql = Output supply Q 2 = Demand f o r land and structures Q3 = Demand for hired labour 0,1+ = Demand for animal stocks Q5 = Demand for farm c a p i t a l In order to estimate the conditional p r o f i t function one can pro-ceed d i r e c t l y to estimate (59), or, equivalently, to estimate the net supply equation system (60). It i s a r b i t r a r i l y chosen to estimate the net supply functions (60). Note that i f producers are p r i c e takers then a constant returns to scale technology i s s u f f i c i e n t to imply that the aggregation conditions are met f o r p r o f i t maximizing firms (Debreu). The i n c l u s i o n of operator's educational l e v e l as an explanatory 3 va r i a b l e i n the p r o f i t equation deserves some further comments. It i s well known that a decision maker's education has a p o s i t i v e e f f e c t on the a l l o c a t i v e e f f i c i e n c y of production (Huffman, 1977). In p a r t i c u l a r , increased education i s usually hypothesized to induce f a s t e r adjustments i n the a l l o c a t i o n of resources to any changes such as i n prices which generate disequilibriums. In other words, an e f f e c t of education would be to increase producers' speed to adapt to changing economic conditions. However, given that the analysis i s more long-run i n nature, t h i s e f f e c t of education i s neglected which implies that producers i n the long run are assumed to be p r i c e e f f i c i e n t regardless of t h e i r l e v e l of education. The i n t e r p r e t a t i o n given to the e f f e c t of education i s as follows: at any point i n time, there e x i s t several production technologies a v a i l a b l e . Some technologies are more complex than others and an appropriate use of them requires d i f f e r e n t degrees of entrepreneurial s k i l l s . For example, 79 a l i v e s t o c k producer may choose from a large number o f - l i v e s t o c k breeds which include some rather r u s t i c breeds and also more sophisticated animals. Although the more sophisticated breeds have a more e f f i c i e n t rate of conversion of feeds into meat, they require more c a r e f u l manage-ment. S i m i l a r l y , higher y i e l d crop v a r i e t i e s coexist with low y i e l d v a r i e t i e s but the former are more s e n s i t i v e to c u l t u r a l p r a c t i c e s , require more f e r t i l i z e r s at precise periods, they require more e f f i c i e n t i r r i g a -t i o n techniques, etc. Thus i t i s hypothesized that more educated entre-preneurs have the management a b i l i t i e s required to handle more complex technologies which are normally the more productive technologies. Hence, producers w i l l choose the .most- e f f i c i e n t technology which t h e i r s k i l l s can handle. Therefore, i t i s assumed that producers are t e c h n i c a l l y (and price) e f f i c i e n t i n the sense that each producer w i l l pick the best tech-nology which has a l e v e l of complexity consistent with h i s s k i l l s . A .. le s s educated producer using a technology which i s not the most produc-t i v e one a v a i l a b l e i s not i n e f f i c i e n t because i f he were to choose a p o t e n t i a l l y more productive and complex technology the end r e s u l t would be a lower p r o d u c t i v i t y i f he i s not able to handle i t i n the appropriate manner. I t i s assumed that the diverse technologies a v a i l a b l e d i f f e r i n t h e i r degree of factor augmentation and t o t a l f a c t o r p r o d u c t i v i t y . The actual technology used by a farmer w i l l depend on h i s s k i l l s which i n turn 4 depend upon h i s educational background. Thus, the l e v e l of education can be interpreted as an index of technology analogous f o r example, to the time trend used i n time seri e s a n a l y s i s . The use of education as an index of technology i n the long run equilibrium model may imply some problems i f education i s indeed an endogenous v a r i a b l e i n the long run. 80 However, a number of authors, p a r t i c u l a r l y G r i l i c h e s , have argued that t h i s i s not a serious problem because, among other reasons, "the large influence of parents, the state, teachers, and classmates on the actual l e v e l of schooling achieved by an i n d i v i d u a l , only part of which can be interpreted as a r e s u l t of h i s own ex-ante optimizing behaviour" ( G r i l i c h e s , p. 13). Hence, one can add terms to the o r i g i n a l general-ized Leontief function (expressed i n terms of input and output prices only) to obtain a f i r s t order approximation i n E: 5 5 ^ 5 . if = E E b..(q.q.) 2 + E E a.q.. 1=1 j = l J J 1=1 This form may be interpreted as a l i n e a r f a c t o r augmentation. The constant returns to scale production technology assumed i s : F(Q*l> Q*2> Q*3> Q*m Q*5» L * l ) = 0 where Q* (1=1 5), L* x i n d i c a t e output i and on-farm operator labour i n e f f i c i e n c y units (using as a base the l e a s t advanced technology). The e f f i c i e n c y l e v e l of a net output Q i s assumed to depend on the technology used which i s determined by the farm operator's educational l e v e l (E), by the actual l e v e l of the net output and by the on-farm (entrepreneurial) work by the operator ( L x ) . A l i n e a r form for the Q* (E,Li,QD functions i s used: Q* = Q i + a^ ^ E Lj and L*1 = Lj_ + yE. The i n c l u s i o n of L x i n the e f f i c i e n c y functions of a l l net outputs is. j u s t i f i e d by considering that the e f f e c t of education i n choosing a production technology w i l l also be influenced by the number of hours i n organizational and entrepreneurial work which the farm operator i s w i l l i n g to provide. The y c o e f f i c i e n t i n would consider the "worker e f f e c t " of education which has been judged r e l a t i v e l y small by most studies ( B a r i c h e l l o ) . Hence, for sim-p l i c i t y i t i s assumed that y = 0 and therefore, L*]_ = . Thus, with 81 the e f f e c t of education s p e c i f i e d i n t h i s form i t can be e a s i l y seen that the net outputs Q^/operator labour r a t i o s w i l l be Q*i Q - ^ + a. E L * l M i which i s the actual s p e c i f i c a t i o n used for the net supply equations i n (60). Note that t h i s s p e c i f i c a t i o n of factor augmentation allows the rate of factor augmentation to be dependent on and E, i . e . , -i ^(Qa./L*!) a. _ 1 x 1 _ x  n i ~ Q*./L*-i 3E Q . / L i + a.E x 1 x 1 x This i s i n contrast with other f a c t o r augmentation indices commonly used, a.E for example, Q*. = e Q . , where n. = a. i s a constant independent of Q. X X I X X and E. F i n a l l y , notice that the technical change induced by education w i l l have a f a c t o r augmenting e f f e c t i f a^ f= 0 for at l e a s t one i = 2,...,5. If a. = 0 for i = 2,...,5 and a x ^ 0 then education w i l l induce a neutral Q ± e f f e c t on the -— ( i = 2,...,5) f a c t o r s . L l In summary, a GPF f u n c t i o n a l form for the i n d i r e c t u t i l i t y function and a generalized Leontief form for the conditional p r o f i t function are chosen. The GPF i s chosen considering that i t i s the most general form which allows for consistent aggregation and where the demand system s a t i s f i e s the i n t e g r a b i l i t y conditions. The generalized Leontief func-t i o n a l form f o r the p r o f i t function has been a r b i t r a r i l y selected from a number of a l t e r n a t i v e f l e x i b l e f u nctional forms. 82 4.2 The Econometric Model In order to estimate the parameters of the i n d i r e c t u t i l i t y func-t i o n and p r o f i t f u n c t i o n i t i s necessary to assume a s t o c h a s t i c s t r u c t u r e f o r (55) and (60). I t i s assumed that the disturbances are a d d i t i v e and normally d i s t r i b u t e d w i t h zero means and p o s i t i v e s e m i d e f i n i t e v a r i a n c e -covariance m a t r i x E. . Thus, i f (55) and (60) are w r i t t e n i n a more com-pact n o t a t i o n and i f the disturbance terms are added then the econometric model i s : ( i ) S i = f 1 ( T f,w 2,p,Z;E,F) + U l ( i i ) S 2 = f2(^^2,v,Z;E,¥) + y 2 ( i i i ) S 3 = f3(ff,w z,p,Z;E,F) + y 3 4 ( i v ) Q /L! = f ^ E ) + E C l k D k + v , i = l , . . . , 5 (61) k=l Note that ft i s now used as a v a r i a b l e r a t h e r than as a f u n c t i o n . Note that 5 Q. 5 ft = E 7 ^ q. = <j>(q;E) + v, where cb(q;E) E E <r1(q;E) • q. i = l 1 i = l and the disturbance term v = E v.q.. The expenditure equations (S.) 1=1 1 1 1 i n (61) are assumed to be dependent on the a c t u a l p r o f i t per hour of work (ft) r a t h e r than on the optimal or expected p r o f i t l e v e l , <j>(q;E). An a l t e r n a t i v e procedure would be to s p e c i f y the expenditure equations as f u n c t i o n s of the optimal or expected p r o f i t and thus the model would be: = f j( (J )(q;E),w 2,p,Z;E,F) + y u ( j = l , . . . , 3 ) (62) The advantage of t h i s procedure i s that the interdependence of u t i l i t y and p r o f i t maximizing d e c i s i o n s ( i . e . , between the equations S_. 83 and Q^/L|) i s more e x p l i c i t i n the econometric model because of the c r o s s -c o n s t r a i n t s between the parameters of the expenditure and c o n d i t i o n a l net output supply equations. U n f o r t u n a t e l y , e s t i m a t i o n of a model based on (62) was i n f e a s i b l e because of the extreme computational d i f f i c u l t i e s and costs i n v o l v e d and, t h e r e f o r e , the econometric s p e c i f i c a t i o n (61) was used i n s t e a d . Under the assumptions used (constant returns to s c a l e and no f i x e d f a c t o r s of production) the v a r i a b l e .fr i s exogenous, independent of household's preferences (independent of Lx.) and hence i t s use as an explanatory v a r i a b l e represents no inconvenience."' The i n t e r p r e t a t i o n of using ft as an explanatory v a r i a b l e i n the expenditure equations i s that households are able to estimate the returns to i t s labour time spent on the f a m i l y farm based on in f o r m a t i o n regarding output p r i c e s , input p r i c e s and knowledge of the production technology they have a v a i l a b l e . Based on t h e i r e s t i m a t i o n of the returns to their.work on t h e i r own farm and c o n s i d -e r i n g the p r e v a i l i n g o f f - f a r m wage r a t e (w 2), cost of l i v i n g index ( p ) , and households f u l l income they decide upon t h e i r optimal expenditures which maximize t h e i r u t i l i t y . Another problem i s the i n t e r p r e t a t i o n of the disturbance terms a s s o c i a t e d w i t h equation system (61). Given that the disburbance terms were assumed s t o c h a s t i c and normally d i s t r i b u t e d , one has to i n t e r p r e t the i n d i r e c t u t i l i t y f u n c t i o n (54) and the c o n d i t i o n a l p r o f i t f u n c t i o n (59), from which the expenditure equations (S..) and the net supply f u n c t i o n s (Q^/I^) were d e r i v e d , as the true f u n c t i o n a l forms. I f (54) and (59) were i n t e r p r e t e d as second order approximations of the true f u n c t i o n a l forms then the disturbance terms added to the e s t i m a t i o n equations would a l s o i n c l u d e e r r o r s of approximation. In t h i s case, given no i n f o r m a t i o n on the true f u n c t i o n a l form i t i s not p o s s i b l e to 84 know the nature of the disturbances and they would be non-stochastic and c o r r e l a t e d w i t h the explanatory v a r i a b l e s . Thus, the disturbance terms can be seen as random e r r o r s i n o p t i m i z a t i o n made by the farm households. Given the budget c o n s t r a i n t i t i s c l e a r that the covariance matrix of the disturbances i s s i n g u l a r and hence one can drop one of the expendi-ture equations. I t i s a r b i t r a r i l y chosen to drop the equation c o r r e s -ponding to expenditures on goods. Hence, the expenditure f u n c t i o n s f o r H-L^ and H-L 2 are estimated (S^ and S 2, r e s p e c t i v e l y ) . The expenditure equations and the net supply f u n c t i o n s i n (61) need to be j o i n t l y estimated d e s p i t e that there are no> parameter r e s t r i c t i o n s across them. The t h e o r e t i c a l model discussed i n Chapter I and I I i s based on the r e c o g n i t i o n of the f a c t that production and consumption d e c i s i o n s are both taken by one i n d i v i d u a l (or household). Hence, a l o g i c a l i m p l i c a t i o n of such a model i s that the e r r o r s made by farmers i n t h e i r production d e c i s i o n s w i l l be c o r r e l a t e d w i t h t h e i r u t i l i t y maximization e r r o r s . I t may be hypothesized that those farmers who make fewer e r r o r s i n t h e i r production d e c i s i o n s w i l l make smaller e r r o r s i n t h e i r u t i l i t y maximizing d e c i s i o n s as w e l l . Given the r e c u r s i v e nature of the model and, i n p a r t i c u l a r , that the expenditure equations are dependent on ff r a t h e r than on a market p r i c e (unrelated to the firm's production technology and q ) , the estimates of the expenditure equations w i l l not only be a s y m p t o t i c a l l y i n e f f i c i e n t but al s o i n c o n s i s t e n t i f the 6 production and consumption se c t o r s are not j o i n t l y estimated. This represents an important d i f f e r e n c e w i t h the conventional model (based on the hypothesis of independent production and consumption d e c i s i o n s ) which ignores the r e c u r s i v e nature of the model by assuming that the shadow p r i c e of on-farm work i n unr e l a t e d to the household's f i r m 85 production technology and net output p r i c e s . In the l a t t e r model the consequences of e s t i m a t i n g the production and consumption s e c t o r s i n a d i s j o i n t manner (as i s u s u a l l y done) are not so s e r i o u s ; the estimates are s t i l l c o n s i s t e n t although some l o s s of e f f i c i e n c y occurs.'' The system of equations (61) i s j o i n t l y estimated, a f t e r dropping the consumption goods expenditure equation, using a F u l l - I n f o r m a t i o n Maximum L i k e l i h o o d Method (FIML). Using a FIML method the c o e f f i c i e n t s estimated w i l l not depend on which equation i s dropped. Thus, the para-meters of the u t i l i t y and p r o f i t f u n c t i o n s which maximize the l o g a r i t h m of the concentrated l i k e l i h o o d f u n c t i o n , L, are chosen T V T T L = - ^ (Jin 2TT + 1) - ^ £n|s| + E £n(abs |B |) (63) N=l where k = number of equations T = number of observations abslB | = absolute value of the determinant of the matrix of d e r i v a -t i v e s of the disturbances w i t h respect to the endogenous v a r i a b l e s , , S 2, and Q./Lj. g I t i s important to i n d i c a t e that the matrix B^ i s t r i a n g u l a r and that abslB I = 1 and hence the l i k e l i h o o d f u n c t i o n L i s reduced to the 1 N 1 f i r s t two terms of (63). The data used on expenditures (S x and S 2 ) , net output supply per u n i t of on-farm work (Q^/Lj), farm operator's education (E) , number of dependents ( F ) , and f u l l income (Z) c o n s i s t of- average household values by census d i v i s i o n r a t h e r than i n d i v i d u a l household's data. On the other hand, i t i s assumed that p r i c e s of consumer goods, on-farm labour r e t u r n s , o f f - f a r m farm wages, and net output p r i c e s are i d e n t i c a l f o r 86 a l l households i n any census d i v i s i o n . I t i s a l s o assumed that the v a r i a n c e of the i n d i v i d u a l household's disturbances are constant. How-ever, given that the number of households v a r i e s across the d i f f e r e n t census d i v i s i o n s , the variances of the disturbance terms w i l l be d i f f e r -ent f o r the d i f f e r e n t observations. Thus, one may expect heteroscedas-t i c i t y which, as i s w e l l known, reduces the e f f i c i e n c y of the estimates and a l s o i n v a l i d a t e s some t e s t s of s t a t i s t i c a l s i g n i f i c a n c e . The l e f t - h a n d - s i d e terms of the net output supply equations are defined by N ( Q i k t / L l k t ) / N t k=l where Q i s the net output i produced by household k of census d i v i s i o n t , L„ i s s i m i l a r l y defined, and N i s the number of households i n l k t t census d i v i s i o n t . S i m i l a r l y , the expenditures a r e defined by S l = * S l k t / N t a n d S 2 = * S l k t / N t k=l k=l Hence, the disturbance term i n equation j f o r a given observation (or census d i v i s i o n ) t , w i l l be: \ - k = i e J k t where e , i s the disturbance term corresponding to household k f o r j k t equation j at observation t . The covariance between the disturbance of equation j and equation s w i l l be cov N N t t E e . E e k-1 J k t k=l S k t 1 ^ 2- E cov(eA., e ) (65) N„ z , , j k t ' slkt t k=l J 87 Note that (65) holds provided the disturbances of the i n d i v i d u a l households are not correlated. Now, i f the covariances between e._ and j k t e . are i d e n t i c a l f o r a l l k then (65) can be written as: skt " N N t t cov . ., i k t . , skt x=l J 1=1 N t > N t N f c 2 * " t " s j N t a . N a . = - v 1 (66) where a . i s the covariance between the disturbances of equations s and j associated with the i n d i v i d u a l households. Thus, the covariances of the disturbance terms w i l l be smaller the larger N i s and, hence, heteroscedastacity i s a problem. In order to tackle t h i s problem one needs: to consider (66) i n the variance-covariance matrix of the l i k e l i h o o d function or, equivalently, one may transform the estimating equations i n such a way that the variance-covariance matrix of the transformed equations be homoscedastic. This i s done by 9 multiplying through equations (61) by the square root of N^. In t h i s case the covariances f o r the averages w i l l be constant equal to a .. S3 If (61) i s m u l t i p l i e d through by N 2 then N t E 6 i k t k=l J k t 2 N 2 t and using expression (66) one obtains that cov(y. , u ) = a .• Thus, j t st sj the above transformation allows to obtain consistent and asymptotically e f f i c i e n t estimates of the expenditure and p r o f i t functions."*"^ The fa c t that the data are aggregated implies some problems leading to the econometric complications discussed above and also to use a more r e s t r i c t i v e f u n c t i o n a l form for the i n d i r e c t u t i l i t y function i n order to obtain consistent and integrable demand functions as discussed i n the 88 previous section. However, Aigner and Goldfeld have shown that under conditions of exactly o f f - s e t t i n g measurement errors i n the microvari-ables, a model using aggregated data w i l l out-perform a model based on microdata. In general, the macrodata have a smaller observation error than the microdata i f the c o r r e l a t i o n among the microdata errors i s not perfect. There i s no reason to assume that measurement errors at the microlevel w i l l not be at l e a s t p a r t i a l l y o f f - s e t t i n g . Thus., although aggregated data are c o s t l y from the viewpoint of requiring more r e s t r i c -t i v e f u n c t i o n a l forms, there are also some advantages concerning smaller observation errors i n using aggregated data. 4 . 3 Testing f or Independence of U t i l i t y and P r o f i t  Maximization Decisions The analysis of Chapters I and II suggested that the hypothesis of independence of u t i l i t y and p r o f i t maximizing decisions i s not l i k e l y to be appropriate f o r modelling farm households' supply and demand responses. I t was also shown that the necessary and s u f f i c i e n t condi-tions f o r independence are quite strong. Based on t h i s , a r e l a t i v e l y more complex model which does not r e l y on the hypothesis of independence has been developed and estimated using Canadian farm data. A relevant question to ask i s whether Canadian farm households' u t i l i t y and p r o f i t maximizing decisions are independent, i . e . , whether the use of a model which assumes interdependence i s j u s t i f i e d i n the case of Canadian a g r i -culture. In other words, i t i s necessary to test the hypothesis that Canadian farm households' u t i l i t y and p r o f i t maximizing decisions are independent; For the purpose of empirically t e s t i n g the hypothesis of indepen-dence one may use a model based on the hypothesis of independence s i m i l a r 89 to the one used by Lau et a l . This model avoids the problem of i n t e r -dependence by assuming that households are i n d i f f e r e n t between working on t h e i r own farms and o f f - f a r m as wage earners."'"''" This allowed the authors to use the o f f - f a r m wage r a t e as the unique exogenous p r i c e of l e i s u r e under the i m p l i c i t assumption that households do some o f f - f a r m 12 work. Thus, such a model i s the f o l l o w i n g : G(p,w2,Z;E,F) = max {U(R-'Ll-Lz,X) :  H—L"j—1*2, X ( i ) px + w Z(H-Li-L 2) <_ Tr(q,w2;E) + w 2H + y s Z ( i i ) X ^ 0; H-L 1-L 2 ^ 0; 1^ ^ 0, L 2 :> 0} (69) T • ~ where Tr(q,w 2;E) E {max q Q - w^j^ : Q J L J e T (E)} i s the u n c o n d i t i o n a l p r o f i t f u n c t i o n , G(-) i s the i n d i r e c t u t i l i t y f u n c t i o n and a l l other v a r i a b l e s have p r e v i o u s l y been defined. N o t i c e that i n t h i s model the assumption of constant returns to s c a l e i s r e l a x e d . R e l a x a t i o n of t h i s assumption i s necessary because the hypothesis of i n d i f f e r e n c e between working on-farm and o f f - f a r m i s not c o n s i s t e n t w i t h constant returns to s c a l e i f L-^  > 0 and L z > G. Hence, given that i n the sample used L-^  and L 2 are both greater than 13 zero, the assumption of constant returns to s c a l e i s not used. Using Roy's i d e n t i t y one can d e r i v e the e s t i m a t i n g u t i l i t y maximizing equations from G(«) and using H o t e l l i n g ' s lemma the u n c o n d i t i o n a l net output supply responses are obtained from T r(q,w 2). Thus, the e s t i m a t i n g model i s : ( i ) H-L 1-L 2 = g 2(p,w z,Z;E,F) + jjj ( i i ) Q i = h 1(q,w 2;E) + \>± ( i = 1,...,5) ( i i i ) L x = h s(q,w 2;E) + v 6 ( i v ) X = g 3(p,w 2,Z;E,F) + y 2 (70). 