COMPARATIVE STATICS AND THE EVALUATION OF AGRICULTURAL DEVELOPMENT PROGRAMS by BARRY THOMAS COYLE B.A., The University of California at Berkeley, 1970 B.Sc. Agric. (Honours), The University of British Columbia, 1974 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES (Department of Agricultural Economics) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October 1979 © Barry Thomas Coyle, 1979 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Depa rtment The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date ii ABSTRACT Perhaps the most important task of any economic analysis of agricultural policy is to estimate the effects of policy on various economic measures such as income and output. This is usually done by combining economic theory with data. However, the economic theory seldom is fully descriptive of the situation and the empirical knowledge generally is far from complete. Thus, even aside from difficulties in aggregating gains and losses over individuals, economic analyses of policies are often unsatisfactory. * r ' / The major purpose of this thesis is to extend economic theory and methods so as to be more descriptive of various agricultural policy situations and to make more appropriate use of available empirical knowledge. This leads us to relax some assumptions in the standard theory of the firm that often seem inappropriate, and to propose a potentially more effective method of incorporating available empirical knowledge of farm structure into economic analysis of policy. In addition, we also attempt to verify the appropriateness of other theoretical constructs of fundamental importance. First, the static theory of the firm is extended to the case of variable factor prices, i.e., factor prices endogenous to the firm. Under these more general conditions, we establish (among other things) (1) the relation between measures of surplus in factor markets and of consumer plus producer surplus, and (2) relations between the slope of a firm's derived demands schedule and various properties of its production function. It is shown that (2) provides additional support for the well-known fact that traditional qualitative com parative static methods can seldom be useful in economic policy-making. Second, we introduce a method of "quantitative comparative statics" that in principle overcomes this defect of established comparative static analysis. This methodology incorporates the available degree of empirical knowledge of the firm's structure without imposing further specification of structure (in contrast to, e.g., the traditional linear and nonlinear programming models of the firm, where a full structure must be specified). This degree of knowledge and its relations to comparative static effects of interest can be expressed as a set of quadratic equalities and inequalities. Then the range of quantitative as well as qualitative effects of policy that are consistent with our degree of knowledge of farm structure and the assumption of static optimizing behavior can in principle be calculated by nonlinear programming methods. Third, we consider the issue of the appropriateness of constructs of static optimizing behavior in predicting farm response to policy. We demonstrate that, by estimating an equilibrium shadow price for an input rather than (e.g.) supply response, one can reduce the significance of many of the problems associated with studies of supply response via representative farm models and investigate this issue more clearly. In this manner, we derive empirical support for the use of the construct of static optimizing behavior in predicting the effects of agricultural policy. iv Thus, by extending economic analysis marginally in the direction of more appropriate theory and more appropriate use of empirical knowledge, we hope to contribute towards the improvement in methodology for evaluating agricultural development programs. Towards this end, the extensions in theory and methods are related to a particular policy situation (evaluation of government funded community pasture programs in British Columbia). V ACKNOWLEDGEMENT I wish to thank Rick Barichello for the opportunity to assist in the evaluation of B.C. ARDA community pasture programs. This experience increased the author's awareness of the problems in the evalua tion of agricultural development programs that are considered here, and hence motivated this study. In addition Professor Barichello has shown considerable patience with the author. I am also particularly grateful to Chuck Blackorby for agreeing to be the "technical expert" on the committee and for suffering through an essentially unintelligible first draft of this thesis. Moreover, his suggestions as to additional readings helped to round out the thesis in many respects. In addition I am grateful to Earl Jenson for agreeing to round out the committee and for providing useful comments. Conversations with Ramon Lopez on general matters of theory and his own research have been very helpful. His comments on parts of an earlier draft of this study have also improved the exposition. Roger McNeil has also made useful suggestions. Many people assisted in the construction of the linear program ming model reported in one section of this thesis. Rick Barichello helped considerably in collecting the data and testing the model. John Kidder provided valuable and necessary assistance in the field. Robert Holtby (B.C.D.A., Prince George), Albert Isfeld (B.C.D.A., Fort vi St. John), Dale Engstrom (B.C.D.A., Dawson Creek) and Gerry Kirtzinger (president of the grazing association of Sunset Prairie Community Pasture) all provided many valuable hours of consultation. Long and helpful interviews were conducted with seven community pasture users at Sunset Prairie, five at Beatton-Doig and twelve at W.M. I am grateful to Maryse Ellis, Wendy Lymer and Carol Clark for typing drafts of this manuscript with skill and patience. Finally, I would like'to thank Phyllis Coyle for her encouragement and support. vii TABLE OF CONTENTS Page Abstract ii Acknowledgement v Table of Contents viList of Tables xiii List of Figures xiv List of Theorems, Propositions, Corollaries and Lemmas xv Chapter 1. Introduction 1 1.1 Overview 2 1.2 The Problem 3 1.3 Statement of Purpose 7 1.1 Research Procedure 9 1.5 Organization of the Study 11 2. Qualitative Comparative Statics and Derived Demand: An Extension 12 2.1 Introduction 3 2.2 Results of Previous Studies 16 2.2.1 Relation Between Surpluses in Factor and Product Markets 17 2.2.2 Slope of the Firm's Derived Demand Schedule 18 2.3 Difficulties in Extending Results via Usual Methods 20 viii Chapter Page 2.3.1 Primal Methods 20 2.3.2 Dual and Primal-Dual Methods 21 2.3.3 Use of Aggregate (Industry) Relations 23 2.4 Extensions to the Theory of Derived Demand 24 2.4.1 Notation and Definitions 25 2.4.1.1 A Definition of Derived Demand 26 2.4.1.2 A Shadow Price Relation Similar to Derived Demand 28 2.4.1.3 List of Major Assumptions 30 2.4.2 Derived Demand as a Schedule of Shadow Prices 31 2.4.3 Relation Between Surpluses in Factor and Product Markets 34 2.4.3.1 Shift in Factor Supply Schedule of Single Firm 35 2.4.3.2 Shift in Industry Factor Supply Schedule 38 2.4.4 Slope of the Firm's Derived Demand Schedule.. 40 2.4.4.1 Relation Between Slopes of Derived Demand and Factor Supply Schedules at Equilibrium 40 2.4.4.2 Relation Between Slope of Derived Demand Schedule and Various Proper ties of the Production Function 42 2.5 Summary of Implications for the Evaluation of Community Pasture Programs 47 2.5.1 Relation Between Surpluses in Factor and Product Markets 47 2.5.2 Slope of Derived Demand Schedule and the Measurement of Distortions 48 ix Chapter Page 2.5.2.1 General Case 50 2.5.2.2 Special Cases 1 2.5.3 Further Research 52 3. Quantitative Comparative Statics and Derived Demand: A Proposed Methodology 53 3.1 Introduction 4 3.1.1 The Problem 55 3.1.2 Outline of Proposed Methodology 57 3.2 Previous Methods of Comparative Static Analysis 59 3.2.1 Qualitative Methods 60 3.2.1.1 Minimal Restrictions 60 3.2.1.2 A Calculus of Qualitative Relations .... 61 3.2.2 Quantitative Methods 62 3.3 Limitations of Previous Methods of Comparative Static Analysis 63 3.3.1 The Qualitative Relation Between Comparative Static and Comparative Dynamic Effects 64 3.3.2 Qualitative Methods 65 3.3.2.1 Primal, Dual and Primal-Dual Methods.. 66 3.3.2.2 A Computational Method of Lancaster... 66 3.3.3 Quantitative Methods 68 3.4 A Proposed Methodology for Quantitative Comparative Statics 69 3.4.1 Restrictions Implied by the Maximization Hypothesis 75 3.4.2 Major Additional Restrictions 76 Chapter Page 3.4.2.1 Model with Output Exogenous 77 3.1.2.2 Model with Output and a Subset of Inputs Exogenous 79 3.4.2.3 Model with a Subset of Inputs Exogenous 82 3.4.3 Minor Additional Restrictions 83 3.4.4 Major Difficulties and Partial Solutions 86 3.5 An Illustration of a Quantitative Comparative Statics Model: Initial Models for Estimating Welfare Effects of Community Pasture Programs 88 3.5.1 Objective Function 83.5.2 Constraints (N = 2,3) 92 3.6 Summary and Suggestions for Further Research 100 4. A Static Linear Programming Model of a Representative Beef Ranch 103 4.1 Introduction .; 104 4.2 Representative Farm Approaches to Estimating Farm Response 106 4.2.1 Studies of Supply Response 106 4.2.2 A Means of Evaluating Constructs of Static Optimizing Behavior 107 4.3 A Static Linear Programming Model of a Represen tative Beef Ranch 113 4.3.1 Methodological Problems 114.3.1.1 Static, Optimizing Behavior Versus Dynamic, Non-Optimizing Behavior 114 xi Chapter Page 4.3.1.2 Endogenous Versus Exogenous Specification of Activities and Combinations of Activities 115 4.3.1.3 The Degree of Aggregation and Supply 117 4.3.2 Summary of Model Structure 118 4.3.3 Limitations of the Model 126 4.3.3.1 Lack of Knowledge of the Production Function and the Extreme Difficulty in Obtaining an Adequate Sensitivity Analysis 127 4.3.3.2 Errors in Simulating Adjustment Costs 128 4.3.3.3 Errors in Specifying Expected Beef Prices 129 4.3.3.4 The Exclusion of Measures of Risk ... 130 4.4 Results and Implications 134.4.1 Results 131 4.4.2 Results of Related Studies 137 4.4.3 Implications 141 4.5 Summary 145 5. Summary and Conclusions 146 5.1 Summary 147 5.2 General Conclusions 152 5.3 Suggestions for Further Research 154 Bibliography 159 xii Appendix Page I. Why Comparative Statics and the Maximization Hypothesis? 170 II. Qualitative Comparative Statics and Derived Demand: Proofs 181 III. Quantitative Comparative Statics and Derived Demand: Details of the Model , , 225 IV. Quantitative Comparative Statics and Derived Demand: Proofs 267 V. Partial Solutions for the Major Difficulties with the Proposed Method of Quantitative Comparative Statics 32VI. Structure of Static Linear Programming Models for Representative Beef Ranches at Peace River Community Pastures .-s 363 V xiii LIST OF TABLES Table Page 1. Feeding and Labor Constraints 123,367 2. Farm Value of Community Pasture and Selected Model Activities: 1975 Market Prices, 1975 Market Prices Plus Subsidies, and Long Run Prices for Calves and Yearlings 133 3. Farm Value of Community Pasture: Selected Calf and Yearling Prices Intermediate Between 1975 Market Prices and 1975 Market Prices Plus Subsidies 135 4. Farm Value of Community Pasture: Selected Calf and Yearling Prices Intermediate Between 1975 Market Prices and 1975 Market Prices Plus Subsidies 136 5. Farm Value of Community Pasture for Extremely High Calf and Yearling Prices . 138 6. Sensitivity of Shadow Prices for Community Pasture With Respect to Profitability of Hay Enterprises 143 7. Definitions of Activities (Columns) of Peace River Income Assurance Model 410 xiv LIST OF FIGURES Figure Page 1. Given a Positively-Inclined Derived Demand Schedule for Pasture, the (inframarginal) Allotment of Community Pasture is Less than the Resulting Change in the Level of Pasture 46 2. Estimating the User Value of Community Pasture Programs in a Market for Pasture 49 3. Summary of Major Constraints for the Quantitative Comparative Statics Model 84 4. Hypothesized Effect of the Community Pasture Program on the Supply Schedule of Pasture to the Farm 90 5. Comparative Static Constraints for Community Pasture Model (N = 2) 93 6. Comparative Static Constraints for Community Pasture Model (N =3) 9 7. Outline of Matrix for the Peace River Income Assurance Farm Model 120,365 8. Input-Output Relations for the Farm Model 121,366 9. A Discrete Analogue to [TT..(X*)] Only Negative Semi-Definite .IJ. 232 10. Model of Disposal of Cows During Feeding Periods Three and Four 384 11. Submatrix "A" of Peace River Income Assurance Model 401 12. Submatrix "B" of Peace River Income Assurance Model.... 402 13. Submatrix "C" of Peace River Income Assurance Model.... 403 14. Submatrix "D" of Peace River Income Assurance Model 404 15. Submatrix "E" of Peace River Income Assurance Model 405 16. Submatrix "F" of Peace River Income Assurance Model.... 406 17. Submatrix "G" of Peace River Income Assurance Model.... 407 18. Right Hand Side of Peace River Income Assurance Model.. 408 XV LIST OF THEOREMS, PROPOSITIONS, COROLLARIES AND LEMMAS Page 32,188 Theorem 1 43,209 Theorem 2 237,285 Theorem 3 307 Theorem 4 228,276 Proposition 1 • 348 Proposition 2 351 Proposition 3 •. 33,190 Corollary 1 • • 33,194 Corollary 2 36,197 Corollary 3 • 41,200 Corollary 4 294 Corollary 5 302 Corollary 6 316 Corollary 7 • 185 Lemma 1 ' 202 Lemma 2 207 Lemma 3 • 272 Lemma 4 1 CHAPTER 1 INTRODUCTION 2 CHAPTER 1 INTRODUCTION 1.1 Overview Perhaps the most important task of any economic analysis of agricultural policy is to estimate the effects of policy on various economic measures such as income and output. This is usually done by combining economic theory with data. However, the economic theory seldom is fully descriptive of the situation and the empirical knowledge generally is far from complete. Thus, even aside from difficulties in aggregating gains and losses over individuals, economic analyses of policies are often unsatisfactory. The major purpose of this thesis is to extend economic theory and methods so as to be more descriptive of various agricultural policy situations and to make more appropriate use of available empirical knowledge. This leads us to relax some assumptions in the microeconomic theory of the firm that often seem inappropriate, and to propose a potentially more effective method of incorporating available empirical knowledge of farm structure into economic analysis of policy. In addition, we also attempt to verify the appropriateness of other theoretical constructs of fundamental importance. Thus, by extending economic analysis marginally in the direction of more appropriate theory and more appropriate use of empirical knowledge. 3 we hope to contribute towards the improvement in methodology for evalua ting agricultural development programs. Towards this end, extensions in theory and methods are related to a particular policy situation (evaluation of government funded community pasture programs in British Columbia). 1.2 The Problem The problems to which this thesis is addressed are essentially three-fold: 1. endogenous factor prices apparently are realistic in many cases but have not been introduced (correctly) into the theory of the firm, 2. comparative static methods that are presently available generally make inadequate use of the degree of knowledge about particular policy situations (at least at the firm level), and 3. the usefulness of the construct of static, optimizing behavior in estimating farm response via representative farm models apparently remains a matter of some controversy. These problems can be elaborated upon as follows. First, the theory of the firm has been formulated under the assumptions of exogenous factor prices.1 On the other hand, there appear to be many situations where factor prices are endogenous at the firm level. For example, Earlier studies by Ferguson (1969, Chapter 8) and Maurice and Ferguson (1971) have tried to analyze the theory of the firm in the case of variable factor prices. However, a series of fundamental errors in these studies will be pointed out in Chapter 2. k situations characterized by a single employer, collusive monopoly, imperfec tions in information or in mobility, or internal labor market structuring all imply a positive slope to the labor supply schedule faced by the individual 2 firm. The most common cause of endogenous labor supply prices within U.S. and Canadian Industry may be due to the idiosyncratic nature of many skilled and semi-skilled jobs, which seem to require a degree of on-the-job training and investment by the firm. Since the probability of quitting is inversely related to salary, in effect the supply price of such labor is endogenous to the firm, i.e., the expected length of the period of return on investment in such human capital increases with the level of 3 earnings offered by the firm. In addition, it has been estimated that on average a firm in Canadian agriculture is either employing its own labor off-farm or hiring non-family labor during only 1/6 of the year. Since the marginal utility of leisure presumably varies with the level of leisure, and leisure in one time period will not substitute perfectly for leisure in a different period (when, e.g., labor is bought or sold by the farm), it follows that the supply price of labor to the farm typically is (to at least some extent) endogenous to the farm. See Addison and Siebert (1979, Chapter 5) for a summary of the literature concerning endogenous supply prices for labor at the firm level. 3 See Stoikov and Raimon (1968), Parsons (1972) and Williamson etal_. (1975). 4 See Statistics Canada ( 1976), chapter on Multiple Job Holdings). 5 Such labor in "developed" countries seems far from the only input that is typically endogenous at the firm level. Endogenous prices at the firm level may well be the rule in most "underdeveloped" countries or g wherever markets are weakly developed. Second, methods of comparative static analysis for the firm that are presently available often appear to make poor use of the degree of knowledge about a particular policy situation. It is well known that qualitative comparative static methods, as embodied in Samuelson (1947) and more recent dual and primal-dual approaches, have led to relatively few determin istic results (signed comparative static effects) under reasonable assumptions.7 Indeed, it seems to be difficult to Incorporate many empirically-based quantitative restrictions on production functions into such methods. Thus, in the absence of such restrictions, relatively few predictions of firm response or testable predictions of firm behavior can be obtained. Unfortunately, the only alternative methods of comparative static analysis that are presently available are essentially dependent upon complete knowledge of the structure of the problem. For example, in * ^This view was implicit in the address by Nerlove at the AJAE meet ings (1979). g In the Peace River region there is a (sparse) market for rented hay-land, and hay and pasture appear to be close substitutes. However, observations of farm behavior suggested that the supply price of pasture is endogenous to the typical user of community pasture in the region. Moreover, in the other region studied for the B.C. ARDA community pastures evaluation (Prince George), the supply price of pasture is clearly endogenous at the farm level due to the absence of any (market-clearing) rental markets for pasture or any close substitutes. See Barichello (1978). 7This point is elaborated upon in Sections 3.2.1 and 3.3.2 of Chapter 3. 6 order to obtain a solution to a static linear or nonlinear programming model of a firm, the entire structure of the production function must be specified; in fact, our knowledge of the production function is generally far from complete. Usually this problem cannot be handled adequately, at least at the firm level. This can be seen most clearly by thinking in terms of local comparative statics: if a twice differentiate objective function Tr(x;a) for the firm has a maximum at x* > 0 (the equilibrium level of inputs), then the comparative static change in x* due to the effect (iTa) of an infinitesimal change in a policy parameter a can be calculated as ^— = - [ TT..(X*) ] 1 IT 3 a ij a (Nxi) (NxN) (Nxl) where [TT..(X*)] * denotes the inverse of the Hessian of TT(X) at x*. N (N +1) Since the majority of the —^—- individual elements of [TT..(X*)] are usually largely unknown (in the case of a firm's production function) and 3 x* the relation between y^— and [TT.^(X*)] is complex, a sensitivity analysis that depends on direct user alteration of structure [TT.J(x*)] or analogous forms is seldom adequate. Third, there is debate as to the utility of such concerns about comparative static theorems and methods. In particular, lists of possible causes of the apparent failure of representative farm studies of supply response typically have included the static, optimizing nature of these models. Indeed, at least some observers have stated that the decision to model static, optimizing behavior rather than dynamic non-optimizing behavior was the major cause of failure for these studies, g See Chapter 4. 7 On the other hand, theory suggests that, given our present state of knowledge, farm response generally can be estimated more effectively from static, optimizing models than from dynamic or non-optimizing models. The essential arguments are that comparative dynamic effects can be differentiated from comparative static effects only on the basis of essentially unavailable knowledge of adjustment cost functions, and that static models are internally consistent and (unlike dynamic models) relatively simple in 9 structure . Given this contrast between opinion and theory and the importance of the i ssue, there appears to be a need to test the relative utility of the construct of static, optimizing behavior in estimating farm response via representative farm models. 1.3 Statement of Purpose The purpose of this thesis is essentially three fold. 1. Extension of the traditional qualitative comparative statics of derived demand at the firm level to the case of endogenous factor prices. This will involve the development of theorems concerning: properties of derived demand schedules and comparative static effects of a shift in a factor supply schedule for an individual firm facing variable factor prices. 2. Extension of comparative static methods of analysis at the firm level so as to incorporate more fully our empirical knowledge about 9 See Appendix 1. 8 parameters without specifying more than this knowledge, i.e., to develop a method of analysis that provides a useful "middle ground" between the (generally underdeterminate) traditional qualitative methods as embodied in Samuelson (1947) et al. and the (generally overdeterminate) quantitative methods as embodied in (e.g.) static linear and nonlinear programming models of the firm. This will involve the development of a method of local comparative static analysis that in principle incorporates additional restrictions on potentially observable parameters of the firm's maximization problem, i.e., restrictions that have not been incorporated into traditional methods of local comparative static analysis, and that leaves the degree of specification of structure as optional to the user. 3. Examining the appropriateness (in a particular case) of constructs of static, optimizing behavior in the estimation of farm response. This will involve the use of a static linear programming model of a "representative" farm for a particular community pasture in British Columbia. Since these extensions are in the direction of making theory more relevant to practice, it is hoped that they will not be "empty" theoretical exercises with zero practical implications. Towards this end, and in addition to (3), an attempt is also made to relate the more theoretical parts (1) and (2) to the problem of predicting farm response to ARDA community pasture programs in British Columbia. However, the major task of obtaining computational and practical experience with the 9 "intermediate" method of comparative statics (part 2 above) will be post poned to a future study. 1 .4 Research Procedure The manner in which these objectives are met can be summarized as follows: 1. The theory of derived demand with variable factor prices is investigated by making explicit use in formal analysis of the following equivalence: a firm's derived demand schedule is equivalent to a schedule of shadow prices for the input. This is simply the "intuitively obvious" equivalence leading to the textbook statement that factor market equilibrium occurs at an intersection of factor demand and supply schedules. Never theless, this equivalence has not been incorporated previously into formal analysis of derived demand, and in effect this equivalence was even labelled as incorrect by a paper in a prominent journal. ^ Implications of this theory for the evalua tion of ARDA community pasture programs are pointed out. 2 . The traditional methodology of local comparative statics for the maximizing firm (e.g., Samuel son, 1947) is generalized by expressing the comparative static implications of the maximiza-10See Schmalensee (1971). 10 tion hypothesis and of many potentially observable parameters of the firm's maximization problem as a set of nonlinear con straints. These constraints define the comparative static effect d X y^- in terms of potentially observable parameters p of the firm's maximization problem. Then reasonable restrictions corresponding to our degree of knowledge about the structure of the problem are specified for p. By solving for the maximum and minimum value 8x of a scalar-valued function z of -1^- over a feasible set 9 a da defined by all of these constraints, we can calculate the range on 3x" the comparative static effects z tnat are consistent with the maximization hypothesis plus the specified restrictions on p. Partial solutions to the major computational difficulties of this method are developed. 3. A static linear programming model of a "representative" multi-product farm using the Sunset Prairie community pasture is presented. Data for the model circa 1976 has been gathered from interviews with local farmers and B.C. Ministry of Agriculture personnel. The resulting estimates of the static equilibrium price of pasture in the region circa 1976 are compared with estimates of the rental price of hayland gathered by Barichello11 and with results obtained by other models. This comparison provides a rough test of the hypothesis that constructs of static, optimizing 11See Barichello ( 1978). 11 behavior are appropriate in estimating farm response via representative farm models. 1.6 Organization of the Study Chapter 1 includes a brief statement of the problems, the objectives and the basic methodology to be followed. Chapter 2 presents a theoretical study of properties of derived demand schedules and comparative static effects for an individual firm facing variable factor prices, and points out the implications of the theory for a methodology of evaluating community pasture programs. Chapter 3 presents a method for calculating the comparative static effects of a shift in a firm's factor supply schedule. This method in principle incorporates verifiable restrictions excluded from traditional comparative static methods without at the same time specifying essentially unknown aspects of structure. Simple (two and three input) illustrative models are constructed. Chapter 4 examines the appropriateness of the construct of static, optimizing behavior in the context of estimating farm response via represen tative farm models, and summarizes the structure of a static linear pro gramming model of a "representative" beef ranch. Chapter 5 summarizes the study and provides basic conclusions. Technical material related to Chapters 1-4 (primarily proofs and details of the method of comparative static analysis and the linear pro gramming model) is presented in the appendices. 12 CHAPTER 2 QUALITATIVE COMPARATIVE STATICS AND DERIVED DEMAND: AN EXTENSION 13 CHAPTER 2 QUALITATIVE COMPARATIVE STATICS AND DERIVED DEMAND: AN EXTENSION 2.1 Introduction In the previous chapter, we pointed out that variable factor prices are not uncommon at the firm level. Indeed, we noted that the supply price of land and (during most of the year) of labor should generally be endogenous to the firm in agriculture. This was confirmed by observation in the case of pasture in the Peace River and Prince George regions of British Columbia. Suppose that the construct of static, optimizing behavior is of value in the applied economics of agriculture — this assumption will be verified in Chapter 4. Then it follows that extending the theory of the firm to the case of variable factor prices should be a small positive addition to the tools of the profession. Since the received theory of the firm is embodied in a set of formal propositions and proofs, extensions to this theory also should be made in a rigorous manner. In this chapter, we shall extend the static theory of the firm to the case of variable factor prices and we shall point out implications for the methodology of evaluating community pasture programs. As we shall see, the farm value of community pasture depends solely on the farm's demand and supply schedules for pasture, and the related comparative 14 static output effect1 can be decomposed as the product of the related com parative static change in total pasture and the comparative static output effect of an exogenous change in the quantity of total pasture employed by 2 the farm. For these reasons, we shall concentrate on extending the theory of derived demand, i.e., price-quantity relations in input markets, to the case of variable factor prices. The theory of derived demand with variable factor prices is investigated here by making explicit use in formal analysis of the following equivalence: a firm's derived demand schedule is equivalent to a schedule of shadow prices for the input. This relation is the "intuitively obvious" principle that underlies the textbook statement that factor market equilibrium occurs at an intersection of factor demand and supply schedules; but this principle has not previously been incorporated into formal analysis of the theory of derived demand. This equivalence implies that the area under any section of the firm's derived demand schedule is equal to the general equilibrium benefits to the firm (gross of supply costs of the input) of employing the corresponding levels of that input, i.e., the gross value to the firm of those levels of input. This in turn implies that the user value of programs shifting factor supply schedules (and. For simplicity, we shall usually refer to "the" comparative static change in beef output. In fact, we can define "short run," "long run," etc. comparative static changes by making appropriate assumptions about the structure of the "stationary state" and the actual underlying adjust ment cost function (Rothschild, 1971). 2 These two statements seem "obviously" true, but the first statement has in effect been the subject of some controversy. 15 in the absence of market "distortions,11 etc., the associated change in consumer plus producer surplus) can be determined directly from knowledge 3 of the appropriate factor market. These results support the approach to evaluation of community pasture programs adopted by Barichello (1978): (in the absence of a commercial market for pasture) the farm value of the program is estimated from observations of a commercial market for an alternative use of improved land (plus a correction for any distortions). In addition, relations between the slope of the derived demand schedule and several properties of the firm's maximization problem are readily established from this equivalence. For example, the derived demand schedule for pasture is necessarily positively inclined given increasing returns to scale and fixed prices for all other inputs, and the schedule can be positively inclined over large areas of its domain given decreasing returns to scale and non-convex isoquants. Moreover, non-convexity of isoquants and increasing returns to scale cannot be ruled out a priori in the case of variable factor prices, and the possibility of non-convexity of isoquants cannot readily be verified or rejected by empirical observation. Therefore, we cannot readily deduce an upper bound on the slope of the derived demand schedule for pasture from this theory of the firm plus empirical observation (except in terms of This relation between surpluses in factor and product markets had at one time been declared incorrect (Schmalensee, 1971), and has been the subject of additional papers that have proved the relation under various special conditions (Panzar and Willig (1978) provide the most general treatment). Here we shall prove the relation under general conditions and by methods that are quite different from those employed in previous studies. 