90 Model (70) i s estimated using the same f u n c t i o n a l forms f o r the i n d i r e c t u t i l i t y f u n c t i o n (Gorman P o l a r Form) and f o r the p r o f i t f u n c t i o n (Generalized L e o n t i e f ) used i n e s t i m a t i n g the model based on the hypoth-e s i s of interdependence defined by (61). As i n model (61) i t i s neces-sary to drop one of the equations of the consumption side i n (70). I t i s a r b i t r a r i l y chosen to drop the equation correspondong to the demand f o r consumption goods ( 7 0 . i v ) . Before proceeding w i t h a d e s c r i p t i o n of the t e s t i n g procedure i t i s convenient to comment on the s t r u c t u r a l d i f f e r e n c e s between the model based on interdependence (model (61)) and the model based on independence (model (70)). The c e n t r a l d i f f e r e n c e between the two models i s that w h ile i n model (61) the labour supply and consumption goods' demand equations j o i n t l y r e f l e c t household's preferences and the fi r m ' s produc-t i o n technology, i n model (70) they are s o l e l y determined by household's preferences. Furthermore, i n model (61) although the net output supply c o n d i t i o n a l on L x are not a f f e c t e d by household's preferences, the i m p l i c -14 i t u n c o n d i t i o n a l net output supply responses ( i . e . , when L x i s con-s i d e r e d v a r i a b l e ) w i l l a l s o be j o i n t l y determined by household's p r e f e r -ences and the firm's production technology. This i n con t r a s t w i t h model (70) where the u n c o n d i t i o n a l net output supply equations are defined independently of the household's preferences. More s p e c i f i c s t r u c t u r a l d i f f e r e n c e s are the f o l l o w i n g : w h i l e i n (61) labour supply on-farm and labour supply o f f - f a r m are considered two d i f f e r e n t "commodities" from the p o i n t of view of the household as a u t i l i t y maximizer, i n model (70) on-farm and o f f - f a r m labour supply are viewed as i d e n t i c a l commodities. In Model (70) there i s a unique r e l e -vant wage r a t e (w 2) which i s independent of the household's f i r m tech-91 nology or net output p r i c e . In c o n t r a s t , i n model (61) there are two wage r a t e s r e l e v a n t to the household: one i s the p r i c e of o f f - f a r m work and the other i s the (shadow) p r i c e of on-farm work which i s dependent on production technologies and net output p r i c e s . The equation f o r L x i n (70) does not correspond to a labour supply equation. I t i s the demand f o r f a m i l y labour determined at the f i r m l e v e l l i k e the demand f o r any other input i s determined. While i n model (70) the l e v e l of on-farm work by household's members i s e n t i r e l y demand determined ( i . e . , i t i s assumed an i n f i n i t e l y e l a s t i c supply of household's labour and hence the household's f i r m labour demand determines the l e v e l of L x ) i n model (61) i s e n t i r e l y supply determined. The assumption of constant r e t u r n s to s c a l e i m p l i e s that the demand for' Lj_ schedule i s i n f i n i t e l y e l a s t i c . R e l a x a t i o n of the constant returns to s c a l e assumption i n model (61) would imply that the e q u i l i b r i u m l e v e l of L j i s determined by both the supply and demand s i d e s . The problem i n f o r m a l l y t e s t i n g the n u l l hypothesis of independence, i . e . , that model (70) holds, against the a l t e r n a t i v e hypothesis of no independence using model (61), i s that the parameter space of (70) i s not contained i n the parameter space of (61). Thus, i f A e fi and A e fi, where A and A are the v e c t o r s of es t i m a t i n g parameters of model (61) and (70) r e s p e c t i v e l y and fi and fi represent the parameter spaces, and i f fi / fi (or fi / fi) then one i s d e a l i n g w i t h separate f a m i l i e s of hypotheses and the standard t e s t s cannot be employed ( G o l d f e l t & Quandt). There are two a l t e r n a t i v e formal t e s t s designed to d i s c r i m i n a t e between separate fam-i l i e s of hypotheses. One i s a t e s t derived by D. R. Cox which i s a modified l i k e l i h o o d r a t i o t e s t and the other one i s a t e s t f o r s p e c i f i -c a t i o n e r r o r developed by Davidson and Mackinnon which i n turn i s a refinement of a t e s t o r i g i n a l l y proposed by Hoel i n 1947. Cox's method i s not used i n s p i t e of i t s r i g o u r and elegance because the computations r e q u i r e d t u r n out to be extremely d i f f i c u l t . The Hoel-Davidson-Mackinnon t e s t , henceforth r e f e r r e d to as the HDM t e s t , allows one to t e s t the t r u t h of l i n e a r or no n l i n e a r and m u l t i v a r i a t e r e g r e s s i o n model, when there e x i s t s a non-nested a l t e r n a t i v e hypothesis. The HDM t e s t , i h - c o n t r a s t -with the Cox's procedure, i s simple and can be e a s i l y imple-mented using e x i s t i n g computer software. For t h i s reason, the HDM pro-cedure i s used i n t e s t i n g the n u l l hypothesis of independent u t i l i t y and p r o f i t maximizing d e c i s i o n s against the a l t e r n a t i v e hypothesis of no independence. Consider an N equation r e g r e s s i o n model, the t r u t h of which i s de s i r e d to t e s t : H 0 : y ± k = z k Q ( X ± , 6 0 ) + e ° l k f o r k = 1,...,N (71) where H 0 stands f o r the n u l l hypothesis, y ^ i s the i observation of the dependent v a r i a b l e of equation k, i s a vector of observations on exogenous v a r i a b l e s , 6 0 I s a v e c t o r of parameters to be estimated and e u ^ k i s the e r r o r term assumed to be normally d i s t r i b u t e d w i t h v a r i a n c e -covariance E. Suppose that an a l t e r n a t i v e hypothesis suggested by economic theory i s : H A : ? i k = Z \ ( Z i ' V + e \ k f o r k = 1,...,N (72) where Z. i s a vect o r of observations on exogenous v a r i a b l e s and 6. i s a i A v e c t o r of es t i m a t i n g parameters. Assuming that H^ i s not nested w i t h i n H 0 and that H 0 i s not nested w i t h i n H^, i . e . , implying that the t r u t h of one hypothesis i m p l i e s the f a l s i t y of the other, the HDM t e s t suggest to estimate the f o l l o w i n g model: 9 3 y i k = ( 1 _ e k ) z k o ( X i ' 6 o ) + \ zkA<Zi'»V + £ i k f o r k = 1>---> N <73> where denotes the estimated <5^  parameter vector. Since the a r t i f i c i a l v a r i a b l e (Z.,6 ) i s independent of e . by the way i t i s constructed, A i A ( 7 3 ) may be estimated l i k e any other regression model. It i s clear that i f Hg i s true, one only needs to estimate ( 7 3 ) and test whether 6 = 0 for a l l k = 1,...,N. In deriving the asymptotic properties of the t e s t , Davidson and Mackinnon have used the following assumptions: Assumptions E: (E.l) E i t h e r the n u l l hypothesis H Q i s true or the a l t e r n a t i v e hypoth-e s i s i s true, ( E . 2 ) the matrices X and Z are nonstochastic, and fi x e d , and ( E . 3 ) a s n + « , the matrices ^ % U 0 ) ] T [ N k Q (6 0 ) J , [ N k A < 6 AX] "T ( N k A (6 A ) ] and ^ [ N \ ( 6 0 ) ] T [ n \ (£0)3 ' w h e r e ^ o ^ O ^ a n d ^ A ^ A ^ A R E T H E k matrices of f i r s t p a r t i a l d e r ivatives of the functions z Q and k z with respect to 60 and 6^ , res p e c t i v e l y , converge to w e l l -defined f i n i t e l i m i t s f o r a l l bounded 6Q and 6^ . Using assumptions E, Davidson and Mackinnon have shown that the t - s t a t i s t i c s f o r g^ . from regression ( 7 3 ) provide ' an asymptotically legitimate test f o r the truth of H Q i n the following sense: 1 . If HQ i s true, plim g^ = 0 (k = 1,...,N) and the variance of g^ i s consistently estimated by ( 7 3 ) ; 2 . If H^ i s true then plim g^ . = 1 (k = 1,...,N), and the variance of g^ i s overestimated by ( 7 3 ) . This implies that i f either H Q or H^ i s true then one can test whether g^ = 0 (k = 1,...,N) using an asymptotic t - t e s t or a j o i n t test such as the l i k e l i h o o d r a t i o t e s t . Notice, however, that i f neither 94 Hg nor H^ i s tru e then the asymptotic p r o p e r t i e s of the t e s t are, i n general, unknown. In using the HDM procedure i n t e s t i n g the hypothesis of independence the n u l l hypothesis i s represented by equations ( 7 0 . i to 7 0 . i i i ) and the a l t e r n a t i v e hypothesis i s embodied i n equations ( 6 1 . i ) , ( 6 1 . i i ) , and ( 6 1 . i v ) . I t i s necessary, however, to introduce some m o d i f i c a t i o n s i n t o the two models i n order to have the same dependent v a r i a b l e s . Thus, the equations to be j o i n t l y estimated are the f o l l o w i n g : ( i ) L x = ( l - g i ) h 6 ( . ) + 3i [H-f 1 (.)/TT| + i i x ( i i ) L 2 = ( l - 3 2 ) [ H - g 2 - ( 0 " h6(.)J + B 2[H-{2(.)/w 2] + y 2 i i -m ( i i i ) Q± - ( 1 - 3 ^ ) ^ ( 0 + B ^ g ^ O • + cov (— , v ± ) : + y 2 + ± ( i = 1,...,5) (74) where a hat (*) above the f u n c t i o n s i n d i c a t e s expected or p r e d i c t e d values. N o t i c e that the second terms of the right-hand-sides represent the p r e d i c t e d or expected values (obtained from model (61)) of L]_, L 2 and ra t h e r than of S^, S 2 and Q •/Lj.''"*' Thus, the n u l l hypothesis that u t i l i t y and p r o f i t maximization d e c i s i o n s are independent ( i . e . , that model (70) i s the true model) i s test e d against the a l t e r n a t i v e hypothesis of i n t e r -dependence represented by model (61) by j o i n t l y t e s t i n g whether = 0 fo r ( k = l , . . . , 7 ) . 1 6 ' 1 7 The f i r s t terms of the right-hand-side correspond to model (70) modified i n order to o b t a i n a s p e c i f i c equation f o r L 2 from (70.i) and ( 7 0 . i i i ) . The i n t e r p r e t a t i o n of L x and L 2 i n (74) should be c a r e f u l l y considered; the model based on independence does not provide two labour supply equations. I t only d e f i n e s one aggregated labour supply and a demand equation f o r Iq determined at the f i r m l e v e l . Hence the equation 95 for L 2 ( i . e . , H-g z(*) - h e ( - ) ) has been obtained from model (70) as a r e s i d u a l reduced form, only f o r the purpose of making model (70) compar-able to model (61). Thus, the equation for L 2 , obtained a f t e r some transformations of model (70) have been made, does not correspond to a household's behavioural equation. The only household's behavioural equation i s the t o t a l labour supply function. The assumption of constant returns to scale used i n the a l t e r n a t i v e hypothesis may cause some problems i n the i n t e r p r e t a t i o n of the t e s t . If the true production technology does not approximately exhibit constant returns to scale then i t i s possible that neither the n u l l hypothesis nor the a l t e r n a t i v e hypothesis are true. As indicated before, i n t h i s case the asymptotic properties of the test are generally unknown and hence i t would be d i f f i c u l t to in t e r p r e t the r e s u l t s of regression (74). The important thing, however, i s that i f Hg i s true then the plim JS^  = 0 (for a l l k) and the variance of (3^ i s consistently estimated by (73) . This implies that the confidence i n t e r v a l f o r g^ i s ; c o r r e c t l y estimated i f Hg i s true and hence the p r o b a b i l i t y of a type I . i r r o f i s c o r r e c t l y given 18 by the l e v e l of s i g n i f i c a n c e chosen. Apart from the formal s t a t i s t i c a l t e s t , an informal test based on comparing the estimates provided by (61) and (70) with respect to con-formity of the estimates obtained with a p r i o r i knowledge i s also per-formed. In p a r t i c u l a r , the emphasis i s placed on whether the estimates of each model s a t i s f y the quasiconvexity and monotonicity properties of the i n d i r e c t u t i l i t y function implied by the u t i l i t y maximization hypoth-e s i s . S i m i l a r l y , s a t i s f a c t i o n of the properties of the p r o f i t function i s also considered, i n p a r t i c u l a r the convexity and monotonicity prop-e r t i e s . Unfortunately, the symmetry r e s t r i c t i o n cannot be tested 96 because e s t i m a t i o n of the u n r e s t r i c t e d model increases the number of parameters to a p r o h i b i t i v e l e v e l . 4.4 The Data The data used were obtained from the 1971 a g r i c u l t u r a l and popula-t i o n censuses. I t i s not p o s s i b l e to have access to household's data because of tax c o n f i d e n t i a l i t y problems. However, aggregated data are a v a i l a b l e at the census d i v i s i o n l e v e l . There are approximately 240 a g r i c u l t u r a l census d i v i s i o n s i n Canada and the data are a v a i l a b l e as t o t a l values per census d i v i s i o n and given that data on number of farm households per census d i v i s i o n are a v a i l a b l e , one can transform the data i n t o averages per household. A l l census d i v i s i o n s were not used, however. Some were excluded because a g r i c u l t u r a l production was n e g l i g i b l e . More imp o r t a n t l y , the number of census d i v i s i o n s corresponding to the d i f f e r e n t regions was not n e a r l y r e p r e s e n t a t i v e of the share of the regions jm a g r i c u l t u r a l produc-t i o n . For example, the Maritime provinces are e q u a l l y represented i n the o r i g i n a l 240 census d i v i s i o n s , as the P r a i r i e provinces d e s p i t e that the Maritimes' a g r i c u l t u r a l output was l e s s than 15% of the P r a i r i e provinces' output. Given that the r e s u l t s are expected to be approximately repre-s e n t a t i v e of Canadian a g r i c u l t u r e , i t was decided to use 95 census d i v i -19 s i o n s . These census d i v i s i o n s were randomly s e l e c t e d from f i v e regions i n such a way that the percentage of census d i v i s i o n s of each region approximately correspond w i t h the importance of the region on a g r i c u l t u r a l output and employment. Thus, the regions and t h e i r approximate share i n the t o t a l sample were: Maritimes (with approximately 7% of the observa-t i o n s ) , Quebec (19%), Ontario (28%), P r a i r i e provinces (40%), and B r i t i s h Columbia (5%) . 97 The required data f o r t h i s study are the number of days of off-farm work by household's members, number of days worked on-farm, off-farm wage rate, farm's net returns per day of work by household members, household's non-labour income, output and input prices faced by the household's firm, farm operator's years of schooling and the number, of family dependents. An aggregated output p r i c e index and three input p r i c e i n d i c e s , namely, h i r e d labour wage rate, animal stock r e n t a l p r i c e index and a land r e n t a l p r i c e index, are needed. The p r i c e index of farm c a p i t a l (machinery, implements and other intermediate inputs) i s not a v a i l a b l e and i s assumed constant across the observations. Farm machinery, f e r t i l i z e r s and spray materials i n contrast with other farm inputs (such as labour, land, and li v e s t o c k ) are traded by large firms, which operate at a natio n a l or even continental scale. It i s reasonable to assume that these firms tend to charge rather homogeneous prices f or t h e i r products i n the d i f f e r e n t regions of the country. Thus, given that these products are traded i n a nation a l market rather than i n segregated regional markets, one can expect a c e r t a i n degree of p r i c e invariance throughout the country and hence the above assumption may not be too u n r e a l i s t i c . The following i s a b r i e f discussion of the data sources and methods 21 used to c a l c u l a t e the s p e c i f i c v a r i a b l e s required i n the analysis. 4.4.1 Off-farm Work and Off-farm Wages The off-farm work ( i n days) of the farm operator i s taken d i r e c t l y from the census of a g r i c u l t u r e data. The population census provides data on off-farm wage income f o r each household's member. Hence, the off-farm wage rate f o r the farm, operator i s calculated by d i v i d i n g the 22 off-farm wage income by the number of off-farm days of work. I t i s assumed that the other male members of the household earn the same o f f -98 farm wage r a t e and hence t h e i r days of o f f - f a r m work a r e c a l c u l a t e d by d i v i d i n g t h e i r o f f - f a r m wage income by the c a l c u l a t e d wage r a t e . F o r female members i t i s assumed a wage r a t e w h i c h i s a f i x e d p r o p o r t i o n of the male wage. The p r o p o r t i o n a l i t y f a c t o r i s based on e s t i m a t e s of p r o v i n c i a l average h o u r l y wages f o r female and male w o r k e r s (Labour F o r c e Survey [ 1 9 7 0 ] ) . The a c t u a l c o e f f i c i e n t s of female wage/male wage assumed v a r i e s from 0.65 t o 0.75. G i v e n t h a t a number of assumptions have been made i n o r d e r t o : c a l -c u l a t e wages and o f f - f a r m work f o r h o u s e h o l d ' s members o t h e r t h a n the o p e r a t o r , t h e s e v a r i a b l e s a re measured w i t h a r e l a t i v e l y l a r g e e r r o r . One c o u l d e x p e c t t h a t t h e s e e r r o r s would not be as l a r g e i n the case of a g g r e g a t e d d a t a as i n the i n d i v i d u a l h o u s e h o l d ' s d a t a . 4.4.2 On-farm Work, R e t u r n s t o Farm Work and Non-Labour Income The p o p u l a t i o n census p r o v i d e s d a t a on t o t a l number of days worked by each h o u s e h o l d member. G i v e n t h a t the number of days of o f f - f a r m work can be c a l c u l a t e d , one can a l s o o b t a i n the number of days worked on the farm by s i m p l y s u b t r a c t i n g the o f f - f a r m work from the t o t a l number of days worked. I n t h i s manner the number of days worked on the farm by each h o u s e h o l d member i s c a l c u l a t e d . The v a r i a b l e t o t a l number of days t h a t h o u s e h o l d ' s members have a v a i l a b l e f o r work and non-work a c t i v -i t i e s (H) was c a l c u l a t e d by s i m p l y adding-up the t o t a l number of days of each h o u s e h o l d member's age 13 o r above, e x c e p t t h a t the women's hours of o f f - f a r m work were w e i g h t e d by the same p r o p o r t i o n a l i t y f a c t o r used 23 t o c a l c u l a t e the female members' o f f - f a r m wage r a t e . I n o r d e r t o c a l c u l a t e r e t u r n s t o farm work i t i s n e c e s s a r y t o f i r s t c a l c u l a t e the net farm income. The net farm income i s e q u a l t o t o t a l f arm s a l e s l e s s o p e r a t i o n a l c o s t s and minus the r e n t a l v a l u e s a s s o c i a t e d 99 24 • with the farm c a p i t a l and land owned by the household. The returns to farm work are thus obtained by d i v i d i n g the calculated net farm income by the t o t a l number of days worked on the farm by household's members. This v a r i a b l e i s used as an explanatory v a r i a b l e ( i f ) i n the expenditure equations. The non-labour income includes two components: (a) the returns associated with f i n a n c i a l assets .(bonds, s e c u r i t i e s , etc.) owned by the household as well as government transfer payments; (b) the returns associated with r e a l assets owned by the household which are used i n the farm operation. Non-labour income component (a) i s d i r e c t l y obtained from the population census data. Non-labour component (b) i s calculated by f i r s t estimating the r e n t a l value of the t o t a l farm c a p i t a l (including land, b u i l d i n g s , machinery, l i v e s t o c k , and equipment). Given that not a l l farm c a p i t a l i s owned by the household, one needs to correct the r e n t a l value obtained by an equity proportion c o e f f i c i e n t . There are no data on equity proportions and hence an average equity proportion r a t i o of 0.92 as calculated by Danielson (1975) for the year 1970 i s used. 4.4.3 Output and Input Prices Data on output prices by province obtained from the S t a t i s t i c s Canada CANSIM data f i l e are a v a i l a b l e . P r o v i n c i a l prices are a v a i l a b l e for major grains, animal products, major f r u i t s , and vegetables crops. A d i v i s i a p r i c e index i s constructed f o r an aggregated output p r i c e v a r i -25--able by province. The d i f f e r e n t p r o v i n c i a l p r ices are then assigned to the census d i v i s i o n observations according to the province where they are located. With respect to input p r i c e s , there e x i s t census data on t o t a l wages paid and t o t a l number of days of hi r e d labour used by census d i v i -sion. Hence, one can obtain an average hired labour wage rate by 100 d i v i d i n g t o t a l wages i n t o the number of days of h i r e d labour used. Data on estimated p r i c e s f o r farm stocks of d i f f e r e n t c a t e g o r i e s of animals are a v a i l a b l e from S t a t i s t i c s Canada by province. A r e n t a l p r i c e index f o r the animal stock as an aggregate i s constructed and the procedure of a s s i g n i n g the corresponding p r o v i n c i a l p r i c e to the census d i v i s i o n s according to the province where they are l o c a t e d i s fol l o w e d . In order to construct an asset p r i c e index f o r land and s t r u c t u r e s , one can use data corresponding to t o t a l market value of land and b u i l d i n g s and d i v i d e i t by the t o t a l number of improved acres. Thus, given that there are no data on p r i c e s of farm b u i l d i n g s , i t i s necessary to assume that improved land and c o n s t r u c t i o n s are i n f i x e d p r o p o r t i o n s . Using the asset p r i c e the corresponding r e n t a l p r i c e of land i s c a l c u l a t e d using some assumptions regarding the value of d e p r e c i a t i o n r a t e s , c a p i t a l gains and discount r a t e s (see Appendix 2). 4.4.4 Other V a r i a b l e s Used and Taxes Data regarding operator's educational l e v e l and number of- /' -household's dependents are a l s o needed. Data on schooling years ' of farm operators are d i r e c t l y a v a i l a b l e from the po p u l a t i o n census. These data are used as a proxy f o r . o p e r a t o r ' s educational l e v e l . The number of f a m i l y dependents v a r i a b l e i s defined as the number of c h i l d r e n age 13 or l e s s l i v i n g on the farm. These data are a l s o d i r e c t l y a v a i l -able from the po p u l a t i o n census. With respect to the tax c a l c u l a t i o n s , i t i s re q u i r e d to know the T marginal tax r a t e (3.) as w e l l as T. and Y . (see Se c t i o n 4.1). Data l i i on average tax p a i d per farm household by census d i v i s i o n are a v a i l a b l e from the p o p u l a t i o n census. Thus using these data one i s able to o b t a i n the marginal tax r a t e f o r a r e p r e s e n t a t i v e household i n the 101 census d i v i s i o n by c o n s u l t i n g the 1970 f e d e r a l and p r o v i n c i a l tax t a b l e under the assumption that the r e p r e s e n t a t i v e household f i l e s a j o i n t tax r e t u r n . 4.4.5 Data Problems The f i r s t data problem i s r e l a t e d to the f a c t that there are no data a v a i l a b l e f o r o f f - f a r m work f o r household's members other than the farm operator. This made i t necessary to f o l l o w the i n d i r e c t procedure described above i n order to c a l c u l a t e o f f - f a r m work, which r a i s e s doubts w i t h respect to the r e l i a b i l i t y of these c a l c u l a t e d v a r i a b l e s . Another problem i s r e l a t e d to the c a l c u l a t i o n of a p r i c e index f o r land and s t r u c t u r e s . The assumption of f i x e d p r o p o r t i o n between land and s t r u c t u r e s may be h i g h l y u n r e a l i s t i c and hence the c a l c u l a t e d p r i c e sub-j e c t to l a r g e e r r o r s . I f land values are p r o p o r t i o n a l l y dominant over the value of the productive c o n s t r u c t i o n s attached to l a n d , then the er r o r s may not be so l a r g e . In general, the l a r g e r the p r o p o r t i o n of land values on the t o t a l value of land and b u i l d i n g s , the smaller w i l l be the e r r o r of the c a l c u l a t e d p r i c e index. Perhaps the most se r i o u s e r r o r s are r e l a t e d to the f a c t that i n order to o b t a i n data f o r a number of v a r i a b l e s i t has been necessary to use combined i n f o r m a t i o n from the Census of A g r i c u l t u r e ' and the Census of P o p u l a t i o n . Although both censuses asked i n f o r m a t i o n concerning the same year (1970) to the same households, there were ho f i e l d cross-checks between.the responses to the po p u l a t i o n census and a g r i c u l t u r a l census q u e s t i o n n a i r e s . Consistency was checked i n each census but there were no cross-checks between the two censuses. This problem may shed some doubts about the r e l i a b i l i t y of the data c a l c u l a t e d by combining data from the two censuses and r e q u i r e s that the r e s u l t s obtained be i n t e r p r e t e d 102 c a u t i o u s l y . 4.4.6 Was 1970 a "Normal" Year? T h i s i s an i m p o r t a n t q u e s t i o n t o answer because i f 1970 was i n d e e d a normal y e a r from the p o i n t of v i e w of weather, i n p u t p r i c e s , and o u t p u t p r i c e s t h e n one can i n t e r p r e t the r e s u l t s o b t a i n e d as b e i n g r e l a t e d t o l o n g r u n e q u i l i b r i u m r e s p o n s e s . On the o t h e r hand, i f t h e r e have been d r a s t i c p r i c e changes i n t h a t y e a r , f o r example, then the l o n g run i n t e r p r e t a t i o n i s more q u e s t i o n a b l e and i t would be p o s s i b l e t h a t p r o d u c e r s be i n p r o c e s s of a d a p t i n g t h e i r e x p e c t a t i o n s t o the new p r i c e s or i f t h e r e a r e a d j u s t m e n t l a g s t h a t i n p u t s demanded would not c o r r e s p o n d t o the l o n g run e q u i l i b r i u m l e v e l s . F i r s t , c o n s i d e r i n g weather , t h e r e a r e two c h a r a c t e r i s t i c s w h i c h a r e c r u c i a l i n farm p r o d u c t i o n , i . e . , t e m p e r a t u r e s and p r e c i p i t a t i o n . C o n s i d e r i n g the p e r i o d of May t o October (which i s the r e l e v a n t c r o p season) average t e m p e r a t u r e s per month i n 1970 as measured i n 25 e x p e r i -m e n t a l s t a t i o n s , s p a n n i n g the major c r o p g r o w i n g a r e a of Canada, were w i t h i n a 10% range of the average monthly t e m p e r a t u r e f o r the 1960-70 decade ( D a n i e l s o n ) . The o n l y e x c e p t i o n was the month of June w h i c h i n 1970 " showed an average t e m p e r a t u r e 45% above the average f o r the decade. A s i m i l a r s i t u a t i o n o c c u r s w i t h r e s p e c t t o monthly p r e c i p i t a t i o n s and a g a i n o n l y the month of June appears t o have been w e l l above the average p r e c i p -i t a t i o n f o r the decade. M o n t h l y d a t a on t e m p e r a t u r e and p r e c i p i t a t i o n show o n l y s l i g h t v a r i a t i o n s w i t h r e s p e c t t o the p r e v i o u s two or t h r e e y e a r s w i t h the o n l y e x c e p t i o n of the month of June. Thus, i n g e n e r a l i t seems t h a t as f a r as the weather p a t t e r n i s c o n c e r n e d the y e a r 1970 appears t o be f a i r l y normal i n c o m p a r i s o n w i t h p r e v i o u s "years w i t h the e x c e p t i o n of the month of June w h i c h shows above normal t e m p e r a t u r e s and p r e c i p i t a t i o n s . W i t h r e s p e c t t o i n p u t p r i c e s , the n o m i n a l average farm wage r a t e i n 103 1970 was s l i g h t l y above the average farm wage r a t e p r e v a i l i n g i n the previous two years although i t had jumped 8.5% i n 1968 (Lopez). Average land p r i c e s i n 1970 remained e s s e n t i a l l y at the same l e v e l as i n 1969 but the nominal land p r i c e had increased by 7% i n 1969 and 3.5% i n 1968. Average farm machinery nominal p r i c e s increased at a very constant r a t e of approximately 3.5% per year between 1965 and 1970. Feed and f e r t i l i z e r p r i c e s both decreased s l i g h t l y i n 1970 and, i n f a c t , they had been sl o w l y decreasing since 1967. With respect to. output p r i c e , one can use the a g r i c u l t u r a l wholesale p r i c e index of farm products, which remained i n 1970 e s s e n t i a l l y at the same l e v e l as i n 1969 although i t had increased by approximately 4% i n the previous years. Thus, i n general input and output p r i c e s i n 1970 remained approximately at the same l e v e l s of the previous years or they continued i n c r e a s i n g at s i m i l a r h i s t o r i c a l r a t e s . With respect to the g r a i n i n d u s t r y however, the s i t u a t i o n was not normal. In 1969 a r a t h e r l a r g e g r a i n inventory was accumulated and i n 1970 the f e d e r a l government implemented a program o r i e n t e d to d i v e r t land from g r a i n production. Given that changes i n i n v e n t o r i e s are not con-si d e r e d i n the income data, the r a t h e r important changes i n g r a i n inven-t o r i e s which took place i n the years 1969 and 1970 may imply that the net farm income v a r i a b l e used i n the . e s t i m a t i o n i s subject to important e r r o r s i n those census d i v i s i o n s where g r a i n production i s dominant. Moreover, the acreage d i v e r s i o n program may have had an important e f f e c t i n g r a i n farmers' production d e c i s i o n s which were not considered i n the e m p i r i c a l study. The use of dummy v a r i a b l e s f o r d i f f e r e n t regions might p a r t i a l l y capture t h i s problem given that g r a i n production tends to be concentrated i n s p e c i f i c r e g i o n s , i n p a r t i c u l a r the P r a i r i e p rovinces. In general one can conclude that the major ab n o r m a l i t i e s i n 1970 104 were r e l a t e d to the g r a i n i n d u s t r y which i s indeed a very important sector i n Canadian a g r i c u l t u r e . With respect to the r e s t of the a g r i c u l t u r a l i n d u s t r y , one may i n d i c a t e that although 1970 was not a p e r f e c t l y normal year i n r e l a t i o n to the previous 5 to 10.year p e r i o d , i t i s at l e a s t pos-s i b l e to i n d i c a t e that t h i s year cannot be s i n g l e d out as a n o t o r i o u s l y abnormal year. Hence, the i n t e r p r e t a t i o n of the r e s u l t s obtained as long run e q u i l i b r i u m supply and demand responses may not be considered t o t a l l y i n a p p r o p r i a t e . 105 Footnotes The e m p i r i c a l work could have been developed us i n g the model presented i n Chapter I I I . The main advantage of that model w i t h respect to the model a c t u a l l y used i s that i t i s more gene r a l , a l l o w i n g f o r f i x e d f a c t o r s of production and non-constant returns to s c a l e . However, given that the data used are c r o s s - s e c t i o n a l i t i s appropriate to p o s t u l a t e long-run e q u i l i b r i u m (see, f o r example, N a d i r i & Rosen) and hence i t i s not a problem to assume that a l l f a c t o r s are v a r i a b l e . Moreover, the use of the model of Chapter I I I would have i m p l i e d that a l l e s t i m a t i n g equations would be h i g h l y n o n - l i n e a r . This i s i n c o n t r a s t w i t h the model i n Chapter I I which allows to estimate l i n e a r c o n d i t i o n a l net out-put supply equations (see s e c t i o n 4.1). Thus, only the labour supply equations are n o n - l i n e a r and, hence, the e s t i m a t i o n procedure i s l e s s expensive and l e s s complex by using the model of Chapter I I . F i n a l l y , the model of Chapter I I allows to disentangle the production technology from households' preferences and a l s o provides an e x p l i c i t equation f o r on-farm work. In c o n t r a s t , the model of Chapter I I I does not a l l o w us to do t h i s . Thus, there are some advantages i n using the simple model of Chapter I I although a major cost i s the n e c e s s i t y of using the assump-t i o n of constant returns to s c a l e , which not only imposes r e s t r i c t i o n s on modelling farm households' demand and supply responses but a l s o leads to some problems i n i n t e r p r e t i n g the r e s u l t s of the t e s t f o r independence of production and consumption d e c i s i o n s (see s e c t i o n 4.3). 2 The four regions considered were: (a) the P r a i r i e p rovinces; (b) the Maritimes; (c) Quebec; and (d) the r e s t of the country. I t was f e l t that each of these regions were more or l e s s homogeneous from the standpoint of weather c o n d i t i o n s and output composition. Given that the use of dummies i s an ad-hoc procedure, they w i l l be considered i n the f i n a l model only i f they are s t a t i s t i c a l l y s i g n i f i c a n t and/or improve the s p e c i f i c a t i o n of the model. 3 N o t i c e that the i n c l u s i o n of E i n the c o n d i t i o n a l p r o f i t f u n c t i o n i s not c o n t r a d i c t o r y w i t h the assumption of no f i x e d f a c t o r s . The l e v e l of education of the operator i s seen as a v a r i a b l e which a f f e c t s the production technology r a t h e r than as a production f a c t o r . Thus, con-s i d e r i n g education, the c o n d i t i o n a l p r o f i t f u n c t i o n can be defined by: TT (q;L i ,E) = {max q T Q : F(Q;L l 5E) = 0} Q where F(Q;L 1,E) i s a transformation f u n c t i o n whose c h a r a c t e r i s t i c s are dependent on E. I t can be shown that T r(q;Li ,E) i s homogeneous degree one i n 1^ provided F e x h i b i t s constant r e t u r n s to s c a l e i n a l l f a c t o r s of production (excluding E). i r(q;XL l sE) =' {max q T Q : F(Q;XL 1 >E) = 0} Q = {max q T (AQ) : F(XQ;AL l i :E) = 0} Q = {max X ( q T Q) : XF(Q;L X,E) = 0} Q 106 = \{ max q Q : "F(Q;Lj,E) = 0} TT (q;XLj ,E) = A i r (q;L x ,E) . Hence, the p r o f i t f u n c t i o n w i l l be homogeneous of degree one i n I4 . 4 C e r t a i n l y , the s e l e c t i o n of a technology a l s o depends on d e c i s i o n maker's farm experience and extension work u s u a l l y made by the government i n making new techniques a v a i l a b l e to farmers. Unfortunately there are no data on farmers' experience and extension expenditures and hence they are ignored. "*It might e x i s t , however, a problem given the method used to c a l c u -l a t e f f. The procedure used was to d i v i d e t o t a l net returns (sales minus t o t a l expenditures, excluding expenditures on fa m i l y labour) i n t o the t o t a l number of household's on-farm work, i . e . , I T / L I . Since the a c t u a l Li used i n v o l v e s a s t o c h a s t i c component r e l a t e d to the disturbance term H i , the c a l c u l a t e d w w i l l be c o r r e l a t e d withu'i, and hence the explanatory v a r i a b l e fr i n (61.1)- would be c o r r e l a t e d w i t h the disturbance term. This problem i s considered i n the es t i m a t i o n method described below. Note that the estimates of the net output supply equations would be co n s i s t e n t but not a s y m p t o t i c a l l y e f f i c i e n t . The i n c o n s i s t e n c y of the -3 estimates of f J ( j = 1, tory v a r i a b l e if i n the .,3) can be v e r i f i e d by no t i n g that the explana-equations i n (61) i s c o r r e l a t e d w i t h the disturbance terms p.; r e c a l l i n g that ff = <j>(q;E) + v where v.q. and v. are the disturbance terms a s s o c i a t e d w i t h the net out-I X x 5 v = E i = l put supply equations: _ E(if • y.) = E[(<j,(q;E) + v ) • "y J = c o v ( v , y ), j = 1, 2, 3 Thus, i f v and y. are c o r r e l a t e d then cov(v,y.) ^ 0 and hence ff and y. are c o r r e l a t e d . ~* • ^ ^ ^Another important d i f f e r e n c e w i t h the c o n v e n t i o n a l m o d e l i s t h a t , although there are no parameter r e s t r i c t i o n s across the production and consumption se c t o r s i n model (61), a l l the cross symmetry r e s t r i c t i o n s between the p r o f i t maximizing and u t i l i t y maximizing equations discussed i n Chapter I I I are i m p l i c i t i n i t . Indeed, these cross r e s t r i c t i o n s are imposed by the s t r u c t u r e of the model ra t h e r than by parameter r e s t r i c -t i o n s . Thus, i n model (61) the compensated constant u t i l i t y e f f e c t of a net output p r i c e increase on L 2 i s •9L2 3TT 3 J _ du=0 9 q i 3ff <j>:L(q;E) du=0 and the e f f e c t on net output supply (Q^) of a change i n w2 i s 3Q, 3iT 3w2 3q. 3 I n 3w2 f 3 L i = * (q;E). du=0 3w2 du=0 but by the w e l l known symmetry r e s t r i c t i o n between u t i l i t y maximizing equations: 107 3L-, 9ff 'du=0 9WT du=0 and, t h e r e f o r e , using t h i s i n the above equations: 3L, du=0 3Q, [9w2J du=0 which i s a cross symmetry r e s t r i c t i o n between the consumption and produc-t i o n s e c t o r s . S i m i l a r l y , i t can be shown that i n (61) that '9X ' | 9Q i 3q. du=0 { 9PJ du=0 which i s the other cross symmetry r e s t r i c t i o n discussed i n Chapter I I I . Thus, a major d i s t i n c t i v e f e a t u r e of a model which assumes interdepen-dence of u t i l i t y and p r o f i t maximization i s the c o n s i d e r a t i o n of these cross symmetry r e s t r i c t i o n s . These r e s t r i c t i o n s may be incorporated v i a parametric r e s t r i c t i o n s or imposed by the s t r u c t u r e of the model as occurs i n model (61). 8 The 7 x 7 B„, ma t r i x i s : N N where a ^ 1 0 0 0 0 0 0 3yi 0 1 0 0 0 0 0 a 1 3 a 2 3 1 0 0 0 0 3Vl a 1 5 a 1 6 a 1 7 a 2 1 + A 2 5 a 2 6 a 2 7 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 3u, 9(Q 1/L 1) 3TT qi ^ 0, and i n general a. k i "i-2 * ° This transformation can be done given that the averaged v a r i a b l e s Z, E, and F enter i n the right-hand-side of (61) i n a l i n e a r form when the values of the other explanatory v a r i a b l e s are f i x e d at t h e i r corresponding census d i v i s i o n l e v e l s . ^ I n maximizing the l i k e l i h o o d f u n c t i o n L the F l e t c h e r a l g o r i t h m as described i n the UBCrNLMON (1975) write-up as w e l l as the non - l i n e a r v e r s i o n of Shazam (1979) were u t i l i z e d . "'"''"An a l t e r n a t i v e would be to t e s t a model which avoids interdepen-dence by assuming that operator and h i r e d labour are p e r f e c t s u b s t i t u t e s i n production and thus using the h i r e d labour wage r a t e as the p r i c e of l e i s u r e , under the assumption that some h i r e d labour i s used by the house-holds. This a l t e r n a t i v e i s not considered because previous e m p i r i c a l s t u d i e s have shown that h i r e d labour and operator labour are not p e r f e c t s u b s t i t u t e s i n production. 108 12 Or u s i n g O l s e n ' s argument t h a t i n the l o n g run e q u i l i b r i u m the on-farm l a b o u r r e t u r n s a r e i d e n t i c a l t o the o f f - f a r m wage r a t e (see f o o t n o t e 6 i n C h a p t e r I I ) . 13 An a d d i t i o n a l advantage i n r e l a x i n g the c o n s t a n t r e t u r n s t o s c a l e a s s u m p t i o n i s t h a t i f t h i s was not done and i f the model based on i n d e p e n -dence were r e j e c t e d the doubt would p e r s i s t whether the r e j e c t i o n was due t o the i m p o s i t i o n of such an a s s u m p t i o n or because of the a s s u m p t i o n of independence. 14 The u n c o n d i t i o n a l net o u t p u t s u p p l y r e s p o n s e s a r e not d i r e c t l y e s t i m a t e d m a i n l y because i t would l e a d t o r a t h e r s e r i o u s e c o n o m e t r i c p r o b -lems a s s o c i a t e d w i t h the n a t u r e of the e r r o r s t r u c t u r e . However, the u n c o n d i t i o n a l net o u t p u t s u p p l y r e s p o n s e s t o any exogenous v a r i a b l e a r e c a l c u l a t e d e x - p o s t u s i n g the e s t i m a t e s f o r ( 6 1 . i ) and the e s t i m a t e s of the c o n d i t i o n a l net o u t p u t e q u a t i o n s (see Chap t e r V ) . ^ T h e e x p e c t e d v a l u e of Q. i s o b t a i n e d from the model based on i n t e r d e p e n d e n c e i n the f o l l o w i n g way: Q i s r L ' L> Q i Q i Q i E(Q._.) = E ( - i ) • E (L l) + E{(L, - E(L,)) • ( - i - E(-±))} X - Li ^  -L I J ^ L I J = • H i l l + G O V , V . ) 1 TT To be c o n s i s t e n t w i t h the model based on i n t e r d e p e n d e n c e ( e q u a t i o n s 5 5 ) , an e x p l o r a t o r y e s t i m a t i o n of the model based on independence (equa-t i o n s 70) u s i n g e x p e n d i t u r e r a t h e r t h a n q u a n t i t y f u n c t i o n s f o r ( 7 0 . i ) and ( 7 0 . i i i ) was p e r f o r m e d . C o n s e q u e n t l y , the t e s t f o r independence was made u s i n g e x p e n d i t u r e e q u a t i o n s f o r L, and L 2 i n ( 7 4 . i ) and ( 7 4 . i i ) . G i v e n t h a t i n t h i s model the r e s u l t s were l e s s c o n s i s t e n t w i t h economic t h e o r y and t h a t the t e s t f o r independence was r e j e c t e d by a v e r y wide m a r g i n , i t was p r e f e r r e d t o use a model based on independence w h i c h e s t i m a t e s q u a n t i t y demand e q u a t i o n s w h i c h p r o v i d e d more r e a s o n a b l e r e s u l t s . That i s , i n t e s t i n g f o r independence the b e s t s p e c i f i c a t i o n f o r the model based on independence has been chosen. ^ T h e r o l e s o f the a l t e r n a t i v e and n u l l h y p o t h e s e s can a l s o be r e -v e r s e d , thus u s i n g i n t e r d e p e n d e n c e as the n u l l h y p o t h e s i s . T h i s t e s t i s a l s o p e r f o r m e d a l t h o u g h one s h o u l d i n d i c a t e t h a t t h i s i s i n d e e d a j o i n t h y p o t h e s i s of i n t e r d e p e n d e n c e - c o n s t a n t r e t u r n s t o s c a l e . Hence, i t can be r e j e c t e d even i f i n t e r d e p e n d e n c e h o l d s i f the t e c h n o l o g y does not a p p r o x i m a t e l y e x h i b i t c o n s t a n t r e t u r n s t o s c a l e . 18 The power of the t e s t may, n e v e r t h e l e s s , be a f f e c t e d , i . e . , the p r o b a b i l i t y of not r e j e c t i n g H 0 when i t i s i n d e e d f a l s e may i n c r e a s e because of the c o n s t a n t r e t u r n s t o s c a l e a s s u m p t i o n i n H . 109 19 The, problem of r e g i o n a l r e p r e s e n t a t i v i t y of the sample i s i m p o r t a n t when one r e c o g n i z e s t h a t t h e r e a r e i m p o r t a n t d i f f e r e n c e s i n weather, s o i l q u a l i t y , d i s t a n c e t o m a r k e t s , e t c . , among r e g i o n s w h i c h may a f f e c t farm s u p p l y and demand r e s p o n s e s . An e f f o r t has been made to c a p t u r e p a r t of t h e s e d i f f e r e n c e s u s i n g r e g i o n a l dummy v a r i a b l e s f o r the i n t e r c e p t c o e f f i c i e n t s of the e s t i m a t i n g net o u t p u t s u p p l y e q u a t i o n s . However, t h i s may not be s u f f i c i e n t g i v e n t h a t r e g i o n a l d i f f e r e n c e s may a f f e c t not o n l y i n t e r c e p t terms but a l s o s l o p e c o e f f i c i e n t s . One c o u l d a l s o use dummies f o r the s l o p e c o e f f i c i e n t s but t h i s would g r e a t l y i n c r e a s e the number of e s t i m a t i n g p a r a m e t e r s . Thus, r e c o g n i z i n g the e x i s t e n c e of r e g i o n a l d i f f e r e n c e s w h i c h a r e not e n t i r e l y c o n s i d e r e d i n the e s t i m a t i n g model one has t o a c c e p t as a second b e s t a l t e r n a t i v e t h a t the e s t i m a t e d c o e f f i c i e n t s would r e f l e c t some s o r t of average t e c h n o l o g y . I f t h i s i n t e r p r e t a t i o n i s used th e n one needs t o s e l e c t a s e t of o b s e r v a t i o n s w h i c h i s a p p r o x i m a t e l y r e p r e s e n t a t i v e of the Canadian a g r i c u l t u r a l s e c t o r . 20 There are o n l y 40 o b s e r v a t i o n s f o r the P r a i r i e P r o v i n c e s . T h i s would have a l l o w e d one t o use a maximum of 100 o b s e r v a t i o n s i f the P r a i r i e P r o v i n c e s a r e t o be r e p r e s e n t e d by 40% of the t o t a l sample. U s i n g o n l y 95 o b s e r v a t i o n s a l l o w e d us t o o b t a i n a sample of 38 o b s e r v a t i o n s from the P r a i r i e P r o v i n c e s i n a random manner. 21 See A ppendix 2 f o r a more e x t e n s i v e d e s c r i p t i o n of the d a t a and methods used i n t r a n s f o r m i n g the d a t a . 22 ' The e c o n o m e t r i c problems r a i s e d by t h i s p r o c e d u r e of c a l c u l a t i n g l a b o u r r e t u r n s are n e g l e c t e d ( A i g n e r ) . 23 T h i s i m p l i e s the a d d i t i o n a l a s s u m p t i o n t h a t the shadow p r i c e s of l e i s u r e , h o u s e h o l d s ' a c t i v i t i e s , and on-farm w o r k i n g a c t i v i t i e s a r e the same f o r each h o u s e h o l d ' s member. T h i s a s s u m p t i o n a l l o w s t o use H i c k s ' a g g r e g a t i o n c o n d i t i o n f o r the v a r i a b l e H. 24 F o r a d e s c r i p t i o n of the r e n t a l v a l u e c a l c u l a t i o n s f o r l a n d , l i v e -s t o c k , and o t h e r farm c a p i t a l see Appendix 2. 25 I n c a l c u l a t i n g a g g r e g a t e ouput as w e l l as i n p u t p r i c e i n d i c e s a q u a d r a t i c mean of o r d e r one ( D i e w e r t , 1977c) i s used as a d i s c r e t e a p p r o x i m a t i o n of a d i v i s i a p r i c e i n d e x . CHAPTER V EMPIRICAL RESULTS In t h i s chapter the e m p i r i c a l r e s u l t s obtained from the e s t i m a t i o n of the expenditure equations (55) and the net output supply equations (60) using Canadian farm census data are reported. The r e s u l t s of the d i f f e r e n t hypothesis t e s t s are a l s o provided. A l l t e s t s have been performed using r e s t r i c t i o n s on equations (55) and (60) w i t h the excep-t i o n of the t e s t f o r independence which uses equations (74) as discussed i n s e c t i o n 4.3. A d d i t i o n a l l y , the r e l e v a n t supply (and demand) response e l a s t i c i t i e s corresponding to the production and consumption sides of the model are presented. 