16 the supply schedule for this input). This in turn implies that we cannot readily deduce in this manner an upper bound on the comparative static chan in total pasture employed by the farm or on the comparative static change in beef output due to the community pasture programs. Thus the usual qualitative comparative statics methods, employed in this chapter, permit us to conclude that the farm value of the community pasture program can be estimated directly from knowledge of the market for pasture or of the market for an alternative use of improved land; but these methods plus empirical observation can seldom lead to an adequate measure of the comparative static change in pasture input or beef output. In the next chapter- we shall present a "quantitative" method of comparative static analysis that, in principle, incorporates restrictions on many empirically observable parameters of the firm's maximization problem. 2.2 Results of Previous Studies In this section we summarize the results of two classes of previous studies: studies concerning the relation between surpluses in product and factor markets, and studies concerning the relation between the slope of the individual firm's derived demand schedule and properties of the firm's maximization problem. The following notation will be used. Define the firm's static (primal) maximization problem as N . j maximize TT(X) = R(x) - I wx ... .(P) i=1 17 where R(x) denotes total revenue as a function of input levels x, and w1 = w'(x'; a) if w1 is endogenous to the firm. Let x* be a solution to problem P. The firm's derived demand schedule for input i is obtained by varying the exogenous variable w1 or a1 and recording the relation i* between x and the marginal factor cost. For simplicity, denote the firm's derived demand schedule for input i as x'(w') if the supply price w1 is exogenous to the firm. 2.2.1 Relation between Surpluses in Factor and Product Markets The relation between surpluses under derived demand schedules and consumer's surplus appear to have been considered too obvious for comment until Schmalensee (1971) argued that the change in surplus between a derived demand and supply schedule for an input, generated by a shift in the supply schedule for that factor, generally exceeds the related change in consumer's surplus. Then a short series of papers verified the equivalence between measures of surplus in product and ' 5 6 factor markets under special conditions. ' This literature has established the following: For example, see Prest and Turvey (1965), p. 691. ^See Wisecarver (1974), Anderson (1976), Schmalensee (1976) and Panzar and Willig (1978). g It should be noted that the controversy has concerned fundamental properties of derived demand schedules rather than difficulties in aggrega ting over firms. By assuming that prices are exogenous to the industry, Schmalensee (1971) in effect denied that surpluses in factor and product markets were equivalent even in the case of a shift in a factor supply schedule of a single firm. Likewise, later papers on this relation typically simplified the problem by ignoring firm interactions (the one partial exception is Wisecarver, who outlines an argument that assumes constant elasticity of aggregate factor supply). 18 1. given perfectly elastic or perfectly inelastic supply schedules of inputs at the firm and industry level, the welfare changes resulting from input price changes can be measured as changes in the surplus under the industry factor demand curve; and 2. given that a change in factor price leads to a change in pro ducer's surplus in that market, the welfare changes (change in consumer plus producer surplus) of course cannot be measured simply in terms of changes in consumer surplus in 7 8 output markets. ' 2.2.2 Slope of the Firm's Derived Demand Schedule Suppose that the equilibrium supply price w' is exogenous to the firm, x* is an interior solution for problem P, ir(x) is twice differentiable at x*, and the Hessian matrix [TT..(X*)] is always negative definite. Then 9x' 'J < 0, i.e., the derived demand schedule for input i is always 3w' negatively inclined. If instead [TT..(X*)] is negative semi-definite, then 3x'* S 0, i.e., the derived demand schedule for input i is never 9w' g positively inclined. Ferguson (1969) and Maurice and Ferguson (1971) attempt to extend the analysis of Samuelson and others to the case of variable factor 7See Panzar and Willig (1978). Q This second point favors the measurement of effects of community pasture programs in the pasture market rather than in product markets. Measurement of these effects via product market calculations generally re quires considerably more information than does measurement via the pasture market (Carlton, 1978). g For example, see Samuelson (1947) or, for a simpler approach using duality, see McKenzie (1956-7), pp. 188-9 and Karlin (1959), p. 273. 19 prices; but their manner of doing this is fundamentally incorrect. Their major errors can be summarized as follows: 1. totally differentiating the first order conditions with respect to the endogenous variable w1 rather than with respect to a' (Ferguson, 1969);10 2. defining the firm's derived demand schedule in terms of equilibrium supply price rather than equilibrium marginal factor cost (Ferguson, 1969 and Maurice and Ferguson, 1971). Given this definition of a derived demand schedule, they conclude that "unique factor demand functions do not exist when factor prices are variable to the firm." (Maurice and Ferguson, 1971, p. 133). On the other hand, suppose that the firm's derived demand schedule x'(a') is defined in terms of equilibrium marginal factor cost. Then the statement quoted above is incorrect provided that w1 = w'(x'; a1) rather than w' = w'(x;a').11 Moreover, even overlooking this uniqueness problem, the concept of a derived demand schedule is a much more useful analytical 12 tool when it is defined in terms of equilibrium marginal factor cost. Silberberg (1971b), p. 738 has criticized Ferguson and Saving (1969) for a similar error. 11See Section 2.4.2. 12See Sections 2.4.3 - 2.4.5. 20 2. 3 Difficulties in Extending Results via Usual Methods The theorems summarized in Section 2.2.2 for the case where w1 is exogenous to the firm can be derived in a straight-forward manner from this restriction. Thus it is not surprising that the slope of the derived demand schedule becomes ambiguous when this restriction is relaxed. Nor should it be surprising that the methods commonly used to sign the slope given fixed factor prices are not appropriate for signing the slope under certain quite different restrictions, e.g., various restrictions on the production function. In Section 2.4.5 we shall employ a slightly different method for this purpose. Here we shall point out that the methods commonly employed in comparative static analysis of the firm — primal, dual and primal-dual methods — are at best clumsy in signing the slope of the derived demand schedule given various restrictions on the firm's production function. 2.3.1 Primal Methods The usual restrictions N i* JY" i i* E TT..(X*) —. W jX =0 j=1 9 a' E Tr.. (x*) =0 all k t i j=1 JK 9 a' [ir..(x*)] negative definite for the primal problem P, where w1 = w'Cx1;^), exhaust the implications 21 3 X * 13 for —j— of the maximization hypothesis, and it can easily be shown that the slope of the derived demand schedule for input i is unsigned by these restrictions. Nevertheless the primal approach becomes messy and complex when restrictions such as increasing or decreasing returns to 14 scale and convexity/non-convexity of isoquants are introduced. 2.3.2 Dual and Primal-Dual Methods For the primal problem N . . . . maximize Tr(x;p,a) = pF(x) - Z aw (x )x , ... .(P) i = 1 where Tr(x;p,a) is linear homogeneous in (p,a), define the dual profit function Tr(p,a) = |all (maxx {TT(X; p, a): p,aeP° j where P° denotes the domain of (p,a). As in the competitive case, Tr(p,a) is convex and linear homogeneous in (p,a). It appears that in general a second order approximation (in x) of Tr(x;p,a) at any solution x*(p,a) to p can be constructed from Tr(p,a)1!5 as in the competitive case.16 This suggests that, in principle, restrictions on F(x) can be incorporated into a dual approach to compara tive statics when factor prices are endogenous to the firm. 13 See Section 2.1 of Appendix 3. 14 For some idea of the complexity of primal methods in such cases, and of the ease with which serious analytical errors can be introduced into such approaches, see Ferguson (1969), Chapter 8. 15See Epstein (1978). 16See Blackorby and Diewert (1979), and Section 1 of Appendix 4 for an alternative proof. 22 However, the dual approach may lose its simplicity even if it is possible to incorporate restrictions on F(x) into our analysis. For example, consider the standard assumption that F(x) is concave (which is not necessarily true when w!j(x';a) > 0). The additional restrictions placed on the dual by this assumption are not obvious —convexity and linear homogeneity of ir(p,a) hold irrespective of F(x) concave. Thus the dual approach to comparative statics is cumbersome even though such restrictions on F(x) apparently can be incorporated into the analysis. Similar problems arise with the primal-dual method of comparative statics suggested by Silberberg (1974a). For any problem of the form maximizex iT(x;a) ... . (P1) define the "primal-dual" problem minimize x aL(x,a) = 7r(x*(a),a) - ,ir(x,a) . . . .(P-D) where x*(a) denotes the solution to P' as a function of a. The second order condition for an interior solution to problem P-D is positive semi-definiteness of the Hessian matrix r \ L* ! L* XX xa L* ! L* xa aa where L* is evaluated at a solution (x*,a*) for P-D. Silberberg shows that many standard comparative static theorems can be immediately 9x * deduced from the positive semi-definiteness of the submatrix L* = u -s . r aa xa 8 a • However, F(x) concave implies simple restrictions only on Fxx, which appears in submatrices other than L*a> Thus, for our purposes, methods based solely on L* positive semi-definite also seem unsatisfactory. 23 2.3.3 Use of Aggregate (Industry) Relations Hicks-type formulas for the industry elasticity of derived demand * are consistent with the assumption of variable factor prices at the industry level; but these formulas are not appropriate for the investigation of relations between elasticity of derived demand and other parameters at the firm level even under the assumption of fixed prices for the product and all other inputs. These formulas are derived independently of the second order conditions for a solution to a firm's maximization problem; whereas, the slope of a firm's derived demand schedule for input i depends entirely upon the Hessian for TT(X) + w'x1.^ Thus relations calculated from these formulas can be more ambiguous than the relations implied by the static maximization hypothesis. This criticism can be verified later 21 22 by comparing Theorem 2 and a formula due to Andrieu (1974). ' In particular, see Hicks (1966), pp. 241-46 and Andrieu (1974). 20 The Hicks-Andrieu formulas express the industry elasticity of derived demand as a function of parameters that (or course) do imply restrictions on the Hessian of TT(X) + w'x'; but these restrictions will satisfy the second order conditions for a maximum only by coincidence. 21 See Appendix 2. The example (where firm and industry analyses are equivalent) shows that the criticism applies at the industry as well as firm level. 22 Diewert (1978) includes an analysis of industry derived demand in terms of duality theory. Since the assumptions employed there essentially imply integrability (Epstein, 1978), the analysis can be "collapsed" to the firm level without encountering the criticism levelled here against interpre ting Hicks-type formulas at the firm level. However, it has already been noted that a dual approach seems inappropriate for analyzing the implications of restrictions on the production function F(x) in the case of variable factor prices (see Section 2.3.2 ). 24 2.4 Extensions to the Theory of Derived Demand Here we present extensions to the theory of derived demand that was summarized in Section 2.2. These extensions, which relax the assumption of fixed factor prices, are developed from the equivalence between a derived demand schedule and a schedule of shadow prices for the input. This equivalence is obviously true and underlies the textbook statement that factor market equilibrium occurs at an intersection of factor demand and supply schedules. Nevertheless this principle does not appear to have been incorporated previously into formal analysis of the theory of derived demand. The main points that are established here can be summarized as follows: 1. A derived demand schedule for a firm and a construction relating the exogenous quantity and corresponding shadow price for an input are equivalent under very general conditions (viz., under essentially all conditions where a derived demand schedule can be defined). 2. (Using 1) in the absence of "distortions" in the economy and the presence of a shift in the supply schedule of factor i for a firm or group of firms, the change in surplus in the firm or industry's market for input i is always identical to the corresponding change in producer plus consumer surplus; 3. (Using 1) the firm's derived demand schedule for input i always inter sects the firm's marginal factor cost schedule for input i "from above" at a (generally unique) equilibrium level of input i; and 4. (Using 1 and 3) the firm's derived demand schedule for input i can be positively inclined over some {x1} even given decreasing returns to scale (provided that some isoquants are not convex), and is positively inclined 25 over all {x1} given increasing returns to scale. Statements 2-3 are "intuitively obvious" applications of statement 1. Nevertheless, statement 3 has not been proved previously and statement 2 has even been the subject of some controversy in the literature. Statement 2 is slightly more general than a theorem presented in Panzar and Willig 23 (1978). 3 The main implications of these theorems for a methodology of evaluating community pasture programs can be summarized as follows. First, by statements 1-2, the farm value of the community pastures program and — in the absence of market "distortions" — the related change in consumer plus producer surplus can be measured directly in the factor market for pasture. Second, by statements 3-4, in the absence of knowledge about the firm's production function we can only infer that the community pasture program does not lead to a comparative static decrease in the level of pasture employed by the firm, i.e., we cannot infer a finite upper bound for the pasture. Thus, even assuming that the ratio of pasture to beef output does not decrease, we cannot infer a finite upper bound for the comparative static change in beef output. 2.4.1 Notation and Definitions The following notation, definitions and conditions are used in the theorems presented here. 23 See Section 2.2.1. 24 This statement assumes that community pasture provides the same services as other types of pasture. In fact, a community pasture typically employs a rider to move and watch over cattle. This difference is incorpor ated into the model that is summarized in Chapter 4 and the related appendix. 26 2.4.1.1 A Definition of Derived Demand In most analyses of derived demand, where the factor supply price schedule to be varied is defined as a price exogenous to the firm, there is no need to distinguish between factor supply price and marginal factor cost in defining a derived demand relation. However, an endogenous supply price implies that factor price and marginal factor cost are not necessarily equal. Since the possibility of a divergence between factor supply price and marginal factor cost is to be incorporated into our modelling, we must distinguish between the two in defining a derived demand relation. A firm's derived demand schedule for an input is defined here as the set of pairs of equilibrium quantity and marginal factor cost (for the input) which are obtained by varying the total cost schedule of the input in an otherwise unchanged producer maximization problem. To be more precise, let x = Nxl vector of activity levels for the N inputs of a firm i 25 c (x) = total cost schedule to the firm for its i1 th input c\x;a) = total cost schedule to the firm for its input 1, as a function of x and a parameter a y = Mxi vector of activity levels for the M outputs of the firm 25 i If c (x) is function only of the level of employment of input i by the firm, then c'(x) can have the following forms: w'x' (supply price exogenous), s'(x')x' (supply price endogenous, and, m general, supply price does not equal marginal factor cost), and ^'s'U'Jdx1 (supply price endogenous, and supply price of ith unit equals marginal factor cost). 27 y = f(x) = production function (vector-valued for M>1) for the firm b(y) = total benefits schedule to the firm as a (scalar-valued) function of its M outputs R(x) = b(f(x)), i.e., total benefits schedule to the firm as a (scalar-valued) function of its N inputs x* = vector of the N input levels employed by the firm at a solution to a maximization problem Then a firm's static maximization problem can be defined as follows. Definition 1. A producer problem P is defined as 1 N i maximize TT(X) = R(x) - c (x;ot) - Z c (x) . . . .(F i=2 for a particular value of the exogenous variable a, and the solution set to *P 26 this problem is denoted as {x (a)}. Given this definition of a firm's static maximization problem, the firm's derived demand schedule for any input 1 can be defined as follows. Definition 2. The firm's derived demand schedule for input 1 is defined {(x1*P(a), MFC^a)) for all a} = DP 1 *P where MFC1 (a) = 3c (x (a);a) . Denote the relation defined by the 9X1 pairs in D as p = p (x ). 26 It can be shown that this model of a firm's static maximization problem formally applies to both single-enterprise and multi-enterprise models (e.g., see footnote to Theorem 1). 28 The derived demand schedule is expressed in the form of the relation p1 = p^x1), which is the inverse of the usual form, for the following two reasons: in this manner the derived demand relation is defined as a function 27 rather than as correspondence, and this form of the relation emphasizes 28 the shadow price interpretation of a derived demand schedule. 2.4.1.2 A Shadow Price Relation Similar to Derived Demand an exogenously determined level of input 1 employed by the firm. In the following problem, a quantity constraint rather than a price constraint is associated with input 1. This device will be useful later in developing properties of derived demand schedules (Definition 2). Definition 3. A producer problem Q is defined as Q N i maximize TT(X) = R(x) - I c (x) i=2 . . . .(Q) subject to x1 = x1 for a particular value of the exogenous variable x1, and the *Q T solution set to this problem is denoted as {x (x ) }. Then the following relation can be formulated. 27 See Corollary 2. 28See Theorem 1. 29 Definition 4. The firm's shadow price schedule for input 1 is defined as (X', ) for all x e X where X 1 _ 1 *P x (a) for al I a The derivative is simply the change in the solution value of the objective function for a problem Q that results from a small change in the exogenous parameter x1. In addition, we can define "corresponding" problems P and Q as follows. Definition 5. Any particular problem P maximize R(x) -c (x;a) - Z c'(x) i=2 is said to " correspond" with a problem of the form Q N maximize R(x) - Z c (x) i = 2 subject to x = x where x 1*P is an element of a solution for the problem P. 30 2.4.1.3 List of Major Assumptions The following assumptions will be made at various times in the theorems to be presented in this chapter. i* Condition 1. For any solution x* to a problem P: x > 0, i = 1,-",N. Condition 2. In the neighborhood of any solution to a problem P: R(x) and all c'(x) are twice differentiable. Condition 3. Condition 4. c1 = c^tx^a), i.e., the total cost of input 1 to the firm is independent of the levels of inputs 1, employed 29 by the firm. 2 j 9 ? (x[ > 0 for all x and i,j,k = 1, • •-,N, dxl 9xK Condition 5. i.e., factor supply prices are non-decreasing in x. 8R(x) ax1 > 0 for all x and i = 1, •• «,N, i.e., inputs are "freely disposable." Condition 6. If the set of attainable u(x) for a problem Q is bounded from above, then the set is also closed from above. 29 For our purposes Condition 3 is the most important of these assumptions and it is "generally" correct. Examples where Condition 3 is likely to be violated include (a) the firm in question is a monopsonist in markets for input 1 and another input, which are supplied by a single industry, and (b) input 1 is an intermediate product of the firm (so that the cost of producing input 1 depends on the level of all inputs employed in producing input 1). 31 Condition 6 simply rules out the unlikely possibility that maximum TT(X)^->- K (a real number), i.e., the set of attainable TT(X)^ for a problem Q is bounded but not closed from above. 2.4.2 Derived Demand as a Schedule of Shadow Prices Given that c1 E c^x^a), i.e., that the total cost of input 1 to the firm is independent of the levels of inputs 2, •••/N employed by the p firm, the firm's derived demand schedule D and schedule of shadow prices D^ for input 1 are equivalent. On the other hand given that c1 = c\x;a), i.e., that the total cost of input 1 to the firm is not independent of the levels of inputs 2, •••,N employed by the firm, in P Q general D and D for input 1 are not equivalent. These relations between derived demand schedules and shadow price schedules are stated more precisely as Theorem 1 and Corollary 1, respect ively. Theorem 1-A is obviously true, and Theorem 1-B follows from 1-A plus the envelope theorem.^ The properties of a derived demand schedule listed in Corollary 2 are deduced from Theorem 1. Note that p1 = p^x1) is a function rather than a correspondence (Corollary 2-B). In addition, the domain of p1 = p^(x^) is a convex set, and p^x1) is continuous and differentiable 31 within its domain. 30 The essential points of the proof can be summarized as follows. c1 = c1 (x1; a) (Condition 3) implies that input 1 can be fixed at the equilibrium level(s) xl*P and this factor supply schedule can be removed from the maximiz ation problem P without affecting the solution(s) x ^, which establishes Theorem 1-A. Then, by the envelope theorem (i.e., given an infinitesimal change in an exo genous variable, the change in the value of the objective function when all endogenous variables vary optimally is equal to the change when all endogenous variables remain fixed), Theorem 1-B is established. 31 . • Theorem 1 and Corollary 2 will be useful in proving the remaining theorem and corollaries in this chapter. 32 Theorem 1 and Corollary 1 provide the following rationale for employing Condition 3, i.e., c1 = c^x^a), in any study of the properties of derived demand schedules. Since the shadow price schedule is by definition invariant to changes in the supply schedule for input 1, p Theorem 1 implies that the derived demand schedule D is invariant to changes jn the supply schedule of input 1 given only that c1 = c^(x;a). p Since there is no economy in defining a derived demand schedule D for each possible specification of the supply schedule for the input, we shall restrict our study of derived demand schedules to cases where 1 _ 1, 1 . c = c (x ; a). Theorem 1. Suppose that conditions 1-3 are satisfied. Then (A) {x*P(a) } <=> (x*Q(x1*P(a)) } for all a, i.e., any problem P and the corresponding problem(s) Q have identical solution sets; and 1*P 1 i*P (B) {(x1 r(a), MFC'(a)) for all a}<=>{(x' r(a), a,(x*Q(x^P(a)))Q) fora|, a} Bx1 i.e., DP <=> DQ. 32 32 Formally Theorem 1 only applies to the case where input 1 is employed in a single enterprise, since the cost schedule for input 1 is defined as a function of only one input. However, Theorem 1 generalizes to the firm that employs input 1 in M enterprises. In this case, we can 1 _ 1 M 1; define c = c ( E x ';a) and the quantity constraint in a corresponding j=1 M ii -T problem Q as E x ' = x . It is easily shown that, with these modifications, Theorem 1 applies to the multi-enterprise firm as well as to the single enterprise firm. Corollary 1. Suppose that conditions 1-2 are satisfied, and that for all a: 1 *P •^-^—(a);a) * 0 for at least one i i 1. Then 3X1 (A) ix*P(a)} fl {x*Q(x1*P(a))} = null set for all a i.e., any problem P and any corresponding problem Q do not have any solutions in common; and (B) for any a: {(x 1*P(a), MFC1 (a)) } n DQ * null set if and only if 9R(x *) £ 9c'(x r) = 9R(x u) E 8c'(x g) 8X1 i=2 3X1 9X1 1=2 9x1 for all (or, equivalently, any) x e {x (a)} *Q , *Q, 1*P, 33 x ^ e ix (x (a))}. Corollary 2. Suppose that conditions 1-3 and 5-6 are satisfied, and denote the domain of p1 = p^x1) as X®. Then 1B (A) if x is included in a solution to at least one problem P, then all x1A such that 0 < x1A < xlB are in XD: (B) p1 is a function of x1, i.e., p1 = p^x1) associates one and only one p1 with any particular x1 in X^; amd Corollary 1-B also assumes that conditions 5-6 are satisfied. 34 (C) p (x ) is differentiable for all x "within" X D for .e / all x such that 0 < x < x 1A and x 1A . is an element of 2.4.3 Relation Between Surpluses in Factor and Product Markets As was stated in Section 2.2.1, previous literature has established under special conditions the equivalence of measures of surplus in a factor market and measures of producer plus consumer surplus. These analyses have assumed that factor supply schedules at both the firm and industry level are either perfectly elastic or perfectly inelastic. We shall establish this equivalence under general conditions by direct application of Theorem 1. The analysis has the following implications for a methodology of evaluating various programs that directly shift factor supply schedules: under quite general conditions, the user value and (in the absence of distortions in other markets) change in consumer plus producer surplus can be measured in commercial markets for the input. Since distortions are common and commercial markets for pasture are uncommon in British Columbia, the preceding comments do not apply to the evaluation of community pasture programs. Nevertheless, the results presented here support the approach adopted by Barichello (1978): estimating the farm value of the program by collecting data from the commercial market for an alternative use of improved land, and arriving at a measure of the change in consumer plus producer surplus by attempting to correct this value for distortions. 35 We shall now show that the usual conception of the relation between changes in surpluses measured in factor and product markets is correct — provided simply that each firm's total cost schedule for the input in question is independent of the levels of other inputs employed by the firm (condition 3). For the case where the shift in a factor 34 supply schedule is limited to a single firm, this will involve demonstrating that the change in surplus calculated in the firm's input market represents the general equilibrium benefits to the firm resulting from the shift in the factor supply schedule (given condition 3), and then noting the conditions under which these private benefits correspond to social benefits (in the sense of consumer plus producer surplus). For the case where the shift in a factor supply schedule is experienced by all firms in a group (e.g., an industry) we need only note (in addition to the above) that an industry demand schedule for an input is a collection of price and quantity combinations for derived demand schedules of individual firms. 2.4. 3.1 Shift in Factor Supply Schedule of Single Firm Given Theorem 1, it seems intuitively obvious that the change in surplus between a firm's derived demand and supply schedule for an input, due to a shift in the supply schedule of that input, is equal to the associated change in equilibrium net benefits for the firm. Likewise, we can prove Corollary 3 directly from Theorem 1. 34 As was stated in Section 2.2.1, this is essentially the case con sidered by previous studies. In other words, by ignoring all interactions between firms and by assuming the existence of an aggregate production function, the "industry" analyses in previous studies were equivalent to analyses of a single firm. 36 Corollary 3. Suppose that conditions 1-3 and 5-6 are satisfied. Then *A A (A) for any solution x to a problem P where a = a , TT(X*A) = TT(X*(0))Q + rx 1*A , K . 1 1, 1*A A, p(x Jdx - c (x ;a ) where N 7r(x*(0))Q = max{R(x) - I c'(x) : x1 = 0} i=2 p1(0) E _^(**(0))Q 3x' 36 and *A *B (B) for a solution x and a solution x to two problems A of the form P that differ only in terms of a = a and a = a , respectively. 7T(X ) - TT(X ) = rx 1*B 1*A 1. 1» . 1 1, 1*B B, p (x )dx - c (x ;a ) 1, 1*A A, - c (x ;a ) Suppose that there are no "distortions" in the economy, i.e., that (a) marginal factor cost is always equal to factor supply price for each firm, (b) marginal revenue is always equal to product demand price for each firm, and (c) government taxes and subsidies are non-existent. 37 In addition suppose that (d) the marginal utility of income is constant for all consumers; so that consumer's surplus can be defined in terms of ordinary (non compensated) product demand schedules. This correspondence between changes in surplus in a firm's factor market and changes in comsumer plus producer surplus can be deduced from Theorem 1 and Corollary 3 as follows. By Theorem 1, p1(xlA} = M 9b(y) 9yh(x*(x1A)) _ £ 3c'(x*(xtA)) _ 35 h=l 9yh 8X1 i = 2 3xJ .(1) 1 1A 1 IA 9 c (x • a) By (1) and assumptions (a)-(d), p (x ) —=—equals the differ-9x ence between the dollar-equivalent benefits received by consumers from the production associated with the marginal unit of input 1 minus the supply costs incurred during this production. Since these costs are equal to the surplus foregone by employing these resources in this 1 IA 9c1(x1A-a) particular manner (given assumptions (a)-(d)), p (x ) —=—is 9x] equal to the change in consumer plus producer surplus resulting from the production associated with the marginal unit of input 1, irrespective of 5The notation x*(x ) simply implies that x*(x ) is a solution x* to the problem Q defined by the constraint x1 = xT^, or to a corresponding problem P. 36Any right hand side derivative lim f(x +Ax )-f(x * for Ax1 >0 is represented here as Ax^O Ax1 3f(x1) I 9X1 38 the number of outputs and inputs involved in production. Likewise, by Corollary 3, IB IB [p'tx1) - dc lx 'a)] dx1 = 1A 9X1 x x [ E 3fa{y> 3y (x*(x') TA h=1 9yh dx1 - 2 "'"^'"id,1. (2, i=1 8X1 By (2) and assumptions (a)-(d), the surplus over the interval (x1^, x1^) in the firm's market for input 1 is equal to the change in consumer plus producer surplus resulting from the associated production. Therefore, any change in surplus generated in the firm's market for an input corresponds exactly to the resulting change in consumer plus producer surplus given assumptions (a)-(d) (and condition 3). 2.4.3.2 Shift in Industry Factor Supply Schedule The above analysis can be generalized as follows to the industry case, where shifts in c^x') for all J firms in the industry typically lead to shifts in other factor supply schedules and product demand schedules faced by the individual firm. Suppose that firms always face identical supply schedules for input 1. Then the set of static general equilibria *1 *J across the J firms {x , "-^x ) } can be expressed as a corresponsence of a single parameter a rather than of J parameters (a*, • ••,a*).. If we make the further assumption that a single static general equilibrium exists for each a, then the industry factor demand schedule can be expressed as 39 where {(X^a), p\a) : all a} 1 1* i X'(a) = Z x1 (a)' . . .