5.1 Hypothesis T e s t i n g The main hypothesis t e s t e d i s that u t i l i t y and p r o f i t maximization d e c i s i o n s are independent, i . e . , that 6^  .= 0 f o r k = 1,...,7 i n (74). A r e l a t e d hypothesis considered i s the one concerned w i t h the c o r r e l a t i o n of the e r r o r s of the labour supply and c o n d i t i o n a l net output.supply equations i n (61). Other hypotheses are r e s t r i c t i o n s on the f u n c t i o n a l • form of preferences, on the e f f e c t s of education and on the e f f e c t s of household's dependents. F i r s t , the hypothesis of a f f i n e h o motheticity i s t e s t e d . This hypothesis i m p l i e s the r e s t r i c t i o n s that = 0 f o r a l l i f j. i n (55). Next, homotheticity to the o r i g i n ' i s t e s t e d i f the a f f i n e h o m o t h e t i c i t y assumption i s not r e j e c t e d . This i m p l i e s that 110 I l l 6 = 0 f o r a l l i , j i n (55). A f u r t h e r hypothesis considered i s that the i n d i r e c t u t i l i t y f u n c t i o n i s independent of the l e v e l of education, that i s , that labour supply responses are not a f f e c t e d by education. This hypothesis r e q u i r e s that £^ = 0 f o r i = 1, 2, 3 i n (55). A f i f t h hypothesis i s that the l e v e l of education exerts a n e u t r a l e f f e c t on the demand f o r f a c t o r s of production, that i s , that a^ = 0 f o r i = 2,...,5 i n (59). F i n a l l y , the hypothesis that the number of f a m i l y dependents does not a f f e c t the i n d i r e c t u t i l i t y f u n c t i o n and hence labour supply i s a l s o t e s t e d . This hypothesis needs that b^ = 0 f o r i = 1, 2, 3 i n (55). The importance of the hypotheses regarding r e s t r i c t i o n s on the f u n c t i o n a l form of preferences:is evident c o n s i d e r i n g that a number of stu d i e s have used them i n a n a l y z i n g labour supply and consumption responses ( i . e . , Lau et a l . ; Barnum & Squire; e t c . ) . The e f f e c t of the d e c i s i o n maker's education may be e s p e c i a l l y important i n the f a m i l y farm where production d e c i s i o n s are u s u a l l y made by the owner himself without the a s s i s t a n c e of p r o f e s s i o n a l management personnel as normally occurs i n l a r g e r corpor-ate f i r m s . The s c a l e of ope r a t i o n of the f a m i l y farm does not u s u a l l y a l l o w h i r i n g of p r o f e s s i o n a l management e x p e r t i s e . Hence, i t i s important to f o r m a l l y t e s t whether t h i s v a r i a b l e has indeed an important e f f e c t on production d e c i s i o n s . In order to c a r r y out the above hypothesis t e s t s , asymptotic l i k e l i h o o d r a t i o t e s t s were performed. The l i k e l i h o o d r a t i o i s the r a t i o of the maximum of the l i k e l i h o o d f u n c t i o n under the n u l l hypothesis to the maximum of the l i k e l i h o o d f u n c t i o n under the a l t e r n a t i v e hypoth-e s i s . Minus twice the lo g a r i t h m of a l i k e l i h o o d r a t i o has a s y m p t o t i c a l l y a chi-square (xz) d i s t r i b u t i o n where the number of degrees of freedom i s equal to the number of r e s t r i c t i o n s imposed by the n u l l hypothesis 112 ( T h e i l ) . Table 1 shows the estimated x 2 values c a l c u l a t e d at 5% and 1% l e v e l of s i g n i f i c a n c e (LOS) f o r the corresponding degrees of freedom. The f i r s t row of Table 1 shows the x2 f° r the n u l l hypothesis that u t i l i t y and p r o f i t maximizing d e c i s i o n s are independent against the a l t e r n a t i v e hypothesis of interdependence, i . e . , that 3, = 0 f o r a l l K. k = 1,...,7 against the a l t e r n a t i v e hypothesis t h a t not a l l 3^ c o e f f i c -i e n t s are zero. The c a l c u l a t e d x i s 127.20 which i s higher than the c r i t i c a l values at 5% and even 1% LOS."*" Hence, the hypothesis that production and consumption d e c i s i o n s are independent i s c a t e g o r i c a l l y 2 r e j e c t e d . The hypothesis of zero c o r r e l a t i o n among the e r r o r s of the labour supply and c o n d i t i o n a l net output supply equations i s a l s o r e j e c t e d although the r e j e c t i o n i s not as strong as i n the previous hypothesis. R e j e c t i o n of t h i s l a t t e r hypothesis i m p l i e s t h a t , even though the constant returns to s c a l e assumption allows to estimate c o n d i t i o n a l net output supply equations which do not have c r o s s - c o n s t r a i n t s w i t h the labour supply equations, these equations need to be j o i n t l y estimated. Thus, there e x i s t s i g n i f i c a n t gains i n explanatory power and e f f i c i e n c y by e s t i m a t i n g the consumption and production s e c t o r s j o i n t l y . The hypothesis of a f f i n e homothetic preferences can be r e j e c t e d at 5% LOS but i s not r e j e c t e d at 1% LOS and homotheticity to the o r i g i n i s c a t e g o r i c a l l y r e j e c t e d at 1% LOS. Thus, one can conclude that Canadian farm households' preferences are not homothetic to the o r i g i n and hence that i t s i m p o s i t i o n may induce s e r i o u s s p e c i f i c a t i o n e r r o r s which l e a d to i n c o n s i s t e n t estimates. Therefore, preferences are b e t t e r s p e c i f i e d i f one allows the reference expenditures to be a f u n c t i o n of commodity p r i c e s . Hypotheses (5) and (6) i n Table 1 are r e l a t e d to the e f f e c t of 113 TABLE 1 Chi-Square S t a t i s t i c s f o r the Various Hypothesis Tests Degrees of C r i t i c a l Values N u l l Hypotheses X 2 Value Freedom 5% LOS 1% LOS, 1. Independence of production and consumption d e c i s i o n s 2. Zero c o r r e l a t i o n among the e r r o r s of labour supply and c o n d i t i o n a l net output supply equations 3. A f f i n e h o motheticity 4. Homotheticity to the o r i g i n 5. No e f f e c t of education on labour supply 6. N e u t r a l e f f e c t of education i n production 7. No e f f e c t of number of fa m i l y dependents on labour supply 127.20** 7 14.07 18.48 24.60** 10 18.31 23.20 9.51* 3 7.81 11.34 202.76** 6 12.59 16.81 28.63** 3 7.81 11.34 53.62** 4 9.49 13.28 62.86** 3 7.81 11.34 Note: * denotes s i g n i f i c a n c e at 5% LOS ** denotes s i g n i f i c a n c e at 1% LOS 114 education on labour supply and i n production decisions, r e s p e c t i v e l y . Both hypotheses are rejected at 1% LOS which implies that education s i g -n i f i c a n t l y a f f e c t s labour supply decisions (and hence i t a f f e c t s the i n d i r e c t u t i l i t y function) and that education plays a non-neutral r o l e i n determining optimal factor demands. F i n a l l y , hypothesis (7) confirms the r e s u l t s obtained in previous studies ( B a r i c h e l l o ; Huffman, 1980; Lau et a l . ; etc.) regarding the importance of the number of family dependents on labour supply decisions. 5.2 Supply and Demand Responses The s t r u c t u r a l parameter estimates obtained by the j o i n t estimation of the consumption and production sides of the model are presented i n Table 2. The asymptotic standard errors of the c o e f f i c i e n t s are pre-sented i n brackets under the c o e f f i c i e n t s . Most c o e f f i c i e n t s i n the consumption .and . production sectors appear to be s i g n i f i c a n t with the exception of the Yi2» D24> b 3 t t, and b 3 5 parameters. There i s one degree of freedom i n the parameters of the CES function which can be exhausted by any s u i t a b l e normalization (Blackorby et a l . , 1978). The normaliza-t i o n chosen i s that the share parameter a 2 i s equal to one. The c o e f f i c i e n t s of the regional dummy var i a b l e s used turned out to be i n s i g n i f i c a n t and, moreover, the i n c l u s i o n of these v a r i a b l e s d i s t o r t s the values of other c o e f f i c i e n t s . Given that the use of dummies i s e s s e n t i a l l y an ad-hoc procedure, i t was decided not to use them i n the model a c t u a l l y reported. The r e s u l t s obtained when the regional dummies were used were les s consistent with economic theory and, i n general, provided e l a s t i c i t y estimates which appeared to be quite u n l i k e l y . A reason f o r the lack of s i g n i f i c a n c e of the regional dummies may be that 115 TABLE 2 Parameter E s t i m a t e s of the Consumption and P r o d u c t i o n E q u a t i o n s • . •• "('Equations.-.55 • and 60) Number of ^ k L e i s u r e L e i s u r e Consumption Family i j ' i ' i 1 2 Goods Education Dependents R 2 I. Consumption Sector p 0.980 a 2 1 (0.086) (-) ax 1.124 a 3 41.45 (0.222) (10.35) L e i s u r e 1 612.5 -9.111 4.749 -14.83 : 160.5 (H-LO (4.591) (3.746) (9.603) (3.205) (7.149) L e i s u r e 2 829.3 60.86 -24.88 166.3 (H-L 2) (15.94) (6.16) (2.534) (5.835) Consumption - - -2418 -2.812 42.76 Expenditures (10.59) (1.055) (1.078) Hi r e d Animal Farm Educa-i j ' i Output Land Labour Stocks C a p i t a l t i o n R' I I . P roduction Sector Output Supply 113.6 147.4 -99.09 -39.61 -233.17 -38.77 0. 835 (7.044) (2.562) (7.455) (2.755) (2.858) (2.276) Demand f o r -147.4 -160.1 68.71 -2.584 150.2 15.56 0. 427 Land (2.562) (1.969) (4.562) (1.683) (4.366) (1.414) Demand f o r 99.09 - 102.6 -37.01 -88.86 -9.124 0. 801 h i r e d labour (7.455) (22.21) (4.702) (9.743) (1.199) Demand f o r 39.61 - - 7.518 2.359 1.011 0. 645 animal stocks (2.755) (4.795) (3.499) (0.346) Demand f o r 233.17 - - - 235.68 -32.48 0. 874 farm c a p i t a l (2.858) (4.674) (1.783) G e n e r a l i z e d = 0.994 116 a s u b s t a n t i a l proportion of the regional differences i n weather and s o i l q u a l i t y are captured i n land p r i c e differences amongst the d i f f e r e n t regions. For example, the average r e n t a l p r i c e of improved land i n the Fraser V a l l e y of B r i t i s h Columbia was approximately 15 times as large as the average r e n t a l p r i c e of improved land i n Saskatchewan. This p r i c e d i f f e r e n t i a l r e f l e c t s to a large extent the better weather conditions and, i n general, better land f e r t i l i t y i n the Fraser Valley, which the dummy var i a b l e s are supposed to capture. A reason f o r the d i s t o r t i o n of the estimates of other c o e f f i c i e n t s induced by the dummy variables may be due to the negative e f f e c t on'the econometric e f f i c i e n c y a s s o c i -ated with the i n c l u s i o n of redundant v a r i a b l e s . Thus, a reason f o r obtaining r e s u l t s which are les s consistent with economic theory might be that the i n c l u s i o n of the dummies decreases the pr e c i s i o n of the estimates. The goodness-of-fit measure used i s the "generalized R 2" which was o r i g i n a l l y proposed by Baxter and Cragg. In systems of equations without intercepts such as the one estimated, the conventional R 2 measure i s no longer appropriate and hence the generalized R 2 measure was used. This c o e f f i c i e n t i s defined as follows: R 2 = {1 - exp f2-(L0 - L )/Nj} u max -1 where L Q i s the value of the logarithm of the l i k e l i h o o d function when a l l parameters are constrained to zero, L i s the maximum when a l l r max c o e f f i c i e n t s are allowed to vary and N i s the t o t a l number of observations. The R 2 c o e f f i c i e n t obtained i s very close to 1 i n d i c a t i n g that the good-ness-of-f i t of the estimation was v e r y good. Raw-moment R 2 c o e f f i c i -ents are provided for the i n d i v i d u a l equations as complementary informa-t i o n . 117 In order f o r the estimated f u n c t i o n G(-) to be a v a l i d i n d i r e c t u t i l i t y f u n c t i o n , i t must be monotonically decreasing and quasiconvex i n p r i c e s . This i m p l i e s that the f u n c t i o n s A(«) and TT(') should be con-cave and monotonically i n c r e a s i n g i n p r i c e s . These p r o p e r t i e s were checked u s i n g the estimated c o e f f i c i e n t s . The monotonicity property was s a t i s f i e d by both f u n c t i o n s at each sample p o i n t . The f u n c t i o n \l>(') i s g l o b a l l y concave, that i s f o r a l l p >_ 0, s i n c e a l l c o e f f i c i e n t s estimated are p o s i t i v e . U n f o r t u n a t e l y , the f u n c t i o n A(«)'is not g l o b a l l y concave and moreover i t does not s a t i s f y t h i s property at 62% of the observation p o i n t s . Therefore, the quasiconvexity property of the i n d i r e c t u t i l i t y f u n c t i o n i s s a t i s f i e d at only 38% of the observations. However, the c a l c u l a t e d matrix of e l a s t i c i t i e s of s u b s t i t u t i o n e x h i b i t s negative own s u b s t i t u t i o n e f f e c t s at approximately 55% of the observa-t i o n s . The c o n d i t i o n a l p r o f i t f u n c t i o n reported i n S e c t i o n 2 of Table 1 should a l s o possess c e r t a i n p r o p e r t i e s which are checked at each of the sample p o i n t s . F i r s t , the c o n d i t i o n a l p r o f i t f u n c t i o n has the c o r r e c t gradients w i t h respect to p r i c e s , that i s , c o n d i t i o n a l p r o f i t increases w i t h increases i n the output p r i c e and i t i s decreasing i n input p r i c e s . Secondly, the estimated c o n d i t i o n a l p r o f i t f u n c t i o n i s p o s i t i v e at each of the sample p o i n t s . T h i r d l y , the estimated c o n d i t i o n a l p r o f i t func-t i o n should s a t i s f y the r e q u i r e d convexity property. The s i g n of the determinants of the p r i n c i p a l minors a s s o c i a t e d w i t h the Hessian matrix of the estimated c o n d i t i o n a l p r o f i t f u n c t i o n were checked. Although 3 t h i s matrix was not convex :at approximately 60% of the observations, i t s diagonal elements were a l l p o s i t i v e at more than 80%.of the sample p o i n t s , which i m p l i e s that the own p r i c e net output supply e l a s t i c i t i e s 118 show the c o r r e c t s i g n when evaluated at most of the observations. Another aspect i n t e r e s t i n g to note i n Table 2 i s that the reference expenditures which depend on the 6 , £^ and b_^  c o e f f i c i e n t s are p o s i t i v e f o r H-Lx and H-L 2. However, the reference expenditures i n consumption goods are negative at a l l sample p o i n t s . This precludes the i n t e r p r e -t a t i o n of VA(p) as subsistence q u a n t i t i e s . I t i s important to note, however, that these reference expenditures are l e s s than a c t u a l expendi-tures at a l l the observations and hence the i n d i r e c t u t i l i t y l e v e l s are p o s i t i v e at a l l sample p o i n t s . Table 3 contains the on-farm and o f f - f a r m labour supply e l a s t i c i t i e s TABLE 3 Labour Supply E l a s t i c i t i e s (at mean values of the v a r i a b l e s ) On-farm Labour Off-farm returns wage r a t e Non-Labour Income On-farm labour 0.119 -0.107 -0.612 supply Off-farm labour -0.259 0.180 -0.539 supply T o t a l labour 0.043 -0.049 -0.237 supply w i t h respect to on-farm returns to farm household labour, o f f - f a r m wage 4 r a t e r e c e i v e d by household's members and household's non-labour income. The own wage e l a s t i c i t i e s of labour supply are both p o s i t i v e when e v a l u -ated at mean values being the o f f - f a r m labour supply e l a s t i c i t y substan-t i a l l y l a r g e r than the on-farm e l a s t i c i t y . However, the on-farm supply e l a s t i c i t y was negative at 8% of the observations and the o f f - f a r m 119 e l a s t i c i t y was n e g a t i v e at 19%. ~* These e s t i m a t e s are not comparable w i t h p r e v i o u s s t u d i e s because p r e v i o u s e s t i m a t e s were o b t a i n e d f o r a g g r e g a t e l a b o u r s u p p l y . One c o u l d c a l c u l a t e the t o t a l e l a s t i c i t y of l a b o u r s u p p l y w i t h r e s p e c t t o a s i m u l t a n e o u s change i n the on-farm l a b o u r r e t u r n s and the o f f - f a r m wage r a t e f o r the purpose of c o m p a r i s o n . T h i s e l a s t i c i t y i s a p p r o x i m a t e l y 0.024 w h i c h i s s u b s t a n t i a l l y lower t h a n l a b o u r s u p p l y e l a s -t i c i t i e s o b t a i n e d by Lau et a l . . u s i n g farm h o u s e h o l d ' s d a t a from Taiwan ( 0 . 1 6 ) , by Barnum and S q u i r e who used s i m i l a r d a t a from M a l a y s i a ( 0 . 0 8 ) . Huffman (1980) u s i n g U.S. farm h o u s e h o l d s ' d a t a o b t a i n e d o f f - f a r m l a b o u r s u p p l y e l a s t i c i t i e s of 0.33 f o r husbands and -0-.06 f o r w i v e s . On the o t h e r hand, Wales and Woodland (1976) u s i n g a sample of U.S. h o u s e h o l d s a l s o found s m a l l p o s i t i v e s u p p l y e l a s t i c i t i e s f o r some h o u s e h o l d s and n e g a t i v e e l a s t i c i t i e s f o r o t h e r s . The average s u p p l y e l a s t i c i t i e s f o r hus-bands was 0.11 f o r t h o s e o n the upward s e c t i o n of the s u p p l y c u r v e and -0.32 f o r t h o s e on the downward p a r t . I t i s i m p o r t a n t t o i n d i c a t e t h a t the l a b o u r s u p p l y e l a s t i c i t i e s a re v e r y s e n s i t i v e t o whether the v a r i a b l e edu-c a t i o n i s used i n the e s t i m a t i n g l a b o u r s u p p l y e q u a t i o n s or n o t . When e d u c a t i o n i s not i n c l u d e d , the t o t a l e l a s t i c i t y of l a b o u r s u p p l y r i s e s t o 0.38 w h i c h i s somewhat l a r g e r t h a n p r e v i o u s e s t i m a t e s o b t a i n e d . T a b l e 3 a l s o shows the cross-wage e f f e c t s on l a b o u r s u p p l y . A 1-percent i n c r e a s e i n the o f f - f a r m wage r a t e i n d u c e s a 0.1%, d e c r e a s e i n the number of days worked on t h e i r own farm by the h o u s e h o l d ' s members. The e f f e c t of on-farm l a b o u r r e t u r n s on o f f - f a r m work i s s t r o n g e r . A 1-percent i n c r e a s e i n farm l a b o u r r e t u r n s w i l l i n d u c e a 0.25 p e r c e n t de-c r e a s e i n the o f f - f a r m s u p p l y of l a b o u r . The e s t i m a t e d e l a s t i c i t i e s o f on-farm and o f f - f a r m l a b o u r s u p p l y w i t h r e s p e c t t o h o u s e h o l d ' s n o n - l a b o u r income a r e -0.162 and -0.539, r e s p e c t i v e l y . The e f f e c t of n o n - l a b o u r 120 income on t o t a l l a b o u r s u p p l y (on and o f f - f a r m ) i s a p p r o x i m a t e l y -0.23. T h i s e s t i m a t e can be compared w i t h p r e v i o u s s t u d i e s . For example, Ashen-f e l t e r and Heckman found e l a s t i c i t i e s o f -0.112 f o r males and -0.594 f o r f e males u s i n g U.S. c r o s s - s e c t i o n a l h o u s e h o l d d a t a , and Horney and M c E l r o y e s t i m a t e s were -0.213 and -0.097 r e s p e c t i v e l y (Heckman e t a l . ) , a l s o u s i n g h o u s e h o l d d a t a . The e f f e c t of e d u c a t i o n on b o t h o f f - f a r m and on-farm l a b o u r s u p p l y i s p o s i t i v e but i t s e f f e c t on o f f - f a r m work i s s u b s t a n t i a l l y l a r g e r t h a n on on-farm work. In f a c t , w h i l e a 1-percent i n c r e a s e i n f o r m a l s c h o o l i n g t r a i n i n g (measured i n y e a r s of s c h o o l i n g ) i n d u c e s a 0.35% e x p a n s i o n of on-farm work, a s i m i l a r i n c r e a s e i n e d u c a t i o n w i l l l e a d t o a 1.25%, e x p a n s i o n i n o f f - f a r m work. Thus, the e f f e c t of e d u c a t i o n on l a b o u r s u p p l y appears to be q u a n t i t a t i v e l y v e r y s t r o n g w i t h a b i a s towards o f f - f a r m a c t i v i t i e s . The e f f e c t of e d u c a t i o n on o f f - f a r m work can be compared w i t h Huffman's e s t i m a t e d e l a s t i c i t y of o f f - f a r m work w i t h r e s p e c t to farm o p e r a t o r educa-t i o n a l l e v e l w h i c h was 1.03. The on-farm and o f f - f a r m l a b o u r s u p p l y e l a s t i c i t i e s w i t h r e s p e c t t o number of f a m i l y dependents were b o t h n e g a t i v e (-0.082 and -0.241, r e s p e c -t i v e l y ) . T h i s r e s u l t i s not c o n s i s t e n t w i t h a p r i o r i e x p e c t a t i o n s . One may be i n c l i n e d t o e x p e c t a p o s i t i v e e f f e c t and i n f a c t most p r e v i o u s s t u d -i e s have o b t a i n e d p o s i t i v e e l a s t i c i t i e s (Lau et a l . , f o r example, o b t a i n e d an e l a s t i c i t y o f 0.20 f o r t o t a l l a b o u r s u p p l y , and Huffman's e s t i m a t e was 0.659 u s i n g c h i l d r e n under 5 y e a r s of age as a p r o x y ) . Thus, i t i s v e r y d i f f i c u l t t o r a t i o n a l i z e t h i s r e s u l t and i t appears to be e r r o n e o u s g i v e n t h a t i t c o n t r a d i c t s not o n l y r e a s o n a b l e a p r i o r i e x p e c t a t i o n s but a l s o p r e v i o u s q u a n t i t a t i v e e s t i m a t e s . T a b l e 4 c o n t a i n s the e s t i m a t e d compensated p r i c e e l a s t i c i t i e s o f 121 demand f o r l e i s u r e and goods where y . . = n . . + E . I J i j 1 2 n. . i s the uncompensated p r i c e e l a s t i c i t y of demand, and E_ z^ i s the e l a s t i c i t y of demand f o r commodity i w i t h respect to f u l l income Z. TABLE 4 Compensated Demand E l a s t i c i t i e s (at mean values) On-farm Off-farm P r i c e of Labour Returns Wage Rate Goods L e i s u r e 1 (H-Li) -0.056 0.040 0.015 L e i s u r e 2 (H-L 2) 0.097 -0.158 0.048 Goods 0.038 ' 0.026 -0.068 The compensated p r i c e e l a s t i c i t i e s i n Table 4 are l e s s than u n i t y i n absolute value. The diagonal elements (the own compensated p r i c e e l a s t i c i t i e s ) i n d i c a t e that the. l e i s u r e a s s o c i a t e d w i t h o f f - f a r m work, H-L 2, i s the most p r i c e responsive and H-Lj i s the l e a s t p r i c e responsive. The o f f - d i a g o n a l elements i n d i c a t e that the two types of l e i s u r e and con-sumption goods are a l l net s u b s t i t u t e s . The l a r g e s t s u b s t i t u t i o n p o s s i b i l i t i e s take place between the two types of l e i s u r e , and l e i s u r e 2 tends to e x h i b i t a l a r g e r s u b s t i t u t a b i l i t y w i t h goods than l e i s u r e 1. The r e s u l t that goods (income) and l e i s u r e are net s u b s t i t u t e s i s c o n s i s -tent with, previous e m p i r i c a l s t u d i e s . For example, Wales and Woodland (1976) found that both husband's l e i s u r e and wife's l e i s u r e were 122 s u b s t i t u t e s w i t h income. Next, the c r o s s - e f f e c t s of changes i n the production s e c t o r on the labour supply responses are considered. As discussed before, a model based on independence of consumption and production d e c i s i o n s imposes a model s t r u c t u r e which does not a l l o w any d i r e c t c r o s s - e f f e c t s from the production to the consumption s e c t o r except through the income e f f e c t . The model presented i n t h i s t h e s i s , on the other hand, does a l l o w f o r c r o s s - e f f e c t s and Table 5 shows the q u a n t i t a t i v e relevance of .|hese. TABLE 5 Labour Supply E l a s t i c i t i e s ' w i t h Respect to Net Output P r i c e s • C a l c u l a t e d u s i n g E q u a t i o n (20) Output Land Hired Labour Animal Stock Farm C a p i t a l P r i c e . P r i c e Wage Rate P r i c e P r i c e On-farm Labour Supply 0.390 -0.046 -0.027 -0.015 -0.145 Off-farm Labour Supply -0.849 0.101 0.059 0.033 0.315 Table 5 presents the estimated labour supply e l a s t i c i t i e s w i t h respect to output and input p r i c e changes evaluated at mean val u e s . These e l a s t i c -i t i e s have been c a l c u l a t e d using equation (20) from Chapter I I . As can be expected changes i n the aggregate output p r i c e index have the l a r g e s t e f f e c t on o f f - f a r m and on-farm labour supply i n absolute v a l u e s . A 1-percent increase i n output p r i c e increases the on-farm labour supply by 0.39% and decreases the o f f - f a r m supply of labour by approximately 0.85%. Among the input p r i c e s the smallest e f f e c t i s the one associated w i t h changes i n l i v e s t o c k p r i c e s which have labour supply e l a s t i c i t i e s of minus 0.015. On the other hand, changes i n farm c a p i t a l p r i c e s do have 123 important e f f e c t s on both on-farm and off-farm labour supply. Notice the quantitative magnitude of the cro s s - e f f e c t s from the production sector on both on-farm and off-farm labour supply. The e f f e c t s of output prices are larger than the e f f e c t s of on-farm labour returns or off-farm wage rates on labour supply. The magnitude of these cros s - e f f e c t s suggests that a'model which does not consider the interdependence between the production and consumption sectors neglects q u a n t i t a t i v e l y important e f f e c t s which may be even larger than the d i r e c t wage rate e f f e c t s on labour supply. Table 6 presents the supply and demand e l a s t i c i t i e s c o n d i t i o n a l on L x for outputs and inputs evaluated at the mean p r i c e s . The co n d i t i o n a l e l a s t i c i t i e s (CS..) are defined as 3(Q./Li) q f C = :L_ 1 C S i j 3 q j Q ±/L! * These e l a s t i c i t i e s can be interpreted as net output supply responses assuming that operator and family labour remain constant a f t e r a net out-put p r i c e has changed. The diagonal elements i n Table 6 show the own p r i c e e l a s t i c i t i e s which are p o s i t i v e f o r output and negative for a l l i n -puts. The off-diagonal elements are the condit i o n a l c r o s s - e l a s t i c i t i e s of supply of output and demand f o r inputs. A l l inputs, except land, are p o s i t i v e l y affected by output prices (as r e f l e c t e d i n the p o s i t i v e e l a s t i c i t i e s of column one or i n the negative values of row 1), being the demand f o r hi r e d labour the most responsive to p r i c e changes. The la r g e s t negative e f f e c t of an input p r i c e increase on output supply i s when the farm c a p i t a l p r ices increase. A land p r i c e increase has a p o s i t i v e e f f e c t on output l e v e l s . This may lead one to beli e v e that land i s an i n f e r i o r input which does not seem a •, . 7 very p l a u s i b l e s i t u a t i o n . 124 TABLE 6 Conditional Net Output Supply E l a s t i c i t i e s (at mean values) Prices  Hired Animal Farm Output Land Labour Stocks C a p i t a l Output 0. ,332 0. ,113 - 0 . ,126 - 0 . ,049 - 0 . 269 Land - o . 912 - 0 , .418 0. ,458 - 0 . .016 0 . 888 Hired Labour 1. 557 0. ,797 - 0 , .420 - 0 , .600 - 1 . 334 Animal Stocks 1. 103 - 0 . ,053 - 1 . .107 - 0 . .006 0 . 063 Farm C a p i t a l 0 . 626 0, ,298 - 0 . .260 0, .005 -.0. 660 TABLE 7 Unconditional Net Output Supply E l a s t i c i t i e s Calculated using Equation (22). (at mean values) Prices  Hired Animal Farm Output Land Labour Stocks C a p i t a l Output 0. ,732 0 . 066 - 0 . ,153 - 0 . ,064 - 0 . 414 Land - 0 . ,522 - 0 . 464 0. ,430 - 0 . ,031 0 . 743 Hired Labour 1, .947 0 . 750 - 0 , .447 - 0 . .666 - 1 . 479 Animal Stocks 1. .493 - 0 . 099 - 1 . .134 - 0 . .021 - 0 . 082 Farm C a p i t a l 1. .016 0 . 251 - 0 . .287 - 0 . .010 - 0 . 835 125 Table 7 contains the u n c o n d i t i o n a l supply and demand e l a s t i c i t i e s as defined by equation (22) i n Chapter I I . These e l a s t i c i t i e s measure the a c t u a l market net output supply responses a f t e r the e f f e c t s of output or f a c t o r p r i c e changes on f a m i l y and operator labour supply have been considered. The output supply e l a s t i c i t y obtained i s 0.73 which i s somewhat lower than supply e l a s t i c i t i e s obtained by previous s t u d i e s f o r a g r i c u l -t u r e . For example, Tweeten and" Quance, using d i f f e r e n t procedures, obtained estimates of 0.31, 1.79, and 1.52 f o r long-run aggregate output supply e l a s t i c i t i e s i n U.S. a g r i c u l t u r e . The e f f e c t s of f a c t o r p r i c e changes on output supply are g e n e r a l l y small w i t h the exception of farm c a p i t a l p r i c e s . A 1-percent increase i n farm c a p i t a l p r i c e s induces a 0.4 percent decrease i n output supply. Changes i n land p r i c e s have a small e f f e c t on the demand f o r a l l inputs, w i t h the only exception of h i r e d labour. Factor demands are not very responsive*to changes i n t h e i r own p r i c e s . A l l f a c t o r s present a r a t h e r i n e l a s t i c demand schedule and the demand f o r animal inputs appears to present the l e a s t e l a s t i c w i t h an e l a s t i c i t y c o e f f i c i e n t of only -0.006. These estimates can be compared w i t h previous r e s u l t s f o r U. S. and Canadian a g r i c u l t u r e . Binswanger's (1974) own f a c t o r demand e l a s t i c i t y estimates f o r U.S. a g r i c u l t u r e were -0.34 f o r l a n d , -0.91 f o r labour, -1.089 f o r machinery, and -0.95 f o r f e r t i l i z e r s . Lopez estimates f o r Canadian a g r i c u l t u r e were -0.52 f o r labour, -0.35 f o r farm c a p i t a l , -0.42 f o r l a n d , and -0.41 f o r intermediate i n p u t s . Thus, although the r e s u l t s are not e n t i r e l y comparable because the inputs disaggregation i s d i f f e r e n t and because these s t u d i e s estimated compensated p r i c e e l a s t i c i t i e s , ( i . e . , f o r a constant l e v e l of outp u t ) , the general p a t t e r n of i n e l a s t i c f a c t o r 126 demands i s consistent i n the three studies. The sign structure of the c r o s s - e l a s t i c i t i e s of demand i s of con-siderable i n t e r e s t . An increase i n the hired labour wage rate leads to a decrease i n output and a reduction i n the demand for a l l other inputs except land. The largest depressing e f f e c t of a wage rate increase i s on the demand f o r animal inputs. This i s consistent with the f a c t that production of animal outputs i s more labour intensive than other a c t i v -i t i e s . An increase i n the p r i c e of farm c a p i t a l (say machinery) w i l l cause a reduction i n output production and an increase i n the demand for land but a decrease i n the use of a l l other f a c t o r s , being the e f f e c t on hired labour the l a r g e s t . S i m i l a r l y , an increase i n l i v e s t o c k prices causes a decrease i n the demand f o r a l l inputs y with the largest depressing- e f f e c t on hi r e d labour demand. The e f f e c t of farm operator's education on the structure of the net output vector can be considered by inspecting the c o e f f i c i e n t s associated with the v a r i a b l e educat ion i n Table 2. The e f f e c t of education on the structure of production can be analyzed, as previously discussed, as a proxy f o r te c h n i c a l change. Changes i n technology may have a neutral e f f e c t on the a l l o c a t i o n of resources, i . e . , the marginal rate of s u b s t i -t u t i o n between factor i and factor j stays constant (at a constant factor i / f a c t o r j r a t i o ) or may also have a biased e f f e c t on resource a l l o c a t i o n i f the marginal rates of s u b s t i t u t i o n do change when education changes. Accordingly, one may use the convention of defining a factor i p o s i t i v e bias of education i f i t induces an increase i n the factor i cost share and a factor j negative bias i f i t induces a reduction i n factor j cost share. Neglecting the e f f e c t of education on labour supply, i t can be shown that the cost share of factor i w i l l always increase i f the 127 g c o e f f i c i e n t a s s o c i a t e d w i t h education (a^) i s p o s i t i v e . However, the converse i s not n e c e s s a r i l y t r u e , i . e . , a negative a_. c o e f f i c i e n t does not n e c e s s a r i l y imply that the cost share of f a c t o r j decreases w i t h education. I f there are more than one negative c o e f f i c i e n t s then a l l one can say i s that education induces a negative b i a s f o r at l e a s t one of the f a c t o r s which e x h i b i t s negative a.. When there are more than J one negative a^, the e f f e c t of education on those f a c t o r shares w i l l depend on p r i c e s and cost shares. In Table 2 there i s a negative e f f e c t of education on the shares of two f a c t o r s w i t h negative a_. c o e f f i c -i e n t s when i t s e f f e c t i s evaluated at mean values of p r i c e s and f a c t o r demands. Hence, given the j o i n t s i g n i f i c a n c e of the v a r i a b l e education one may conclude that the e f f e c t of operator's education on resource a l l o c a t i o n i s non-neutral and biased towards l i v e s t o c k forms of c a p i t a l and land and against a l l other f a c t o r s . The negative e f f e c t of education on output l e v e l s i s q u i t e s u r p r i s i n g . I t i m p l i e s that farm operator's increased education leads to reduce the s c a l e of production and farm expenditures perhaps towards other investment o p p o r t u n i t i e s outside a g r i c u l t u r e . Education would a l l o w farmers to consider a l t e r n a t i v e , perhaps more p r o f i t a b l e , sources of investment. Thus, the main e f f e c t of education would be to induce cost savings r a t h e r than an output expansion. I t i s important to note, however, that the negative e f f e c t of education i s on c o n d i t i o n a l output supply and that i n order to c a l -c u l a t e the t o t a l e f f e c t one needs to consider the e f f e c t of education on on-farm work which i s p o s i t i v e . This e f f e c t reduces the s i z e of the negative e f f e c t but i s not s u f f i c i e n t to reverse i t . A 1-percent increase i n education induces a 0.09% r e d u c t i o n i n farm output when t h i s e f f e c t i s evaluated at mean val u e s . In any case, these r e s u l t s should 128 be i n t e r p r e t e d c a u t i o u s l y because the v a r i a b l e education used as a proxy f o r t e c h n i c a l change may not be e n t i r e l y a p p r o p r i a t e , c o n s i d e r i n g that other v a r i a b l e s such as extension expenditures and farm operator's 9 experience have been neglected. F i n a l l y , the cross e f f e c t s of changes i n some parameters of the consumption s e c t o r on the s c a l e of production are considered. Thus, using r e s u l t s obtained i n Chapter I I (equation (24)), the e l a s t i c i t y of the s c a l e of production w i t h respect to the non-labour income v a r i a b l e i s i d e n t i c a l to the e l a s t i c i t y of on-farm labour supply w i t h respect to the same v a r i a b l e . Hence, a 1-percent increase of non-labour income would lead to reduce on-farm labour supply and the s c a l e of production by 0.16% (see Table 3). Thus, i n c r e a s i n g farmer's assets which y i e l d higher non-labour returns l e a d to q u i t e an important c o n t r a c t i o n i n the s c a l e of a g r i c u l t u r a l production ( i n c l u d i n g output supply and input demand). This e f f e c t has been ignored i n previous s t u d i e s which have assumed that changes on the consumption s i d e have no e f f e c t s on net output supply. To assess the importance of the e r r o r made by n e g l e c t i n g these e f f e c t s one can compare the e f f e c t of changes i n non-labour income on output supply w i t h the e f f e c t of input p r i c e s on output supply; the non-labour income (or wealth e f f e c t ) i s indeed l a r g e r i n absolute terms than any of the input p r i c e e f f e c t s , except farm c a p i t a l , on output supply. S i m i l a r l y , i n Chapter I I i t was shown, that the impact of the o f f -farm wage r a t e on the s c a l e of production i s i d e n t i c a l to i t s e f f e c t on on-farm labour supply (equation (25)). Thus, the e l a s t i c i t y of on-farm labour supply w i t h respect to the o f f - f a r m wage r a t e i s -0.107 and, t h e r e f o r e , the e l a s t i c i t y of net output supply w i t h respect to a change i n w 2 w i l l be the same. Hence, a 1-percent increase i n o f f - f a r m wages 129 r e c e i v e d by farmers w i l l cause a c o n t r a c t i o n i n net output supply of approximately 0.1%. Although t h i s e f f e c t i s not as l a r g e as the e f f e c t of non-labour income i t i s by no means n e g l i g i b l e and i s more important than the e f f e c t of land p r i c e s and animal inputs on output supply. 5.3 Further I m p l i c a t i o n s of the Results The most important r e s u l t of t h i s chapter i s the c a t e g o r i c r e j e c t i o n of the hypothesis of independence of consumption and production d e c i s i o n s when a model based on interdependence i s used as the a l t e r n a t i v e hypoth-e s i s . The wide margin by which the n u l l hypothesis has been r e j e c t e d i s c e r t a i n l y q u i t e s t r i k i n g . I t would be i n t e r e s t i n g to repeat t h i s t e s t using household's microdata and r e l a x i n g the assumption of constant returns to s c a l e imposed i n the model based on interdependence used as the a l t e r n a t i v e hypothesis. I n t u i t i v e l y , one may argue that i m p o s i t i o n of the constant returns to s c a l e assumption on the a l t e r n a t i v e hypothesis, w h i l e t h i s r e s t r i c t i o n i s not imposed on the n u l l hypothesis, increases the p r o b a b i l i t y of accepting the n u l l hypothesis. Thus, i f t h i s i s t r u e , r e l a x a t i o n of constant returns to s c a l e should l e a d to an even stronger r e j e c t i o n of the n u l l , hypothesis. Apart from the formal econometric r e j e c t i o n of the hypothesis of independence i t i s important to note that the estimates of the model based on independence conform l e s s w i t h a p r i o r i knowledge than the model based on interdependence. In p a r t i c u l a r , the estimated i n d i r e c t u t i l i t y f u n c t i o n under independence does not s a t i s f y the r e q u i r e d q u a s i -convexity p r o p e r t i e s at any of the sample p o i n t s although i t does s a t i s f y monotonicity. The estimated p r o f i t f u n c t i o n shows the expected gradients w i t h respect to the net output p r i c e s but i t i s convex at only 45% of the 130 sample p o i n t s and, i n contrast - "• w i t h the main model estimated i n t h i s t h e s i s , the own p r i c e net output supply e l a s t i c i t i e s have the c o r r e c t signs at only 32% of the sample p o i n t s . Thus, the e m p i r i c a l evidence shows that a model based on independence of production and consumption d e c i s i o n s should be s t a t i s t i c a l l y r e j e c t e d and a l s o that i t s estimates conform l e s s w i t h economic theory than a model based on interdependence. Another r e s u l t which i s i n t e r e s t i n g i s the one r e l a t e d to the t e s t f o r a f f i n e h o m o t h e t i c i t y . The f a c t that t h i s hypothesis cannot be r e j e c t e d at 1% LOS suggests that the'hypothesis of a f f i n e homothetic preferences, which i s so convenient i n e s t i m a t i n g aggregated labour supply and commodity demand r e l a t i o n s v i a l i n e a r expenditure systems, might not be as bad an approximation of Canadian farmers' preferences a f t e r a l l . However, the strong r e j e c t i o n of homotheticity to the o r i g i n i m p l i e s that econometric models based on such hypothesis should be discarded. With respect to the q u a n t i t a t i v e r e s u l t s of the consumption s i d e , i t appears that the l a r g e m a j o r i t y of Canadian farmers are s t i l l on the upward pa r t of both the on-farm: and o f f - f a r m labour supply.schedules. Hence, one can expect a p o s i t i v e on-farm and o f f - f a r m labour supply response to wage r e l a t e d i n c e n t i v e s . A r e s u l t c o n s i s t e n t f o r a l l obser-v a t i o n s i s that o f f - f a r m work i s more responsive to o f f - f a r m wages than on-farm work i s to on-farm labour r e t u r n s . However, while farmers' t o t a l labour supply (on-farm plus off-farm) response to changes i n the o f f - f a r m wage i s negative, t o t a l labour supply i s p o s i t i v e l y a f f e c t e d by an increase i n on-farm labour r e t u r n s . Farmers' non-labour income has a very strong negative e f f e c t on both on-farm and o f f - f a r m work. How-ever, the negative e f f e c t on o f f - f a r m work i s s u b s t a n t i a l l y l a r g e r than 131 the e f f e c t on on-farm work. An important i s s u e discussed i n Chapter I and I I was whether farmers view o n-farm and o f f - f a r m work - a s p e r f e c t s u b s t i t u t e s . I f t h i s were the case one would expect an - i n f i n i t e A l l e n e l a s t i c i t y o f "substi-t u t i o n between H-L^ and H-L 2. The f a c t that the compensated cross demand e l a s t i c i t i e s between H-Iq and H-L 2 (see Table A) are so low i s i n i n d i c a -t i o n that indeed farmers do not regard on-farm .and o f f - f a r m work a s p e r f e c t s u b s t i t u t e s . A l s o , the average A l l e n e l a s t i c i t y of s u b s t i t u t i o n between H-Lj and H-L z was only 0 .12 and, moreover, i t s l a r g e s t value when eval u -ated at any sample po i n t was l e s s than 0 . 2 . The labour supply responses estimated may have important i m p l i c a -t i o n s f o r the implementation of p o l i c i e s aimed to r a i s e incomes of r u r a l f a m i l i e s (which has been an e x p l i c i t p o l i c y . o b j e c t i v e i n Canada). Farmers' incomes can be increased e s s e n t i a l l y v i a threes mechanisms: (1) to increase on-farm returns to t h e i r work ( i . e . , to increase the p r o f i t l e v e l c o n d i t i o n a l to hours of on-farm work, i f ) ; (2) to s u b s i d i z e o f f - f a r m work, i . e . , to increase the e f f e c t i v e o f f - f a r m wage, w2; (3) to increase the amount of d i r e c t government t r a n s f e r s , that i s , to increase.the non-labour income r e c e i v e d by. farm f a m i l i e s . Each of these mechanisms has a d i r e c t impact on t o t a l farm household's income and a l s o an i n d i r e c t e f f e c t v i a induced changes on labour supply o f f - f a r m and on-farm. An important p o l i c y question i s which of these three mechanisms i s more e f f e c t i v e i n r a i s i n g r u r a l f a m i l i e s ' income. Using the e s t i -mated c o e f f i c i e n t s i t can be shown"'"'"' that f o r each d o l l a r spent by the government v i a mechanism (1) the average farm f a m i l y increases i t s income by $ 1 . 