(3) j=1 1 _ 1 1 1 * i i P (a) = maximum p in {p (x (a)J)J : j = 1, •••,J } - p (x (a) ) Thus the change in surplus in the industry market for input 1 resulting from A a AS = B A . . a - a can be expressed as aB 1, 1*, ,lvkM . Jv 1, 1*, BJ B, p (x (a) ) da - E c (x (a )';a ) a j=1 + Z c1(x1*(aA)j;aA) j = 1 (4) Statement (4) implies that the argument presented in the previous paragraph can be applied to the case of a surplus generated in a factor market by a number of interacting firms. Therefore, statement (4) implies that, given the assumptions of no "distortions" in the economy and of a constant marginal utility of income for all consumers, the change in surplus between the industry's demand and supply schedules for any input 1, resulting from a shift in c^x1) for all firms in the industry, is exactly equal to the 37 associated change in consumer plus producer surplus over all markets. 37 If firms do not face identical supply schedules for input 1, then statement (3) does not necessarily define the industry demand schedule for input 1. However, the industry demand schedule will still be a collection of price and quantity combinations from the derived demand schedules of individ ual firms; so a change in surplus (correctly measured) in the factor market due to shifts in the factor supply schedules for each firm will still correspond to the associated change in consumer's surplus. 2.4.4 Slope of the Firm's Derived Demand Schedule 40 Here we present a corollary and a theorem concerning the slope of a derived demand schedule, and effects of a finite shift in a factor supply schedule, that are independent of any assumption of a fixed price for the input. These statements imply the following: in general, an upper bound for the comparative static increase in the quantity of total pasture employed by a single recipient of community pasture cannot be deduced without incorporating empirical knowledge of the firm's production function into the analysis. The reason for this negative result is that the firm's derived demand schedule can be positively inclined under conditions that are reasonable a priori. Thus, even if we assume that the community pasture program does not increase the ratio of beef out put to the quantity of pasture employed by the user, we cannot deduce an upper bound for the comparative static change in beef output with out incorporating empirical knowledge of the firm's production function into the analysis 2.4.4.1 Relation Between Slopes of Derived Demand and Factor Supply Schedules at Equilibrium Textbook diagrams typically show that a firm's derived demand schedule intersects the supply schedule for the input 'from above' at an equilibrium in the factor market, and the analogue of this condition in 38 the product market was proved long ago. Nevertheless, this statement 38 For example, see Samuelson (1947), pp. 76-77. 41 apparently has not been proved in the past. Given the fundamental nature of this proposition and the controversy that eventually arose over the "obvious" relation between surpluses in factor and product markets, a proof of this statement appears desirable. Given Theorem 1 and conditions 1-3, it seems intuitively obvious that the following condition 1, K . . . 3c\x1;a) , . . 1A I, 1, p (x ) intersects 1—^—- from above at x , or p (x ) Sx1 3 c1 (x1 • oc) 1A 1 coincides with ——- at x and some level x in the 3x' 1A 1A neighborhood of x is necessary for activity level x to be included in a global solution to the firm's maximization problem P and 1A (almost) sufficient for x to be included in a local solution to P. Like wise, Corollary 4 can be proved from Theorem 1. Corollary 4. Suppose that conditions 1-3 are satisfied for a problem P. (A) If x is included in a local solution to P, then 1, 1A* 3c^(x^;a) p (x ) = i— 3x' „ 1, 1A, .2 1, 1A . 3p (x ) < 3 c (x ; a) 3x1 3X1 2 (B) lfp1(x1A) = 3c1(x1A;a) 3X1 » 1. 1A, .2 1, 1A , 3p (x ) 3 c (x ;a) 1 1 2 3x 3x 1A then x is included in a localsolution to P. 42 2.4.4.2 Relation Between Slope of Derived Demand Schedule and Various Properties of the Production Function The relations between slopes of derived demand schedules and certain additional properties of the firm's maximization problem (especially returns to scale in production, and convexity or non-convex ity of isoquants) can be developed essentially from Theorem 1 and 39 Corollary 4. These relations are summarized as Theorem 2. By statements A and B of the theorem, the firm's derived demand schedule for input 1 is never positively inclined when R(x) is concave, irrespective of the slope of c^x1). By statement D, the firm's derived demand schedule is perfectly elastic when R(x) shows constant returns to scale and the price for each input (other than 1) is exogenous to the firm, irrespective of substitution possibilities (shape of isoquants) and the slope of c^x1). ""By means of Corollary 4, we are able to equate the comparative static problem of determining the direction of change in equilibrium level of input 1, resulting from a change in the factor cost schedule c1(x1), to a problem of determining the existence of an equilibrium for particular specifications of c^x1). Since set-theoretic concepts, such as quasi-concavity and returns to scale, are readily incorporated into analyses of the existence of equilibrium, we are able to relate the direction of slope of an individual firm's derived demand schedule to such properties by these methods. An overview of the method of proof for Theorem 2 (as well as the proof itself) is presented in Appendix 2. 43 Theorem 2. Suppose that conditions 1-6 are satisfied. Denote the domain of p^fx1) as X^, and denote a wage or rental rate that is exogenous to the firm as w1. Then the slope of the firm's derived demand schedule is related to certain properties of R(x) and c'(x) (i = 2, •••,N) as follows: (A) If R(x) is strictly concave,40 then 9p (x * < 0 and 8X1 pVx1) > pVx1+e) for all (x1 ,x1+e) in XD, where e > 0. (B) If R(x) is concave,, then 8P < 0 for all x1 in XD. 3X1 (C) If R(Ax) i AR(x) for all A > 1 and x > 0 but R(x) is not concave, then: (1) 9P (x * ^ 0 always for at least some x1 in XD but 8x N . 1 1. (2) for some R(x) and Z c'(x) : P lx ' > 0 for some 1. YD 1 = 2 9x x in X . (D) If R(Ax) = AR(x) = AR(x) for all (x. A) > 0 and c1 = w'x' >V 9x for i = 2, ...,N, then i^li^ll = o for all x1 in XD. (E) If R(Ax) > AR(x) for all A > 1 and x > 0 and c1 = w'x1 for i = 2, -,N, then 8p (x ) ^0 and p1 (x< p1 (x1 +e) foralt ax1 (x1^1 + e) in XD, where e > 0.41'42 (on fo,lowin9 Pa9e) 40 The firm's total benefits function R(x), which is simply a total revenue function if the firm maximizes profits, is strictly concave if and only if (1) R(Ax) < AR(x) for all A > 1 and x > 0, and (2) all isoquants of R(x) are strictly convex for x > 0. Likewise, R(x) is concave if and only if (1) R(Ax) < AR(x) for all A > 1 and x > 0, and (2) all isoquants of R(x) are convex for x ^ 0. 44 However, a derived demand schedule may be positively inclined under many possible conditions. By statement C, a derived demand schedule may be positively inclined at some points even if R(x) shows decreasing returns to scale, provided that at least some isoquants are not convex. By statement E, a derived demand schedule is positively inclined over the entire domain when R(x) shows increasing returns to scale and the prices of all other inputs are exogenous, irrespective of substitution possibilities for the firm. Theorem 2 has important implications for the evaluation of community pasture programs. By Theorem 2: a positively inclined derived demand schedule is consistent with the notion of equilibrium for the firm, provided that R(x) shows increasing returns to scale or non-convex isoquants. Overlooking mathematical details (concerning inflection points) that are of no economic significance, conditions in A and E imply that ARVi < o and ^1 > ° • Sx1 9x' respectively. 42 1 1 > 1 ] Note the asymmetry between statements C and E: p (x ) < p (x +e) for decreasing returns to scale and fixed factor prices (i£l) whereas, p^x1) < p^x1 +e) for increasing returns to scale and fixed factor prices (i£l), where e > 0. 45 Neither of these conditions can be ruled out a priori. Therefore, even if we assume that the firm is at long run equilibrium before and after the introduction of the community pasture programs, we cannot rule out a priori the possibility than an inframarginal shift, of magnitude x0'3 units, in the firm's supply schedule of pasture leads to an increase in the level of pasture employed by the firm that is greater than x0'3 (see Figure 1). In addition, we can demonstrate that, for a downward shift in a firm's factor supply schedule cVx1), dx1* > dR(x*) i * < * x1 R(x ) 44 depending on properties of the particular maximization problem. In other words, under reasonable assumptions the percentage change in (farm value of) output can be either more or less than the percentage change in total pasture that is due to the community pasture program. This defines a second source of difficulty in calculating a finite upper bound for the comparative static change in beef output associated with community pasture programs (in addition to problems in calculating a finite upper bound for the change in total pasture). If the supply schedule for any input is positively inclined, then R(x) may show increasing returns to scale and/or non-convexity of iso-quants in the neighborhood of an interior long run equilibrium, i.e., these conditions are consistent with the maximization hypothesis. Moreover, increasing returns to scale and non-convexity of isoquants cannot be ruled out a priori as "unreasonable" properties of a production process relevant to comparative static modelling. The argument can be summarized as follows Divisibility of the production process would imply decreasing or constant returns to scale, and additivity and divisibility together would imply con vexity of isoquants (and also constant returns to scale) (see Malinvaud, 1972, pp. 51-3 for definitions). However, divisibility may be unrealistic, and additivity is reasonable only for models of change in long run equilibrium (in any "short run," certain changes in input levels are likely to be, in effect, infeasible due to high adjustment costs). 44 (on the following page) 46 c P x = number of units of community pasture a Noted to the firm AxP E change in the number of units of pasture employed by the firm (due to the community pasture program) P p(x ) = the firm's derived demand schedule for pasture s(xP) E the firm's supply schedule for pasture, prior to the community pasture program x(xP)' E the firm's supply schedule for pasture, as a result of the community pasture program Figure 1 Given a Positively-inclined Derived Demand Schedule for Pasture, the (inframarginal) Allotment of Community Pasture is Less than the Resulting Change in the Level of Pasture. 47 2.5 Summary of Implications for the Evaluation of Community Pasture Programs In this section, we summarize the major implications of the theory presented in this chapter for evaluations of community pastures programs. The restrictions on comparative static effects of community pastures programs that are implied by this theory have been shown to be extremely weak. Indeed, these restrictions may be considerably weaker than the reader had previously considered possible, or at least reasonable. Thus, the discussion here should help us to avoid errors in our a priori theorizing about "likely" comparative static effects of community pastures programs, and underscores the importance of incorporating into our analyses greater knowledge of the producer problem(s) faced by users of community pasture. 2.5.1 Relation Between Surpluses in Factor and Product Markets The analysis of the relation between surpluses in factor and product markets (Sections 2.2.1 and 2.4.3) implies that, for various programs shifting factor supply schedules, the user value and (overlooking ^ it ic U4Silberberg (1974b) shows that > dR(xJ for firms that x1 R(x ) minimize average cost (as in long run competitive equilibrium). Assuming 2 2 2 that the number of inputs equals 2 and that c = w x , we can also show 1 * that ^r*- > ^(x ^ if R(x\x2) is homogenous of degree less than 1 and x R(x*) 1* * all isoquants are convex, and that -y^- < tx ' if R(x ,x ) is homogeneous x1 R(x*) of degree greater than 1. distortions in other markets) change in consumer plus producer surplus could be measured directly in commercial markets for the input (see Figure 2). This analysis, plus the presence of market distortions and absence of commercial markets for pasture in British Columbia, suggests the following approach to the evaluation of community pasture programs. The farm value of the program is estimated from data concerning the commercial market for an alternative use of improved land (e.g., as hay land), and the related change in consumer plus producer surplus is 45 estimated as this farm value plus or minus corrections for distortions. 2.5.2 Slope of Derived Demand Schedule and the Measurement of Distortions However, the theory presented in this chapter does not provide any useful restrictions concerning the comparative static effects of community pasture programs on distorted markets. In particular, the theory presented in this chapter does not provide any useful restrictions concerning the comparative static change in beef output for the representative farm. Indeed, this theory does not determine an upper bound for the change in total pasture employed by the farm, which in itself precludes the calculation of an upper bound for the change in beef output for the farm. This last statement can be elaborated upon as follows. See Barichello (1978). price 49 p(xP) quantity of pasture services xcp = number of units of community pasture alloted to the farm pcP = rental price of community pasture p(xK) = firm's derived demand schedule for pasture s(xP) = firm's supply schedule for pasture, prior to community pasture program s(xP)' E firm's supply schedule for pasture, as a result of community pasture program pn E equilibrium price of pasture, prior to community pasture program P 1 equilibrium price of pasture, as a result of community pasture program net user benefits associated with the increase in employment of pasture net user benefits associated with the land freed from use as pasture Figure 2 Estimating the User Value of Community Pasture Programs in a Market for Pasture.^ 46, This diagram is discussed in Barichello (1978), pp. 28-32. 50 2.5.2.1 General Case As shown in Theorem 2, derived demand schedules can be pos itively inclined under reasonable conditions. In the absence of considerable knowledge of the farm's production function, we can only infer than the derived demand schedule for pasture cuts the supply schedule for pasture "from above" (Corollary 4). However, this condition seldom places any useful restrictions on the change in total pasture employed by the farm. Assume, for simplicity, that the aggregate cost schedule of pasture to the farm can be written as c1 = c^fx^a), where x1 denotes the total quantity of pasture employed by the farm. The community pasture can be represented as a small change in the parameter a that leads to a decrease in the marginal cost of pasture for some levels of x1, i.e., 9 C^fx^a) . » , 1 . B^C^fX^a) n r n *u 1 if ^—'-— < 0 for some x and ^ —- = 0 for all other x . If 9x 9a ' 9x 9a this cost schedule is continuous at the farm's pre-community pasture equilibrium x* and the farm employs pasture at this equilibrium, then the following condition is satisfied: PV*) - 9cl(*1*;a> = o . ... .(5) 9X1 Totally differentiating (5) with respect to a for the change in the level of total pasture employed by the farm: 2 1 1* 9 c (x ;a) i * 1 9x' _ 9x 3a 2 11* 11* .IDj 3a 3 c'(x' ;a) _ 3p'(x' ) 3X1 2 3x1 51 2 1 1 * Therefore, given (6) and - 9 c (x—> 0 , the condition 9x 9 a 1 1 * 2 1 1 * 9p(x).9c(x;a) . . . .. A . • —C : < =—T simply implies the restriction 9x 9x' < < +CO ... .(7) 9 a which is not helpful. By Theorem 2, a restriction stronger than (7) cannot be obtained without knowledge of the farm's production function. 2.5.2.2 Special Cases Nevertheless, note that we could obtain meaningful results if the community pasture program did not affect the marginal cost of pasture at equilibrium: „ 1* .2 1, 1* . 9x A . 9 c (x ;a) n -s = 0 given \ '—— = 0 . 9 a 3 • « *"\ 9x 9 a This case is accurate when the farm can rent additional pasture or hay land at a constant price at equilibria both before and after introduction of the community pastures program. However, commercial rental of pasture does not appear to occur in the Peace River and Prince Ceorge regions, and appears to be accurate for only a minority of beef ranches in other areas. Moreover, rental of hay land seems uncommon in Prince George, and the rental price of hay land seems essentially endogenous to the firm in the Peace River. 52 Allowing for discontinuities in the aggregate supply schedule of pasture to the farm implies only one modification of the above conclusions. If the supply curve is vertical at the equilibria established both before and after introduction of the community pastures program, then the resulting change in total pasture is identical to the quantity of community pasture rationed to the farm by the program. Such a supply schedule may be accurate for some short run producer problems P. However, farms using community pasture in the Peace River and Prince George regions typically allocated own land to both pasture and hay (and often grain as well) prior to and after introduction of the pastures program. This suggests that the assumption of a vertical supply schedule is inappropriate even in the short run for these evaluations. 2.5.3 Further Research In sum, we have shown that we cannot obtain useful restrictions on the comparative static changes in pasture (and in beef output) in the absence of knowledge about the production function of a farm. However^ our knowledge of such production functions is uncertain. Moreover, this knowledge is largely expressed in terms of parameters (e.g., a set of "reasonable" values for a factor substitution effect) that have not been directly incorporated into the traditional methods of qualitative comparative statics. In the next chapter, we shall develop a technique for incorporating such knowledge into comparative static analysis. In principle, this method will enable us to place many restrictions on the structure of a producer problem P without at the same time specifying the entire structure of the problem. CHAPTER 3 QUANTITATIVE COMPARATIVE STATICS AND DERIVED DEMAND:A PROPOSED METHODOLOGY 54 CHAPTER 3 QUANTITATIVE COMPARATIVE STATICS AND DERIVED DEMAND: A PROPOSED METHODOLOGY 3.1 Introduction In this chapter, we shall introduce a method of "quantitative" com parative statics that is designed to incorporate many empirically-based restrictions into the theory of the firm without specifying essentially unknown parameters. A detailed presentation of the method necessarily includes many equations, and computational experience to date has been minor. For these reasons, details of structure and means of reducing computational problems have been relegated to appendices. It has been the author's conviction that, in the initial stage, research related to this method of quantitative comparative statics should emphasize clarification of logical structure and means of facilitating compu tation rather than the collection of numerical results. The theory and methods to be presented in this chapter and related appendices suggest that this methodology will be useful in predicting farm or firm response in various policy situations. Likewise, this methodology may well be useful in generating testable hypotheses of farm or firm behavior. 55 3.1.1 The Problem In the previous chapter, we stated that the traditional methods of comparative static analysis, as embodied in Samuelson (1947) and more recent dual and primal-dual approaches, cannot readily incorporate many quantitative restrictions on production functions. We were able to incorporate various properties of production functions into our analysis of derived demand, but the results largely served to emphasize the value of including many empirically based restrictions on production functions in comparative static analysis. These results showed that, in the case of endogenous factor prices, not even the slope of the individual firm's derived demand schedule can be signed unless the analysis incorporates empirically based restrictions that are sufficient to determine convexity or non-convexity of isoquants and decreasing or increasing returns to scale. These latter properties are not easily observed directly. In addition, we saw that the usual qualitative analysis (supplemented by empirical observation) is seldom able to place any meaningful restrictions on the comparative static effects of community pasture programs. In particular, such an analysis plus empirical observation of the factor supply schedule can seldom lead to a finite upper bound on a comparative static change in beef output (or even pasture) due to a community pasture program. These results serve to complement previous observations on the difficulties of obtaining many useful results (signed effects) from tra ditional methods of comparative static analysis. It is well known that 56 relatively few comparative static effects can be signed from the maximiz ation hypothesis plus qualitative knowledge of the elements of the firm's Hessian [TT,..(X*)]. Although signed results could be obtained by incorpor ating quantitative restrictions on the elements of [ir..(x*)], there does not appear to be any empirical basis for placing such restrictive conditions directly on the elements of the Hessian [TT.J(X*)]. Thus such restrictions would have to be derived indirectly from other (more empirically based) restrictions. In the absence of such restrictions, relatively few predictions of farm response or testable predictions of farm behavior can be obtained. Given our assumption that'constructs of static, optimizing behavior have utility in the applied economics of agriculture (an assumption to be verified in the next chapter), alternative methods of comparative static analysis at the firm level seem desirable. The commonly employed alternative has been to specify exactly the structure of the individual firm's problem "maximize TT(X)," to compare the solutions x* and x** for two different values of the exogenous variable a, and to perform a sensitivity analysis by repeating this procedure for alternative structures TT(X). However, this second approach also has serious draw backs in the absence of fairly complete knowledge of the correct structure for TT(X). Since the number of possible structures is infinite and the Ax * relation between structure and comparative static effects ^ is likely to be complex, any procedure that relies on specifying exactly alternative structures TT(X) can bound the set of "reasonable" comparative static 57 effects only if the set of "reasonable" structures TT(X) is quite small. In general, there appears to be considerably more knowledge of the structure of TT(X) than has been incorporated into qualitative com parative static methods; but knowledge of TT(X) is far from complete. Thus there is need for a method that incorporates many restrictions on the structure of the firm's maximization problem into comparative static analysis without specifying an exact structure for TT(X). Moreover, in order to be most useful as a tool in applied economics, this method should be capable of placing quantitative as well as qualitative bounds on comparative static effects of interest. 3.1.2 A Proposed Methodology In this chapter, we shall introduce a method for incorporating empirically based quantitative restrictions on TT(X) into the traditional qualitative comparative static analysis of the firm. Quantitative as well as qualitative bounds on comparative static effects can be calculated by this method. Thus this method can, in principle, calculate a "reasonable" finite upper bound on a comparative static change in beef output from empirically based restrictions on a beef ranch's production function and price schedules. A detailed discussion of the method and of partial solutions to the important computational problems are presented in accompanying ^ee Section 1. 2 of Chapter 1. 58 appendices. In addition, a simple illustrative model for community pasture programs is presented in this chapter. The major task of accumulating computational and practical experience with the method has essentially been postponed to a further study. We shall express our quantitative restrictions, and the usual restrictions implied by the maximization hypothesis, as (a) a set of equations relating the comparative static change in the firm's activity levels 9 x -r— to potentially observable properties p of the firm's maximization o Ot problem, and (b) a set of empirically derived restrictions on p. By 9x' calculating the maximum and minimum values of -5— (or of a scalar 9x function z(.-~—)) over this feasible-set (a and b above), the range of a Ot 9 x' 3 x comparative static effects •* (or z[-z—)) that is consistent with the a Ot 0 Ot maximization hypothesis and the specified restrictions on p can be 9 x determined. These upper and lower values of z(-~—) can in principle be calculated as solutions to corresponding (non-linear) programming problems. The set of variables p largely consists of various factor sub stitution and scale effects defined for various subsets of fixed inputs. The rationale for emphasizing such variables p in a comparative static model is essentially as follows: our knowledge (from direct and econometric observation of firm behavior and physical processes) typically is in a form more closely related to such parameters p than to the elements of the Hessian [TT:.]. 59 This method of quantitative comparative statics is not without its drawbacks. In particular, a local solution to the above nonlinear programming problems is not necessarily a global solution, and the number of equations in the set of constraints increases exponentially with the number of inputs included in the firm's maximization problem. However, there appear to be somewhat adequate methods of coping with both problems. 3. 2 Previous Methods of Comparative Static Analysis Methods of comparative static analysis can be classified as either "qualitative" or "quantitative": qualitative methods incorporate restrictions primarily on the sign of parameters, whereas, quantitative methods incorporate many restrictions on magnitudes as well as signs of parameters. Thus qualitative methods can only lead to qualitative restrictions on com parative static effects, whereas, methods classified as quantitative can lead to either quantitative or simply qualitative restrictions on comparative static effects. Qualitative methods have been developed for uses of both minimal and exhaustive restrictions on the signs of parameters; but quantitative methods have been applied essentially only in cases where the structure of the maximization problem (or equilibrium system) is completely specified. 60 3.2.1 Qualitative Methods 3.2.1.1 Minimal Restrictions Primal, dual and primal-dual methods of comparative static 2 analysis have been applied to models of the firm or group of firms where only the maximization hypothesis, i.e., the existence of an interior static equilibrium where TT(X) is twice differentiable, and competitive conditions 9x[ 8oJ 3 ax1 are assumed. However, any comparative static effect —— for a primal problem maximize ir(x; a) is signed unambiguously by the maximization hypothesis if and only if 2 9 TT(X*) i i ——r- is signed and (x , or) are "conjugate pairs," i.e., —j-—:- = 0 for all k ± i. This condition for signing —r is not ax^ce1 See1 altered by the assumption of perfect competition. Similar comments must apply to dual and primal-dual methods based solely on the maximization hypothesis and competitive conditions. 2 See Sections 2.3.1 and 2.3.2. 3 The assumption [TTJJ] negative definite at x* actually imposes quantitative as well as qualitative restrictions on [TTJJ]; but comparative statics based solely on the maximization hypothesis has nevertheless been defined as a "qualitative" method. 4See Samuelson (1947), pp. 30-33 and Archibald (1965). 61 3.2.1.2 A Calculus of Qualitative Relations The relation between the signs of the comparative static effect and the signs of the elements of the primal structure [TT..] has also been investigated. The model considered by this literature can be expressed as r i 3x 1 .1a in the case of a shift in the supply schedule of input 1 for a single firm, or more generally as [A] dx The problem posed by this literature can be expressed as follows: when 3x 3a x can the signs of the elements of -r— (or dx) be deduced from knowledge of the signs of the elements of [TT..] and cja (or [A] and b)? Thus the central problem considered in this literature is the deducation of the signs 'j -1 -1 of elements of [TT..] (or [A] ) from knowledge of the signs of elements of [TT..] (or [A]). 5ln this chapter, partial derivatives will usually be specified in sub script for m with arguments omitted. For example, -^0- = and ^(x';") E c\ . In addition, the structure of the firm's objective function will be specified as TT(X) or equivalently Tr(x;a), and the total cost schedule for input 1 will be specified as c'lx'ja) (when a is clearly a scalar)or equivalently c1 (x1;a). 62 Sameulson (1947, pp. 23-29) pointed out that the sign of an element of [TT..] 1 can be deduced solely from qualitative knowledge of the elements of [TT..] only under unusual conditions, and these conditions have been formulated more precisely by others.6 Moreover, combining the maximization hypothesis with such qualitative knowledge does not significantly reduce this indeterminacy of comparative static effects.7 In addition, Lancaster (1965, 1966) developed a computational procedure for determining the qualitative solution of such systems. Given a system of equations expressed in the form [B]y = 0 and knowledge of the signs of the elements of [B], the sign pattern of the vector y (where an element is either +, - or indeterminate) can be calculated by a method presented in Lancaster (1966). 3.2.2 Quantitative Methods Quantitative comparative statics has been employed only in the case where the structure of the primal problem maximize TT(X; a) or of the equilibrium system is completely specified. Lancaster (1965) has pointed out that his general approach can incorporate partial quantitative g See Lancaster (1962) (1964), Gorman (1964) and Bassett et al_. (1968). 7See Quirk and Ruppert (1968). 63 information about [ TT - j ]; but his method does not provide an adequate basis for a quantitative comparative statics of microeconomic units (see Section 3.3.2.2). 3. 3 Limitations of Previous Methods of Comparative Static Analysis In general, there appears to be considerably more knowledge of the individual firm's structure TT(X) (or, equivalently, tTTJ-]) than has been incorporated into qualitative comparative static methods. In particular, there is often considerable knowledge of TT(X) that is quantita tive in nature and difficult to incorporate into established methods of qualitative comparative statics. Moreover, incorporation of quantitative restrictions on TT(X) into comparative static analysis may permit the calculation of quantitative restrictions on comparative static effects as well as lead to greater qualitative determination of comparative static effee ts. On the other hand, knowledge of TT(X) is far from complete and the relation between structure and comparative static effects is likely to be complex. Thus the established quantitative comparative static methods, which rely on specifying the entire structure of TT(X), cannot readily bound the set of comparative static effects that is consistent with the set of "reasonable" structures for TT(X). 64 3.3.1 The Qualitative Relation Between Comparative Static and Comparative Dynamic Effects Lancaster (1962, p. 100) presents essentially the following argument g for calculating only qualitative properties of comparative static effects: the comparative dynamic effects over time of a unidirectional change in a parameter almost always have the same sign as the corresponding compara tive static effect (whereas, the effects obviously have different magnitudes). It is often asserted that this last statement about dynamics follows from the static Le Chatelier principle. However, this argument is incorrect: comparative dynamic effects of a unidirectional change over time in a parameter can easily vary in sign as well as in magnitude over time. For example, if capital stocks adjust somewhat in the short run, then the short run (impact) effects may differ in sign from the long run (comparative static) effects.10 The short run and long run effects necessarily have the same sign only if (in the short run) one set of inputs remains fixed at the initial equilibrium levels and all other inputs adjust so as to attain a new full equilibrium given the levels 8Lancaster also states that incorporation of quantitative restric tions greatly complicates the analysis. However, in Section 3.4 we shall out line a comparative static method that can (at least in principle) incorporate quantitative restrictions on TT(X) at a reasonable cost. q To be more specific, Lancaster states that the signs of the impact (short run) effect and (assuming stability) long run effect of a change in a parameter can always be calculated correctly by comparative static methods, and that intermediate run effects of a (unidirectional) change in a parameter can generally be signed by comparative static methods. 