1 0 . I f mechanism (2) i s used i n s t e a d then one d o l l a r spent w i l l y i e l d only 94 cents increase i n the farm f a m i l y income. F i n a l l y , i f 132 mechanism (3) i s used then income w i l l i ncrease by 98 cents. The reason f o r t h i s i s that t o t a l labour supply decreases when mechanisms (2) and (3) are used w h i l e i t increases when mechanism (1) i s used."'""'" Thus, on the b a s i s of purely the cost to the government of r a i s i n g farm f a m i l i e s 12 income, mechanism (1) should be p r e f e r r e d . Using the estimates of the i n d i r e c t u t i l i t y f u n c t i o n one can a l s o determine which of the above three mechanisms should be chosen i f the goal i s to improve farm f a m i l i e s w e l f a r e r a t h e r than merely t h e i r income. One may rank the e f f e c t of one d o l l a r spent v i a each of the mechanisms on the l e v e l of G(ff ,W2,p;Z) , i . e . , which of the above mechanisms w i l l y i e l d the highest increase i n ' t h e i n d i r e c t u t i l i t y f u n c t i o n of the average farm 13 household. The s m a l l e s t e f f e c t on f a m i l y welfare takes place i f mechanism (2) i s chosen w h i l e mechanism (1) and ( 3 ) , being s u b s t a n t i a l l y more e f f e c t i v e than ( 2 ) , y i e l d very s i m i l a r r e s u l t s . This p a t t e r n was c o n s i s t e n t when the three mechanisms were compared at a l l sample p o i n t s . Thus, mechanism (2) i s the l e a s t e f f i c i e n t using e i t h e r the income or w e l f a r e c r i t e r i o n and hence i t should not be used. Mechanisms (1) and (3) are e q u a l l y e f f e c t i v e i n improving w e l f a r e but i t i s reasonable to argue that mechanism (3) might be r e l a t i v e l y simpler to a d m i n i s t r a t e . Hence, i t appears that i f the purpose i s to increase farm f a m i l i e s w e l f a r e one might p r e f e r to increase the amount of d i r e c t government t r a n s f e r s r a t h e r than s u b s i d i z i n g labour r e t u r n s . The e f f e c t s of an income tax cut on farm households' labour supply and on the s c a l e of farm production can a l s o be analyzed using the e s t i -mated c o e f f i c i e n t s . A l l r e d u c t i o n i n the marginal tax r a t e ( i . e . , a 1% r e d u c t i o n i n the 3 c o e f f i c i e n t i n (58)) w i l l induce the average farm household to decrease both on-farm and o f f - f a r m labour supply. The 133 e f f e c t on o f f - f a r m work, however, i s s u b s t a n t i a l l y l a r g e r than the e f f e c t on labour supply to farming a c t i v i t i e s . Thus, w h i l e a 1% tax cut would decrease o f f - f a r m labour supply by 0.155%, a s i m i l a r tax cut would reduce 14 on-farm work by only 0.036%. The e f f e c t on the s c a l e of production i s al s o negative and equal to the e f f e c t on on-farm labour supply. I t i s important to poi n t out that the e f f e c t of an income tax cut on labour supply i s c o n s i s t e n t l y negative when evaluated at a l l sample p o i n t s and at a l l tax brackets. A reason f o r the negative e f f e c t of a red u c t i o n i n the tax r a t e on labour supply i s that the income e f f e c t of the tax cut i s stronger than the e f f e c t of the induced increase i n the a f t e r tax returns to on-farm and o f f - f a r m work. A tax cut w i l l induce an expansion of " f u l l income" which w i l l have a strong depressing e f f e c t on labour supply which w i l l more than o f f - s e t the p o s i t i v e e f f e c t a s s o c i a t e d w i t h the increase i n a f t e r tax returns to labour. The s m a l l r e d u c t i o n i n the s c a l e of pro-duction i s due to the c o n t r a c t i o n of on-farm work. Thus, an income tax cut w i l l have an important depressing e f f e c t on labour supply, a f f e c t i n g mainly the l e v e l s of o f f - f a r m work, and i t w i l l a l s o induce some negative e f f e c t s on farm production. In connection w i t h the production s e c t o r , i t i s important to note the q u a n t i t a t i v e d i f f e r e n c e s between the c o n d i t i o n a l and u n c o n d i t i o n a l net output supply equations (Tables 6 and 7, r e s p e c t i v e l y ) . The impact of on-farm households' labour supply response on output supply, demand f o r animal inputs and farm c a p i t a l i s q u i t e impressive; the output supply e l a s t i c i t y i s doubled and the own demand e l a s t i c i t y f o r animal inputs i s more than t r i p l e d when the e f f e c t of output and input p r i c e changes i n labour supply i s considered. On the other hand, the e f f e c t of 134 households' labour on land demand and hi r e d labour demand.is very small as r e f l e c t e d by the small changes i n t h e i r own conditional and uncondi-t i o n a l e l a s t i c i t i e s . The generally i n e l a s t i c output supply and input demand patterns are rather s u r p r i s i n g considering that, as discussed i n the previous chapter, the estimates can be considered as long-run e l a s t i c i t i e s . In p a r t i c u l a r , the low e l a s t i c i t y of demand f o r hi r e d labour suggests that observed trends towards farm workers unionization i n Canada may lead to higher returns to farm workers with a r e l a t i v e l y small decrease i n hired labour employment, and hence to increase t h e i r share i n t o t a l farm income.""""' The f a c t that the e l a s t i c i t y of demand f o r land with respect to the output p r i c e i s negative i s quite s u r p r i s i n g and may suggest that land i s an i n f e r i o r input. One i n t e r p r e t a t i o n could be that as output prices increase and hence as output l e v e l s are expanded, the output composition changes towards outputs which are les s intensive users of land, i . e . , from crops to poultry and hog production. Thus, although the pure out-put scale e f f e c t may be p o s i t i v e , the negative e f f e c t on demand for land due to changes i n the composition of outputs might predominate. I t i s also possible that the p o s i t i v e e f f e c t of land prices on output supply (which i s equivalent to say that output prices induce a negative e f f e c t on land demand) may be due to the fac t that land q u a l i t y differences among the observations were ignored i n the estimation process. Thus, a sub s t a n t i a l part of the land p r i c e v a r i a b i l i t y may be associated with changes i n land q u a l i t y . Higher land prices may also imply better land and, hence, t h i s may have a p o s i t i v e e f f e c t on output supply. The results obtained as well as the i r p o l i c y implications discussed above should be interpreted only as a preliminary approximation of the 135 p r o b l e m of me a s u r i n g s u p p l y and demand r e s p o n s e s of the s e l f - e m p l o y e d producer-consumer. The major l i m i t a t i o n s of the a n a l y s i s a r e the f o l l o w i n g . 1. A p a r t from the d a t a l i m i t a t i o n s d i s c u s s e d i n Chap t e r IV one needs to c o n s i d e r the problems i m p l i e d by the f a c t t h a t l a c k o f d a t a on o u t p u t p r i c e s a t the census d i v i s i o n l e v e l made i t n e c e s s a r y t o a l l o c a t e p r o v i n -c i a l p r i c e s t o the d i f f e r e n t census d i v i s i o n s a c c o r d i n g t o t h e i r g e o g r a p h i c -a l l o c a t i o n . I t i s c l e a r t h a t many f a r m e r s b e l o n g i n g t o census d i v i s i o n s l o c a t e d i n b o r d e r i n g areas might have s o l d t h e i r p r o d u c t s i n the n e i g h b o u r -i n g p r o v i n c e r a t h e r t h a n i n t h e i r own p r o v i n c e . T h i s would i m p l y t h a t an a d d i t i o n a l s o u r c e o f e r r o r i s g e n e r a t e d by a l l o c a t i n g commodity p r i c e s a c c o r d i n g t o the p r o v i n c e where the census d i v i s i o n s a r e l o c a t e d . 2. The f a c t t h a t a l l o u t p u t s a r e a g g r e g a t e d i n t o one c a t e g o r y i s a l s o a problem, p a r t i c u l a r l y i n c r o s s - s e c t i o n a l a n a l y s i s . Output b u n d l e s a r e r a t h e r heterogeneous i n the d i f f e r e n t r e g i o n s of the c o u n t r y . The d i v i s i a p r i c e i n d e x a p p r o x i m a t i o n used may c o n s i d e r t h e s e d i f f e r e n c e s i n o u t p u t c o m p o s i t i o n v i a changes i n the w e i g h t s o f the i n d i v i d u a l o u t p u t s . I n any c a s e , i t i s w e l l known t h a t the r e s t r i c t i o n s on p r o d u c t i o n t e c h n o l -o g i e s ( i . e . , s e p a r a b i l i t y o r f i x e d p r o p o r t i o n s ) o r on r e l a t i v e p r i c e s ( i . e . , H i c k s ' a g g r e g a t i o n c o n d i t i o n ) r e q u i r e d f o r c o n s i s t e n t commodity a g g r e g a t i o n , a r e v e r y s t r o n g . A b e t t e r p r o c e d u r e would have been t o d i s a g g r e g a t e o u t p u t s i n t o , f o r example, g r a i n s , a n i m a l p r o d u c t s , and o t h e r c o m m o d i t i e s . U n f o r t u n a t e l y , t h i s would have g r e a t l y i n c r e a s e d the number of p a r a m e t e r s t o be e s t i m a t e d i n an a l r e a d y complex model. 3. The a s s u m p t i o n of c o n s t a n t r e t u r n s t o s c a l e i s a l s o r e s t r i c t i v e . M oreover, t h i s a s s u m p t i o n combined w i t h the as s u m p t i o n of no f i x e d f a c t o r s l e d t o an e m p i r i c a l model i n wh i c h the i n t e r d e p e n d e n c e between u t i l i t y 136 and p r o f i t m a x i m i z a t i o n d e c i s i o n s a r e not as prominent ( a l t h o u g h s t i l l i m p o r t a n t ) as i n a model where e i t h e r o f tho s e a s sumptions are r e l a x e d . 4. The a s s u m p t i o n r e g a r d i n g the e x i s t e n c e of a c o n s i s t e n t h o u s e h o l d u t i l i t y f u n c t i o n , a l t h o u g h w i d e l y used, a l s o r e q u i r e s r a t h e r s t r o n g and u n r e a l i s t i c c o n d i t i o n s . By p o s t u l a t i n g a h o u s e h o l d u t i l i t y f u n c t i o n i t i s assumed t h a t the h o u s e h o l d makes c o n s i s t e n t c e n t r a l i z e d d e c i s i o n s . " ^ T h i s i s an a d d i t i o n a l r e s t r i c t i o n o f the e m p i r i c a l a n a l y s i s w h i c h i s i m p o r t a n t t o c o n s i d e r i n e v a l u a t i n g the r e s u l t s and p o l i c y i m p l i c a t i o n s . 5. F i n a l l y , a n o t h e r l i m i t a t i o n of the approach i s the l a c k o f c o n s i d e r a t i o n of changes i n the q u a l i t y of i n p u t s among the o b s e r v a t i o n s . T h i s problem i s p a r t i c u l a r l y i m p o r t a n t f o r the l a n d i n p u t but i t c o u l d a l s o be r e l e v a n t f o r o t h e r i n p u t s such as h i r e d l a b o u r . Thus, p a r t of the o b s e r v e d p r i c e v a r i a b i l i t y on i n p u t s may be due t o d i f f e r e n c e s i n q u a l i t y o f the i n p u t s among census d i v i s i o n s . I n summary, the above l i m i t a t i o n s suggest t h a t the r e s u l t s o b t a i n e d might be s u b s t a n t i a l l y a f f e c t e d by the r a t h e r u n r e a l i s t i c a ssumptions used and, c o n s e q u e n t l y , they s h o u l d be c o n s i d e r e d o n l y as a f i r s t a p p r ox-i m a t i o n t o the pro b l e m and s h o u l d be c a u t i o u s l y i n t e r p r e t e d . 137 Footnotes ''"It i s important to p o i n t out that a l l the c o e f f i c i e n t s , except those a s s o c i a t e d w i t h the demand f o r h i r e d labour and demand f o r farm c a p i t a l equations, were s i g n i f i c a n t at 5% LOS when the t - s t a t i s t i c values were c a l c u l a t e d f o r each c o e f f i c i e n t . 2 However, when the r o l e s of the n u l l and a l t e r n a t i v e hypotheses were reversed i n (74), i . e . , when the n u l l hypothesis was interdependence under constant returns to s c a l e , the c a l c u l a t e d ^ 2 value was not s u f f i c i -e n t l y l a r g e to r e j e c t i t at 1% LOS. 3 Given that at many sample p o i n t s the determinants of most minors were very small negative numbers, one could expect that a s t a t i s t i c a l t e s t would not r e j e c t the n u l l hypothesis that convexity holds. A t e s t f o r convexity using a method suggested by Lau (1978) which uses the f a c t that any p o s i t i v e s e m i d e f i n i t e Hessian.matrix has a Cholesky f a c t o r i z a -t i o n was t r i e d . U n f o r t u n a t e l y , given the l a r g e number of parameters being estimated, the added computational burden i m p l i e d by the method made the e s t i m a t i o n very d i f f i c u l t and expensive and convergence could not be obtained. 4 The means, standard d e v i a t i o n s and the extreme values of the e l a s -t i c i t i e s when evaluated at the d i f f e r e n t sample points are presented i n Appendix 4. This i n f o r m a t i o n i s provided i n Appendix 4 not only f o r the labour supply e l a s t i c i t i e s but a l s o f o r a l l other e l a s t i c i t i e s estimated. ^The average o f f - f a r m labour supply e l a s t i c i t y was -0.126 f o r those on the downward pa r t of the supply curve and 0.231 f o r those on the up-ward s i d e . The average on-farm e l a s t i c i t i e s were -0.017 and 0.165 f o r those on the downward and upward segments, r e s p e c t i v e l y . ^This can be seen more c l e a r l y by i n s p e c t i n g the A l l e n c r o s s -p a r t i a l e l a s t i c i t i e s of s u b s t i t u t i o n , 3 P J denotes compensated Z • (3x*./3p.) A. . = ^— where i i x. • x. or u t i l i t y constant change i n x^ when p_. changes. The estimated A l l e n e l a s t i c i t i e s are A 1 2 = 0.120, 3 = 0.011 and A 2 3 = 0.054. ^ C e r t a i n l y t h i s r e s u l t may a l s o be r e f l e c t i n g unaccounted improve-ments i n the q u a l i t y of land as i t s p r i c e i n c r e a s e across the observa-t i o n s . g In compact n o t a t i o n one may w r i t e the f a c t o r demand equation as: .Q = L x . <j,(q) 4- a J ^ E where a. i s the c o e f f i c i e n t a s s o c i a t e d w i t h education. _ l The cost share equations are: 138. q ± Q ± . q J - i ' K q ) a i L i E q i S i = c (q,E) = c (q,E) + c (q,E) where c(q,E) i s the t o t a l cost evaluated at optimal output l e v e l Q * ± ( q,E). D i f f e r e n t i a t i n g with respect to E 3s.. a.Lip. - — . • .s . 1 x x x 3E x 3E c(q,E) 3c Given that — <_0, i . e . , i t i s assumed that increased education 3 E — . . 9s. does not increase cost, i f a. > 0 then — — > 0. Thus, i n t h i s case x o E education i s fac t o r i biased. However, i f a. < 0 the e f f e c t of E on s. 3 3 i s ambiguous and w i l l depend on p r i c e l e v e l s , educational l e v e l s and on the factor share. 9 Education may also be negatively correlated with farm operator's age, and hence i t may also r e f l e c t the e f f e c t of younger farmers on production decisions. This may be consistent with the i n t e r p r e t a t i o n of education as a proxy for technical change i f one can reasonably argue that younger farmers are more receptive to new technologies. "^The t o t a l income of a farm household ( Y ) can be defined as follows: Y = TTL^ + w 2L 2 + y Hence an increase i n Y (AY) can be decomposed i n the following way: A Y = f f A L j + L 1 A f f + w 2AL 2 + L 2Aw 2 + Ay Al t e r n a t i v e (1) implies to spend one d o l l a r i n increasing on-farm returns, f f . Given that the average farm family works 320 days on-farm, i t implies that A f f = for the average farm family. Hence, using the 9 L i 9L 2 A Y A L 1 A L z estimates f o r -—-— and -rz~ , -rrr = f f —z 1- L,i + w? — z - = 1.10. 3TT 3TT ATT ATT 1 z A i r Thus, the e f f e c t of an increasing of ff by $320* l - e - > °f spending $1 i n subs i d i z i n g if is^ an increase i n the average family's income of $1.1. Si m i l a r l y , the e f f e c t of subsidizing w2 can be calcu l a t e d : A Y A L X A L 2 ~ — = ff + w2 - — + L 2 = 0.94 A w 2 A w 2 ^ A w 2 and, f i n a l l y , the e f f e c t of increasing non-labour income by $1 i s : A Y A L i A L 2 A i + TT — + w2 +1 = 0.98 . Ay Ay z Ay 139 Perhaps the e a s i e s t way of s u b s i d i z i n g ir i s by s u b s i d i z i n g output p r i c e s . I f one d o l l a r per f a m i l y i s spent on s u b s i d i z i n g the output p r i c e r e c e i v e d then the average f a m i l y income w i l l i ncrease by $8.8. This e f f e c t i s s u b s t a n t i a l l y l a r g e r because i n t h i s case not only labour supply inc r e a s e s but a l s o the l e v e l and composition of output and inputs are o p t i m a l l y changed. 12 N o t i c e , however, that these r e s u l t s do not take i n t o account p o s s i b l e e f f e c t s of the subsidy on wages and p r i c e s faced by the farm f a m i l i e s . 13 The e f f e c t of i n c r e a s i n g the net on-farm labour returns on the u t i l i t y l e v e l of the average farm f a m i l y i s : 3U 3?f 3 G 3 G 3 Z 3 if 9Z 3 i f 3U where U i s u t i l i t y l e v e l . Thus, -^r r e f l e c t s the increase on we l f a r e 3TT when if increases by $1. I f $1 per f a m i l y i s spent on r a i s i n g on-farm labour returns then, given that the average f a m i l y works 320 days on-farm, i t means that if increases by $ 3 ^ 0 ' Thus the e f f e c t of spending $1 per 3U 1 fam i l y on i n c r e a s i n g on-farm labour returns i s g~ 320 " -3U = 3G_ 3 G _3_Z d. 3 U . = 3 6 ' J ^ 3w2 3Z 3w2 a n 3y 3Z 3y S i m i l a r l y , using 3w- one can c a l c u l a t e the e f f e c t s of mechanisms (2) and (3) on the l e v e l of u t i l i t y . 14 9 L i B i D e f i n i n g e = ——- -— as the r a t e of change i n due to a 1% L1 P . 3 3 . L i increase i n the marginal tax r a t e , 3 ^ , one can v e r i f y t h a t : " L i 3 . -3 L x 3if 3L T 3w„. 3L X 3Z. + — — + ± _ 3 T r ± 5 3 , ' 3w 2 i 3g ± 3Z ± sg±_ Also 3 if. 8P± 3w 21 = - w.. i - e : w 2 i 1 " i and using the d e f i n i t i o n of Z. i n equation (58): 3Z, ^ = Y ± - y - H(w 2 + if) + Ex. then ^ i - y - H(w 2 4- ft) + Ex. ~Ln w + l w 2 i 1 1 140 where e ~ i s the on-farm labour supply e l a s t i c i t y w i t h respect to TT. V i and e and e • are s i m i l a r l y defined. Using the same procedure one J- . ] W „ . L i . / can f i n d that Y T - y - H(w 2 + rf) + Ex. - e. ~ - e + £, 'L?^:. Lzw„. L]Z. Z. 1 2 i 1 I • l - 3 ± S u b s t i t u t i n g the values f o r the e l a s t i c i t i e s evaluated at mean p r i c e s and using the mean l e v e l s f o r Y^, y, H, w2, TT, Ex and one obtains that e T „ 0.036 and that e_ . . = 0.155. ^"^If the change i n h i r e d labour wage r a t e does not induce any change on output and input p r i c e s then i t i s easy to c a l c u l a t e the e f f e c t of an increase i n the wage r a t e (w) due to workers u n i o n i z a t i o n on t h e i r share i n the t o t a l value of output, d e f i n i n g the labour share by: wL where then S - 0 Q L = h i r e d labour used q = output p r i c e Q = output l e v e l s 3S L fi , \ 1 -z— = — (1 + e ) - — e 3w qQ Lw qQ Qw where e T i s the own wage e l a s t i c i t y of labour demand and i s the out-Lw J Qw put e l a s t i c i t y w i t h respect to w. Given that e„ < 0 then a s u f f i c i e n t Qw SS c o n d i t i o n f o r -r— to be p o s i t i v e i s that e T > -1. Given that £ T i s dw Lw = Lw 8S approximately -0.25, then — i s unambiguously p o s i t i v e . The problems involved i n p o s t u l a t i n g a household u t i l i t y f u n c t i o n are discussed by Samuelson ( 1 9 5 6 ) . CHAPTER VI SUMMARY AND CONCLUSIONS In this f i n a l chapter a summary of the study and the main conclusions are presented. The main purposes of this research have been to analyze the economic behaviour of self-employed farm producers, to provide an empirically feasible model to estimate farm households supply and demand responses and to formally test the often used hypothesis of independence of production and consumption decisions. It has been shown that an important peculiar-ity of self-employed households is that the allocation of resources takes place with reference to market prices for some commodities and to internal non-market prices for other commodities. In particular, the household's labour allocated to i t s own firm (which is normally the most important factor of production used by this type of firm) and the existence of non-traded goods produced and entirely consumed by the household are perhaps the most important examples of commodities which are traded within indi-vidual households and which do not have a "visible" price. The shadow price of these commodities are internally generated and may be endogenous with respect to the households. The importance of these internally generated prices is that they certainly affect the allocation of resources of not only labour and non-traded goods but the allocation of resources and commodities which do have exogenous market prices as well. Another implication is that production and consumption decisions are not li k e l y to 141 142 be independent and hence in order to estimate households' supply and demand responses i t is necessary to adapt the conventional household's model and to modify the conventional econometric framework which is mainly oriented to estimate the consumption and production equations in a disjoint manner. In Chapter II ".a general static model for the self-employed house-hold was developed and, using some simplifications based on assumptions which appear reasonable for Canadian farm households, a model appropriate to empirically analyze supply and demand responses of self-employed households was derived. In Chapter III a model which does not rely on the above simplifications and that can be used in estimating preferences and production technologies of self-employed producers under general conditions was discussed. The use of a dual representation of prefer-ences for a given non-linear budget constraint which contains the produc-tion sector embodied in a conditional profit function was discussed. Moreover, some specific comparative static effects were analyzed and an econometric framework for the estimation of the supply and demand rela-tions was proposed. In Chapter IV an econometric framework to estimate the simplified model developed in Chapter II was developed and a procedure to formally test the hypothesis of independence of production and consumption deci-sions was discussed. In Chapter V the empirical results obtained when the above model was used to estimate preferences and production technol-ogies of Canadian farmers have been presented. The major empirical findings were: 1. The hypothesis of independence between u t i l i t y maximizing and profit maximizing decisions was categorically rejected. Moreover, i t was 143 shown that there are important gains i n explanatory power by estimat-i n g the production and consumption sectors j o i n t l y . These gains occur d e s p i t e that the use of the constant returns to s c a l e assump-t i o n allowed to estimate c o n d i t i o n a l net output supply equations which had no parameter c r o s s - c o n s t r a i n t s w i t h the consumption s e c t o r . 2. The c r o s s - e f f e c t s between the u n c o n d i t i o n a l net output supply equa-t i o n s and the labour supply responses were q u a n t i t a t i v e l y very strong. 3. The f r e q u e n t l y used hypothesis of homotheticity of preferences has been r e j e c t e d . However, the t e s t of the hypothesis of a f f i n e homo-t h e t i c i t y provided l e s s c o n c l u s i v e r e s u l t s ; a f f i n e h o motheticity i s r e j e c t e d at 5% l e v e l of s i g n i f i c a n c e but not at 1%. 