10See Nagatani (1976) and Yver (1971). 65 of the fixed inputs, i.e., these effects necessarily have the same sign only if the short run and long run correspond exactly to the static models con sidered in the Le Chatelier principle. Thus comparative statics can in general be employed correctly only in the estimation of long run effects or of essentially static short run or intermediate run effects, which in turn implies that qualitative properties of comparative static effects are only as valid as the quantitative properties of comparative static effects.11 3.3.2 Qualitative Methods Qualitative comparative static analysis is known to be unsatisfactory: 8 x' i i —i- is signed by the maximization hypothesis only if (x , cr) are conjugate 9 a1 3x' pairs, and —r is signed by qualitative knowledge of the elements of 8a1 [TT..(X*)] only under unusual circumstances. Moreover, the elements of [TT..(X*)] are not generally observable; so there is seldom an empirical basis for expressing quantitative restrictions directly on the elements of 13 [TT..(X*)] and incorporating these into the analysis. If the restrictions employed in a qualitative analysis are a subset of the restrictions employed in a quantitative analysis, then of course the set of results obtained by the qualitative method are more (or at least not less) ambiguous —and therefore more likely to include measures of compara tive dynamic effects within its range — than are quantitative (or qualitative) results obtained by the quantitative method. However, these qualitative results are "superior" to the quantitative results only in this trivial sense, i.e., only in the sense that an arbitrary marginal relaxation of quantitative restrictions necessarily leads to "superior" (more ambiguous) results. 12 See Section 3.2.1. 13 Note that, if the comparative static problem is [A] dx=b where [A] is (e.g.) a matrix of first derivatives of net aggregate supply schedules, then quantitative information about the elements of [A] may be directly available. However, such information is unlikely to be available if [A] dx =b describes microeconomic units. 66 3.3.2.1 Primal, Dual and Primal-Dual Methods However, quantitative restrictions are not readily incorporated into available methods of qualitative comparative statics that utilize the maximization (or cost minimization) hypothesis. Traditional primal methods are known to be quite messy in the case of such restrictions, and dual and primal-dual methods cannot readily incorporate the restriction of concavity 1 a for F(x) in the presence of variable factor prices. 3.3.2.2 A Computational Method of Lancaster The computational approaches of Lancaster (1965 , 1966) also fail to provide a satisfactory means of incorporating quantitative restrictions on microeconomic units into comparative static methods. His first method (1965) calculates whether or not all elements of a vector y are qualitatively determined by a system of equations and particular convex cones as feasible sets for the columns of [B]. These convex cones can in principle incorporate quantitative as well as qualitative restrictions. Given the system of equations [B]y = 0 .(1) .(2) See Sections 2.3.1 and 2.3.2 of Chapter 2. 15Convex cones are the class of convex sets such that Ax is included in the set for all A Z 0 if x is included in the set. and particular convex cones as feasible sets for the columns of [ TT-.], Lancaster's method can in principle be used to calculate the sign pattern 9 x of -r—. His second method (1966) is more general in that it directly a Oi signs all elements of Y(+, - or indeterminate) from (1); but knowledge of [B] is generally restricted to signs on elements of [B]. However, it appears that Lancaster's approaches cannot incorporate many empirically-based quantitative restrictions at the firm level. The argument for this statement can be sketched as follows. First, we shall later argue that most of our relevant quantitative knowledge is not directly expressed in terms of the elements of 1^.]. For example, there may be considerable knowledge about the elements of a matrix K that is related to [TT..] by the system of equations iT c. K (3) where c. = i 1 1 * rdc (x ;a 9x 9 c (x ) \l N and I = an identity matrix. Second, Equations (2)-(3) cannot be expressed in the form of (1) where y 9 x includes all elements of — and excludes all elements of [TT..]. Thus 9a IJ Lancaster's approaches are inappropriate as a method for quantitative 16 comparative statics at the firm level. 16 Likewise, empirically-based quantitative knowledge of the elements of a matrix B seem unavailable for other microeconomic units. In addition, Lancaster's approaches are not intended to incorporate second order con ditions implied by the maximization hypothesis. 68 3.3.3 Quantitative Methods The established quantitative approach is to specify ?r(x) precisely and to calculate solutions (x*, x**) to the problem maximize TT(X; a) for two different values of a, and to perform a "sensitivity analysis" by repeating this procedure for a limited number of alternative specifications of TT(X) . However, knowledge of TT(X) is far from complete and the relation between structure and comparative static effects is likely to be complex; so an adequate sensitivity analysis seems improbable or at least very costly. Thus this quantitative approach cannot readily bound the set of comparative static effects that is consistent with the set of "reasonable" structures of TT(X). In principle, the above procedure can be modified by replacing the fully specified structure TT(X) with a flexible functional form TT(X)^ whose structure is not entirely specified.17 However, this approach also appears unsatisfactory. 17Such an approach of directly calculating x*(a) and x**(a + Aa), if successful, would have the following advantages over the traditional approach of directly calculating 3x* : the social effects of many programs 3a may depend on the levels x*(a) and x*(a + Aa) rather than simply on the 3x* difference -z / and some of the parameters of ir(x) at x* may depend a a critically on the level x* (as in the case of a constant elasticity of factor substitution). For example, suppose that the procedure is to solve a pair of problems of the form 69 maximize TT(X; q) subject to E(x, p) = 0 L < < U P ^ o < o minimize TT(X ; a) subject to E(x,p) = 0 P ^ P ^ P for one value of a, and then for a second value of a. The equations E(x,p) = 0 express restrictions on the structure of TT(X)^ in terms of a vector p of observable parameters, and the inequalities p*~ ^ p £ p^ specify "reasonable" restrictions on p. However, this approach to quantitative comparative statics has the following serious defects: (a) the solution values for p in any two problems with different values for a will almost always be different, whereas, p is (by definition) to be unchanged by changes in a; and (b) even overlooking (a), this approach cannot bound any comparative static effects other than the change in TT(X*)1 or in variables that are in fixed proportion to rr(x*)^. 3.4 A Proposed Methodology for Quantitative Comparative Statics Users of qualitative comparative statics methods have sought to supplement the meagre content of the maximization hypothesis largely by attempting to calculate the signs of elements of [Tr..] 1 from restrictions 1 g placed directly on the elements of the Hessian matrix f TT.. 1. However, 18See Section 3.2.1. 70 this approach has not been successful in the modelling of the firm or other microeconomic units: such qualitative knowledge of [TT..] is seldom sufficient to determine the signs of elements [TT..] 1 and direct knowledge of the magnitudes of elements of [TT..] is in general unavailable since these elements are essentially unobservable. Moreover, these methods of qualitative comparative statics cannot readily incorporate additional quantitative restrictions, and established methods of quantitative comparative statics cannot readily incorporate anything less than a full 19 specification of the structure TT(X). Thus, there is need for an additional method of comparative statics analysis that incorporates quantitative restrictions on many potentially observable parameters of the firm's static maximization problem(s) without specifying an exact structure for TT(X). Since qualitative comparative statics is only as valid as quantitative comparative statics, results obtained with such a method would in principle have the same status as results 20 obtained with methods of qualitative comparative static analysis. Here we shall propose such a method of comparative static analysis and shall illustrate how this method can be applied in principle to the evaluation of community pasture programs. In contrast to the usual comparative static approaches, which attempt to deduce knowledge 19 See Section 3.3.2. 20 See Section 3.3.1. 71 of [TT..] 1 from restrictions placed directly on tTT-j 1, we shall place restric tions directly on the inverse of matrices that are essentially sub-matrices of [TT..]. 'J In this manner we shall arrive at a system of equations and in equalities which incorporate the restrictions on the comparative static 125 9 a g X effect -r- that are implied by (a) the maximization hypothesis, plus (b) "reasonable" restrictions on potentially observable parameters p of the structure of [ TT.. 1. i j Given an interior solution to the producer's static optimization problem "maximize Tr(x;a)," the restrictions implied by the maximization hypothesis are the total differential of the first order conditions, i.e., , , 9x 21 ij 9 a ia (NxN) (NX1) (Nx1) plus the second order condition [TT..] negative definite. (NxN) 9 x Thus the range of comparative static effects z = z(-^—) that is 20 See Section 3.3.1. 21 In order to make the discussion less abstract, we shall assume that the exogenous change experienced by the firm i for input 1 c = c^x^a), i.e., s a shift in its supply schedule - TT. ia 0 where c! •= ^V*;^ 'la _ 1 -9x 9 a 72 consistent with (a) and (b) can in principle be calculated from the two programming problems maximize z 9x 9a minimize z 9x 9 a subject to [TT..] |£ = -Trja subject to [TT..] || = -Trja [TT..] negative definite [TT..] negative definite CCEiTj.], p) = 0 L . . U P ^ P ^ P C([TT..], p) = 0 P ^ P ^ P where C([iTj.],p) =,0 denotes the relations between the Hessian [TTJ.] and 9 x the more directly observable parameters p, the variables (g-jp [ TT..], p) are endogenous to the problems, and (TTJ , pK p^) are exogenous to the 22 L U problems. The restrictions p ^ p £ p denote our degree of empirical knowledge about parameters p of the structure of [TT..] . In the case of 9 x community pasture programs, the scalar-valued function z = z(-r-) may (e.gO a a define the comparative static change in producer plus consumer surplus as a function of the comparative static change in the firm's input levels that is induced by a community pasture program. Note that these two programming problems are essentially analytic rather than simply behavioral in nature. The behavioral implications of the maximization hypothesis are defined there by the relations 22 The most important types of equations and inequalities for such problems are summarized in Figure 3. 73 3 x [TT..] -x— = - TT. [TT..] negative definite ij 3a ia ij a 3 x where-r— and [TT..] are treated as endogenous variables. These equations, 3a ij 3 ^ plus the restrictions G([TT..], p) = 0 pL < p < pU where p is an additional set of endogenous variables, define the analytical 3 x relations between the variables ( , [Tr-], p) that are consistent (feasible)with the behavioral assumptions and the degree of empirical knowledge p*" ^ p ^ p^. Thus, solving the two programming problems above is a purely analytical procedure for obtaining the extreme values of 3 x the set of values for z = z(.-z—) that are consistent with the maximization a a hypothesis and the degree of empirical knowledge p'" 2 p £ p^. The vector of parameters p typically includes measures of the following types of properties of E TT-.3: (a) possibilities of factor substitution within any subset of inputs, and (b) scale effects (changes in input levels and "profits") for a given change in output when any subset of inputs is held constant and all other inputs vary optimally in the static sense. In these cases, the relations G([TT..],P) = 0 in effect decompose the Hessian matrix [TT..] into a set of more directly observable parameters p. A priori knowledge of a range of "reasonable" values for some of these 74 parameters presumably is available in most cases, in contrast to the 23 essentially unobservable elements of [m.] per se. This knowledge would be derived from observation of physical processes, from econometric esti mation of physical processes and "short run" behavior, and from observation of firm behavior that approximates various "short run" comparative static effects. By formulating these restrictions on p as confidence intervals or 9 x 9 x as Bayes intervals, the corresponding feasible set for and z —) can o Ot o Ot also be interpreted as a confidence-Bayes interval. Thus the values of 9 x Z(T;—) at the solutions to the two programming problems presented above O CX 9 x define the confidence-Bayes interval for z(y^) that is implied by the maximization hypothesis and the empirically-based restrictions (confidence-Bayes intervals) p*" ^ p ^ p^ in the model. This method of comparative static analysis is not without its drawbacks. In particular, a local solution to either of the above programming problems is not necessarily a global solution, and the number of equations in these models increases exponentially with the dimension of the input vector x. However, there appear to be somewhat adequate methods of coping with the local-global difficulty and of aggregating inputs 24 and enterprises. Further research on these matters seems desirable. 23 Moreover, since the elements of [TT..] as well as of p are included as endogenous variables in the above programming problems, any direct qualitative or quantitative knowledge of the elements of [TTJJ] can easily be incorporated into our model as restrictions of the form TT..L ^ TT.. £ TT..u (i,i = 1, •••,N). IJ ij ij 1 ^See Appendix 5. 75 3.4.1 Restrictions Implied by the Maximization Hypothesis It can be shown that the assumption of maximizing behavior is essentially as realistic as the results of comparative static analysis, and that comparative static methods usually are more appropriate than comparative 25 dynamic techniques for the evaluation of community pasture programs. Thus it is important to incorporate the restrictions implied by the maximization hypothesis, i.e., by the existence of an interior static maximum, into our methodology. However, in order to avoid placing arbitrary restrictions on the structure TT(X), we should model in this manner only those restrictions that correspond exactly to the comparative static implications of the maximization hypothesis. The task of determining the precise comparative static implications of the maximization hypothesis has been labelled the "integrability problem" in comparative statics (Silberberg, 1974a), and has been largely solved in the case of the dual approach to comparative statics (Epstein, 1978). It can also be shown that, for problem P 11 N i i maximize TT(X;OC) E R(X) - C (X ;a) - E c (x ) i=2 the usual set of primal restrictions 1 .(P) [77ij ] ITa 'la 25, See Appendix 1 and Chapter 4. 76 [>..] symmetric and negative definite corresponds exactly to the implications of the maximization hypothesis for primal comparative statics in the case of a shift in a single firm's supply 1 11 26 schedule c = c (x ;a) for input 1. Thus the "integrability problem" is solved in this special case. In addition, the restriction [TT..] negative definite can be expressed in a form that is more appropriate for our (primal) quantitative comparative statics model: Theorem. A real symmetric matrix A is negative definite if and only if there exists a real lower triangular matrix H with positive diagonal elements such that A = - HH^". 3.4.2 Major Additional Restrictions Here we shall outline how the Hessian matrix [TT..] at a solution x* to the producer's static optimization problem "maximize Tr(x;a)" can be decomposed into more readily observable factor substitution effects and 27 scale effects within any subset of inputs. These and other relations were denoted in the two programming problems above as G([TT..],P) = 0. In contrast to the usual comparative static methods which place restric tions directly on the elements of [TT..(X*)], these relations shall place restrictions on the inverse of matrices that are essentially submatrices of [Tf..(x*)]. 20 Our quantitative comparative statics analysis could be extended easily to the case of a shift in the firm's product demand schedule (see the related section of Appendix 4). 27 The relations between [TT..(X*)] and the more readily observable properties presented here are fairly' obvious, and are detailed within Section 3.1 of Appendix 3. 77 3.4.2.1 Model with Output Exogenous Given the firm's static maximization problem N . . . maximize TT(X; a) = R(x) - Z c (x ; a) ... .{P) i=1 with solution x*, we can arbitrarily define the related problem where output is treated as exogenous to the firm N . . . maximize TT(X; a) = R(x) - I c (x ;a ) i=1 ... .f4) subject to R(x) = R(x*) This problem (4) will enable us to decompose [TT..(X*)] for the producer's static optimization problem P into substitution and scale effects with all inputs variable. Problem (4) can be expressed in Lagrange form as maximize 7r(x;a) - X(R(x) - R(x*)j . . . .(5) where the endogenous variables are (x,X) and the exogenous variables are (a,R). Suppose that the differentials of the interior first order conditions for (5) with respect to each of (a,R) yield a unique solution for all comparative static effects ' dx** 3 X dx** d X 8A ' DA' 3R ' 3R 28 28 It can be shown that this assumption of uniqueness is correct whenever any such comparative static effects exist for a problem (5). 78 This assumption is equivalent to the restriction that this system of differen tials can be expressed in the form [A] [K] = I . (6) where the matrices [A], [K] and I are as defined in Theorem 3 of Appendix 3. [A] is the Hessian matrix [Tfj.(x*)] bordered by marginal factor costs J _ f„1, 1* U N, N* N„T 29 c. = (c^x , a ),••••, cN(x ; a )) , [A] = [lT..]| C. II 1 I -fT 1 0 i i (7) 30 [K] is a matrix of all the comparative static effects for problem (5), and I is an identity matrix. 9x** 9A 9x** 9A 9a' 9 a' 9 R 9 R Nevertheless, in many situations knowledge of the comparative static '9x** 9x**' substitution and scale effects when all inputs are variable 9a ' 9R > 29n The symbol "T" denotes the transpose of a (column) vector. j** 30 9 x The "revenue effect" —— is related to the corresponding out-i** S R i** i** * « • 9x . . , „ 9x 9x 9R(y*) put effect —-— simply as follows: — - —-— • ±i—L (by the 9F F 9R 9y chainrule where y = F(x) and R(y) = R(F(x)). Likewise, 9TT(X*) _ 9TT(X*) # 9R(y*) 9F 3R 9y 79 may be almost as scarce as knowledge about the comparative static total 3x * 31 effect itself. Considerably more knowledge about substitution and 8 a1 scale effects may be available for cases where subsets of inputs are fixed for the firm. 3.1.2.2 Model with Output and a Subset of Inputs Exogenous (3x** dx** 9a ' df is quite For many situations where knowledge about weak, a narrower range of "reasonable" values for substitution and scale effects when some inputs are fixed may be readily available. Such information could be obtained (e.g.) from engineering or field studies of physical processes or econometric estimation of "short run" equilibrium 32 models of the firm. Moreover, this knowledge of substitution and scale (Footnote 30 continued) . d 2TT(X*) 3(3TT/3F) 3R(y*) . . . . . . 3 X 3 A f3R(y*)]2 and so ^— ~ ——~ ' which yields —— = • 1Z—- . 3F 3R 3y 3F 3R <• 3y > 31 A potentially important exception to this statement occurs when the econometric estimation of cost minimizing via the dual approach is more appropriate than the estimation of maximizing behavior. (On the advantages of estimating production functions by a dual approach, which involves the estimation of maximizing or cost minimizing factor demand functions, see Varian, 1978, Chapter 4 for an introduction, and Fuss and McFadden, 1979). Even if adjustment costs of varying inputs per se are low, observed activities may not correspond to maximizing behavior due to adjustment costs of searching for an optimum (see Appendix 1). Since cost minimization is a weaker condition (involving less search) than maximization and also defines conditional factor demand functions, this assumption is often preferred to maximization. Moreover, to the extent that production approximates constant returns to scale, estimation errors due to endogeneity of output (e.g., when output is adjusted in the short run but not too long run equilibrium levels) can be avoided by estimating conditional factor demand in terms of unit output: i, ,- S+1 N» - i, , x x , x (a,F,x ,'*',x )=F«x(a,1, -p -p-). 32 A subset of inputs (S+1, N) may be relatively fixed in the short run due to: concavity of adjustment cost functions (Rothschild, 1971), im perfect rental and used capital markets, and indivisibilities. In this case. 80 effects when various subsets of inputs are fixed may imply strong restric-8 x * tions on the comparative static effect -r for problem P. This statement can be elaborated upon as follows. Given the firm's problem P, we can arbitrarily define the related "short run" static maximization problem N I i=1 maximize Tr(x;a) = R(x) - £ c'(x'; a1) • (8) subject to R(x) = R(x*) xJ = xJ j = S + 1, •• «,N where output and an arbitrary subset of inputs are "exogenous" to the firm at the equilibrium levels for P. This problem can be expressed in Lagrange form as N . . — maximize TT(X; a) - A(R(x) - R(x*J) - I. yJ(xJ-xj ) ... .(9) j=S+1 where the endogenous variables are (x V • • *,x^, X, y^+\ • • •,y^) and the ... i o S+1 NA exogenous variables are (a,R,x ,'*«,x ). Suppose that the differentials of the interior first order conditions for (9) with respect to each of (a, R) yield a unique solution for the com parative static effects 9x**S 8 AS 9x**S 9 XS1 33 9a ' 9a 9R ' 9R > This assumption (Footnote 32 continued) there is at least a theoretical argument for statistically estimating maximizing or cost-minimizing factor demand functions with inputs (S+1, •••,N) treated as predetermined. Then, in the absence of specification errors, the structure of problems of the form (8) and (12) is being estimated. **s „ s **s 9Af 9R J 33 There is no loss in generality in assuming that is uniquely defined for a given problem (9). 9x 9v~ 9x 9a ' 9a ' 9R ' is equivalent to the restriction that this system of differentials can be expressed in the form 81 [A^] [L] = I . .(10) where the matrices lA^], [L] and I are as defined in Corollary 5 of A Appendix 3. [A^] consists of (a) the principal submatrix [ TTJ. ] of [TT.J(X*)] that is formed by deleting rows and columns (S + 1,««*,N) from 1, 1* 1. iA S, [ir..(x*)] and (b) the subvector c; = (c^x ;a ),•••,c<-(x ;a )) on the 'J borders of [TI.. ], i.e., ij r Ai IJ [An] iA iA c. .(11) [L] is a matrix of the comparative static effects and I is an identity matrix. **c c **c c 3x a 8 Xs 3x 3 9 AS 3 a 3 a 3R 3R iA By (10) — (11), knowledge of the elements of [L] and c. places restrictions oh [m.(x*)]. Thus knowledge of the comparative static effects 3x 3x 3 X* for problem (9) and of equilibrium marginal factor 3x* 3a 3R 3R costs places restrictions on the "long run" comparative static effect for problem P. 34 3 a1 34,., Since [L] is symmetric and knowledge of S 3x 3R "' C1al,*",CSos presumably is greater than knowledge of 3X* 3 a per se, restrictions on 3 X-3 a seldom would be specified (see Corollary 5-A) 82 9x' Moreover, the comparative static effect for problem P can 9 a1 almost always be defined precisely in terms of a set of comparative static effects 9x 9x **« 9 a' 9R for an appropriate set of problems (9), and these relations are implicit in our standard quanti tative comparative com parative statics model. The comparative static effects included in this set will differ in terms of the partition into fixed and variable inputs and the choice of shift parameter a . This important relation between various sets ( 9x 9x k*s0 9x* 9 a1 and 9a' 9R is demonstrated in Appendix 3. In sum, restrictions on 9x 3 9x 3 9X 9a1 9R 9CC1 for various problems (9) and {aJ} plus the relations (10) may imply considerable restrictions on 9x — for problem P. Since knowledge of substitution and scale effects will 9 a be defined primarily in terms of problems (9) with various subsets of fixed inputs, these restrictions derived from a model with output and a subset of inputs exogenous are a very important aspect of our quantitative comparative statics model. 3.4.2.3 Model With a Subset of Inputs Exogenous In addition, direct knowledge about the total effects of da1 when certain subsets of inputs are fixed may be available. Such knowledge can be specified as restrictions on comparative static effects various "short run" static problems of the form 9x 9 a for 83 N . . . maximize Tr(x;a) = R(x) - E cl(xl;al) 1=1 ... .(12) i i* subject to x' = xJ j=S+1,«*',N. These restrictions plus the following relations can be incorporated into our quantitative comparative statics model: [TTj*] [P] = I where [TL. ] is the principal submatrix of [TT..(x*)] obtained by deleting the rows and columns (S+1,«*',N) from [TT..(X*)], [P] is symmetric and i*S / 9x / i P.. = :— / c. : . In this manner, knowledge about a "reasonable" range 'J 9aJ / JoJ 9x*S of values for — corresponding to any problem (12) places restrictions on 9x* 9a' for problem P. 9a 3.4.3 Minor Additional Restrictions Other forms of knowledge about the structure of the firm's static maximization problem P may be available and useful in defining "reasonable" 9 x* limits on the comparative static effect —j- for problem P. These additional 9 a forms of knowledge are of at least two types. First, there may be knowledge of the comparative static effect of a change in the demand schedule for the firm's output or in the firm's production function. Including the correspond ing restrictions in our standard quantitative comparative statics model seems 9 x* likely to lead to a small reduction in the range of feasible values for —-. 9 a If such comparative static effects and its "short run" variations with fixed inputs are included in our model, then our model incorporates knowledge of (First page of Figure 3) (A) first order conditions for a maximum 38 [TT..] ^ 3al 1 Sa1 N quadratic equations (B) second order conditions for a maximum = [H] [H]' N(N + 1) quadratic equations H. . > 0 1.1 (j = 1, -".N) N bounds (C) long run decomposition (see Theorem 3) I (K) = I (N+1) x (N+1) (N+2MN+1) independent quadratic equations "TL . i „ ~~ru c. i c. S c. i i (i = 1, • • *,N) K. L < c! , • K. . < 3al '-I J") '-I M 3x ** 3F 11 3F K. .L < R • K. , < K. .U '.I y i.J ".J '.1 $ -R2 • K. . < K. .U y '.J '.) R L < R < RU (i.j = 1. •••,N) (i = 1.---.N and j = N + 1) (i.j = N+1) 2(N + 1) bounds 2 bounds (D) decompositions, given fixed inputs (see Corollary 5); for each decomposition with N-S fixed inputs: '"ij ' iA cr j o [L] = I (S+1) x (S + 1) (S+2)(S+1) 2 independent quadratic equations i**S L. .L £ c' , • l_. . £ L 7~D 3ol ' U " i°J i.J i.J i**S ^r- : L L £ R • L < L. . 3F M y i.i I.I (i.J = 1.---.S) (I = 1, •••,N and j = S+1) 2(S+1) bounds FIGURE 3 Summary of Major Constraints for the Quantitative Comparative Statics Model 39 38 The mark " — "is placed above any symbol that refers to a constant rather than an endogenous variable in the model. 39For definitions of the symbols used here, see Theorem 3 and Corollary 5. R = 3 ^(y) where y £ F(x) and R(y) = R(F(x)). V V (Second page of Figure 3) 8 5 (D) (continued) ,S r 3 A' 3F R 2 • L. . < L. U y '.j I.J (i.j = S + 1) 2(S + 1) bounds (E) non -decompositions (output exogenous), given fixed inputs (see Corollary 6)- for each non-decomposition with N-S fixed inputs: ] [P] = 1 (SxS) (S*S) 1 (S+2)(S + 1) . | 2 independent quadratic equations 3x i*S : P. .L < c! • • P. . < P (i.j = i. ---.s) totals: (N+2)(N+1) N(N + 1) + N quadratic equations (N+2)(N+1) N(N + 1) + + 2N + 1 variables (S+2)(S + 1) additional quadratic equations and variables for each decomposition 2 non-decomposition (C,D,E) with N-S fixed inputs Summary of Major Constraints for the Quantitative Comparative Statics Model 39 (Footnotes 38 and 39 are the same as the previous page) 86 all types of comparative static effects that can occur realistically at the 35 level of the single firm. Second, there may be specific knowledge about the functional form of the firm's static maximization problem P. The following examples are con sidered in Appendix 3: separability of fr(x;a) in x, linear homogeneity of Tr(x;a) in a, fixed factor proportions for R(x), and homotheticity of Tr(x;a) in x. The first two properties, and presumably many other special properties of Tr(x;a), are easily incorporated into our quantitative compara tive statics model. Such restrictions will be useful when (a) observation and/or theory suggests that such a property is closely approximated, or (b) sensitivity of comparative static results to such properties is an 36 important issue. In these circumstances, the imposition of such proper ties or of limits on the "degree of deviation" from such properties can be useful in our quantitative comparative statics models. 3.4.4 Major Difficulties and Partial Solutions The two major difficulties with the proposed method of quantitative comparative statics concern the identification of a global solution and the incorporation of a reasonable number of inputs and outputs into the model. 37 Partial solutions for these overlapping problems are suggested here. First, 35 See Section 6 of Appendix 3. 36 For example, calculating the sensitivity of comparative static results to the property of separability may provide a rough estimate of errors due to inappropriate aggregation of inputs in a quantitative comparative statics model (see Sections 3.2 of Appendix 3 and 3.1 of Appendix 5). 37 See Appendix 5 for details of these partial solutions. 87 given an algorithm that is reasonably effective in finding local solutions for a quantitative comparative statics model, we can tentatively conclude 9 X that there are "relatively few" feasible values for z( —j) that are outside 9 a of the observed range. This observed range forms an (X-Y) percent 9 x confidence-Bayes interval for z (—when the constraints 9a L < , U P ^ p S p form an X percent confidence-Bayes interval for the observable parameters p, and it becomes approximately an X percent confidence-Bayes interval 9 x for z(—-) as the search for feasible solutions becomes sufficiently 9 a detailed. More precise estimates of confidence-Bayes intervals for 9 x observed ranges of feasible z(—^) depend largely upon the ability to 9 a approximate random sampling of the feasible set. Second, computational difficulties increase exponentially with the number of inputs included explicitly in a quantitative comparative statics model; so procedures for aggregating such models within and across enterprises are presented here. These aggregation procedures generally lead to some error in characterizing the disaggregate model: correct aggregation of inputs within an enterprise depends on satisfaction of appropriate Leontief separability conditions or fixed factor proportions within the disaggregate enterprise, and correct aggregation across enterprises depends essentially on exogenous marginal factor costs for each enterprise. The aggregation procedures suggested here are shown to have certain optimum properties. In addition, aggregation errors can be crudely estimated by observing the effects of aggregation errors in small models. 88 3.5 An Illustration of a Quantitative Comparative Statics Model: Initial Models for Estimating Welfare Effects of Community Pasture Programs In order to illustrate the structure of a quantitative comparative statics model and the potential relevance of this approach to the evaluation of community pasture programs, we shall present a very simple model of the comparative static welfare effects of supplying community pasture to a single farm. First, an objective function denoting the comparative static change in producer plus consumer surplus is derived for this situation. Then the structure of the constraints is illustrated in the case of 2 and 3 input models of the farm. For simplicity, procedures for aggregating over inputs and enterprises are not illustrated here. 3.5.1 Objective Function Since the change in consumer plus producer surplus due to an input subsidy can be measured correctly in either input or output 40 markets, our analysis can be restricted to the market for pasture and to corrections for "distortions" in other markets. Considerably more information would be required for estimation of the change in surplus directly in output markets. To a first approximation, the change in producer plus consumer surplus due to the employment of dA units of community pasture by a given farm during a given time period can be expressed in the following form: 40See Section 2.4.3 of Chapter 2. 89 dSW = p1 • dA + MSB6 • |jL • dA ... .(13) or equivalently 9SW 1 ^ MCQB 9B ,,,,, = p + MSB • JJ^ . ... .(14) Here p1 = farm demand price for an additional unit of community pasture, 9 B = comparative static change in the output of the beef enter prises (measured as total revenue of beef sales), and B MSB E marginal social value of beef output minus the marginal value 41 42 to the farmer of beef output. By the definition of 9_B , and assuming an interior solution where the 9A farmer's objective function TT(X; a) is differentiate. 9B _ 1 9X1 * i 9x' M_. 9A " P 9A~ + .f2Ci * 9A~ * ' ' ' '(15) In addition, suppose that the community pasture program simply shifts the farm's pasture supply schedule to the right, as shown in Figure 4. Then The costs of supplying dA units of community pasture are not considered in (13), since these costs are essentially exogenous to the farms utilizing the pasture. Farm response to these pasture programs is essen tially independent of these costs, which are largely borne by the public rather than by the users. 42 B MSB > 0 due to market "distortions" resulting from import and export taxes or subsidies. Other "distortions" related to the B.C. commun ity pasture programs appear to be minor. In this case, MSB^ = .13 circa 1975. See Barichello (1978). 90 x = farm's equilibrium quantity of pasture prior to community pasture program p^x1) = fa rm's derived demand schedule for pasture cjfx^a ) = supply (marginal factor cost) schedule for pasture prior to community pasture program cjfxScJ+da1) = supply (marginal factor cost) schedule for pasture during community pasture program pc'P' E price at which dA units of community pasture are offered to the farm Figure 4 Hypothesized Effect of the Community Pastures Program on the Supply Schedule of Pasture to the Parm 91 1,1* 1. c cn(x ;a ) = - ... .(16) dA/x = Ic^tx^chdal/c . ... .(17) By (16)-(17), |c- i(x ;a ) «da | dA = ^ pt—^ . .., .(18) c^fx ;a ) Moreover, a shift dA > 0 is equivalent (in terms of comparative static 1 1*1 1 effect) to a shift c^al(x ;a ) • da < 0, and our convention is to inter-i 43 pret dA and daA as +1. Thus, from (18), 11*i 11*1 _ 1_44 clai(x ;a]) = -c^fx ;a ) given dA = 1, da = 1 . . .(19) In sum, the comparative static change in consumer plus producer surplus due to supplying the farm with one unit of community pasture can be written as 9SW _ 1 iirpB , 1 9X1 "J! J 9x' (jn, — = p + MSB (p • — + £ c. ... . C 20J 3A 9a1 i=2 1 9a1 by (14)-(15) and (18)-(19). 43 1 1* 1 1 Since the shift dA is equivalent to a shift c,j i(x ;a ) «da , which is a product of two terms that are independent of the structure of the farm's static maximization problem, the conventions dA = 1 and da1 = 1 are consistent. 44 From (18), it is obvious that the comparative static effects dx* are linear homogeneous in dA just as dx* is linear homogeneous in 1 1* 1 1 c. i(x ;a )»da . Thus dA = 1 would lead to a doubling of comparative 1 1* 1 static effects for a given [Tr..(x*;a)j. In other words, c. i(x ;a ) = 1 1*1 ' 1 -2cj (x ;a ) given dA = 2, da = 1 . 92 Thus the quantitative comparative statics model will have a linear 8x1 i objective function (20) in —x if p and all marginal factor costs c. (i^1) 8a 1 are also treated as exogenous. Presumably, fairly tight bounds on many equilibrium marginal factor costs can often be derived from observation. In addition, the marginal value of community pasture p^ prior to the pasture program seems to be well estimated by the methods to be described in Chapter 4. Thus, if the quantitative comparative statics model is employed in conjunction with a model similar to the linear programming model to be presented in Chapter 4, the objective function (20) can often be 8 x treated as linear in the endogenous variables —-. 8a 3.5.2 Constraints (N = 2,3) Suppose for simplicity that we can construct an "extreme short run" static model with two variable inputs, one enterprise (cow-calf) and a one month time frame (a summer month when pasture is grazed). Of course, such a model is very unrealistic. Let x^ = animal unit months of pasture employed by the farm during the month 2 ... .(21) x = hours of labor supplied to the cow-calf enterprise during the month. The comparative static implications of the maximization hypothesis for this 1 1 * 1 model are specified in Parts A and B of Figure 5. Here c^ai(x ;a ) = 11*1 - 45 -c^fx ;a ), i.e., it is assumed without loss of generality that dA =1. 45See the discussion of (19) in the previous section, (A) ^ l+7T 1 2 3x 8x 1 •c. . - 12 - i ~11 8 a 3 a 8X1 a 8x2 _ . ^12—T+ ^22—T " ° 8 a 8 a (B) -TTN = h,2 "TT12 = h11h21 "TT22 = h22+h22 hn £ 0.01 h22 > 0.01 (C) ir11K11 +TT12K12 = 1 ^K^ + IT^K^VK^ = 0 TrNK13 + TT12K23 ^^33 = 0 1T12K13 + ^22K23+C2K23 = ° PLK13 +C2K23 = 1 7T12K12 + TT22K22 +c2K23 = 1 (D) 5 Sp1 2 10 _ 0 S cjj £ 0.5 2 £ c2 < 4 100 £ R £ 120 -24£KNS0 -80<K22£0 0<K12<32 0.9<RYK13^1.1 0.5<RYK23£2 0 < RYK33 £ 2 Figure 5 Comparative Static Constraints for Community Pasture Model (N = 2) 94 The one decomposition (output exogenous) is specified in Part C. These 6 "upper triangular" equations of the 9 equation matrix system A K = 1 46 completely describe the comparative static content of this system. Also, p1 rather than c| is specified in Part C in order to emphasize the difficulty in obtaining direct market observations of the equilibrium marginal factor cost of pasture. A set of constraints pL ^ p ^ pU on p1, c^, c2, Ry and [K] are presented in Part D of Figure 5. Given that these constraints form an X percent confidence-Bayes interval for p, the feasible set defines (at gSW 47 least) an X percent confidence-Bayes interval for ^^ . The constraints 112 on p , c,|j, c2 and Ry have been derived as follows. First, the analysis to be presented in Chapter 4 strongly suggests that the farm value of pasture for the Peace River region circa 1975 typically was between $5 to $10 per animal unit month at equilibrium. This estimate defines the bounds on p1 in the quantitative comparative statics model. Second, suppose that the (inverse) elasticity of factor supply for pasture del / . 1 1 II / dx _ x 1 1/1 " 1 ' c11 c1 ' x c1 (22) 1 * 11*1 is between 0 and 2 and x ~ 40, c^x ;ct ) ^ 10. Then the bounds on the slope of the pasture supply schedule at equilibrium (c]j) can be specified as 0 and 0.5. Third, interviews with ranchers in the region 46 See Section 7 of Appendix 3. ^See Section 5 of Appendix 3. 95 suggested that the opportunity cost of supplying labor to the farm (in terms of foregone leisure or off-farm employment in the case of own labor, or wages for hired labor) typically varied between $2 and $4 per hour 2 circa 1975. These are the bounds on c2< Fourth, the total revenue per calf sold by users of British Columbia community pasture circa 1975 typically varied between $100 and $120. These define the constraints on R . y The constraints on the elements of [K] are related to the static problem N . . . maximize R(y) F(x) - Z c (x ;a) - X(F(x) - F(x*)) . . . .(23) i=1 where x* solves the related problem "maximize TT(X;CX)." Denote the com parative static effects for (22) as dx** dx** dx** dx „ . dx dx —I—, -——, —_ _ / —r ana -—-—-z da da df 9F 9a 9 a Then (for N = 2) j** 1** 9x . 1 — is dx .2 — is r/c1al-K11 ' C2a2 ~ K12 9 a 2** ,1 _ K ~7~T 1 cia1 " K21 9 a 9a , 2 _K C2a2 ~ K22 9a 9x ** 9F y 13 dx 2** 9F / R = K,_ y 23 r / c, i — K-, / C-2 - K,, / R = — K_-i la 31 „_2 2a 32 n7= y 33 9a 9 a 9F (24) and (by the maximization hypothesis) [K] is symmetric . 48 48, See Theorem 3 of Appendix 3. 96 Thus the direct constraints on [K] are defined by "reasonable" restrictions on comparative static effects for problem (23) and the 49 assumption of symmetry for [K]. These constraints have been derived as follows. First, suppose that the own price elasticity of factor sub stitution for input 1 1 dc1 c1 1** dx , 1 _ 1 1 9x . . 1 ' 1 - 1 " 1 • . . .Ubj x c x c, i 3 a is between -3 and 0 (over a year or, equivalently, over a "typical" one 1* 1 month period). Then, for x s 40 and c^ > 5, -24 < Kn £ 0 for a one month model. Likewise, if the own price elasticity of factor 2* 2 substitution for input 2 is between -1 and 0, x = 160 and s 2, then -80 £ K22 £ 0 . In addition, if the other price elasticity for input 1 is between 0 and 2, and for input 2 is between 0 and 1, then 0 < K12 S 40 0 £ K21 £ 32 . However, K12 = K2^ by the maximization hypothesis. Thus assuming that 0 < K12 £ 40 and 0 £ K21 < 32 define X percent confidence-Bayes 49 9 X 9 X Knowledge about —-, —^ seems essentially redundant and is 9a 9a therefore ignored. See Section 3.4.2.1. intervals for these parameters and that the maximization hypothesis is correct, the range 0 ^ K12 S 32 also defines an X percent confidence-Bayes interval for K17. 1** 7** 3 X 3 X Second, suppose that —— and 3— , which denote the change 3F 3F in level of inputs 1 and 2 for the month that would be associated with an 50 exogenous increase of one calf of output for the year, are between 0.9 and 1.1, and 0.5 and 2, respectively. Then 0.9<R • K,_ < 1.1 - 0.5<R • K_, ^ 2 . y 13 y 23 g 2^/ x *) 3 A Likewise, if s —— , which denotes the second derivative of maximum 3F 3F "profits" Tr(x*;a) for the month with respect to an exogenous increase of one calf of output for the year, is between -2 and 0, then 0 * Ry ' K33 S 2 . Alternatively, suppose that we can construct a static model with three variable inputs that is similar to the above two input model, where 3 x = expenditures on capital services supplied to 51 ... .(26) the enterprise during the month. ^^Since Ry has been defined as the revenue received from the sale of one calf at the end of the year and 3x'** _ 3x'** „ , _ x. _ „ _ ,x 7 ——- = —• R (see Section 3.4.2.1), 3xi** 3F 3R y —^— is to be measured as the change in level of input i for the month that 3F is associated with an exogenous increase of one calf of output for the year. (Footnote 51 on the following page). 98 Such a simple model is very unrealistic. Constraints for the three input model are presented in Figure 6, and are constructed in essentially the 52 same manner as are constraints in Figure 5. Knowledge of j** ~T ' HoJ S Kij U,J = l.-.3) often may be relatively scarce; so these restrictions are excluded from Figure 6 in order to emphasize this point. In addition, note that Figure 6 incorporates (a) knowledge of comparative static effects for a "short run" decomposition (D), with output and capital fixed, that is equivalent to the decomposition A K = I in Figure 5, and (b) knowledge of comparative static effects for a "short run" 53 decomposition (E), with capital fixed. Short run decompositions with pasture and/or labor fixed are excluded from the model in order to emphasize the following: comparative static behavior 51 3 Input x is implicitly defined as an aggregate of various capital inputs. This aggregate model can be derived from knowledge of a more disaggregate comparative static model by aggregation procedures presented within Appendix 5. 52 The restrictions presented here correspond to the "major" restric tions on comparative static effects (see Section 3.4.2). For a sample of "minor" restrictions that could be included in the model, see Section 3.2 of Appendix 3. *S *S 53 9 x 9 x The comparative static effects , — in Part E of Figure 6 refer to a "short run" problem 9a 9a maximize TT(X;a) subject to x3 = x3* . 9 9 A. it 3x' + Tt 3x2 + Tt 3x3 » -c' 3oT TT 3x^ + Tt 3x2 + TI 3xS = 0 3a' 22 3a' 3a> Tt 3x' + TI 3x2 + Tt 3x3 ""SaT ""SaT "Sta B. -TI = g -TI on n " 11 12 UU921 9..9. -TI = g 2 + q 2 22 21 a22 -TI =gg +gg -TT =q2 + q2 + q2 21 M21M!1 M22M»2 IS Hll S32 M9S g > O.Ol q > 0.01 g > 0.01 "- a22 — Mu -C. Tt K +TlK +TIK +p'K =1 TI K +TIK +TtK + p]K =0 11 11 12 12 19 13 Ik 11 12 12 22 13 23 2k TtK +TtK +TIK + c2K = 1 12 12 22 22 23 23 2 2k TTK +TIK +TTK +p'K "0 TtK +TIK +TtK +p'K =0 1 1 1 3 12 2 3 1 3 9 3 SO 11 Ik 12 2k IS 3k kk TtK ti K + it K + c2K • 0 wK + Tt K + TT K + C2K = 0 12 13 22 23 23 S3 2 Sk 12 Ik 22 2k 23 3k 2 kk TtK +TtK +TtK + c'K «1 TIK +TtK +TIK + c'K = 0 13 13 23 23 39 33 3 9k IS Ik 23 2k 33 3k 3 kk p'K + c2K + cJK Ik 2 2k 3 3k D. Tt L + ¥ L + p'L =1 TI L + Ti L + DlL =0 •I 11 12 12 19 11 12 12 22' P 23 TI L + Tt L + c2L = 1 12 12 22 22 2 23 TI L + Tt L + p'L = 0 1113 12 23 33 Tt L +TTL +c2L =0 12 1 S 22 23 2 33 p'L + C2L 1 3 2 2 3 E. Ti 3x'«s + Tt 3x*«s = c1 • 11 "So"1- 12 "ssr- la Tt 3x'«s + TI 3x2«s = 0 12 ~iZr- 22 laT-3x'«s + TI 3x2«s = 0 ""5a5- 12 -5S5-12 -j^i— 22 -552- 2Q' F. 5 < p1 < 10 0 <_ cj, < 0.5 2 < c\ < 4 cj = 1 100 < Ry < 120 0.7 <_ RyK < 1.05 0.3 < RyK^ < 1.5 1 '< RyK^ < 2 0 < 1.5 • 24 < L < 0 — ii--80 < L < 0 0 < L < 33 — . 22 — — 12 — 0.9 <_ RyLi s < 1.1 0.5<RyKjs£2 nlRy2Ljs<2 c* S 1 c» . ia1 2a' -20 < a*'*s i 1 8x2>s 3a' / c1 . - 3x'«s / c2 53 / l« ~5S3~ / 2a < 3x'»s / c1 . < 0 -20 < 3x2«s / c' < 15 -90 < 3x2tlS / c2 , < 0 3a2 / FIGURE 6 Comparative Static Constraints for Community Pastures Model (N = 3) This symmetry condition follows from the maximization hypothesis (see footnote to Theorem 3 in Appendix 3). 100 is least likely to be observed for decompositions with relatively non-durable 55 56 inputs defined as fixed. 3.6 Summary and Suggestions for Further Research In this chapter, we have introduced a method of comparative static analysis that is intermediate between traditional qualitative and quantitative methods in terms of the degree of structure imposed on the firm's static maximization problem. Traditional qualitative comparative static methods have incorporated the implications of the maximization hypothesis, signs on elements of [TT..(X*)], and additional elementary restrictions such as fixed factor prices. The method presented here can incorporate these and many other restrictions. We have emphasized restrictions corresponding to various "short run" (fixed input) comparative static effects, especially factor substitution and scale effects. These restrictions can be derived from (e.g.) engineering or field data on physical processes, and from econometric estimation of production processes or of comparative static effects. Since this information can be incorporated into the model in the form of confidence-Bayes intervals, this method avoids the major drawback of traditional quantitative comparative static methods: results obtained by Also note that labor can be viewed as a capital asset (rather than as an input whose level can be adjusted quickly without incurring signifi cant adjustment costs) if costs of initial training are borne by the firm; but such costs appear to be minimal for most of the labor employed in a cow-calf enterprise. 56 Nevertheless, there may be significant knowledge of physical pro cesses that corresponds to comparative static effects for decompositions with pasture or labor fixed. 101 these determinate methods are dependent upon essentially arbitrary specification of many aspects of the firm's static maximization problem. The manner in which this method can be applied to the estimation of comparative static effects for community pasture programs is illustrated in terms of two simple models. To the extent that this chapter has been successful in laying the main theoretical foundations for such an intermediate method of comparative static analysis, future research should explore the computa tional problems and practical significance of this methodology. Computation problems are considered in Appendix 5 of this thesis, and concern difficulties in solving nonlinear programming models that are associated with this method. The material presented there appears to solve these problems in part; but further research along these lines is needed. In particular, the problem of approximating random sampling procedures for obtaining feasible solutions to the nonlinear programming models must be considered more carefully, and experience in solving such models should be accumulated. As a byproduct, such experience should provide some measure of the practical importance of the methodology, i.e., of the extent to which "reasonable" knowledge can define the comparative static effects of interest. It is expected that the degree of success will vary with the comparative static effect of interest and also across "reasonable" degrees of knowledge of the structure of various firms. Thus, as has been noted previously, 57 this intermediate method and traditional methods of quantitative comparative statics to some extent 57See Section 3.5. 102 complement each other. Traditional quantitative methods are useful in estimating comparative static effects that depend primarily upon aspects of structure that are known with considerable precision or in estimating comparative static effects that can be compared with alternative measures of the effect. In the next chapter, we shall illustrate both of these points by means of a static (deterministic) linear programming model of a beef ranch. In the process, we shall also gather support for the hypothesis that the comparative static paradigm is useful in the evaluation of community pasture programs. These results will suggest that the essentially theoretical exercises of the last two chapters can make a positive contribu tion to the applied economics of agriculture. CHAPTER H A STATIC LINEAR PROGRAMMING MODEL OF A REPRESENTATIVE BEEF RANCH 104 CHAPTER 4 A STATIC LINEAR PROGRAMMING MODEL OF A REPRESENTATIVE BEEF RANCH 4.1 Introduction The purpose of this chapter and the accompanying Appendix 6 is threefold. First, we shall present a static linear programming model of a "representative" beef ranch for users of community pasture in the Peace River region of British Columbia circa 1975. Such deterministic models can often be useful complements or even substitutes for the method of comparative static analysis that was developed in the previous chapter. In particular, knowledge of the equilibrium shadow price of pasture for the farm is extremely important in the evaluation of community pasture programs, and this equilibrium shadow price depends primarily upon knowledge of parameters that can be specified with some degree of confidence. Moreover, other microeconomic models that have been adapted to the study of derived demand for pasture in Western Canada appear to have been either non-optimizing or partial equilibrium in nature; whereas, the model presented here is explicitly static general equilibrium and optimizing in nature.1 Second, solutions to this model are consistent with the assumption that constructs of static, optimizing behavior are ^See Department of Regional Economic Expansion (1976) and Graham (1977), which are mentioned in Section 4.4.2. 105 often appropriate for microeconomic models designed to estimate farm response. Estimation of aggregate farm supply response via construction and solution of linear programming models of representative farms was a major area of research in the profession in the late 1950's and the 1960's. Unrealistic assumptions of static optimizing behavior have been included on the list of possible explanations for the apparent failure of these studies, and at least some observers have speculated that these have been the primary reasons for failure. However, by attempting to estimate an equilibrium shadow price for an input rather than response, we are able to reduce the significance of many of the problems associated with studies of supply response and focus more clearly on the appropriateness of con structs of static, optimizing behavior. By comparing solutions to the static optimizing model to be presented here with calculations based on direct observations of hayland rental markets and beef ranch activities, we derive empirical support for the major assumption underlying this thesis: the comparative static paradigm and maximization hypothesis are often useful constructs, and often more useful than alternative constructs, in the empirical estimation of micro-economic behavior. These results are consistent with the theoretical discussion of static, optimizing behavior versus dynamic, nonoptimizing behavior that was summarized earlier in this thesis (Appendix 1). Third, in this process we demonstrate that the comparative static models and methods of this thesis should be relevant to the 106 problem of predicting farm response to community pasture programs. Thus the theoretical content of this thesis should have constructive (i.e., real world) uses. 4. 2 Representative Farm Approaches to Estimating Farm Response 4.2.1 Studies of Supply Response In the late 1950's and the 1960's, there were many studies attempt ing to estimate aggregate short run or long run supply response by aggregating estimates of supply response for "representative" farm models. Apparently these studies are considered in large part to have been un-2 successful. Lists of potential causes of this failure have emphasized the following: (a) use of the unrealistic assumptions of (short run or long 3 run) static equilibrium and optimizing behavior, (b) poor knowledge of the relevant structure of an individual farm, (c) poor knowledge of differences in structure among farms, and (d) poor knowledge of correct procedures for aggregating response over farms, i.e., for modelling interactions of farms. For discussion of these studies, see Nerlove and Bachman (1960), L. Day (1963), Carter (1963), Barker and Stanton (1963), and Sharpies (1969). Indeed, the stronger assumptions of static and profit-maximizing behavior were generally employed. 107 Furthermore, it has been speculated by at least some observers that the primary reason for failure in these studies was the common assumption H of static, optimizing behavior. 4.2.2 A Means of Evaluating Constructs of Static Optimizing Behavior Here we shall formulate a means of investigating the appropriate ness of constructs of static, otpimizing behavior in the estimation of farm response via representative farm models. Results will be presented in Section 4.4. In the previous section, it was pointed out that the apparent failure of microeconomic supply response studies has occasionally been attributed to erroneous assumptions of static and maximizing behavior. On the other hand, theory summarized earlier in this thesis suggests that static optimizing models will often provide the most effective means of estimating supply response. This is primarily because the difference between static and dynamic response presumably depends critically upon essentially unknown parameters of the firm's adjustment cost functions, and static models at least have the virtues of internal consistency and relative simplicity of structure.5 Thus it seems reasonable to suppose that dynamic and/or non-optimizing representative farm models are even less effective than static, optimizing models in predicting farm response. See Carter (1963) especially pp. 1455-64, White (1969) and, in the same spirit. Smith and Martin ( 1972). 5See Appendix 1 for details. 108 Given this contrast between theory and the opinions of at least some observers on this fundamental issue, it seems desirable to consider empirically the effectiveness of constructs of static, optimizing behavior in predicting farm response. However, an obvious problem here is to separate the effects of such constructs from other potential sources of error in estimating response. In order to obtain a clearer picture of the effects of such constructs of static optimizing behavior, we shall focus on the equilibrium shadow price of pasture in representative farm models rather than on measures of the change in levels of input (output) for a given change in input (output) price schedules. . In this manner, we can significantly reduce the effects of other potential sources of error in the estimation of farm response. The effects of poor knowledge of the individual farm's static structure Tr(x;a), of the individual farm's adjustment cost functions, of differences in structure among farms, and of interactions of farms all appear to be less important in estimating shadow prices for inputs than in estimating other types of farm response. These points can be elaborated upon as follows. First, the equilibrium shadow price or value of input (e.g., pasture) is more dependent than most other measures of farm response on relatively observable aspects of the structure of an individual farm. For example, in the case of a differentiable function R(x) and a static problem N . . . maximize Tr(x;a) = R(x) - Z c (x';a') . . . .(1) i=1 109 with a known interior solution x , the shadow price (demand price for an additional unit) of pasture at equilibrium is defined by i i * * p'fx1 ) = R^x ) whereas (2) 8x* 8 a1 [TTj.fx*)] -1 la] (3) Since knowledge of the first derivatives of R(x) at x* is likely to exceed the knowledge of the second derivatives of Tr(x;a) at x*, the shadow price 1 1 * for pasture p (x ) can be estimated with more confidence than can effects such as 3x' 8 a This statement remains true but in a slightly weaker form when x* is unknown; to the extent that x* is estimated with E 1 E 1 1 * error as x and R (x*) t R^x ), the accuracy of the estimate for p (x ) depends on second (and higher) derivatives of R(xE) and the degree of E 3 x * knowledge of R.(x ); whereas, the accuracy of the estimate for % I a Ot depends on third (and higher) derivatives of Tr(xE;ct) and the degree of knowledge of [TT..(XE)]. However, unless x* is estimated with very larger error or second derivatives of TT(X) are roughly comparable in magnitude to first derivatives, the equilibrium shadow price can still be estimated with more confidence than can effects such as x 1 8 a In addition, the analysis is essentially unchanged in the case where the model utilized for estimation exhibits a non-differentiable x* is defined as an interior solution if x >0, i = 1, •••,N. no production function. For example, in the linear programming case, the equilibrium price of pasture is calculated from pV*i E isJ^ . iSisH. x cjcx'V) 5*11 («) 3x' 9X1 i=2 9x 8 x * rather than from (2) per se, where defines the fixed factor proportions 3X1 at x*. Given that the "true" model is differentiable (or approximately differ entiate), we should construct the programming model so that the parameters , -9-X— , c.' (i = 1, •**,N) vary in a piecewise manner over x in rough dx1 9x1 1 accordance with our estimates of R, and [TT..] over x. Thus the analysis .1 ij presented above for a differentiable model carries over directly to the non-differentiable case. Second, the equilibrium shadow price is less dependent than most aspect of farm response on the largely unknown structure of the adjustment cost functions of the individual farm. This follows essentially from Equation (2). By definition, R^(x*) is independent of the ease in adjusting various inputs. On the other hand, observed changes in input levels resulting from shifts in price schedules are very dependent upon the particular adjustment cost functions for the farm. The argument generalizes to the case of non-differentiable R(x) in roughly the same manner as above. Third, equilibrium shadow prices of pasture presumably are more uniform across farms than are most measures of response. Suppose that transportation costs of incorporating off-farm improved land into the farm Ill enterprise are negligible, that (equilibrium) rental prices are equal across farms, and that equilibrium exists in the available rental markets for improved land. Then the equilibrium shadow price of pasture will be identical across all farms that trade in these markets and employ improved land as pasture. Although these assumptions may not be realistic, the argument does suggest that market forces tend to reduce the variation in equilibrium shadow price of pasture across farms in the Peace River region of British Columbia. On the other hand, market forces presumably do not tend to reduce the differences between farms in "higher structure" that is analogous to [iijj(x*)] and adjustment cost functions. Moreover, differences in such structure apparently have significant effects on most 6' aspects of farm response. Thus market forces tend to reduce the 1 1 * differences between p (x ) across farms without influencing the variation in response of input and output levels to shifts in price schedules. Fourth, the farm value of pasture is influenced less by farm inter-relations than are most aspects of farm response. This is because any particular unit of pasture is generally employed only by a single farm, whereas, an exogenous shift in a factor supply or product demand "This seems obvious from Equations (3) above. In addition, see Day (1963) and Paris and Rausser (1973) for studies that are formulated specifically in the context of linear programming. On the other hand. Day (1969) has also noted that many farmers may tend to imitate the response of managers who are recognized as relatively effective decision-makers. If farm response happens to consist primarily of such behavior, the variation in structure across farms is unlikely to lead to large differences in either shadow prices or other aspects of response within a region dominated by a single manager with recognized decision-making skills. However, results to be presented in Section 4.4 support the hypothesis that such imitation, which is not static and optimizing behavior, is relatively unimportant. 112 schedule is generally experienced simultaneously by a large number of farms. Thus shifts in factor supply and product demand schedules of farm A due to activities of other farms have less influence on farm A's shadow price for pasture than on farm A's response to an exogenous shift in a price schedule that is also experienced by many other farms. In sum, there is reason to investigate the assumption that constructs of static, optimizing behavior are relatively useful in farm response studies, and the effects of alternative sources of error (b)-(d) can be controlled more effectively by focussing on equilibrium shadow prices of pasture than on most other types of farm response. Moreover, the marginal value of pasture can also be estimated in the Peace River region from observed rental prices for hay land, given the substitution relationship between hay and pasture observed on farms in the region. Since these estimates are derived from real world data without imposing any significant assumptions about static, optimizing behavior, they provide a criterion for evaluating the appropriateness of constructs of static, optimizing behavior in estimating equilibrium shadow prices for pasture . In addition, the results of such an examination of shadow prices also provides some information about the appropriateness of the constructs in estimating other types of farm response that are long run or intermediate run in nature. For example, in the case where alternative models have identical differentiable production (revenue) functions R(x), the likelihood of accurately predicting 113 RI(X*(CXQ)) within a given model presumably increases with the likelihood of accurately predicting x*(aQ). In addition, the likelihood of predicting the farm responses x*(a1) -x*(aQ) presumably increases with the ability to predict x*(aQ) and x*(a.j). Since any actual equilibrium activities x*(a) are a composite of adjustments over time, it follows that the same model tends to be most appropriate in predicting R^(x*} and long run and intermediate run farm response. 