4. I t has been shown that farm operator's educational l e v e l has a s i g n i f i c a n t non-neutral e f f e c t on demand f o r i n p u t s . Moreover, education a l s o had a s i g n i f i c a n t e f f e c t on labour supply responses and induced a r e - a l l o c a t i o n of household's labour from on-farm to o f f - f a r m work. 5. F i n a l l y , the number of f a m i l y dependents a l s o had an important e f f e c t on labour supply responses. A d d i t i o n a l l y , i t i s noted that the model estimated e x p l a i n s farm households' consumption and production d e c i s i o n s reasonably w e l l . The model generates r e s u l t s which are c o n s i s t e n t w i t h economic theory on the consumption side,and, although there are some problems w i t h the production s i d e of the model, one may c l a i m that i t represents a s u b s t a n t i a l improve-ment w i t h respect to previous studies.''" In the e m p i r i c a l implementation of the model, a major problem was the use of aggregated r a t h e r than house-hold's data. A number of d e t a i l e d farm households surveys have been r e c e n t l y c a r r i e d out i n Canada and elsewhere y i e l d i n g data which are 144 2 a l r e a d y a v a i l a b l e or w h i c h w i l l be a v a i l a b l e i n the near f u t u r e . T h i s may a l l o w a more p r e c i s e e s t i m a t i o n of f a r m e r s ' p r e f e r e n c e s and farm p r o d u c t i o n t e c h n o l o g i e s . The r e s u l t s o b t a i n e d may be used i n d e r i v i n g p o l i c y i m p l i c a t i o n s . The e f f e c t o f a t a x c u t ( o r n e g a t i v e income t a x schemes) on l a b o u r s u p p l y d e c i s i o n s and the d e r i v e d c r o s s - e f f e c t on o u t p u t s u p p l y and f a c t o r demands has been a n a l y z e d . S i m i l a r l y , t h e r e s u l t s can be used t o a n a l y z e the e f f e c t s of changes i n government t r a n s f e r payments to farm f a m i l i e s on l a b o u r s u p p l y and p r o d u c t i o n d e c i s i o n s . T h i s can be done by u s i n g the l a b o u r s u p p l y e l a s t i c i t i e s w i t h r e s p e c t t o n o n - l a b o u r income c o n v e n i e n t l y a d j u s t e d by the w e i g h t of such t r a n s f e r payment on t o t a l n o n - l a b o u r income. Ano t h e r p o l i c y i m p l i c a t i o n a s s o c i a t e d w i t h the l a r g e d i f f e r e n c e s i n l a b o u r s u p p l y r e s p o n s e s to changes i n on-farm l a b o u r r e t u r n s and o f f - f a r m wage r a t e s was d i s c u s s e d i n C h a p t e r V. The f a c t t h a t the t o t a l l a b o u r s u p p l y e l a s t i c i t y w i t h r e s p e c t t o on-farm l a b o u r r e t u r n s i s p o s i t i v e w h i l e the t o t a l l a b o u r s u p p l y e l a s t i c i t y w i t h r e s p e c t t o o f f - f a r m wages i s n e g a t i v e , l e d t o the c o n c l u s i o n t h a t a p o l i c y o r i e n t e d t o i n c r e a s e on-farm r e t u r n s may be more e f f e c t i v e and l e s s c o s t l y i n i n c r e a s i n g f a r m e r s ' income th a n one o r i e n t e d t o expand o f f - f a r m l a b o u r r e t u r n s . I n c l o s i n g t h i s c h a p t e r i t i s w o r t h w h i l e t o i d e n t i f y t h e s p e c i f i c t h e o r e t i c a l and e m p i r i c a l c o n t r i b u t i o n s of t h i s t h e s i s . (1) T h e o r e t i c a l c o n t r i b u t i o n s : U s i n g r e c e n t developments i n d u a l i t y t h e o r y , a s e t of e m p i r i c a l l y t e s t a b l e p r e d i c t i o n s have been o b t a i n e d f o r the s e l f - e m p l o y e d h o u s e h o l d producer-consumer. The problem of o b t a i n i n g t e s t a b l e p r e d i c t i o n s f o r the f a r m - h o u s e h o l d as a u n i t of p r o d u c t i o n and consumption has been f r e q u e n t l y a n a l y z e d by a u t h o r s c o n c e r n e d w i t h a g r i c u l t u r e i n d e v e l o p i n g economies. T h e i r f o c u s of a t t e n t i o n , however, has c e n t e r e d almost 145 e x c l u s i v e l y i n o b t a i n i n g p r e d i c t i o n s f o r o u t p u t s u p p l y r e s p o n s e s ( s e e , f o r example, Sen or Hymer and R e s n i c k ) . G i v e n t h a t o u t p u t s u p p l y r e s p o n s e s t o p r i c e changes a r e i n g e n e r a l ambiguous i n t h e t h e o r e t i c a l models, i t has been v e r y d i f f i c u l t t o e m p i r i c a l l y t e s t the v a l i d i t y o f the f a r m -h o u s e h o l d model. A t h e o r e t i c a l c o n t r i b u t i o n of t h i s t h e s i s has been to show the e x i s t e n c e of o t h e r p r e d i c t i o n s w h i c h can be used t o e m p i r i c -a l l y t e s t the farm h o u s e h o l d model. In, p a r t i c u l a r , a c o n s t a n t - u t i l i t y o u t p u t s u p p l y e x p r e s s i o n has been d e r i v e d and i t has been shown t h a t , i n c o n t r a s t w i t h the v a r i a b l e - u t i l i t y ( o r " M a r s h a l l i a n " ) o u t p u t s u p p l y r e s p o n s e , i t i s unambiguously n o n - n e g a t i v e . S i m i l a r l y , c o n s t a n t - u t i l i t y c o m p a r a t i v e s t a t i c e x p r e s s i o n s f o r changes i n the e q u i l i b r i u m l e v e l of (on-farm) h o u s e h o l d work w i t h r e s p e c t t o net o u t p u t p r i c e s and f i x e d c ap-i t a l have been d e r i v e d and shown to be n o n - n e g a t i v e . I t was a l s o p r o v e d t h a t the b e h a v i o u r a l e q u a t i o n s f o r net o u t p u t s u p p l y , shadow p r i c e of farm c a p i t a l , and demand f o r consumption commodities a r e homogeneous of degree z e r o i n consumption good p r i c e s , net o u t p u t p r i c e s , and n o n - l a b o u r income. I n p a r t i c u l a r , the net o u t p u t s u p p l y e q u a t i o n s are no l o n g e r homogeneous of degree z e r o i n net o u t p u t p r i c e s as i n the c o n v e n t i o n a l case and the e q u a t i o n f o r the e q u i l i b r i u m l e v e l o f h o u s e h o l d work ( o n -f a r m ) , i n g e n e r a l , does not s a t i s f y any homogeneity c o n d i t i o n . F i n a l l y , a l l symmetry c o n d i t i o n s p r e v a i l i n g i n the c o n v e n t i o n a l models of the house-h o l d and of the f i r m were found t o h o l d i n the f a r m - h o u s e h o l d model. Moreover, i t was a l s o p r o v e d t h a t cross-symmetry c o n d i t i o n s between the p r o d u c t i o n and consumption s e c t o r s c o n s t i t u t e an a d d i t i o n a l d i s t i n c t i v e f e a t u r e of the f a r m - h o u s e h o l d model. I n p a r t i c u l a r , the compensated e f f e c t of a change i n n e t o u t p u t p r i c e on demand f o r consumer goods i s i d e n t i c a l t o the c o n s t a n t - u t i l i t y e f f e c t of a change i n consumer good 146 prices on net output supply. S i m i l a r l y , compensated changes in consumer goods due to changes in fixed c a p i t a l are i d e n t i c a l to minus the constant-u t i l i t y change in the shadow price of fixed c a p i t a l associated with changes in consumption good pric e s . (2) Empirical Contributions: Two non ad-3 hoc empirically estimable models of the farm-household behavioural equa-tions have been derived. A s i m p l i f i e d model which r e l i e s on the assump-tions of constant returns to scale and no fixed factors was estimated using Canadian farm census data. The advantages of this model with re-spect to a more general model which do not use these assumptions are that standard dual i t y theory can be used in deriving the estimating behavioural equations of the farm-household, that i t allows to disentangle household preferences from the farm production technology and that i t i s r e l a t i v e l y easier to estimate since the equations associated with the production sector are a l l l i n e a r in the parameters. A problem of this model i s that, although some basic linkages between the u t i l i t y and p r o f i t maximiza-tio n equations e x i s t , these linkages are less important than in a model which does not r e l y on the constant returns to scale and/or no fixed fac-tors assumptions. A more general model which does not rely on any r e s t r i c t i v e assump-tions regarding production technologies or household preferences was also developed and a f e a s i b l e procedure to estimate i t was indicated. Major advantages of this model with respect to the previous one are that the interdependences between production and consumption decisions become now more apparent and that i t i s more general. It was shown that even the s i m p l i f i e d model appears to be more approp-r i a t e to estimate consumption and production technologies than the conven-t i o n a l model based on independence of production and consumption decisions. 147 A f o r m a l e c o n o m e t r i c t e s t of the h y p o t h e s i s of independence was p e rformed r e s u l t i n g i n a c a t e g o r i c r e j e c t i o n of such h y p o t h e s i s . Moreover, the s i m p l i f i e d model was used t o j o i n t l y e s t i m a t e the e q u a t i o n s of l a b o u r c h o i c e between on-farm and o f f - f a r m employment and the c o n d i t i o n a l net o u t p u t s u p p l y f u n c t i o n s of C a n adian f a r m - h o u s e h o l d s . I n summary, the major e m p i r i c a l c o n t r i b u t i o n s of t h i s t h e s i s have been to p r o v i d e a l t e r n a t i v e e m p i r i c a l l y e s t i m a b l e models of the farm-house-h o l d w h i c h e x p l i c i t l y account f o r i n t e r d e p e n d e n c e of p r o d u c t i o n and consump-t i o n d e c i s i o n s , t o e s t i m a t e one of t h e s e models u s i n g Canadian census d a t a , thus q u a n t i f y i n g l a b o u r c h o i c e r e s p o n s e s , net o u t p u t s u p p l y r e s -ponses and the c r o s s e f f e c t s of changes i n the p r o d u c t i o n s e c t o r on c o n -sumption d e c i s i o n s and v i c e - v e r s a . An a d d i t i o n a l e m p i r i c a l c o n t r i b u t i o n has been t o f o r m a l l y t e s t the h y p o t h e s i s of independence of p r o d u c t i o n and consumption d e c i s i o n s . To the b e s t of the a u t h o r ' s knowledge, no one of the above-mentioned a n a l -y s e s have p r e v i o u s l y been p e r f o r m e d and hence t h e y may be r e g a r d e d as o r i g i n a l c o n t r i b u t i o n s . ^ 148 Footnotes ^"Danielson, f o r example, estimated a p r o f i t function for Canadian a g r i c u l t u r e assuming independence of production decisions from consump-t i o n decisions. His estimated p r o f i t function did not s a t i s f y the convexity property at any sample points. Moreover, he obtained the "correct" sign f o r the own p r i c e e l a s t i c i t y for two out of three outputs and the corresponding negative sloping demands for only one out of four inputs considered. 2 When t h i s thesis was at a l a t e stage, data from a farm household's survey c a r r i e d out i n the province of Saskatchewan were made a v a i l a b l e . 3 The term non ad-hoc i s used here to s i g n i f y that d i r e c t linkages between the t h e o r e t i c a l model and the estimating equations e x i s t . In p a r t i c u l a r , the r e s t r i c t i o n s implied by the optimization hypothesis con-sidered are f u l l y used either by imposing them on the estimating equations or by testing them. 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Proof of Proposition 1 Proposition 1 refers to the case when (a) the household members are indifferent between working in the family firm or elsewhere, and (b) family labour and hired labour are not identical factors of production. Thus, one may formulate the problem in the following manner: max U(H-L1-L2,X) X,L l 5L2 (i) px ^  TrCq;]^) + w zL 2 + y (i i ) L 2 ^  0 ( i i i ) L x ^ 0, X :> 0 where a l l symbols are defined in Chapter II. Assuming differentiability and an interior solution for X and 1^ , the Kuhn-Tucker f i r s t order conditions are: 9TT (q; ) u i " x - I L T " = 0 ( A ) Ux - X w2 - y = 0 (b) U 2 - Xp = 0 (c) L 2 ^ 0; y L 2 = 0; y ^  0 (d) -px + Tr(q;L1) + w 2L 2 + y = 0 (e) Where X and y are the Lagrangean multipliers associated with con-straints (i) and ( i i ) , respectively. Now, i f y = 0, i.e., i f L z > 0, outside salaried work takes place then using (a) and (b) one obtains that = W 2 ' Thus, the shadow price of household's members working in the family firm is equal to the exogenous wage rate received by family members while working outside the family firm, w2. Hence, the unconditional profit function TT°(q,w 2), can now be defined as: Tr°(q,w 2) 5 {max i^q;!}) - w2L2} which implies that 7T°(q,w2) is independent of consumer preferences, con-sumption good prices and non-labour income. To show that when L 2 > 0 the consumption side of the model is inde-pendent from the production side except through the income effect, an indirect u t i l i t y function G can be defined: 1-5.5 G(p,w2;Z°) = { max ITCH-I^-I^,-X) ; H - L ^ L ^ X (i) px + w 2(H - L 1 - L 2 ) . 4 w2H + y + Tr°(q,w2) = Z° ( i i ) H - L T _ - L 2 ^ 0; X ^ 0} . Thus, given that and L Z have the same price, w2, the household maxi-mizes for H - L 1 - L 2 rather than for and L 2 as i t occurs when their prices differ. Hence, the only effect of the production sector on the consumption decisions takes place through the income effect via Z?. If L 2 = 0 and hence u > 0 then 9 T r ( q ; L 1 ) = W 2 + Ji . 9 L 1 X 1 . e., the shadow price of Lq is in general greater than w2. Therefore, w2 cannot be used as the price of and hence the profit function would be: ". U VI \ ir(q,wz + j^) = {max ^(qjL]^) - (w2 +-^)L 1} . L l Where y* and X* represent optimal values. However, Vi* = y*(p,q,y,w2) and X* = X*(p,q,y,w2) and hence the profit function Tr(q,w2,p,y) w i l l not be independent of the consumption decisions. On the other hand, using the f i r s t order conditions (a) to (e) i t is clear that when L 2 = 0 then X -^-j— = Xw2 + y and the household solves for H-L^Pjqjy), X(p,q,y), X(p,q,y) using equations (a), (c) and (e). Hence the indirect u t i l i t y function w i l l now be G(p,q,y) and the effect of changes on the production side of the model on the consumption side w i l l not only be through the income effect. Note that the actual function G w i l l not only depend on the structure of the u t i l i t y function but also on the structure of the profit function 2. Proof of Proposition 2 Proposition 2 refers to: (a) household's members are not in d i f f e r -ent between working in their own firm and elsewhere, (b) family labour is an identical input as hired labour. Now the problem would be: max fKH-Lj ,H-L2 ,X) : (i) px <^  TT (q 0 ;Lg+L]_)+w2L2 + y - w0Lg H-Ll5H-L2,X ( i i ) L 0 > 0, ( i i i ) L 1,L 2;X> =0 where Lg = hired labour used by the family firm TT = profit function conditional on Lg and L^ q 0 = net output price vector excluding w0. Assuming that Lj > 0, L 2 > 0, X > 0; the Kuhn-Tucker f i r s t order condi-tions are A + e = 0 (c') 156. . 3 T f ( q 0 ; L 0 + L x) I W W l = 0 (a') U 2 - A w2 + y = 0 (b') 'STTCqojLo+I^) ; W L Q > 0; L Qe = 0; E ^ 0 (d') U z - AP = 0 (e*) - px + u ( q 0 ; L o + L i ) + w 2 L 2 + y " W 0 L 0 = 0 where £ is the Lagrangean multiplier associated with constraint ( i i ) . Notice that BTrCqo^o+L!) 9 T r ( q 0 JLQ+LT) 3LQ 9L^ and hence i f L Q > 0, then E = 0 and hence using (c') 3-fr = w = 3L 0 °. 9L X Hence, the unconditional profit function w i l l be 7r(qg-.WQ), independent from the consumption sector of the model. Also, the indirect u t i l i t y function G w i l l be: G(p,w0,w2;Z) E { max U(H-L1,H-L2,X):px + wQ(H-I^) H—L^,H—L2,X + w2(H-L2) H(w0+w2) + 7r(q0.,w0) + y E Z} . Thus, when L 0 > 0 the indirect u t i l i t y function w i l l be affected by the production technology only through the income effect via Z. Notice, that the functional form of G depends only on the structure U, being totally independent from the production technology represented by TT-On the other hand, i f = 0 then E > 0 and hence the profit func-tion and the indirect u t i l i t y function w i l l be functionally related. In this case in order to obtain solutions for H-LT, H-L2, X, A and E i t is necessary to use the f i r s t order conditions (a'), (b'), ( c T ) , (e'), and (g 1) and hence G w i l l be dependent on the parameters of TT not only through the income effect but via direct price effect. Similarly, TT w i l l also depend on the household's preferences and on p, w2 and y. 3. Proof of Proposition 3 Proposition (3) assumes the following model: max U(H-L1-L2,X) : (i) px •< T T ( q 0 ;-Ln+L0) - w 0L Q + y + w 2L 2 ( i i ) L 0 > 0, L 2 > 0, Lj > 0 ( i i i ) X ^  0. Assuming an interior solution for X, the Kuhn-Tucker f i r s t order conditions are: .157 U i - A 3TT 3LT_ U i - X W2 + P u 2 - x p = o + n = o 0 r 3TT 3L + e = 0 L 0 > 0 1 2 > 0 L x >: 0 eL 0 = 0; TILI = 0; (a (b (c (d e >. 0 (e 0 (f 0 (g 0 L 0 - y - W2L2 = 0 (h where n i s the Lagrangean m u l t i p l i e r a s s o c i a t e d w i t h the c o n s t r a i n t Li >. 0. From (a") and (b") we have that 3TT 3LI = W2 + y - n and from (d") we have that 3TT 3L Note that by assumption = w 0 + -3TT 3L then e 0 3TT 3Ln I f L y > 0 and L x > 0, but L 2 = 0, n = 0 and hence 3TT 3TT 3L 0 3 L l = w 0 = w 2 + y Thus the p r o f i t f u n c t i o n (unconditional) w i l l be Tr(qrj,wo) and the i n d i r e c t u t i l i t y f u n c t i o n w i l l be G(p,w 0,Z) being TT and G independent. I f L 2 > 0 and L : > 0 but 0 then n = y = 0 and hence the u n c o n d i t i o n a l p r o f i t f u n c t i o n w i l l be T r (qo , W 2 ) and the i n d i r e c t u t i l i t y f u n c t i o n w i l l be G(p,W2,Z) again being i r and G independent. 4. Proof to C o r o l l a r y 3.1 From ( a " ) , (b") and (d") i n the proof of p r o p o s i t i o n (3) one obtains that n - £ - y W2 Wn + '0 ' X hence i f W2 > WQ then n - £ - u > 0 necessary that n > 0. 5. Proof to C o r o l l a r y 3.2 Again using the r e l a t i o n n - £ - y Given that £ ^  0 and p > 0 i t is This i m p l i e s that L j = 0 (using g"). w = w 0 4-I f W2 < WQ then n - £ - y < 0. Now, suppose that the f i r m uses h i r e d labour and the household's members do work outside the f a m i l y farm, i . e . , that Lg > 0 and L 2 > 0. Equation (e") and ( f " ) imply that £ = y = 0 and given that n ^  0 one obtains that n - £ - y >: 0, which 158 contradicts the original assumption that w2 < WQ. Hence, i f L 0 > 0 and L 2 > 0 then w2 cannot be less than wg. Proof Of Proposition 5 Define G(p,y) = {max U(x) : E(x,p) < y} x where E(x,p) <^  y is a non-linear budget equation. It is assumed that E(x,p) = y at the optimum. The variables x, p, y are defined as follows: x = consumption bundle p = a price vector y = "income" Proposition 5 says that G(p,y) w i l l be quasi-convex in p i f and only i f E(x,p) is concave in p. Define B E X : E(x,p) <_ y B' E X : E(x,p') ^ y B" E x : E(x,p") £ y where p" = tp + (l-t)p' and 0 < t < 1. It is necessary to show that G(p,y) £^  k and G(p',y) <^  k w i l l imply that G(p",y) < k i f and only i f E(x,p) is concave in p. This is equivalent to show that i f E(x,p) is concave in p then B" w i l l be contained in either B or B' or both, i.e., B" <= BUB'. Suppose E(x,p) is concave in p but B" <f: BUB' then E(p",X) = E[pt + p'(l-t),xj < y but some of the x w i l l not be contained in BUB', hence: (i) E(p,x) > y ( i i ) E(p',x) > y. Now multiplying (i) by t and ( i i ) by (1-t) and adding: (i) tE(p,x) > ty ( i i ) (l-t)E(p',x) > (l-t)y ( i i i ) tE(p,x) + (l-t)E(p',x) > y. But i f E(p,x) is concave in p then: tE(p,x) + (l-t)E(p',x) 4 E ( t p + (1-t) p',x). Hence E(tp + (1-t) p',x) = E(p",x) > y which contradicts our original assumption ( i t contradicts A). This implies that i f E(p,x) is concave in p then B" cB'UB, i.e., that G(p,y) is quasi-convex. Proposition 7. If the production technology exhibits constant returns to scale and i f there are no f i x inputs then the conditional profit function is homogeneous of degree one in L]_ and hence i t can be written as Tr(q,wA;Li) = L 1if(q,w A). 159 Proof. TT(q,w A;Li) = (max qQ - wAz : F(Q,z;Lx) = 0} Q,z ^(c^w. ;tL) = {max qQ - w.z : F(Q,z;tLi) = 0} A A Q,z = {max q(tQ) - w A(tz) : FCtQjtzjtLi) = 0} tQ,tz = {max t(qQ - w z) : t F C Q j Z j L x ) = 0} Q,z = t{max qQ - w.z : F^z;!^) = 0} Q,z = tir(q,w A:L 1) . Hence ir(q,wA;L1) is homogeneous of degree one in L i and can be written as L i*fr(q*w A). ,160 APPENDIX 2 Data Transformations 1. Rental P r i c e C a l c u l a t i o n s In c a l c u l a t i n g r e n t a l p r i c e s of durable f a c t o r s of production one may f o l l o w Diewert's (1972) method which i n t u r n i s based on Walras' approach. 1.1 Rental P r i c e of Land and B u i l d i n g s . 1 - 6 , PT = PT - Pt (A.l) where p T = r e n t a l p r i c e of land P T = current asset p r i c e of land P T = expected p r i c e of land at the end of the p e r i o d 6,j, = d e p r e c i a t i o n r a t e of land r = i n t e r e s t r a t e F o l l o w i n g Danielson (1975) i t i s assumed that r =0.05 and s^, = 0. In t h i s case A . l can be w r i t t e n as P - P T T . P + r T  PT 1 + r T and d e f i n i n g P T = PT - PT  PT Thus where p T i s the expected r a t e of growth of the asset p r i c e . PT = 1 7 7 ^ ' <A-2> B a r i c h e l l o (1979) has estimated the r a t e of growth of l a n d p r i c e s i n 1970 at 3%. Hence, i t i s assumed that = 0.03. 1.2 Rental P r i c e of Animal Stocks I t i s assumed that the expected r a t e of change i n the asset p r i c e of l i v e s t o c k s i s zero. Furthermore, a long run steady-state e q u i l i b r i u m , i n the sense that producers are s a t i s f i e d w i t h t h e i r animal stocks and do not intend to expand i t , i s assumed. F i n a l l y , i t i s assumed that each census d i v i s i o n i s e n t i r e l y s e l f - s u f f i c i e n t i n producing the animals to r e p l a c e those which are s o l d . In t h i s case the animal stock r e n t a l p r i c e (q^) formula i s simply: 16T qA 1 + r (A.3) where P^ i s the asset p r i c e of the animal stock. 1.3 Rental Value of Machinery and Equipment I t i s assumed that the r e n t a l p r i c e of machinery and equipment i s constant across the observations given that there are no data on asset p r i c e s . However, i n order to c a l c u l a t e net p r o f i t i t i s necessary to estimate r e n t a l values of machinery and equipment. Given that there are data on the asset value of machinery and equipment one can c a l c u l a t e i t s r e n t a l v a lue, v^. Thus, assuming that the asset p r i c e s are not expected to i n c r e a s e , VM =" r + 6 M 1 + r V M (A.4) where VM E V„ = p^'M i s the r e n t a l value of machinery and equipment P^«M i s the asset value of machinery and equipment M 6„, = d e p r e c i a t i o n r a t e . M A d e p r e c i a t i o n r a t e 5^  = 0.13 as c a l c u l a t e d by Danielson i s used. Data A v a i l a b l e (Data Code) XI = Number of farm households i n each census d i v i s i o n X2 = Operator's o f f - f a r m labour income X3 = Spouse's o f f - f a r m labour income X4 = Other household's members o f f - f a r m labour income X5 = Operator's o f f - f a r m work ( i n days) X6 = Operator's t o t a l number of days worked (on-farm and off-farm) X7 = Spouse's t o t a l number of days worked (on-farm and off-farm) X8 = Other household's members t o t a l number of days worked X9 = T o t a l non-labour income f o r a l l household's members ( i n t e r e s t s , d i v i d e n d s , government t r a n s f e r s , excludes returns to farm c a p i t a l ) X10 = T o t a l number of people age 13 above l i v i n g i n the farm X l l = T o t a l taxes paid by the household's members X12 = T o t a l value of farm s a l e s X13 = T o t a l o p e r a t i o n a l costs per farm: f e r t i l i z e r s , spraying m a t e r i a l s , machinery and land r e n t a l s , f u e l , seed and wages X14 = T o t a l value of land and b u i l d i n g at 1970 estimated market p r i c e s X15 = T o t a l value of machinery and equipment at 1970 estimated market p r i c e s X16 = T o t a l value of l i v e s t o c k s on the farm at 1970 estimated market p r i c e s X17 = Number of improved acres of land X18 = Output p r i c e s by province i n 1970 (major g r a i n s , animal products, and f r u i t and vegetables) X19 = Estimated p r i c e s of major l i v e s t o c k s on the farm ( c a t t l e except cows, m i l k cows, hogs, and sheep) by province X20 = T o t a l h i r e d labour ( i n days) used by the farms 162 X21 = T o t a l wages paid to h i r e d labour X22 = Operator's s c h o o l i n g years X23 = Average d i s t a n c e to urban centers (miles to the c l o s e s t m e t r o p o l i t a n area w i t h a pop u l a t i o n of 100,000 or more) X24 = T o t a l value of s a l e s by each census d i v i s i o n . Notes: 1. A l l data correspond to the year 1970 and are a v a i l a b l e by average farm household per census d i v i s i o n . 2. V a r i a b l e s X2, X3, X4, X5, X6, X7, and X8 are al s o disaggre-gated by household's members sex. 4. Data Required Inspecting equation (55) i n Secti o n (4.1) one can v e r i f y that the f o l l o w i n g data are r e q u i r e d : 1) T o t a l number of days of on-farm work f o r a l l household's members ( 1 ^ ) . 