4.3 A Static Linear Programming Model of a Representative Beef Ranch The purpose of this section and the accompanying Appendix 6 is to detail and to explain farm programming models developed for an economic evaluation of British Columbia ARDA community pasture programs in the Peace River region.7 These models were developed as an alternative to the available non-static or partial equilibrium beef ranch models for Western Canada. 4.3.1 Methodological Problems Here we outline issues that were considered to be particularly important in choosing a structure for the farm model. In sum, the model (1) specifies static, optimizing behavior rather than dynamic, non-optimizing behavior; (2) generally defines the levels and ratios of various capital 7The material presented in this section and Appendix 6 overlaps considerably with Coyle and Barichello (1978). 114 stock activities (number of cows on farm and disposition of calves) and enterprise and feeding combinations as endogenous rather than as fixed; and (3) disaggregates labor requirements and supplies over the model year. 4.3.1.1 Static Optimizing Behavior versus Dynamic, Non-optimizing Behavior First, a model can be static and optimizing or dynamic and/or non-optimizing in nature. In Appendix 1, it has been argued on theoretical grounds that static, optimizing models should be preferable for estimating farm response. In summary, deviations from static, optimizing behavior are due essentially to the existence of "adjustment costs," and our present knowledge of adjustment costs enables us to estimate comparative dynamic and non-optimizing effects only as a series of comparative static and optimizing effects. In addition, static equilibrium models are internally consistent (unlike most dynamic models) and can more easily accommodate a complex structure of production within the unit time period. Thus it was decided that a static programming model was most appropriate. "Short run" o and "long run" equilibrium versions were constructed. g The choice of a one year static model is satisfactory for the purpose of estimating long run equilibrium and response, which are our main concerns. Given the 2.5 year lag in buildup of the beef herd, a static model with a three year time period would be most appropriate for estimating "short run" comparative static effects. 115 4.3.1.2 Endogenous versus Exogenous Specification of Activities and Combinations of Activities^ Second, livestock capital activities, enterprise combinations, etc. can be specified either as endogenous or as exogenous (fixed) in the static model. In the presence of uncertainty about "true" farm structure, i.e., errors in specifying the production function or price schedules, an endogenous specification is not necessarily appropriate. For example, let all such activities be subsumed in the vector x of all farm activities, and let Xg denote any sub-vector of x that may be treated as exogenous to the farm model (all other activities x^ will always be treated as endogenous to the model). Then, in the differentiable case, the problem is to choose between the following estimates of the true equilibrium shadow price R1(X*(Y0);Y0): R1(X*(Y0 + AEY); YQ + AEY) . • • .(5) R1(X*(Y0 + AEY);X*(Y0)+AEXB, Y0 + AEY) . • • -(6) w here the true parameters YQ (of price schedules and the production function) and true equilibrium levels x^(Yg) are estimated with error AEyand AExg, respectively, and X^(YQ + AEY) is the equilibrium when Y = YQ + AEy and Xg is fixed at its true level X^(Yq). The appropriate choice between (5) and (6) depends essentially on the derivatives IT , TTx (and also R , RX ) over the relevant region and the prior B B q This section is the most technical part of this chapter, and can be omitted by the reader without seriously affecting his comprehension of the remainder of this chapter. 116 distributions of the errors AEy ar>d AEXg. Thus an endogenous specifi cation of capital activities, etc. is not necessarily appropriate for estimating the shadow price of pasture in the presence of errors .E 10,11 A - Y-An endogenous specification of livestock capital activities, enter prise combinations and feeding combinations was selected for the model 12 essentially on the basis of the following extremely crude argument. Suppose that the errors AEy are "small" relative to the errors A^x„. Then, in the absence of further information, the expected error in estimating R (X*(Y0); Yq) is 'ess for method (5) than for method (6). More over, the parameters seem reasonably specified for the purpose of 13 estimating shadow prices, and current activities x may be quite differ ent from static equilibrium (especially "long run equilibrium") levels. In 14 15 this case, x should be specified as endogenous. ' ^Similar statements hold when R(x) is non-differentiable and the shadow price is estimated for a discrete change in the level of pasture. 11 Whether or not an endogenous specification of Xg tends to stabilize equilibrium TT (and hence the equilibrium shadow price) in the presence of errors AEy depends on the (essentially unknown) direction of the errors: A^Y > 0 implies that an endogenous specification is destabilizing essentially * i due to convexity of TT(X (y); Y) in Y« For example, if y is the output price for enterprise j, then the firm will maximize the increase in equilibrium TT in response to (AEy)' > °« Thus, if there are no other errors E E EE TT(X*(Y0); Y0) < TT(XA*(Y0 + A Y);XB*(Y0)/Y0+ A Y) < TT(X*(Y0+A Y);Y0+A Y). 12 Interviews with farm and B.C.D.A. staff suggested that possibil ities for factor substitution could be estimated with at least some accuracy across enterprises and feeding possibilities, but could not be estimated with any accuracy within any particular combination of enterprise and manner of feeding. Thus factor ratios within each of the various enterprises are gen erally specified as fixed at the observed levels. ( Footnotes 13, 14, 15 on the following page) 117 4.3.1.3 The Degree of Aggregation and Endogeneity of Labor Demand and Supply Third, and related to the second issue, labor requirements and supplies can be specified at various levels of disaggregation and endogeneity in the model. It had been suggested that community pasture programs could have various labor-saving effects of considerable value to users, and reasonable point estimates of labor requirements for the various enter prises over the year were obtained. Thus, by the argument for treating capital levels, etc. as endogenous, labor requirements for each enterprise are disaggregated over the model year, and point estimates of the labor-output ratio within each enterprise are specified. Likewise, the supply 13 See Section 4. 2. 2 above. 14 E Even though given errors A y have a greater effect on estimates 3 x * of other aspects of farm response such as ——, an endogenous specifi-9a' 3x* cation of x„ also seems preferable for estimating —— . By the Le 3 a' Chatelier Principle, 3x(x*(a); a) 3 a1 3x(xA*(a);xB*(a), a) 3 a1 where a is a subset of y. Thus fixing xD at xD* leads to errors in estimating 3x* E when A Y = 0, and on the other hand fixing xD at xD* reduces the error „ j D O E i in estimating xA*(YQ) 'n tne presence of any error (A y) . However, this advantage of (correctly) specifying the level of xR in the model should be less 3 x* important in estimating the difference in equilibrium levels. 15 3a' It should also be noted that the "best" estimates of y for the model are not the expected values of y. Since TT(X*(Y);Y) is a convex function of y provided only that TT^ does not change sign over (x*(y),Y) (e.g., (Footnote 15 continued on the following page) 118 schedules of family and hired labor to the farm are disaggregated over time 16 and exogenous supply prices are specified. As noted in Chapter 1, the equilibrium supply price of labor appears to be endogenous to the farm during many periods of the year. Nevertheless, the directions of bias on changes in labor use and value of community pasture due to this mis-specification are readily determined in this model, and the magnitudes of errors can be estimated simply by varying the supply price of labor in the appropriate directions.17 Attempts at direct modelling of endogenous supply prices for labor have been avoided here precisely because neither evaluation of direction of bias nor sensitivity analysis could then be done so easily. 4.3.2 Summary of Model Structure Here we summarize the basic single year linear programming model of a "representative" farm using community pasture in the Peace River region of British Columbia. "Long run" and "short run" variations of the model were constructed. The structure of both versions and sources of data for the models are detailed in Appendix 6. (Footnote 15 continued) McFadden,l978), y should generally be defined in the model at less than its expected value in order to obtain unbiased estimates of equilibrium IT before and after a shift in the supply schedule of pasture. However, we shall ignore this problem on the grounds that the estimated difference in these equilibrium levels of TT should be less sensitive to such difficulties. 16 The exception to this statement is that upper bounds are placed on the supply of family labor available in each time period. 17See Section 1.7 of Appendix 6. 119 The structure of each model can be disaggregated into the following groups of constraints and activities: (1) land, (2) cattle numbers, (3) cattle feeding, (4) labor, (5) income assurance, and (6) income. Each model has the same objective function. The relations between these groups of constraints and activities are summarized in matrix form in Figure 7. In addition, a flow diagram of the model is presented in Figure 8. Each model farm has 350 acres of improved land that can be used as pasture, or in production (and establishment) of hay, barley or oats, and 150 acres of unimproved land that can only be used as range. Each farm can rent up to 300, 50 and 75 animal unit months of range in summer periods June 1 to September 1, September 1 to September 15 and September 15 to October 7, respectively. Each farm can also rent up to 180, 30 and 18 45 animal unit months of community pasture in the same summer periods. In addition, a farm can rent up to 50 acres each of hay, barley and oat land during the year. Three-quarters of own acres in hay and in grain are in production during the year. Otherwise, quantities of on-farm and off-farm land in the various uses are free to vary, subject to the supply constraints mentioned above. However, the structure of the cattle numbers subsection of the models depends on the variant of the model. In a "short run equilibrium" 18 In the models, one animal unit month (AUM) is equal to one yearling month of grazing as well as one cow (plus calf) month of grazing. Although one yearling presumably requires less grazing than does one cow (plus calf), it had been suggested that a yearling exhausts about the same quantity of pasture as does a cow plus calf (due to greater trampling of grass by yearlings). In fact cows and yearlings were charged at the same rate on the observed community pastures. Own Farm Land Rented Hay and Brain Land Cattle on F arm AUH's Grazing Hay and Grain Labor Use Labor Hired Roundup Transfer Activities Cattle Sales Income Opening Stock Added Cattle Closing Stock Own Land Rented Land Fad to Cattle Acres Harvested Sales Purchases Custom Roundup Surplus Total Subsidized 1. Land 1 + + • II. Cattle + + -III, Fcedlno IV. Hay and Grain + * + -V. Labor —r + V 1 + + - + - - + -VI. Income Assurance VII. Income VIII. Objective Function —r~ T""' 1—* r 1 - ——f T + + + + Right Hand Side* "b7~ Indicated otherwise Figure 7. Outline of Hatrlx for the React River Income Assurance Fern Hodel For details of this matrix, see Appendix6. RESOURCES OUTPUTS VALUATION OF ACTIVITIES Land (Own improved and unimproved, community pasture, rented range, rented hay and grain land. Cattle (cows, calves) Available Income Assurance Subsidies (incidental costs) Cattle Replacements (cows, calves) Cattle Sold (calves, yearlings) (rents, fees ^capital costs per acre for each land use) Income Hay and Grain Purchased Available Labor (own family, hired labor) Cattle Sales Subsidized Hay and Grain Sold Custom or off-farm work • (wage) Lei sure Dollar Val ue of Leisure Figure 8. Input-Output Relations for the Farm Model 122 model, the farm number of cows is defined as greater than or equal to 40, which is approximately the average number of cows on sampled farms. In a "long run equilibrium" model, bounds are never placed on the number of cows. However, both short run and long run models are generally specified such that net investment in cows and calves over the year must be equal to zero. Cross investment in cows then consists solely of replacing cows lost during the year through culling (10% of the opening stock) or through death (2% of the opening stock). Cow replacements must come from the on-farm herd, i.e., cannot be purchased. Likewise, gross investment in calves consists solely of accumulating a stock of calves at the end of the year that is equal to the stock of calves held at the beginning of the year (and sold as yearlings towards the end of the year). In contrast to the level of cows, the opening and closing stock of calves is unbounded in short run as well as in long run models, and the calf replacements (closing stock of calves) can be purchased as well as 19 raised on-farm during the year. For feeding purposes, the year is divided into six periods, as 20 shown in Table 1. During the first two periods, November 1-June 1, 19 Notice that none of the constraints discussed here fixes the levels or ratios of (a) calves sold at the end of the year, (b) calves held over for sale as yearlings in the following year, and (c) calves purchased at the beginning of the year for sale as yearlings towards the end of the year. The levels and ratios of these activities are endogenous to all programming models. A lower bound on cow numbers is generally included in short run models because of their apparent short run fixity (see Section 1.2 of Appendix 6). 20 The year is defined within the model as beginning and ending November 1. The selection of a starting and terminal date is simply a matter of convenience, provided that the short run or long run equilibrium assumptions on which the model is based are realistic. Table 1. Feeding and Labor Constraints FEEDING CONSTRAINTS Feeding Period Feeding Period No. Manner of Feeding Labour Period Labour Period No. Labour Supply (excluding hired labour) hrs./wk. I. Dry fed LABOR CONSTRAINTS Sept. 15 Dry fed Apr. 7 Apr.21 Cattle feeding and management, off-farm or custom work Cattle feeding and management, off-farm or custom work, calvi ng Cattle feeding and management, off-farm or custom work Cattle feeding and management Grazed on pasture or range July I Grazed on (pasture or range Cattle feeding and management, hay and grain culture Cattle feeding and management, hay harvest Sept. I Cattle feeding and management. hay harvest Oct. 7 Grazed on pasture, range, hay aftermath or zero- j grazed ' 6. Grazed on hay aftermath or zero-grazed Sept. 15 Cattle feeding I and management, hay and grain harvest, roundup Oct. 7 Cattle feeding and management hay and grain harvest, roundup Cattle feeding and management 124 all cows and yearlings must be fed hay (produced on-farm or purchased) at a fixed rate; yearlings also receive barley. During the next two periods (June 1-September 15), cows and yearlings must be grazed, either on own pasture, range or community pasture. Weight gains for calves and yearlings are specified as being lower on range than on pasture by 15 pounds per cow (with calf) AUM on range and 21 pounds per yearling AUM on range (in the standard case). Grazing supplies of rented range and pasture cannot be substituted between periods. However, grazing capacities per own acre in pasture or range are defined as fixed aggregates over these two periods, i.e., it is assumed that the total quantity of grazing available on an acre of own pasture or range is invariant with respect to the grazing schedule over these two periods. In the fifth period (September 1 5-October 7), all cows must be grazed, either on own pasture, range, community pasture, or hay aftermath. Yearlings must be grazed or zero-grazed (fed hay), and also require barley. Weight gains on range are lower than gains on pasture and hay aftermath by 30 pounds per cow (with calf) AUM on range and 42 pounds per yearling AUM on range (in the standard case). Grazing capacity in the fifth period is not transferable to or from other periods. During the sixth feeding period, cows and calves require grazing on hay aftermath, and yearlings must be grazed on hay aftermath or zero grazed (and require barley). For purposes of labor accounting, the year is divided into nine periods, as also shown in Table 1. A schedule of on-farm labor supply has been estimated (for a work force of one operator, wife, and two school children), and it is assumed that additional labor can be hired at 125 any time at a constant wage rate. During the winter (November 1-May 10) to 30 hours of an on-farm labor supply of seventy-five hours per week can be allocated to off-farm employment or custom work. Cows and yearlings require labor at fixed rates within winter and spring periods. In the three labor periods during which community pasture is available (June 1-October 7) cattle demands on farm labor vary with method of feeding (lowest on community pasture) and time of roundup from community pasture and rented range. In the first of these three periods (June 1-July 1), labor also is required for cultivation of hay and grain land. Harvesting of hay can occur within any of the three labor periods from July 1 to September T5, on "appropriate" days (60% of days within the period determined by the vagaries of weather). Thus approxi mately 60% of the labor available from the farm family in a particular period can be utilized for harvesting. This is the labor constraint on harvesting in the models. Grain can be harvested on appropriate days within the two labor periods from September 1 to October 7. Labor requirements per acre harvested do not vary with the period of harvest; but the yield of hay per acre and grain per acre decreases with the delay in harvesting. The resulting hay and grain can be either sold or fed to animals during the year. In the final period (October 7-November 1), 21 cows, weaned calves and yearlings require labor at fixed rates. That component of leisure which is the unemployed surplus of on-farm labor supply is valued in the models. Values are highest during the two week calving period in April and days appropriate for harvesting. 126 Several Income assurance-related constraints and activities are 22 included in income assurance versions of the "short run" model. This subsection determines the number of beef pounds from calf and yearling sales that qualify for income assurance subsidies to the farm and also determines the level of subsidy. There is an upper bound to the number of qualifying pounds. Farm income is specified simply as the total revenue for the year from sales of calves, yearlings, hay and grain, plus revenue earned by farm family labor in off-farm or custom work, minus the sum of purchase costs of farm inputs, depreciation and interest costs of capital (excluding land) for the year. In some short run models, costs of maintaining the stock of cows and/or capital in hay and grain enterprises are not specified, i.e., negative net investment is permitted occasionally. 4.3.3 Limitations of the Model These models of a "representative" farm have many limitations. The most important of these appear to be (1) errors in specifying production functions and extreme difficulties in performing an adequate sensitivity analysis; (2) errors in simulating the effects of adjustment costs (except in long run equilibrium models); (3) errors in specifying expected prices for beef; and (4) failure to incorporate risk into the The B.C. Farm Income Assurance program subsidizes ranchers in terms of their beef output. 127 model. The first two points appear to be the most serious weaknesses of these models and of many other models that are designed for estimating farm response. 4.3.3.1 Lack of Knowledge of the Production Function and the Extreme Difficulty in Obtaining an Adequate Sensitivity Analysis The most serious problem with the model from the viewpoint of estimating "long run" comparative static effects appears to be the diffi culty in accurately specifying the farm's production function and in performing an adequate sensitivity analysis for the effects of this uncertainty. Long interviews with farmers and consultations with district agriculturalists led to a rough consensus on current (circa 1975) input-output ratios in various enterprises for the "average" user of community pasture in the Peace River region of British Columbia. However, reliable estimates of possibilities of factor substitution within an enterprise or of returns to scale were not obtained. Moreover, estimates of equilibrium shadow prices should be somewhat sensitive to such uncertainty, and estimates Ax* of other comparative static effects of the form — should be particularly Aa1 sensitive to mis-specifications of possibilities for factor substitution-and 23 of returns to scale. Since there is considerable uncertainty about the appropriate structure of the production function and the relation between structure See Section 4.2. 2. 128 and comparative static effects is generally complex, there should be con siderable difficulty in performing an adequate sensitivity analysis for the effects of such uncertainty. This can be seen most clearly in the case of local comparative statics and a twice differentiable production function. Then 3xj 3 a1 tyx*)] 1 ^a1 . . .(7) i.e., any comparative static effect 3x' 3a' depends on the values of N (N + 1) all 2 elements (assuming symmetry) of the Hessian [ TT..(X*) ]. For any reasonable number of inputs N and a realistic degree of uncertainty about the structure of the production function. Equations (7) virtually preclude the possibility of doing an adequate sensitivity analysis by vary ing directly the values of elements of [TT..(X*)]. Moreover, a linear programming model cannot even incorporate many reasonable conditions on the production function (increasing returns to scale and non-convex isoquants) and also has difficulties in handling many other reasonable properties (decreasing returns to scale and smooth strictly convex isoquants). 4.3.3.2 Errors in Simulating Adjustment Costs For the purpose of estimating "short run" comparative static effects, the most serious problem with this model or perhaps any other farm model may be the difficulty in accurately simulating the effects of 24 adjustment costs. As has been noted previously, our poor knowledge of adjustment costs generally implies that comparative static analysis is our most effective means of estimating real response, i.e., real comparative dynamic effects. Nevertheless, even if a static model with a three year time period had been constructed, errors in simulating the effects of adjustment costs presumably would lead to considerable errors in 8 x* estimating "short run" response of the form —T- . aa 4.3.3.3 Errors in Specifying Expected Beef Prices All comparative static effects, including the shadow price of community pasture, will be sensitive to errors in estimating ranchers' expected beef prices. Since there is considerable variation in calf and yearling prices over the ten to eleven year beef cycle and the process of expectations formation for these ranchers has not been quantified, these errors are likely to be significant. On the other hand, a sensitivity analysis for expected calf and yearling prices (two parameters in a one year model), for a given production function, is much more manageable than a sensitivity analysis for the elements of the equilibrium Hessian [R.j(x*)] of the production function. Thus, in terms of formulating appropriate confidence intervals for comparative static effects, lack See Appendix 1. 130 of knowledge about expected prices poses less of a problem than lack of knowledge about second order properties of the production function. 4.3.3.4 The Exclusion of Measures of Risk Although risk is consistent with the use of static, optimizing models and incorporation of risk would lead to more realistic modelling of behavior, farmer's uncertainty about prices, etc. has not been incorporated directly into the model. The main reason for this is that—in a sensitivity analysis — the effects of risk can be incorporated in terms of variations in expected input and output prices. 4.4 Results and Implications The equilibrium farm value of community pasture has been estimated under various conditions for the above static linear programming models, and these results shall be summarized here. These results will also be compared with estimates of equilibrium shadow prices for pasture that have been obtained by other methods. Of most importance, the results obtained here are similar to estimates of the marginal value of pasture that were obtained by Barichello (1978) from actual hay market data under essentially independent assumptions. On the other hand, other farm models simulating non-optimizing or partial equilibrium behavior led to quite different results. The conclusion is that the results of these studies are consistent with the argument of Appendix I: assumptions of static. 131 optimizing behavior are more appropriate than alternative constructs in the estimation of supply response or other response via representative farm models. 4.4.1 Results Solutions obtained from various specifications of the linear pro gramming model strongly suggest that static equilibrium values of pasture in the Peace River region of British Columbia under 1975 conditions typically would be between $5 and $10 per AUM. Some of the results supporting this conclusion are presented in Tables 2 to 4. In addition, results presented in Table 5 suggest that (as expected) significantly higher shadow prices for pasture are implied by static equilibrium and extremely high (circa 1979) expected real beef prices. The results presented in Tables 2-5 illustrate the variation in equilibrium farm value of pasture with respect to expected beef prices, bounds on cow numbers and the possibility of backgrounding (purchase 25 of calves for subsequent sale as yearlings). Best estimates for other parameters of a linear model of a "representative" user of community "Community pasture differs from on-farm pasture in the following respects: cattle on community pasture are tended by a rider employed by the grazing association, and cattle must be moved to and from the community pasture within specified periods. Model results suggested that the net effect of these differences on income and the dollar-equivalent value of leisure is negligible. Thus we can assume that the marginal products of community pasture and on-farm pasture are equivalent. 132 pasture in the Peace River region of British Columbia circa 1975 have been 26 employed in obtaining these particular results. These estimates of representative parameters circa 1975 were gathered from interviews 27 with farmers and B.C. Ministry of Agriculture personnel. Table 2 shows the variation in equilibrium farm value of pasture and several aspects of model solutions with respect to (a) three important combinations of expected beef prices, (b) a lower bound of 40 cows, and (c) the possibility of backgrounding. The following combinations of beef prices are employed: the 1975 market prices of $30 and $36 per cwt. for calves and yearlings; the 1975 market plus Farm Income Assurance sub sidy prices of $56 and $50 per cwt'. for calves and yearlings; and the average real prices over the previous beef cycle of $50 and $45 per cwt. for calves and yearlings. Since 1975 market prices represent the bottom of the beef cycle and the anticipated subsidies for 1975 were presumably less than the subsidies that were subsequently legislated, the calf and yearling prices for 1975 that were most commonly expected at the beginning of the year should be bounded by the first two combinations. '"The effects of alternative "reasonable" values for some additional parameters (e.g., variable cost and yields of the hay enter prise, hayland rental rates, difference in calf and yearling weight gains on pasture and range, dollar-equivalent value of leisure) have also been considered. These variations do not alter the basic conclusions presented here. 27 See Section 4 of Appendix 6 for details. In addition, it should be noted that the Income Assurance section of the model is not employed here, and that the expected price for cull cows is varied pro portionately with the expected price of calves. For simplicity, these two prices (per cwt.) are equated here (other results show that this assumption does not affect our conclusions). TABLE II Farm Value of Community Pasture and Selected Model Activities: 1975 Market Prices, 1975 Market Prices Plus Subsidies, and Long Run Prices for Calves and Yearlings Calf Yearling Cow Backgroundhg CATTLE HAY AUM'|28) A Beef 3.90 + 3.90 +'29' Price Price Bound Bound cows calf yearling own rented tons sold (+) community pounds A income A OBJ ($/cwt.) ($/cwt.) (lower) (equality) sale sale acres acres purchased (-) pasture per AUM per AUM per AUM 30 36 0 0 123 0 0 -230 _ _ _ _ ' - - 0 0 181 0 0 -339 255 85.8 9.05 7.91 40 - 40 0 98 43 50 -224 - - - -40 - 40 0 122 1 57 0 -216 255 32.4 7.27 6.21 - 0 0 0 0 189 0 + 177 - - - -- 0 0 0 0 189 0 +177 0 - - -40 0 40 0 28 154 0 -48 - - - -40 0 40 0 28 154 0 -48 11 19.6 5.99 6.26 56 50 - - 0 0 112 0 0 -209 - - - -- - 0 0 170 0 0 -318 255 85.8 8.17 7.03 40 - 40 0 98 43 50 -232 - - - -40 - 40 0 120 43 50 -267 255 37.5 7.22 6.93 - 0 35 0 24 135 0 -42 - - - -- 0 51 0 35 189 0 -66 208 43.0 6.70 4.71 40 0 40 0 28 111 0 -93 - - - -40 0 51 0 35 189 0 -66 208 29.6 6.85 4.79 50 45 - - 0 0 42 67 0 -9 - - - -- - 0 0 42 67 6 -9 0 - - -40 - 40 27 1 189 0 +31 - - - -40 - 40 2 26 189 0 -11 90 104.4 5.13 4.30 - 0 20 0 14 77 0 -24 -- 0 20 0 14 77 0 -24 0 - - -40 0 40 10 19 139 0 -46 -40 0 40 0 28 154 0 -48 68 52.8 5.79 4.64 Average 30) 6.91 5.86 Either 0 ("-") or 255 AUM's of community pasture are supplied to the farm at $3.90 per AUM. 'oBJ = income plus dollar-equivalent value of leisure at solution. 'These are simple averages of values over all situations where community pasture is utilized. 134 For our purposes, the most important point to notice about Table 2 is the stability of the equilibrium farm value of pasture (as measured in either of the last two columns) relative to (e.g.) the number of yearlings sold, acres employed in hay, or comparative static change in beef pounds produced on-farm per AUM of community pasture. For these combinations of beef prices, bounds on cow numbers and backgrounding options, the estimated change in farm income (farm income plus dollar-equivalent value of leisure) varies from $5.13 to $9.05 ($4.30 to $7.91) per AUM of community pasture for farms using community pasture, and has a simple mean value of $6.91 ($5.86) per AUM of community pasture. Thus, to the extent that the shadow price of pasture depends on absolute beef prices rather than relative calf and yearling prices, these values should bound the most common equilibrium shadow prices of pasture in the region circa 1975. Table 3 illustrates the relation between the equilibrium farm value of pasture and calf and yearling prices intermediate between $30-$36 per cwt. and $56-$50 per cwt., in the absence of bounds on cow numbers or possibilities for backgrounding. These results, together with the results presented in Table 4 (where backgrounding is excluded from solution), suggest that the equilibrium farm value of pasture is highly sensitive to relative calf and yearling prices if and only if backgrounding is defined as feasible in the model. Since backgrounding was observed to be less common than cow-calf or cow-yearling enterprises in the Peace River circa 1975, it seems reasonable to suppose that high prices for yearlings relative to calves were not commonly expected for 1975. In this case, the most common static equilibirum values of pasture in the 135 TABLE III Farm Value of Community Pasture: Selected Calf and Yearling Prices Intermediate Between 1975 Market Prices and 1975 Market Prices Plus Subsidies 31 Calf Price ($/cwt.) Yearling Price ($/cwt.) 3.90 + A Income per AUM (32) 3.90 + A OBJ per AUM 35 35 _ 35 40 8.82 9.05 40 40 6.60 4.96 40 45 12.02 11.32 45 40 - -45 45 8.03 6.87 45 50 16.21 15.50 50 40 - -50 50 10.52 10 74 55 40 6.90 4.00 55 45 6.90 4.00 Average^ 9. 50 8.30 For all results reported here, cow numbers and numbers of calves purchased for backgrounding were endogenous to the model, i.e., not bounded. OBJ = income plus dollar-equivalent value of leisure at solution. 33 These are simple averages of values over all situations where community pasture is utilized. 136 TABLE IV Farm Value of Community Pasture: Selected Calf and Yearling Prices Intermediate Between 1975 Market Prices and 1975 Market Prices Plus Subsidies 34 Calf Price ($/cwt.) Yearling Price ($/cwt.) 3.90 + A Income per AUM 3.90 +^35^ A OBJ per AUM 40 45 - -45 50 6.99 4.63 50 50 6.26 4.24 A (36) Average 6.63 4.44 34 For the results reported here, cow numbers are endogenous (ranging between 0 and 49); but the number of calves purchased for back grounding is defined as 0. 35 OBJ = income plus dollar-equivalent value of leisure at solution. 36 These are simple averages of values over all situations where community pasture is utilized. 137 Peace River region of British Columbia under 1975 conditions should not differ greatly from the averages for Tables 2-3. In sum, static equilibrium values of pasture under 1975 conditions (among users of community pasture in the region) should be between $5 and $10 per AUM. Finally, results presented in Table 5 show that high equilibrium values of pasture can arise from high beef prices irrespective of the relative levels of calf and yearling prices. However, such high expected prices, which are realistic assumptions circa 1979, would have been very unrealistic in 1975 at the low point of the beef cycle. H.H.2 Results of Related Studies Here we summarize the results of some other studies that have been designed to calculate the farm value of pasture in British Columbia and other western provinces. For our purposes, the most important of these is a study by Barichello that is based on observations of hayland rental 37 prices. Eleven observations on cash rent paid for hayland were obtained for the Peace River region of British Columbia during 1975-1976. These observations ranged from $13.50 to $8.00 per acre with a mean value of $11.39 and a variance of $2.77. Moreover, land suitable for the production of hay also was commonly employed as pasture. Thus, given negligible costs of transacting rental agreements and a static equilibrium (with improved land receiving equal rents at the margin in its alternative 37 See Barichello (1978) for details, especially pp. 30-33. The linear programming model presented here and the study of hayland rental prices were designed as complements in the evaluation of British Columbia ARDA community pasture programs. 138 TABLE V Farm Value of Community Pasture for Extremely High 38 39 Calf and Yearling Prices ' Calf Price ($/cwt.) Yearling Price ($/cwt.) Backgrounding Bound (equality) 3.90 + A Income per AUM 3.90 +(40) A OBJ per AUM 70 60 - 8.07 9.17 • 0 8.24 8.51 80 60 - 9.94 10.27 0 9.94 10.27 80 70 - 17.18 16.48 0 12.23 12.19 90 70 - 14.81 14.72 0 14.82 14.72 90 80 - 25.55 24.85 0 17.85 13.91 A (41) Average 13.86 13.51 These beef prices, in 1975 dollars, correspond to considerably higher prices in 1979 dollars. 39 For results reported here, cow numbers are endogenous (ranging between 24 and 140). 40 OBJ = income plus dollar-equivalent value of leisure at solution. 41 These are simple averages of all situations where community pasture is utilized. 139 uses as pasture and hayland), the marginal value of pasture on a "represen-tative"farm in the Peace River region during 1975-1976 can be estimated as 0.71 x $11.39 = $8.09 per AUM.42 An earlier set of calculations based directly on observation also yields estimates of the farm value of pasture that are consistent with our static equilibrium beef ranch model. Wiens (1975) calculated a partial budget for a "typical" beef ranch in the Saskatchewan parkland region circa 1975. Given a "typical" set of farm activity levels and observation of a corresponding level of gross farm receipts and all non-grazing costs related to a cow-yearling operation, the value of pasture was estimated as a residual of $8 per AUM of grazing. On the other hand, a non-optimizing simulation model that was developed for the evaluation of ARDA community pasture programs in 43 the parkland region of Saskatchewan led to quite different results. First a large non-optimizing simulation model was adapted to conditions According to the best information obtained from farmers and B.C.D.A. extension staff, three acres of "average" quality pasture are required to summer one animal unit (cow with calf) over the typical grazing season of June 1 to October 7 (4.25 months), i.e. one AUM of grazing capacity corresponds to 0.71 acres of pasture. See Department of Regional Economic Expansion (1977). This evaluation of Prairie community pasture programs was undertaken simultaneously with the evaluation of British Columbia programs that is reported in Barichello (1978). 140 in the parkland area of Saskatchewan. Then three sets of farm behavior (resource allocations) in the presence and absence of community pasture were specified for a "representative" farm, and the associated cash flows were generated from the technical coefficients included in the simulation model. For these three sets of base simulations in the absence of community pasture and farm responses to community pasture, the farm value of community pasture (excluding its supply price) varied from $12 to $30 per AUM with a reported "weighted" average of $23 per AUM. However, when the most important of the data gathered for the Saskatchewan simulation model was incorporated into the British Columbia optimizing model with the assistance of the person responsible for the data, the estimated farm marginal value of community pasture was less than the corresponding values calculated with British Columbia data (see previous section). This result is not too surprising: pasture appears to be a scarcer resource in British Columbia than in Saskatchewan, and one would expect (ceterus paribus) a higher marginal value for pasture in the region where pasture is relatively scarce. In addition, studies by Graham (1977) and Harrington (1976) have presented estimates of the farm marginal value of pasture within sections of Western Canada. In a preliminary study with a beef farm linear programming model that in effect specifies beef capital activities and many enterprise combinations as exogenous to the model (i.e., as fixed), Graham (1977) obtained estimates of the shadow price of pasture for three British Columbia farms. These estimates varied between $26.00 and $0.62 per AUM for calf-yearling prices between 141 $40-$43 and $30-$33 per cwt. (backgrounding was excluded44''15 In contrast to the above studies, highly aggregated provincial data related to forage, including both pasture and range resources, was included in a large programming model of the Western Canada beef economy by the Economics Branch of Agriculture Canada. The average estimated value of an AUM of forage throughout British Columbia, as reported in Harrington (1976), was between $10 and $11. 4.4.3 Implications Here we note that the results reported in the previous two sections are consistent with the hypothesis that constructs of static, optimizing behavior are most appropriate in estimating farm response. The close similarity between the value of community pasture for the static linear programming beef ranch model and the results of the hay market study (Barichello, 1978) and the partial budget analysis (Wiens, 1975) suggests that these constructs would be somewhat realistic in the absence of adjustment costs. Since farm adjustment cost functions are essentially unknown at present, it follows that these results lend support to the hypothesis that static, optimizing models are most appropriate in estimating farm response. ""This wide variation in shadow prices presumably can be interpreted in part as empirical support for our decision to specify various capital activities and enterprise combinations as endogenous to the Peace River beef ranch model (see Section 4.3.1.2). 45 Additional prices were also considered by Graham (1977). 142 As has been noted previously, the hay market study for the Peace River region of British Columbia and the partial budget analysis for the Saskatchewan parkland both estimated the value of pasture circa 1975 as $8 per AUM. Since there was evidence of substitution between hay and pasture use of land in the Peace River and the variation in market rental prices for 46 hayland was relatively small, the estimates of the hay market study may well be realistic. In addition, given a "typical" set of farm activity levels and observation of corresponding receipts and costs, the estimate of the partial budget analysis would be realistic. This "realistic" estimate of $8 per AUM for the shadow price of pasture circa 1975 in the Peace River also approximates a "most likely" value in the reasonable range of $5 to $10 per AUM for the static equilibrium beef ranch programming model. Moreover, the results of the linear programming model are essentially independent of the calculations in the hay market study. This independence is demonstrated in Table 6: a $1.00 change in the variable cost of hay production on own land and rented land always leads to considerably less than the corresponding $0.71 change in the shadow 47 price of community pasture. The endogeneity of the equilibrium shadow price reflects the resource constraints and substitution possibilities that 48 are incorporated into the model. Thus the close similarity between our linear programming results and calculations based directly on observations strongly suggests that 46 See Barichello ( 1978), Chapter 4. 47 See footnote 42 earlier in this chapter. ^See Appendix 6. TABLE VI Sensitivity of Shadow Prices for Community Pasture With Respect to Profitability of Hay Enterprises 49 Calf Price ($/cwt) Yearling Price ($/cwt) Variable Cost Own Hayland ($/acre) Rental Price Hayland ($/acre) Acres Own Hay Acres Rented Hay 3.90 + A income per AUM c.p. 3.90 +50 A OBJ per AUM c.p. Without Com.Pas. With Com. Pas. Without Com. Pas. With Com.Pas. 30 36 21.25 41.5 304 350 50 50 9.10 8.38 26.25 46.5 95 122 50 50 9.35 8.00 31.25 51.5 0 0 0 0 9.05 7.91 36.25 56.5 0 0 0 0 9.05 7.90 41.25 61.5 0 0 2 2 9.05 7.91 56 50 21.25 41.5 122 136 50 50 9.01 8.31 26.25 46.5 75 122 50 50 8.96 7.44 31.25 51.5 0 0 0 0 8.17 7.03 36.25 56.5 0 0 2 2 8.23 7.07 41.25 61.5 0 0 2 2 8.23 7.06 50 45 21.25 41.5 350 350 0 0 7.93 6.57 26.25 46.5 189 189 0 0 6.73 5.53 31.25 51.5 67 67 0 0 - -36.25 56.5 3 3 0 0 6.08 4.63 41.25 61.5 0 0 0 2 6.08 4.63 Cows and calves purchased for backgrounding are unbounded here. OBJ = income plus dollar-equivalent value of leisure at solution. 144 (a) models of static, optimizing behavior are appropriate for the estimation of equilibrium shadow prices, and (b) the particular structure of the Peace River linear programming model is adequate for this purpose. These results also imply that models of static, optimizing behavior are most appropriate in estimating farm supply response given our present state of knowledge of farm adjustment cost functions. As has been noted in Section 1.2.2, adjustment costs (and many other factors) are less important in determining equilibrium shadow prices than in determining supply response. Other aspects of dynamics (price expectations and biologically-determined time lags in beef production) presumably play an important role in determining shadow prices as well as supply response. Since an ability to estimate equilibrium shadow prices with accuracy also suggests an ability to estimate supply response in the absence of significant adjustment costs,51 our empirical results are consistent with the hypothesis that errors in using constructs of static, optimizing behavior in the estimation of various types of farm response arise essentially from the importance of adjustment costs. Likewise, since farm adjustment cost functions are essentially unknown, our empirical results are consistent with the hypothesis that has been derived from the theory in Appendix 1: constructs of static, optimizing behavior are most appropriate in estimating farm supply response given our present state 52 of knowledge of farm adjustment cost functions. In addition, our results also suggest that, in some respects (e.g., estimation of shadow prices), these seemingly most appropriate constructs can closely approximate real behavior. 51See the last paragraph in Section 4.2.2. 52 For a first attempt to estimate adjustment cost functions statistic ally, see Berndt et al. ( 1979). 145 4.5 Summa ry In this chapter we have (a) outlined a static linear programming model of a "representative" beef ranch for users of community pasture in the Peace River region of British Columbia, (b) formulated a means of examining the appropriateness of constructs of static optimizing behavior in the estimation of farm response, and (c) observed that solutions (farm value of community pasture) to the linear programming model are consistent with these constructs. Thus we have (a) provided an example (estima tion of the farm value of community pasture) where such deterministic models are adequate and most appropriate, and (b) in the process gathered empirical support for the hypothesis that the major abstractions from reality that are employed in this thesis, i.e., the assumptions of static, maximiz ing behavior, are at present most appropriate for the estimation of farm response at the microeconomic level. CHAPTER 5 SUMMARY AND CONCLUSIONS 117 CHAPTER 5 SUMMARY AND CONCLUSIONS 5.1 Summa ry The main purpose of this thesis has been to extend comparative static theory and methods of the firm so as to be more useful in agricultural policy analysis. The traditional static theory and methods of the firm, which remains largely embodied in Samuelson (1947), has the following defects from the viewpoint of application in agriculture. 1. Endogenous factor prices, i.e., factor prices that are variable to the individual firm, apparently are realistic in many cases but have not been introduced (correctly) into the theory of the firm, 2. Comparative static methods that are presently available generally make inadequate use of the degree of knowledge about particular policy situations. Traditional qualitative methods (e.g., as in Samuelson, 1947) cannot readily in corporate our full degree of knowledge about a firm's production function. In part for this reason, these qualitative methods have led to relatively few useful results. On the other hand, traditional quantitative methods (e.g., use of programming models with a fully specified farm 148 structure) are usually too restrictive, i.e., they typically derive results that are dependent in unknown ways on a large number of essentially arbitrary assumptions that can only be partially accommodated in a sensitivity analysis. 3. The assumptions of static optimizing behavior that underlie the traditional theory of the firm may not be appropriate. Thus the main purpose of this thesis has been more specifically three-fold: 1. To extend the traditional qualitative comparative statics of derived demand at the firm level to the case of endogenous factor prices; 2. to extend comparative static methods of analysis at the firm level so as to incorporate more fully our empirical knowledge about parameters without specifying more than this knowledge, i.e., to introduce a method of analysis that provides a useful "middle ground" between the (generally under-determinate) traditional qualitative methods as embodied in Samuelson (1947) et al. and the (generally overdeterminate) quantitative methods as embodied in (e.g.) static linear and nonlinear programming models of the firm; and 3. to examine the appropiateness of constructs of static, optimizing behavior in the estimation of farm response. 149 These objectives have been pursued in Chapters 2-4, respectively, and in related appendices. In order to make the discussion more concrete and the applications more obvious, the material in each chapter was related to the problem of predicting response to government funded (ARDA) community pasture programs in British Columbia. In Chapter 2, the theory of derived demand with variable factor prices was investigated by making explicit use of the following "intuitively obvious" equivalence: a firm's derived demand schedule is equivalent (under very general conditions) to a schedule of shadow prices for the input. In Chapter 2 we demonstrated that a failure to recognize the implications of this equivalence has been responsible for a controversy in the American Economic Review during the last ten years concerning the relation between measures of consumer's surplus in product and factor markets, and also in part responsible for the serious errors committed in the previous attempts to incorporate variable factor prices into the theory of the firm (Ferguson, 1969, and Maurice and Ferguson, 1971). Utilizing this equivalence between derived demand and shadow prices, the following statements (among others) were established for the first time. 1. In the absence of market "distortions," the welfare changes (changes in consumer plus producer surplus) of agricultural policy affecting factor supply schedules can always be measured correctly in the related factor market. 2. The derived demand schedule for an input is necessarily positively inclined given increasing returns to scale and fixed prices for all other inputs, and the schedule can be positively inclined over large areas of its domain given decreasing returns to scale and non-convex isoquants. 150 It was demonstrated that (2) implies that comparative static effects of policies influencing factor supply schedules can seldom be predicted by traditional methods. In Chapter 3 and accompanying appendices, we introduced a method that in principle overcomes this defect of established comparative static methods by incorporating empirically based quantitative restric tions into the traditional qualitative comparative static analysis of the firm (e.g., Samuelson, 1947). This methodology incorporates the available degree, of knowledge of the firm's structure (production function and price schedules) without imposing further specification on the firm's structure (in contrast to, e.g., the traditional linear and nonlinear programming models of the firm, where a full structure must be specified). Then the range of quantitative as well as qualitative predictions of comparative static effects of policy that are consistent with our degree of knowledge can in principle be calculated. This methodology of "quantitative comparative statics" consists essentially of two nonlinear programming problems each characterized by an identical system of equations and inequalities which incorporate the implications of (a) the standard assumption that the firm is at a static optimum, plus (b) "reasonable" restrictions on the firm's production function and price schedules. 151 The comparative static effects, and the potentially observable parameters of the firm's production function and price schedules on which we have placed "reasonable" restrictions (upper and lower bounds), are all treated as endogenous variables in these two problems. These problems also have the same objective function, which is the comparative static effect of interest, and differ only in the sense that one is a maximization problem and the other is a minimization problem. Thus the solution values of the objective function for these two problems define the range of values for the comparative static effect of interest that are consistent with (a) the assumpt ion of a static optimum and (b) the "reasonable" restrictions on the firm's production function and price schedules.1 The empirical knowledge embodied in the restrictions (b) typically would be derived from observation and/or econometric estimation of physical processes and behavior, and would be expressed in the form of confidence-Bayes intervals for these potentially observable parameters of the firm's production function and price schedules. In this case, the range of com parative static effects defined by the solution values to these two problems can also be interpreted as a confidence-Bayes interval for the comparative static effect of interest. In Chapter 4, we (a) outlined a static linear programming model of a "representative" beef ranch for users of community pasture in the Peace River region of British Columbia, (b) formulated a means of examining the A simple schematic model of the methodology of quantitative com parative statics was presented on pages 69-74 in Section 3.4 of Chapter 3. 152 appropriateness of constructs of static, optimizing behavior in the estimi-ation of farm response, and (c) observed that solutions (farm value of community pasture) to the linear programming model are consistent with these constructs. We demonstrated that, by estimating an equilibrium shadow price for an input rather than other aspects of farm response, one can reduce the significance of many of the problems associated with studies of supply response (e.g., the effects of poor knowledge of the individual farm's production function) and fucus more clearly on the appropriateness of constructs of static optimizing behavior. By comparing solutions to the static optimizing Peace River model with calculations based on direct observation of hayland rental markets and beef ranch activities, we derived empirical support for the major assumption underlying the theoretical work in Chapters 2 and 3: models of static optimizing behavior are often the most useful constructs in the prediction of microeconomic behavior. In addition, these results showed that models of microeconomic behavior with a fully specified structure, such as the static Peace River programming model, can be useful in estimating some aspects of farm response that are of importance to policy (e.g., farm value of community pasture programs) although they generally seem to be unreliable in estimating changes in input and output levels. 5.2 General Conclusions The specific conclusions summarized in the previous section support the following broad statements: 153 1. Assumptions of static optimizing behavior are at present generally more appropriate than alternative constructs in predicting farm response to agricultural policy (Chapter H and Appendix 1). 2. Traditional qualitative methods of comparative statics lead to rel atively few predictions that are useful in formulating agricultural policy (Chapter 2). In addition, we have noted that there generally are extreme difficulties in reliably estimating comparative static effects from models of farm behavior with a fully specified structure, e.g., traditional linear and non-linear models of the firm (Section 1.2 of Chapter 1). The above statements imply that an "intermediate" method of comparative static analysis making full use of our degree of knowledge of structure without being dependent on the specification of more than this degree of knowledge would be very useful in the evaluation of agricultural development programs. Thus the most important conclusions of this study are as follows: 3. Traditional qualitative methods of comparative statics (as in Samuelson, 1947) can be extended to incorporate our degree of knowledge of farm structure (production function and price schedules) without becoming dependent on a specification of more than this degree of knowledge (Chapter 3 and Appendix 3). 154 4. This method of "quantitative comparative statics" may well be at least somewhat operational now for "small" (or highly aggregated) models of the farm (Chapter 3 and Appendix 5). Given the promise of this method of quantitative comparative statics, the initial work reported here in Chapter 3 and accompanying appendices should be followed by further studies. 5.3 Suggestions for Further Research Here we shall point out some areas of future research that are suggested by this study. Since the most important and most experi mental part of this thesis concerns the proposed methodology of quantitative comparative statics, we shall limit our comments to that section of the study. First, there are major unresolved computational difficulties with this proposed method of comparative static analysis. In particular, due to the presence of quadratic equality constraints in the underlying programming models, local solutions are not necessarily global solutions for these models. Thus we cannot estimate with any accuracy the confidence-Bayes interval corresponding to the observed range of local solutions for the maximization and minimization problems unless (a) there is a procedure for identifying a finite set of points that contains the global solutions to the maximization and 155 minimization problems (so that the global solutions can be calculated from a comparison of these points), or (b) there is a procedure for obtaining an approximately random sample of the feasible set of the programming problems (so that confidence-Bayes intervals can be estimated for the observed range of comparative static effects). Thus the first priority for research related to this study should be to reduce computational problems associated with this method by developing somewhat adequate procedures of the form (a) or (b) above. The author speculates that this will be possible in the immediate future for small models involving a few inputs.2'3 For a very preliminary discussion of approaches other than (a) for estimating confidence-Bayes intervals of the observed range of solutions for comparative static effects, see Section 2.2 of Appendix 5. 3 For an optimal procedure of aggregating large quantitative comparative static models into smaller models when the restrictions on the firm's production function imply that there exists an approximately correct aggregation procedure, see Section 3 of Appendix 5. Unfor tunately the conditions for correct aggregation of large quantitative comparative static models to a more manageable size presumably introduces errors into the calculation of the global solutions and feasible set for the comparative static effect of interest. For a procedure that may become somewhat useful in estimating such aggregation biases for particular models, see the end of Section 3.1 of Appendix 5. 156 Given that these computational problems are handled somewhat adequately, there appear to be many applications in policy research and other empirical work for such a method of quantitative comparative statics. Here we shall simply point out several examples in order to illustrate the diversity of potential applications. First, the methodology could be employed to estimate the range of comparative static effects of community pasture programs that is consistent with our degree of knowledge about the structure of farms receiving this pasture. For example, we could construct quantitative comparative static models roughly similar in type to those presented in Section 3.5 of Chapter 3. Then we could obtain estimates of the range of "reasonable" comparative static changes in producer plus consumer surplus, i.e., of the confidence-Bayes interval for this effect that corresponds to the specified degree of knowledge of farm structure. For the reasons that have been specified, these results would be superior to those obtained by traditional methods of qualitative and quantitative comparative statics. 157 Second, this method of quantitative comparative comparative statics could be used to investigate the relation between the slope of a firm's derived demand schedule and properties of the firm's production function and price schedules in more detail than was possible with the qualitative methods employed in Chapter 2. Purely qualitative methods do not generally lead to empirically based restrictions on the slopes of derived demand schedules (that are independent of the slope of the factor supply schedule), and knowledge of this slope can be important for policy (see Chapter 2). Third, it appears that interactions between firms can be incorporated into this methodology. In this case we would obtain a synthesis of maximiz ing behavior, firm interactions and empirical knowledge for the theory of the firm. With such an extended methodology, we might well be able to predict the effects of, e.g., national or provincial price support programs on farm output and other activities more effectively than in the past.4,5 Fourth, this method of quantitative comparative statics may well lead to more effective testing of various theories of firm behavior. Since traditional qualitative and quantitative methods have led to relatively few 4 The literature on integrating maximizing behavior and firm inter actions seems to be summarized entirely in Silberberg (1974b). Since incorpor ation of interactions actually increases the ambiguity of results obtained by qualitative methods, there is an even greater need to incorporate empirical information into comparative static methodology for this case than for the standard (no firm interactions) qualitative theory of the firm. 5As pointed out previously (Section 2.3.3 of Chapter 2), the standard (Hicks-type) methods of industry analysis, which do not incorpor ate the implications of the maximization hypothesis, can easily lead to even qualitative errors in predicting comparative static effects. 158 reliable predictions of comparative static effects, these methods have also led to relatively few reliable tests of hypotheses concerning firm 6 7 behavior. ' On the other hand, the method of quantitative comparative statics introduced here in principle makes more effective use of the available degree of empirical knowledge than do these traditional methods. Thus this methodology should lead to a greater number of testable hypotheses than do traditional methods of qualitative comparative statics, and these hypotheses should discriminate between various theories of behavior more effectively than hypotheses derived from traditional quantitative methods. In sum, there is a wide array of potential applications in policy research or other applied work for such a method of quantitative comparative statics. In turn, the potential value of such a methodology in the formula tion of agricultural policy justifies further research to make it operational for a wide variety of problems. For example, the comparative static predictions of two theories X and Y will generally vary with the structure of the firm's production function. 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"On the Cost of Adjustment," Quarterly Journal of Economics, 75 (1971): 605-622. Sahi, R.K. and Craddock, W.J. "Estimation of Flexibility Coefficients for Recursive Programming Models - Alternative Approaches," American Journal of Agricultural Economics, 56 (1974): 344-351 . Sakai, Y. "An Axiomatic Approach to Input Demand Theory," International Economic Review, 14 (1973): 735-751. Samuelson, P.A. Foundations of Economic Analysis. Cambridge, Mass. Harvard University Press, 1947. . "The Problem of Integrability in Utility Theory," Economica, "16 ( 1950) : 355-385. . "Abstract of a Theorem Concerning Substitutibility in Open Leontief Models," Activity Analysis of Production and Allocation. T.C. Koopmans, ed. New York: John Wiley, 1951. 168 Schmalensee, R. "Consumer's Surplus and Producer's Goods," American Economic Review, 61 (1971): 682-687. . "Another Look at the Social Valuation of Input Price Changes," American Economic Review, 66 ( 1976): 239-243. Sharpies, J.A. 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"Consumer's Surplus Without Apology," American Economic Review, 66 ( 1976): 589-597. Wisecarver, D. "The Social Cost of Input Market Distortions," American Economic Review, 64 ( 1974): 359-372. Yver, R.E. "The Investment Behavior and the Supply Response of the Cattle Industry in Argentina." University of Chicago, Depart ment of Economics, (1971) (Mimeo). APPENDIX I WHY COMPARATIVE STATICS AND THE MAXIMIZATION HYPOTHESIS Page 1.1 Static vs. Dynamic Models 171 1.2 Optimizing vs. Non-Optimizing Models 177 171 APPENDIX I WHY COMPARATIVE STATICS AND THE MAXIMIZATION HYPOTHESIS? 