2) T o t a l number of days of o f f - f a r m work f o r a l l household's members ( L 2 ) 3) A f t e r tax household's " f u l l income" ( Z ± ) . 4) Net returns to household's work on the farm ( ff ) . 5) Off-farm wage r a t e (per day) earned by the household's members who work o f f - f a r m (w 2). Using the data code of s e c t i o n 3 of t h i s appendix the above v a r i -ables are defined i n the f o l l o w i n g way: The on-farm work i s : Lj. = (X6 - X5) + (X7 - — ) + (X8 - — ) (A. 5) w° w° X2 where w^  = TTT- i s the operator's o f f - f a r m wage r a t e . X5 For female household's members we use w i n s t e a d of w°, where w = E'w°, where e i s a f a c t o r of p r o p o r t i o n of average women's wage/ men's wage, which i s c a l c u l a t e d by province using data from the Labour Force Survey f o r average wage of males and females. The o f f - f a r m work i s c a l c u l a t e d : v c , X3 + X4 L 2 = X5 + (A. 6) w° where we a l s o use w° in s t e a d of w^  f o r female members. The net returns to household's members' work on the farm or e q u i v a l e n t l y the farm's p r o f i t per u n i t of f a m i l y labour i s defined by: X12 - X13 - V TT = (A.7) where V i s the r e n t a l value of a l l durable f a c t o r s of production (land and s t r u c t u r e s , l i v e s t o c k c a p i t a l , and machinery and equipment), i . e . , f r - p r V 1 + r V m + T 1 + r r.+ 6 V A + 1 X V M (A. 8) A 1 + r M where V T i s the asset value of land and b u i l d i n g s V i s the asset or stock value of animal stocks on the farm A V i s the asset value of the machinery and equipment stocks on M the farm. 163 . The average o f f - f a r m wage r a t e received by household's members i s c a l c u l a t e d as a weighted average of the male and female wage r a t e s where the weights are a l l o c a t e d according to the p r o p o r t i o n of o f f - f a r m wage income of males and females. The a f t e r tax " f u l l income" i s (see equation (48) i n Chapter I V ) : Z ± = 365'X10(ft4v 2)+(l-8 )(X9+Y'V)+g Y T ± r- T (A.9) where y i s the p r o p o r t i o n of farm c a p i t a l owned by the household and T 3., Y . , and T. stand f o r the marginal tax r a t e at tax bracket i , the i i i s m a llest taxable income at tax bracket i and taxes p a i d at income T Y ^, r e s p e c t i v e l y . 4.2 Data f o r the C o n d i t i o n a l Net Output Supply Equations The data r e q u i r e d f o r e s t i m a t i n g the c o n d i t i o n a l net output equa-t i o n s are the f o l l o w i n g (see equation (60) i n Chapter I V ) : 1. 2. 3. 4. 5. 6. Aggregate output p r i c e index ( q i ) . Rental p r i c e f o r land and b u i l d i n g s ( p ^ ) . Wage r a t e p a i d by the household to the h i r e d labour used by the farm (q 2) . Rental p r i c e of l i v e s t o c k c a p i t a l ( q s). Operator's s c h o o l i n g years (E). Net output supply per day of on-farm work 0^, ^ ) L l L l L l L l L l An aggregate p r i c e index by province i s c a l c u l a t e d and then these p r i c e s are assigned to the d i f f e r e n t census d i v i s i o n s according to the province where they are l o c a t e d . The output p r i c e index i s c a l c u l a t e d as f o l l o w s : a quad r a t i c mean of order one p r i c e index, which i s a super-l a t i v e index number, i s used. This index i s exact f o r a ge n e r a l i z e d L e o n t i e f cost f u n c t i o n as has been shown by Diewert (1977c): qO.YO ' l l l l TV 0! l i 9l = l i q l k l k TV 3 i (A.10) l k l k where x l i q l i TV, are the d i f f e r e n t outputs considered are the p r i c e s of the outputs stands f o r the t o t a l value of output i n each province. The s u p e r s c r i p t 0 i n d i c a t e s the value of the v a r i a b l e s i n the benchmark province (where q^ = 1) and the s u p e r s c r i p t j stands f o r the value of the v a r i a b l e s i n each of the other provinces. 164-Th e specific outputs Y ^ considered were wheat, barley, oats, other grains, various f r u i t crops, vegetables, poultry, eggs, dairy products, beef, hogs and sheep and lambs, corn, potatoes, and tobacco. Data on prices and outputs per province were obtained from different Statistics Canada publications, including catalogue numbers 23-203, 23-201, 21-513, 23-202, and 23-203. In order to calculate the rental price of land and buildings, the asset price of land, PT, was f i r s t estimated: Pip = x]_7 ' (A.11) Next P T is used in (A.2) in order to obtain p T, i.e., the rental price of land and buildings. The wage rate (per day) paid for hired labour is directly obtained from the census data: q 2 = fi • (A.12) For calculating the rental price index of livestocks f i r s t the rental prices for the different livestock categories are calculated using (A.3). The following livestock categories are considered: cows, cattle except cows, pigs, sheep and lambs. Provincial prices obtained from Statistics Canada publications nos. 21-514, 21-513, and 23-203 are used. Next an aggregate price index is calculated using a formula equivalent to (A.10) where now the q correspond to the rental prices of the di f -ferent livestock categories and the Y,__^  correspond to the quantity of animals per category. Thus, an aggregate price index per province is calculated and then the corresponding provincial price is assigned to the census divisions. In calculating the net output supplies per day of on-farm work the following procedure was used: the total value of output sales by census division (x 24) is divided by the total number of on-farm days of work (Li) per census division as estimated by (A.5). This operation yielded the total value of output per day of on-farm work per census division. Next the value of output L x was divided by the output price index as calculated in (A.10), which gave a quantity index of output per day of on-farm work. Finally, this index was transformed in terms of average farm household by dividing by the total number of farms in each census division. A similar method was used in obtaining the quantity indexes for demand for land and structures, hired labour, livestocks on farm and farm capital. The variables X14, X15, X16, and X13 were used for this purpose. Finally, the variable operator's schooling years are directly obtained from census data (variables X22 and X23, respectively). 4.3 Summary Statistics of the Data Used Table A.l contains the mean values, standard deviations and extreme values of the most important variables considered in the analysis. It is interesting to note the large differentials between the mean values of the off-farm wage rate received by farmers and their on-farm labour returns. Off-farm wage rates are substantially larger than on-farm 1 6 3 TABLE A . l Mean, Standard D e v i a t i o n and Extreme Values of Some Important V a r i a b l e s Considered Standard Mean D e v i a t i o n Minimum Maximum 30.00 6362.00 5.50 93.00 Number of farm households per census d i v i s i o n 2090.00 1492.00 Before tax returns to f a m i l y ... labour farm work ($ per day) 22.20 13.60 Before tax o f f - f a r m wage r a t e ($ per day) 50.90 15.30 Days of on-farm work by the operator and h i s f a m i l y 320.40 Days of o f f - f a r m work by the operator and h i s f a m i l y 80.60 Value of output per farm ($) 16707.50 Rental value of land per farm ($) 2016.00 Hir e d labour expenditures per farm ($) 1193.50 Rental value of animal stocks ($) per farm 651.50 Rental value of farm c a p i t a l ($) 6240.00 Rental p r i c e of land ($ per acre) 6.03 8.41 Hir e d labour wage r a t e ($ per day) 12.65 2.69 Farm operator's years of sch o o l i n g 8.6 1.05 Distance to urban centres (miles) 20.90 12.00 23.60 101.00 29.50 253.40 380.90 18.50 53.70 122.60 7819.00 7877.00 49683.00 1274.00 530.00 7050.00 1556.00 214.00 7286.00 300.60 102.00 1747.00 2505.00 2439.00 16281.00 1.24 6.32 6.00 5.00 49.50 20.10 10.20 148.40 166 labour returns i n a l l census d i v i s i o n s . Moreover, the d i f f e r e n c e s are s t a t i s t i c a l l y s i g n i f i c a n t at 95% even i f i t i s assumed that t r a n s p o r t a -t i o n costs from the farm to the o f f - f a r m working place are 25% of the o f f - f a r m d a i l y wage r a t e . This i s another i n d i c a t i o n that o f f - f a r m work and on-farm work are not s u b s t i t u t e s i n consumption, i . e . , that there are u t i l i t y d i f f e r e n c e s between the two works. Table A.2 and A.3 show the d i f f e r e n t expenditures as a p r o p o r t i o n of f a m i l y labour farm returns and the cost shares of the v a r i o u s i n p u t s . An important i m p l i c a t i o n of these data i s the l a r g e weight of f a m i l y labour i n the cost s t r u c t u r e . Thus, land costs are l e s s than 30% of the f a m i l y labour (and operator) costs and moreover, land c o s t s , h i r e d labour c o s t s , and l i v e s t o c k costs together are l e s s than 50% of the operator and f a m i l y labour c o s t s . The only input which approximates the importance of f a m i l y labour (88% of i t s cost) i s farm c a p i t a l which in c l u d e s machinery, equipment, and a l l intermediate i n p u t s . Family and operator labour c o n s t i t u t e s 41% of t o t a l production costs and hence the importance of the cross e f f e c t s between production and consumption d e c i -sions may be expected to be considerable. TABLE A.2 D i f f e r e n t Expenditures as a P r o p o r t i o n of Family Labour Farm Returns Farm Sales =2.35 Land Costs =0.28 Hire d Labour Costs = 0.168 L i v e s t o c k Costs = 0.092 Farm C a p i t a l Costs = 0.879 TABLE A.3 Cost Shares of the D i f f e r e n t Factors Land = 0.117 Hired Labour = 0.069 L i v e s t o c k = 0.038 Farm C a p i t a l = 0.363 Family and Op. Labour = 0.413 168 APPENDIX 3 The Data Used The transformed data set used in the study are provided in Tables A.4 and A.5. The variables are identified with the same symbols used in the text (see chapter IV). Table A.4 presents the net output price indexes (qi to qt*), the after tax on-farm and off-farm returns to labour ( T L and w^ ,^ respectively), the levels of expenditures (Si and S^) and farm operator's education (E). Table A.5 shows the net outputs per unit of family labour for each of the five net outputs considered. TABLE A.4 Price Indexes of Net Outputs and After Tax Labour Returns, Expenditures and Farm Operator's Schooling Years 5 6 7 8 9 10 12 13 1* 15 It 17 18 IP 20 21 22 73 24 75 26 27 28 29 30 31 32 33 3* 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 5? 53 54 55 56 57 5P — q i — 10.000 10.000 14.348 14. 348 14. 348 10. 490 10.490 10.490 11.758 11.758 11.258 11.258 11.258 11.258 11.258 11.75 8 11.258 11.758 11.258 11.258 _13.12C_ 13.120 13.120 13. 120 13.120 13. 120 J 3 . 120 13.120 13.120 13.120 13.120 13.120 13.120 13. 120 13.120 13.120 13.120 13.120 13. 170 13.120 13.120 13.120 13.120 13.120 13.120 9.0455 9.0455 9.0455 9.0455 9.0455 9.0455 9.0455 9.0455 9.0455 9.0455 9.0455 9.C455 — q 2 — 10.357 3.9957 4.7324 6.4178 q3 7.0886 2.7965 3.3817 4.5425 2.5996 4.4442 2.0030 3. 5309 5. 9951 7.5402 4.9725 5.5614 2.7827 2.9387 3.1541 6.7343 7. 22 50 13.457 7.B618 11.276 10.576 4.0326 5.6936 22.306 5.0988 5.3397 6.3000 4. 3836 8.5050 23.527 21. 46 6 7.4961 21.547 6.4310 3.3609 3.4130 12.170 11.6 78 13.115 13.214 13.446 14.811 9.5715 15.7 75 9.8383 13.018 10.151 10.890 11. 215 10.071 12.002 14.136 11.090 10.632 10.517 15.496 10.807 20.124 13.6C8 16.374 19.476 11.897 14.586 17.662 11.266 13. 968 15.820 1 1.551 17.043 16.053 257. 23 14.887 17.0C8 14.361 11.755 9. 2479 — q 4 IO.OOO 10.000 10.149 10.149 10.149 9.4654 9.4654 9 . 46 54 10.668 10.668 10.668 10.668 10.668 10.668 10.668 10.668 10.668 10.668 10.668 10.668 10.668 12.618 12.618 12.618 12.618 12.618 12.618 12.618 12.618 12.618 12.618 12.61t 12.618 12.618_ 3.9299 1 2 . 1 7 3 6 . e 5 7 1 8.403 5 16.438 24.486 10.973 15.3 89 11.973 13.615 15.928 17. 114 12.618 12.618 12.618 12.618 12.61e 12.618 TT . 1 1.5655 1.C179 2.0262 1.6592 2i 1.6 72 8 2.7178 1.5477 3.1 599 1.4305 0.91301 0.81795 2.0928 1.1762 1.4812 3.1508 2.4531 4.2025 2.7Q71 3.5882 2.8056 0.73567 0.90901 0.80059 0.64707 1.4367 0.81783 4.4064 3.8936 3.9839 3.3759 3.9164 3.4613 0.77040 0.94922 1.0206 1.3307 1.36 33 1.7107 3.4630 3.0 244 2.4738 4.7899 3.7107 2.7025 1 .4938 1.3663 1.4047 1.5081 1.1996 1.6391 1.9495 2.1819 2.3636 3.4992 2.1947 1.9767 1.3286 1.5319 1 .5664 0.96977 1.9803 1.6568 2.9074 2.0404 1.9352 2.9784 2.3817 2.3410 "~ST 1105.9 81ft.73 1621.0 111 5. 4 ••"TO6.T" 744.89 619.76 2012.7 1142.1 1392. 8 ~T7?.49~ 938.96 789.35 674.85 1353.5 861.48 7 73.1 8 901.32 1079.0 1185. 1 1419. 1 1207.0 1 ( 5 5 1 . 1151.2 1015.6 1176.2 857.95 123 2. 3 979.03 1134.7 1221.5 680.70 1397.9 1374.9 S J — 1538.3 2793.9 1478. 0 7877.9^ 3184.6 2676.9 4779.3 3313.8 4401.5 3283.J5_ 5780.3 4969.2 5042.7 4382.2 4748.9 4459. 2 44 59. 0 3 564. 3 3054.7 5541.0 4619.3 2575. 2_ ^ 7 6 7 . 2 2208.0 2298. 2 3711.1 1992. 6 1 8 85.5_ 2762.7 2007. 2 1771.7 2712.4 2210.0 2276. 1 6.00 8.90 9.50 9.20 9.70 8.50 8.70 7.50 6.50 T.PO, 6.40 7.10 6.20 6.40 7.30 6.40 6.60 6.80 6.70 7.30 6.80 9.60 9.00 9.40 9.40 8.60 8.70 9.90 1. 7725 1.2586 3.3512 1.5607 0.81786 1.0177 4.6352 2.660 2 3.6360 1.9406 4.9807 2-8736 157  5 905.10 7567.3 1106.9 651.86 815.43 12.618 12.618 12.618 12.618 12.618 12.618 1.1065 1.3349 1.0763 1.7217 1.3645 1. 2002 3.5 82 4 2.3618 2.873 3 1.9854 2.8157 3.4641 2.6252 2. 5394 2.1774 1.6531 7.2872 _7. 0632 i.135 8 " 2.5374 1 . 5767 2.0055 1 .7445 13.057 1 3.450 10.454 11.913 7.3708 10.0<6 11.758 11.758 11.758 11.758 11.758 JJ_.7_58_ 1.4148 I. 2214 0.98730 0.83309 1. 11 60 1.2492 2.840 7 2.4701 1.9716 3.5795 2.5988 10.456 7.7580 7.56?7 1 1. 7C8 1 1.965 11.758 11.758 11 .758 11.755 11.758 11 .75 8 0.88780 0. 59439 0.70838 0. 700 33 1 .1407 0 . 6 9 0 74 3.0352 2.7347 2.2871 2.8546 1.7069 2.2073 1069.5" 1050.2 823.31 1349.2 1107. 8 897^ 85 118 8. 4 959.71 697.45 578.74 910.60 909. J 1 583.13 810.54 487.91 511 .98 781.04 66 7?46 34.7 2531.5 3610.2 1895.3 5416.6 2783.3 8.90 9.00 9.00 9.20 9.50 9.70 4304. 6 2373.0 2597.0 1957.7 2830.5 3400.1 30'0.5 2 494.9 1975. 3 3543. 1 2690.1 22 21.2_ 2 890.7 2861.3 2187. 2 2860.0 16*6.6 21 1 7. 8 8.60 9.50 10.20 8.90 7.90 8.30 8.10 9.20 9.10 9.20 9.50 9.70 7.80 8.40 8.90 9.30 9.00 8.80 8.60 8.90 8.40 7.30 8. 10 8.00 vO I— q i qz . 93 ff . 1 W 2 i " s i s 2 E . t r 59 9.0455 1.6910 12.742 11.758 0.76824 3.6163 542.05 3541.8 • 8.20 60 9.0455 1.4530 1 2. 1 76 11.758 .0.62761 2.7461 413.67 2513.5 7.40 61 9.4898 1.4204 1 1. 9 80 12.401 0.S6029 2.9090 670.11 2809.4 8.90 .... ., I 62 " 9.4 898 "•" 1.4242 11. 8 06 12.401 0.85230 2.5349 608.80 2409.3 9.10 63 9.4898 1.3555 12.161 12.401 0.96862 2.4664 719.40 2391.1 9.20 6* 9.4898 1.3636 5.5659 12.401 1.0624 3,3077 . 713.23 323 6. 5 9.00 j « f 65 9.4898 1.5958 U.9C4 12.401 0. 72076 3.089 1 477.97 2916. 0 8.40 66 9.4 89 8 1.7973 12.589 12.401 0.95014 2.999 3 683.48 2927. 8 9.20 67 9.4898 1.5742 10.112 12.401 1.0605 2.2196 762.42 2148.8 9.20 < 68 9.4898 1.5980 12.0S3 12.401 1.1583 2.5254 849.6 7 2411.0 9.30 69 9.4898 1.565 7 11. 826 12.401 0.582 26 2.8011 386.41 2656.1 7.80 70 9.4898 1.5350 12.952 12.401 0.79615 2.9877 545.28 2 884.9 8.30 • 71 9.4398 1.4626 12.757 12 .401 0.79153 3.1913 5 70.80 3098. 7 9.20 72 9.4898 1.5201 11.001 1 2 .401 0.97290 2.4849 693.10 23*0. 2 9.30 73 9.4898 1.5549 10. 162 12.401 1.0729 3.7775 775.37 3845.8 9.20 74 9.4898 1.7845 13.510 12.401 0.83179 2.9 841 592.43 2821.1 8.50 " 75 9.4898 1.9157 1 1.952 12.401 0.77914 2.8138 578.67 2811.3 8.30 76 9.4898 1.4885 10. 9(4 12.401 0.82382 2.6326 564.62 2488.1 8.10 < 77 9.4898 1.7477 12.648 12.401 0.84542 2.8719 566.86 2 764. 0 8.BO 78 9.5478 2.0468 13.518 11.794 1.3446 2.0432 954.51 2C04. 7 9.40 79 9.5478 2.71H 13.777 1 1.794 1.6337 2.0120 1247.3 2067.3 9.70 « 80 9.5478 3.0865 8.6540 11.794 1.6529 3.4163 1066.4 3172. 6 10.00 81 9.5478 1.2402 6.3241 11.794 0.92580 3.5653 628.37 3539.1 9.30 82 9.5478 2.2511 11.148 11.794 1.3744 2.6150 943.44 2541.0 9.90 83 9.5478 4.0103 11.330 11.794 1.5785 3.0533 1075. 1 2917.2 10.00 84 9.5478 I.7435 10.384 11.794 1.1950 2.5854 806.36 2528.1 9.20 85 9.5478 2.7179 12.915 11.794 1.3181 2.4349 920.15 2369. 3 9.10 ( 86 9.5478 1.9440 10.649 11.794 0.90574 2.94Q9 584.31 2825.6 8.60 87 9.5478 3.6783 13.479 11.794 1.0183 2.4 241 725.90 2305.8 8.70 88 9.5478 1.6073 11.324 11 .794 0.77068 3.1672 571.25 3201.8 8.30 I 89 9.5478 2.0492 11.858 11.794 0.89906 3.2658 612.35 3138.5 8.40 90 9.5478 1.5869 13.803 11.794 0.77934 2.5495 612.19 2394.7 8.50 91 15. 180 49.470 16. 528 12.502 1.8902 3.5773 1 553.6 3644. 7 9.40 « 92 1 5. 180 9. 7605 19.372 12.502 1.0676 3.5239 859.72 3517.0 9.90 93 15.180 37.70 2 16. 258 12.502 0.6Q459 3.6220 493.86 3542.9 10.20 94 15.180 36.743 2C.910 12.502 1.5720 3.7404 1222.0 3756.4 9.10 95 15.180 2.1987 14.712 12.502 0.36802 3.3420 279.99 3050.6 9.40 - - -- — _ O * TABLE A.5 Quantities of Net Outputs by Census Division Ql/Ll . Q 2 / L l Q 3 / L 1 Q 4 / L 1 Q 5 / L 1 1 749.15 18.556 100.so 12.1~30 349.79" " " 2 388.17 41.642 27. 929 13.375 174.68 .... — • - — K | 3 425.39 46.995 38.4e9 23.348 9 1 . 4 6 9 \— _ 4 613.88 37.867 78. 3 1 T 22.436 3 26. 73 5 785.0? 42.444 142.43 23.004 522.46 : ^ ' 1 6 471.57 52.09 8 62.461 9.6621 224.54 1 7 396.66 34.471 23.612 19.538 231.53 n ! 8 886.42 4 4 . 4 4 9 105.37 3.6968 263.62 " " ,... 9 386.85 47.453 13.949 20.596 204.77 \ 10 492.57 39.673 11.356 20.913 247. 46 * ! 11 224.56 41.398 6.3444 12.469 127.41 12 265.91 32.296 13.463 13.107 132.12 13 319.00 16.524 6.4757 15.451 216.24 1* 203.94 31.61 7 . 6.8895 12.767 115. 31 ' ' * -15 534.08 42.815 21.025 24.895 218.65 16 334.95 3C.540 11.925 16.830 211.89 17 247.72 34.331 7.6269 15.353 138.B5 18 319.55 39.294 12.912 17.038 177.15 19 379.63 42.642 16.421 22.331 212.96 20 559.48 30.B35 33.6C2 1 7.657 285.11 21 363.77 43.213 11.505 16.733 160.03 22 690.69 42.295 61.088 10.793 254.68 23 463.72 50.795 13. 894 20.189 201.38 24 486.83 47.055 39.323 19.764 221.27 25 591.39 43.140 52.8C2 9.3769 242. 18 «t 26 419.15 40.473 22.393 16.402 163.02 27 320.58 44.112 11.573 18.089 150.16 28 733.67 39.191 94.824 16.154 273. 82 O 29 337.10 45.677 14.9«0 17.650 151.62 30 475.80 48.385 9.6801 19.615 2 24.62 31 527.59 60.153 14.3 80 16.041 257.78 9 32 298.03 44.004 21.481 18.420 167.55 33 660.51 47.545 2 5.760 17.820 251. 78 3* 667.18 25.92 5 79.925 10.697 315.61 ® 35 679.84 26.916 7.3087 3.7211 1 91 . 74 36 398.35 42.906 30. 818 17.059 175.94 37 1210.7 52.300 136.49 19.617 258. 19 38 490.86 39.461 14.401 18.106 236.84 30 280.49 48.458 19. 135 14.304 175.62 40 262.05 56.029 13.473 19.472 125.01 41 321.19 48.737 14.851 18.347 171.49 42 506.39 40.553 35.246 19.472 238.07 *3 319.06 50.819 11.698 22.932 149.82 44 568.33 45.273 16.578 21.998 288.53 45 597.40 32.723 59.864 11.881 271.94 46 637.65 42.163 69. 4e<? 16.416 278.35 9 47 662.80 107.47 19.128 15.761 269.81 48 568.14 139.15 18.846 7.8195 199.12 49 431.49 142.04 8.74 26 15.790 157.58 O " 51 386.25 180.22 7.5668 17.092 137.36 51 574.23 176.11 39.108 10.778 218.42 _ 52 564.90 139.14 20. 145 15.487 177.56 • O i 53 391.20 155.50 10.305 17.049 127.83 54 520.89 135.07 26. 330 18.045 209. U 55 342.33 144.44 16.952 13.650 143.33 56 321.91 9 2 . 1 6 0 8.26C8 22.421 126.13 1 57 4 3 5 . 3 8 1 53.97 10. 3 98 15.728 124.98 1 - 58 360.IP 146. 44 6.7561 9.9259 135.81 V 1 Q l / L l Q2/L1 Q3/L1 Q4/L1 59 60 61 62 63 64 65" 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 334.77 274.20 383.55 356.40 386.21 459.30 277.92 364.86 417.82 481.99 243.13 291.15 360.09 391.99 414.66 344.35 339.76 333.13 361.33 706.30 911.49 717.50 482.97 741.21 826.05 493.33 566.01 339.91 464.16 302. 53 339.72 383.70 802.16 360.16 383.13 601.74 128.99 111.75 214.60 255.03 291.11 261.54 151.71 213.89 250.56 318.51 130.81 168.66 254.00 252.30 244.51 178.70 163.38 177.53 166.10 242.03 195.05 127.60 265.32 258.28 152.39 194.42 114.78 128.13 106.04 12 0.0.9. 114.06 220.36 14.270 50.481 23.503 18.011 7.8246 8.3358 8.6026 8.2772 8.0823 22.557 11.784 19.167 17.029 13.677 16.041 27.827 6.0232 9.2167 11.8*7 13.167 8.9403 13.732 n.6e9 5.5849 5.8260 14 .003 9.8553 8.8954 9.6017 9.4004 11.0 74 7.6212 8.5925 8.9002 9.7336 13.245 13.817 7.4433 9.5817 16.614 9.51 10 23.456 35. 1C7 2 5.4 41 31.340 20.182 19.116 34.733 34.381 33.783 40.022 23.664 31.474 12.793 11.586 7.5905 15.886 7.5321 39.514 25.256 25.255 17.035 19.163 18.951 5.96e? 7.9976 58.299 29. 399 114.CI 36.975 95 270.59 269.70 21.859 16.536 9.6131 24.355 14.254 4.8822 19.565 11.142 QS/L 1 125.83 104.37 .124.16 124.46 129.92 133.14 103.44 121.04 128.41 149.86 98.496 106.91 137.15 134.83 127.91 136.90 131.67 118.37 120.95 189.11 280.99 143.40 141.20 212.66 215.61 149.06 163.40 118.89 171.08 111.49 126.94 171.68 550.80 212.01 192.65 341.67 217.47 9 Q 0 173 APPENDIX 4 Dispersion of the Estimated E l a s t i c i t i e s  Across the Sample Points In this appendix the means, standard deviations, minimum and maxi-mum values of the various e l a s t i c i t i e s when evaluated at the sample points are provided. The purpose of presenting this information i s to give an indication of how the calculated e l a s t i c i t i e s change when they are eval-uated at the different sample points. The symbols used in table A.4 are defined as follows: E ~ i s the supply elasticity of on-farm work with respect to f f , E L]_TT L 2 W 2 is the supply ela s t i c i t y of off-farm work with respect to w2 and in general the E symbols indicate labour supply e l a s t i c i t i e s , the f i r s t subscript represents the dependent variable and the second subscript stands for the independent variable. Similarly, CS. , for example, Q i q i represents the conditional net output, Qi, supply elasticity with respect to the price of output, qj_. In general, the subscripts can be inter-preted in the same way as the subscripts of the labour supply e l a s t i c i t y . TABLE A.6 Mean E l a s t i c i t i e s , Standard Deviations, Minimum and Maximum Values Standard Mean Deviation Minimum Maximum L i ff L 2w 2 T 1 2 L]_y L z f f L 2p L 2y cs n Q 2 q 2 Q393 cs. Qi+qit Qsqs C S A Q i q z cs n Q i q ^ cs A cs_ Q 2 q i Q2q3 cs A Q 2 q 4 Q2q5 Q 3 q i cs n Q 3 q ^ Qsqs CS A ( M i ( M i 0.113 0.088 -0.034 0.478 0.166 0.136 -0.384 0.781 -0.101 0.032 -0.194 0.062 0.171 0.073 0.076 0.390 -0.159 0.076 -0.296 -0.014 -0.262 0.081 -0.465 -0.124 0.313 0.279 -0.225 1.023 -0.498 0.212 -0.853 -0.154 0.321 0.153 0.039 0.690 -0.778 0.453 ^2.489 -0.193 -0.237 0.194 -1.268 0.153 -0.004 0.014 -0.302 0.071 -0.698 0.198 -1.546 -0.292 0.102 0.040 -0.043 0.203 -0.119 0.043 -0.229 -0.046 -0.049 0.018 -0.094 -0.016 -0.267 0.098 -0.538 -0.084 0.931 0.612 0.267 3.012 0.441 0.250 0.152 1.421 -0.018 0.009 -0.052 -0.005 0.931 0.512 0.303 2.942 : 1.790 1.124 3.536 0.151 0.829 0.349 0.085 1.733 -0.671 0.461 -1.712 -0.055 -1.459 1.009 -3.517 -0.130 1.151 0.383 2.739 0.222 -0.040 0.029 -0.228 ^0.019 Table A.6 continued Standard Mean D e v i a t i o n Minimum Maximum CS n 0.4 5 CS CS CS CS Q592 Qsqs Qsqtt -0.998 -1.339 0.707 0.264 -0.289 0.007 0.415 0.656 0.232 0.098 0.094 0.003 -3.221 -3.580 1.526 0.120 -0.556 0.002 -0.154 -0.566 0.260 0.755 -0.098 0.013 

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