1.1 Static vs. Dynamic Models The purpose of this section is to point out the difficulties in modelling stock adjustments to a change in policy, and to argue that the comparative dynamic effect of the community pastures programs apparently can be estimated as accurately by the use of static models as by the use of dynamic models. An example of a static model is the linear programming model of a beef ranch that will be presented in Chapter 4: this model has a time horizon of one year, and the endogenous opening and closing stocks are restricted to be equal. Therefore, by comparing model solutions in the absence and in the presence of community pasture (ceterus paribus), we can calculate a "comparative static effect" for the community pasture program. By defining an appropriate structure for the model, this effect can be "short-run," "long-run," or whatever. However, such comparative static calculations can be, at best, only a very rough guide to the comparative dynamic effect of the pastures program. This is because a truly dynamic response primarily results from an effective cost of stock adjustment constraint,1 and is a function *ln the absence of effective cost of stock adjustment constraints and overlooking the inherent lags in the production process, a comparative dynamic effect is simply a series of instantaneous adjustments to changing conditions, i.e., a series of comparative static effects. This series of comparative static effects is defined by the change in the supply schedule (Footnote 1 continued on following page) 172 of (la) the non-stationary environment (e.g., the different product prices expected over time) perceived by the firm; (lb) the initial stocks held by the firm; and (1c) the cost of stock adjustment schedule Ct = C(lt) faced by the firm where lt is the level of net investment by the firm at time t. (Id) inherent lags in production. Given an effective cost of stock adjustment constraint Ct = Cdt), none of these influences (1a)-(1d) on a comparative dynamic effect can be modelled correctly by a series of comparative static calculations. In spite of these weaknesses of comparative static methods, dynamic models do not seem to be much more helpful (and may often be less helpful) than static models in estimating real-world dynamic response of firms to changes in the supply schedule of community pasture, or to changes in exogenous variables in many other situations. This is due to the following: (Footnote 1 continued) of community pasture and by changes in the firm's environment over time. In the absence of adjustment costs, the effects of inherent lags in production can be roughly accommodated in static models. For example, given the 2.5 year lag between the change in the beef herd and the resulting production of beef, and overlooking adjustment costs, farm response presumably could be simulated with reasonable accuracy by constructing a static model with a time period of three years. Stocks would be freely variable over the period subject to the constraint of equality at the beginning and end of the three year period. 173 (2a) the magnitudes of C(lt) and C'(It) for users of community pasture (and in general) seem essentially unknown; (2b) estimates of initial stocks, by themselves, generally provide little knowledge of comparative dynamic effects; and (2c) errors in the valuation of the terminal stock occur in non-static models, and lead to errors in the estimation of response. These points will be elaborated upon in the above order. The significance of and arguments for statement (2a) are as follows. The difference between the comparative dynamic effect of a change in the supply schedule of pasture and a related series of comparative static effects depends critically upon the cost of adjustment function 2 3 ct = C(lt)/'J The following generalizations seem correct (Rothschild, 1971). For a highly non-stationary environment, the "average" length of delay in response to pasture depends primarily upon the magnitude of C(lt)(>0) and the sign and magnitude of C'Ut.)/ and the degree of fluctuation about this average depends upon the sign and magnitude of C"(lt). On the other hand, for a highly stationary environment, the average length of delay depends primarily upon the sign and magnitude of C"(lt) (<0 implies an extremely rapid, non-periodic response). 3 It should be noted that adjustment costs should also play a role in comparative static calculations: adjustment costs should be incorporated into first and second order conditions for an equilibrium, whether dynamic or static (see Treadway, 1970). However, adjustment costs presumably have considerably more influence on the dynamics of response than on the change in static equilibrium. In this thesis we shall follow the usual procedure of deleting adjustment costs from comparative static calculations. 174 Hence, any claim for superiority of dynamic models over static models, as predictors of real-world comparative dynamic effects, presumably depends largely upon an ability to estimate the function Ct = C(lt) with some accuracy. However, an ability to predict the magnitudes of C(lt) and C'(lt) seldom seems to exist at present. = C(lt) appears to be in large part a complex, and so far unidentified, function of such variables as 4 education, elasticities of product and factor supply and demand, and the particular exogenous change. Thus, it is not surprising that we seem to have very little knowledge of these magnitudes for the adjustment-constraining components of Ct = C(lt), and this in itself suggests that dynamic models seldom will be a significant improvement over static models 5 6 as estimators of real-world comparative dynamic effects. ' 4See Petzel (1976). Presumably some components of Ct = C(lt) are more easily quanti fied than is indicated here. Perhaps the most obvious example concerns the effect of the firm's debt-equity ratio on its marginal cost of borrowing: MCBt = M(D/Et), using obvious notation, and D/Ej = D(Kt), i.e., in the short run the debt-equity ratio increases with the firm's capital stock. However, if the investments being considered by the firm involve only minor changes in techniques and provide relatively quick payoffs, then the firm is likely to face a constant marginal cost of borrowing schedule (M1 = 0) and cash and credit costs of adjustment CJIt) = M'D1 will be zero. This appears to be largely the case for users of B.C. ARDA community pastures. More generally, adjustment will be affected significantly by cash and credit costs C(lt) presumably only if the firm would be expanding its enterprise in the absence of such costs and C(lt)' E M"D" > 0. g For a first attempt at the statistical estimation of adjustment cost functions (in manufacturing), see Berndt et^aL (1979). 175 Statement (2b) can be explained briefly as follows. We are interested in the comparative dynamic effect of the pasture program, i.e., the difference between the time paths in the presence and in the absence of community pasture. This difference presumably is considerably less dependent on the level of initial stocks, and more dependent on the spec ified adjustment costs, than are these two time paths. Moreover, time paths are known to be highly sensitive to errors in specifying initial conditions.7 Hence, even though initial conditions can be incorporated more correctly into dynamic models than into one period models, in general this does not appear to provide dynamic models with a significant advantage over static models as predictors of comparative dynamic effects. The argument for and significance of statement (2c) is as follows. In dynamic models, capital accumulated at the horizon must be assigned an exogenously-determined per unit value in the objective function, which represents an estimate of the capital's discounted net value in production beyond the horizon. Since the farm value of used g capital is in fact endogenous to the farm plan, this procedure inevitably leads to errors in specifying the terminal value of capital. Given such a 'The sensitivity of time paths to initial conditions is documented in growth theory literature, and has been confirmed by simulations with multi-period farm planning models (Boussard, 1971, pp. 475-7). g Due to serious imperfections in markets for used capital (except for the regularly-traded fully depreciated capital, such as cull cows), the farm value of capital at the end of the model year seldom corresponds to the market price. Moreover, even if capital markets accurately reflect current farm value of capital, we would still not be able to compute the farm value of capital that would be consistent with a particular altern ative set of expected prices, etc. 176 mispecification, the time horizon of the dynamic model must be considerably longer than the average life of capital even if the sole intent is to obtain reasonably accurate estimates of activities in the initial time period. Since cows have a productive life of approximately eight years, a dynamic model intended for use in estimating effects of community pasture programs would in general have to be unwieldy, or else extremely simplified within most years, in order to reduce the effects of such a mispecification to insignificant levels. Corresponding problems never occur with static models.1** Thus, in the presence of very limited knowledge of the magni tude of C(lt) and C'(lt) for the firm's cost of stock adjustment function Ct = C(lt), the solutions of various static models may well provide more information about the comparative dynamic effects of the community pasture programs than will the solutions of dynamic models.11 By defining a highly simplified structure for all but the first year in a dynamic model with a long time horizon (and estimating a dynamic response as the series of first year solutions obtained from recursive runs of the model), we will in general simply be exchanging errors due to a , mispecified terminal value of capital for errors due to excessive aggregation. 10This statement is justified simply as follows. If the firm's environ ment and actions in the one year time period of a. static model are in effect repeated in all other one year time periods, then the actions that maximize the value of the objective function (flow of farm benefits) in the one year model will also maximize the discounted sum of flows of farm benefits over time. ^Since the "flexibility constraint" approach to dynamics (Sahi and Craddock, 1974) incorporates historically observed measures of response over time rather than adjustment cost functions per se, it is not a satisfactory approach to dynamics. In other words, the dynamics of response is not specified as endogenous to the farm in the flexibility approach. 177 In sum, apparently the best that we can do in estimating the real world responses to community pasture programs is to calculate various comparative static effects for the programs. For simplicity (and also in part due to a lack of confidence in any estimates of the rate of change in the rate of change of the firm's environment at any particular time), these calculations can be limited to "short run" and "long run" comparative static effects. Then the estimated comparative dynamic effect would simply be the straight line connecting the "short run" and "long run" 12 comparative static effects. 1.2 Optimizing vs. Non-Optimizing Models It is sometimes stated that non-optimizing simulation models are superior to optimizing (or, equivalently, maximizing) models as predictors of farm behavior because "farmers do not optimize." However, we will 13 now argue that this conclusion is incorrect. ,zEven this procedure of calculating "short run" and ''long run" comparative static effects often may be based on inappropriate assumptions. In particular, a comparative dynamic change at time t, will be similar to a comparative static effect only under certain conditions, e.g., certain properties of adjustment cost functions (Rothschild, 1971), indivisibil ities and imperfect capital markets. If these conditions are not sufficiently realistic, then the comparative dynamic change at t, may even have opposite signs from a "short run" comparative static effect calculated for t^, and the comparative dynamic change over time may bear no resemblance to the time path calculated from the "short run" and "long run" comparative static effects (Nagatani, 1976). 13 It is known that purposive behavior of microeconomic units can in principle be described by optimization techniques when the decision making unit's preferences are "consistent," and that inconsistency of preferences can arise when behavior is governed by rules-of-thumb (or is determined collectively). See, e.g., Samuelson (1950). Here we note that 178 It is a tautology to state that an individual decision-maker always obtains a constrained maximum defined by his preferences, resources and the external environment. In other words, the assumption of purposive behavior implies the existence of a constrained maximum, properly defined. Therefore, such farm behavior can always be described analytically in terms of an appropriate optimization model, and "non-optimal" aspects of such behavior (use of rules-of-thumb in decision-making rather than a "global" search procedure) can always be interpreted as reflections of various types of human capital adjustment costs. However, adjustment costs are by definition zero in a stationary state model, and the influence of human capital adjustment costs on behavior in a non-stationary model 14 cannot at present be predicted with any accuracy. Therefore, 1. the "non-optimal" aspects of behavior cannot be predicted at present with any degree of accuracy by farm planning models, and 2. actual behavior can be "approximated" by use of a static equilibrium optimization model, with the extent of the approximation depending upon the relevant adjustment cost functions and the rate of change in the firm's environment. Footnote 13 continued (a) rules-of-thumb can in principle be incorporated into optimization models as adjustment costs, and (b) static equilibrium optimization models are in practice often superior to static or dynamic non-optimization models as pre dictors of microeconomic behavior. See the previous section of this appendix. 179 Thus there appears to be no point in attempting to incorporate "non-optimal" behavior into models designed to predict farm response to changes in policy. Moreover, non-optimizing models appear to be in principle inferior to optimizing models as predictors of such response precisely because the "optimizing" central tendency of behavior, in contrast to the "non-optimizing" aspects of behavior, can on occasion be modelled with some accuracy. A comparison of the marginal value of community pasture obtained by the static linear programming model to be presented in Chapter 4 (and Appendix 6) and by other means supports these theoretical arguments, and also suggests that this particular static optimizing model provides reasonably accurate measures of this value. In order to carry out an evaluation of ARDA community pasture programs in Saskatchewan, a large non-optimizing simulation model was adapted to conditions there.15 This study led to considerably different (generally higher) estimates of the farm marginal value of community pasture than did the British Columbia study to be reported in Chapter 4. However, when the most important of the data gathered for the simulation model was incorporated into the British Columbia optimizing model, the estimated farm marginal value of community pasture was less than the value calculated with British Columbia data. Since pasture appears to be a more scarce resource in British Columbia than in Saskatchewan, one would expect (ceteris paribus) a higher marginal value for pasture in British Columbia than in Saskatchewan. See Department of Regional Economic Expansion (1977). Moreover, the marginal value of community pasture estimated by using British Columbia data in the optimizing model was consistent with an essentially independent measure derived from data in the hay market. 181 APPENDIX II QUALITATIVE COMPARATIVE STATICS AND DERIVED DEMAND: PROOFS Page 1. Preliminaries 182 2. Lemma 1 185 3. Theorem 1 188 4. Corollary 1 190 5. Corollary 2 4 6. Corollary 3 197 7. Corollary 4 200 8. Lemma 2 2 9. Lemma 3 207 10. Theorem 2 9 11. On the Hicks-Andrieu Formula for the Elasticity of Derived Demand 221 182 APPENDIX II QUALITATIVE COMPARATIVE STATICS AND DERIVED DEMAND: PROOFS 1. Preliminaries Let x B Nx1 vector of activity levels for the inputs of a firm c'(x) = total cost schedule to the firm for its itn input (i ^ 1) c^[x;a) = total cost schedule to the firm for its input 1, as a function of x and a parameter a y E Mxi vector of activity levels for the M outputs of a firm y = f (x) E production function (vector-valued for M>1) for the firm b(y) E total benefits schedule to the firm as a (scalar-valued) function of its M outputs R(x) E b(f(x)) x* E N x l vector of the input levels employed by the firm at a solution to a particular maximization problem x1 = an exogenously determined level of input 1 employed by the firm .183 Definition 1. A producer problem P is defined as P 1 N i maximize TT(X) = R(x)-c (x;a)- E c (x) ... .(P) i=2 for a particular value of the exogenous variable a, and the *P solution set to this problem is denoted as {x (a)}. Definition 2. The firm's derived demand schedule for input 1 is defined as {(x1*P(a), MFC^a)) for all a} = DP where 3X1 P 111 Denote the relation defined by the pairs in D as p = p (x ) Definition 3. A producer problem Q is defined as Q N i maximize TT(X) = R(x) - E c (x) i=2 . . . .(Q) subject to x1 = x1 for a particular value of the exogenous variable x1' and the solution set to this problem is denoted as (x^Cx1)}. 184 Definition 4. The firm's shadow price schedule for input 1 is defined as 9TT(X*Q(X1))Q -. . (X* , for all x eX ) 9x' where X1 E {x1*P(a) for all a } Definition 5. Any problem P maximize R(x) - c^[x;a) N i Z c'(x) i=2 is said to "correspond" with the series of problems of the form Q N maximize R(x) Z c (x) i=2 N j maximize R(x) - Z c (x) i=2 1 T*~A~ subject to x = x , 1 F*z" •, subject to x = x where {x1*A, x1*2} = {x^*P} for the problem P. Denote the union of solution sets for this series of problems of the *Q c form Q as {x } . 185 i* Condition 1. For any solution x* to a problem P: x > 0, i = !,•••,N Condition 2. In the neighbourhood of any solution to a problem P: R(x) and all c'(x) are twice differentiable. Condition 3. 1 11 c (x;ct) <=> c (x ;a), i.e., the total cost of input 1 is independent of the levels of inputs 2, •••,N. Condition 4. d 2<T (x* > 0 for all x and i,j ,k =1, •••,N, i.e., factor 9xj3xk supply prices are non-decreasing in x. Condition 5. 9 Rfx^ S 0 for all x and i = 1, N , i. e., input are 3x' "freely disposable." Condition 6. If the set of feasible Tr(x)y for a problem Q is bounded from above, then the set is also closed from above. 2. Lemma 1 Lemma 1. Suppose that conditions 1-2 are satisfied for a problem Q. Then 9TT(X _ 3R(x ) £ 9c'(x*) 3? 3X1 i=2 8xx for any x* e {x*Q} 186 Proof 1.2. Construct the problem Q Q N i maximize TT(X) E R(X) - Z C (X) i=2 . . . .(a) subject to x = x By conditions 1-2, R.(x ) - Z c!(x ) = 0 for all j t I . . . . (b) 1 i=2 J which are first order conditions for a solution. By definition 3, 3 TT( X*1 ^ y '— for this problem Q can be calculated as 3X1 3TT(X*)Q = R (X*} + E R (x*)_3xL- z c\ (x*) 3x* j*1 ' Bx1 m j* - Z Z c! (x*) — . . . .(c) i*1 j^l 1 3xi j* R.(x*) - Z c'(x*)+ Z ^* [R.(x*)- Zc!(x*)] 1 m 1 j#1 3xi J i=1 i (d) by rearranging (c). Substituting (b) into (d). 3tt(x*)Q = R.(x*) - Z c!.(x*). . . . .(e) 3x^ 1 i*1 1 187 9TT(X*)Q Q 9TT(X*)Q —==— exists by (e) and {x* } not null, and —-===z— 3x- ax is unique for Q by Definition 3; so (e) holds for any x*e{x*^}. Q 1. Lemma 1 can be deduced almost directly from the Viner-Wong envelope theorem of Samuelson, which states that the first order change in the value of the objective function Tr(x;a) as x varies optimally (from an initial interior solution) in response to a change in an exogenous variable a is equal to the change in fr(x*;a) for dx = 0, i.e., 9Tr(x*(a),a) 9TT(X"*', a) 9 a 9 a where 9x* = 0 (Samuelson, 1947, p. 34). 9a 2. In proofs, partial derivatives will generally be denoted by subscripts. For example, 9R(x) 9c'(x) . = R.(x) and *— = c.(x) 9xx 1 9xJ 1 188 3. Theorem 1 Theorem 1. Suppose that conditions 1-3 are satisfied. Then (A) {x*P(a)> <=> |x*Q(x1*P(a))| for all a i.e., any problem P and the corresponding problem (s)Q have identical solution sets; and (B) |(x1*P(a), MFC1(a)) for all aj <=> {(x1*^). 3iT-(x*Q(x1*P(a))] ° -)- for all a} i.e., DP <=> D0.1 9X1 *p / Proof. By condition 1 and definitions 1-2, x is a solution to the problem P 1 1 N i maximize TT(X) = R(x) -c (x ;a)- E c (x) . ... .(a) i=2 Construct the related series of problems P P maximize TT(X) maximize TT(X) . . . , 1 1*A . . . . 1 1*z subject to x = x , •, subject to x =x (b) formally Theorem 1 only applies to the case where input 1 is employed in a single enterprise, since the cost schedule for input 1 is defined as a function of only one input. However, Theorem 1 readily generalizes to the firm that employs input 1 in M enterprises. In this case, we can define . . M and the quantity constraint in a corresponding c = c ( E x1J;a) j=1 M ii ~ Producer Problem Q as E x ' = x . It is easily shown that, with these j=1 obvious modifications. Theorem 1 applies to the multi-enterprise firm as well as to the single enterprise firm. 189 where (x ,'",x ) = {x } for problem (a), and problems (b) and problem (a) have identical objective functions. By (a) and (b). {x*P} = j{x*}j . -(c) where |{x }| = the set of solutions for series (b). Since (b) implies that ~ 1 * 1 {x }> is exogenous to problems (b), and therefore r i ~i* ) <{c (x ;a)H is exogenous to problems (b), |{x }> is independent of the specification of . . . .(d) c^{x^;a) in problems (b), which includes the specification c1(x'';a) = 0 for all x1. By (c) and (d), for any a {x*P} = {x*Q}C . . . .(***) where *0 c {x } = the set of solutions for the series of problem Q's corresponding to problem (a) (see Definition 5), which is statement A of the Theorem. By conditions 1-2 and (***), Lemma 1 can be used to calculate 190 3ir(x*P)Q *p N . * P i *P = R (x r) - Z c'(x r) ... .(e) 3X1 1 i=2 1 *P at any solution x for any problem Q corresponding to problem (a). By conditions 1-2, N *P 1 1*P i *P R^x r) - cj(x ; a) - Z c!,(x ) = 0 ... .(f) which is a first order condition for an interior solution to problem (a) By (e)-(f). *P Q 9TU ) 1 1 *P =— = c,(x ;a) . ... .(g) 3X1 1 By (***), (g), and Definitions 2 and 4, DP = DQ which is statement B of the Theorem. • 4. Corollary 1 Corollary 1. Suppose that conditions 1-2 are satisfied, and that for all a: 1 *P 3 c (x (a); a) —: =fc 0 for at least one i t 1, 3x' Then (A) {x*P(a)} fl {x*Q(x1*P(a))}C = null set for all a , i.e., any problem P and any corresponding problem Q do not have any solutions in common; and (B) for any a: {(x1*P(a), MFc\a)) } fl DQ * null set if and only if *P, N . i, *P, .D, *Q, N 'i. *CK 3 R(x ) _ „ 3c (x ) 3R(x ) _ y 3c (x ) 1 1 1 1 3x i=2 3x 3x i=2 3x for all (or, equivalently, any) x e {x (a)i *Q r *Q( 1*P, ^ic 2 x ^ e {x ^(x (a)) } *P Proof. By conditions 1-2, x is any solution to the problem N j maximize ir(x)P = R(x) - Z c (x) i=1 where 2 Corollary 1-B also assumes that conditions 5-6 are satisfied. 192 x'*P > 0 i = l,-,N ... .(b) *p N : *p R.(x r) - Z c!(x r) = 0 j = I, -• N (c) J i=1 J By assumption, c^xia) has input 1 and at least one other input (e.g., N) as its arguments, where cN(x*P;a) * 0 . ... .(d) Construct the problem Q N . maximize R(x) - Z c (x) i = 2 ... .(e) 1 i*p subject to x = x By (b), *P x is a solution to problem (e) only if *p N : *p R.(x *) - I c (x ) = 0 J i=2 J By (c) and (d), N RN(x r) - Z cj^(x ) * o . ... .(g) By (f)-(g), ... .(f) j = 2,-.-,N. 193 1 *p for any P where c.(x ; a) ± 0 for some i i 1 . . . .(***) *P *P *Q c and all x {x } fl {x } = null set. which is statement A of the Theorem. By (c), IM *P i *P 1 *P RAx ) - Z c.(x ) = c^x ; a) ... .(h) i=2 for problem (a). By Lemma 1, 3l"**)Q = R,(x*«)- E cUx*Q) (I) ax1 i=2 for problem (e). By statement (f) in the proof of Corollary 2, * O d TT( x 1 3 —-—— is single-valued for a given problem (e). . . .(j) 3X1 By (h)-(j), , 1*P won I v r '1 i*p 1 *p P for a given (x , c.(x ;a)) on D and a related solution *P x to a P, there exists an identical fl*Q d TT( x *) Q 8X1 on if and only if *P N ; *p *n ^ i *o R.(x *)- Z c!.(x v) = RAx y)- Z c'(x w) 1 i=2 1 i=2 3 Statement (f) in the proof of Corollary 2 depends on conditions 2, 5 and 6 but not on Theorem 1. for all (or, equivalently, any) solutions x to the corres ponding Q's. ED" 5. Corollary 2 Corollary 2. Suppose that conditions 1-3 and 5-6 are satisfied, and denote the domain of p1 = p^x1) as X^. Then 1B (A) if x is included in a solution to at least one problem P, then all x1A such that 0 < x1A < x1B are in XD, (B) p1 is a function of x'\ i.e., p^x1) associates one and only one p1 with any particular x1 in X^, (C) p1(x1) is differentiable for all x1 "within" XD, 1 1 1A IA i.e., for all x such that 0 < x < x and x is an element of X^. Proof. By condition (5), , 1A,Q . , 1B.Q IA . IB max TT(X ) £ max TT(X ) if X < X . . . .( where max TT(X')v is the maximum attainable value of the objective function R(x) - Z c'(x) for the problem Q defined i*1 by the constraint x1 = x1' By (a). 1 1 g if Q with the constraint x = x is bounded. _ 1 IA . IA . IB then Q with x = x , where x < x is also bounded where Q is defined as bounded for x1 if and only if max TT(X •) = k or max TT(X ) -»- k , for a real number k. By condition (6) and (b). 1 1 g if Q with the constraint x = x has a solution. _ .iU 1 IA . 1A ^ IB then Q with x = x , where x < x , also has a solution. By condition (2) and (c). 1 1B if Q with the constraint x = x has a solution. * 1A then 9tt(x 1?—11 is defined for all 8x1 1A ^ IB x < x By (d) and Theorem 1, ( 196 if x is included in a solution to at least one P, 1A 1A 1R then all x such that 0 < x < x are in the . . .(***) domain of p^(x^) which is statement A of the Corollary. By Definition 3, 1 Q * 1 Q max TT(X ) = TT(X (X )) exists and is single-T * valued for each x where a solution x exists . . . .(e) for the Q. (f) By (d), (e) and statement A(***), 3TT(X (x_jj— js sjnq|e_va|uec| for each x1A = x1 ix^ such that 0 < x1A < xlB and Q has a solution , ~~T _ IB for x = x i.e., for each x1A = x1 an element of XD. By (f) and Theorem 1, p^x1) is a single-valued for all x1 an element of XD . . .(***) which is statement B in the Corollary. By (f), condition (2) and Lemma 1, * ~~T Q 3TT(X (x ))_ js a differentiab|e function of 3X1 x1 Ex1 for all x* . . . . (g) 197 (g continued) 1 1 A IA such that 0 < x < x and x is an element of X1 i.e., for all x1 within XD. By (g) and Theorem 1, p^x1) is differentiable for all x1 within X^ • • • .(***) which is statement c of the Corollary. • 6. Corollary 3 Corollary 3. Suppose that conditions 1-3 and 5-6 are satisfied. Then *A A (A) for any solution x to a problem P where a = a , *A P * O TT(X A)K = TT(X (0)P /•x 1*A 1, 1, . 1 1, 1*A A, p (x )dx - c (x ; a ) where * Q N j i TT(X (0))w = max{R(x) - E c (x):x = 0} , i=2 pi(0) a aTr(x*(o))Q dx1 H. . . . .. . . .. lim ftx1 + Ax1)-fix1) Any right hand side derivative for Ax1 > 0 is represented here as Ax1 -*• 0 aftx1) Ax-3X1 t 198 and i, *A *B (B) for a solution x and a solution x to two problem P's A B that differ only in terms of a = a and a = a , respectively. *B,P , *A,P 7T(X ) - TUX ) 1*B r x 1, K . 1 1, 1*B B, p (x )dx -c (x ;a ) 1*A x 1, 1*A A, + c (x ; a ). *A Proof. Let x be a solution to a P. By conditions 1-2 and Theorem 1-A/ *A 1 1 *A r I x is a solution to the corresponding Q (x = x ). . . . . laj By (a). Theorem 1-B, Corollary 2-A and 2-C, * ~~T Q 8TT(X (X ))_ js defined and continuous for all Sx1 0 < x1 < x^A . ... .(b) By (a), and by (c) in the proof of Corollary 2, TT(X (0))^ exists, ... .(c) By (b)-(c), 3x~ exists . . . .(d) 199 where 9TT(X*_(0))Q dx1 lim Tttx^Ax1))0 - TT(X*(0))Q Ax1" for all Ax1 > 0 By (b)-(d) and the definition of * 1 O 8^ TT(X*A)Q - TT(X*(0))Q + 1 ^ m <^ fX 8TT(X (X'))U dxl 9x^~ .(e) where 9TT(X*(0))Q ^ 9TT(X*(0))Q dx1 By (f), Definition 1 and 3, TT(X ) * O •nix (0))y + rx 0 1*A * i 0 9TT(X (X]))U^1 dx 9x-1f 1*A s A, - c (x ; a ) . . . .(f) By (f). 1*B TUX ) , *A,P TT(X ) * i O 9TT(X (x^r^i 1 *A 8 x 1, 1*B Bw 1, 1*A A> -c(x ;a)+c(x ;a) . . . .(g) 200 By (f), (g), condition 3 and Theorem 1, 1*A TT(X*A)P = TT(X*(0))Q + fX p^xbdx1 -c 1(x1*A;aA) TT(X ) - TT(X J 1*B fX 1/ 1,^ 1 ^ 1*B B, p(x)dx -c(x ;a) 1*A x A 1, 1*A A, + c (x : a ) which are statements A and B of the Corollary. • 7. Corollary 4 Corollary 4. Suppose that conditions 1-3 are satisfied for a problem P. 1A (A) If x is included in a local solution to P, then PVA) - 9cl(x'A;a) = 0 3x' „ 1, 1A> „.2 1, IA , 3p (x ) _ 8 c (x ;g) < Q 9X1 3x12 (B) If p\x*A) - 8c1(x^;a) = 0 3x . 1, IA, „2 1, IA . 3p (x ) _ 3 c (x ;a) . 1 » Sx1 3x12 1A then x is included in a local solution to P. 201 Proof. By conditions 1-2 and Definition 2, A x included in a local solution to P => pVx1^} - cj(xA; a) = 0 ... .(a) A P [TTJ.(X ) ] negative semi-definite, and p^x1*) - c](xA;a) = 0 and [TT..(XA))] negative definite (b) => x a local solution to P Since N Z Vx ) 4£ = 0 for i = 2, ...,N , N r * P 9x'* 9xj* N , * P 9X1* Dx'* E E TT..(x ) TT -= = E TU.(X ) . , . . ij 9a 9a . , Ii 9a 3a 1=1 j=1 ' j=1 ' N 1 * i* 1 * ™ , * P 9x' 3xJ 9x' . . . .(c) by condition 3 and Theorem 1-A. By (c) and Theorem 1-B, !* i* MM ; :2 j _ ,..*,P 9x' 9xJ _ ,_1,..1*, _1 ,..1 i=1 j=l z z Vx r fir Sr- = (Pi(x' )-cii(x" ;a)) 1*2 r. X • ... .(d) 9 a 202 By (a)-(b) and (d). Corollary U-A and 4-B are established. • 8. Lemma 2 Lemma 2. Consider a problem Q maximize TT(X)^ subject to x1 = x1 (a) where TT(X)^ is twice differentiable in the neighbourhood of an interior * solution x (not necessarily unique). Also construct the related problems maximize TT(X) subject to x1 = x1 + Ax1 and maximize (Ax1) ll 2 1 1 subject to Ax = Ax , (b) maximize A2TT(X*Z)Q (Ax1) ll 2 . .(c) subject to Ax1 = Ax1 where A2TT(X*)Q = • Z Z TT..(X*)Q AX' Axj, i = 1 j=1 ,J *1 *Z 2 N { x ,**\x }is the solution set to problem (a), and (Ax ,*«\Ax ) r 203 is the vector of endogenous variables for problems (c) Then *b *a (A) (x_x } - {4^1 i Ax1 ) Ux1 i as Ax1 0 where x = a solution to problem (a) *b x = a solution to problem (b) *c Ax = a solution to a problem (c), and * (B) (even if x for problem (a) is not unique) 9 27T(X )' ix^2 for problem (a) is equal to the maximum A2TT(X*)Q (AX1)2 for any problem (c) (Ax1^ 0) Proof. Construct the problem Q maximize TT(X)^ ^ —Y ....(a) subject to x = x which is assumed to have an interior solution x (not necessar ily unique). Construct the related problem Q 204 maximize TT(X) y J y~ • • • • (D) subject tox=x+tAx where t is a given scalar. Problem (b) can be expressed equivalently as * Q maximize TT(X + tAx) —=• . . . -(c) subject to Ax = Ax where (Ax2, •••,Ax^) are the endogenous variables. Given that TT(X)^ * * is twice differentiable in a neighbourhood of x which contains x + tAx, * Q we can express TT(X + tAx) as a second order Taylor expansion * about x : TT(X* +tAx)Q = TT(X*)Q + t Z TT.(X*)Q Ax1 i = 1 1 2 N N n ' • -— Z Z TT..(X)U AX'AX1 . . . . (d) 2 i=1 j = l IJ where x is some point between x and x + tAx. Substituting the interior first order conditions TT.(X*)^ = 0 (i = 2,«««,N) for (a) into (d). * O * Q * Q 1 TT(X +tAx) = TT(X ) + tTT^X )HAx 2 N N . . + — Z Z TT..(X)U AX'AXJ . ... .(e) 2 i=l j=1 IJ Construct the related problem 205 N N . . maximize E E TT..(X)W AX'AX' i=1 j = 1 ,J ... .(f) subject to Ax1 = Ax1 2 N where (Ax ,'««,Ax ) again are the endogenous variables. By (e), problems (c) and (f) have the same set of (primal) solutions. . . . .(g) By the definition of x and the assumption that TT(X) is twice * differentiable at x , TTj.fx)0 + Trj.(x*)Q (i, j, = 1, •••,N) as t -> 0. ...(h) By (h). as t + 0, the limiting (asymptotic) form of problem (f) is problem A: N N * Q i i maximize E E TT..(X )^ Ax Ax' . . . .(i) i=1 j = 1 ' subject to Ax1 = Ax1 Construct the related problem N N * Q i i 12 maximize E E TT..(X )v Ax AxJ / (Ax ) 1=1 j=1 ,J . .(j) subject to Ax1 = Ax1 206 Since Ax1 is exogenous to problems A and (j), problems A and (j) have the same set of (primal) solutions. . . . .(k) By (g), (i) and (k), r 2* N * as t -+ 0, i{ TAX , • • •, ) J. for all (c) 1 tAx1 tAx1 t ***•» 2* N * > .... i j ( A2L_ A*—)} for all (j) defined by {x*} for (a) 2* I1 Ax Ax — • • • Ax1 ' - ' Ax1 * defined by {x } for (a) which is statement A of the Lemma. In addition, I N N * o i i 1 N N —T 1 I TT..(X y* Ax Ax' = • T £ 2 (Ax1) i = 1 j=1 IJ (XAx1) i = 1 j = l TTj-Cx*)0 (XAx')(AAxj) for all (A,Ax) (I) By (I), the solution value of the objective function for problem (j) is invariant with respect to the constraint Ax1 = Ax1 {t 0). . . . .(m) 207 Assuming that a solution or solutions exist for problem (a), * Q . . . .(n) l* 2— 's uniquely defined for problem (a). d2TT(x")Q 9x By (***), (m)-(n) and the definition of 927r(x*)Q 9x I 2 ' 927T(X*)Q ^2— f°r problem (a) is equal to the solution value 9x-of the objective function for any problem (j) (Ax^ 4 0) defined by {x } for problem (a) which is statement B of the. Lemma. • 9. Lemma 3 Lemma 3. Suppose that conditions 1-3 are satisfied. Then, for IA 11 A any x in the domain of p (x ) and the related a and any *A global solution x , * 1f 1A^ .2 1, IA A, .2 9 p (x ) 9 c (x ;g ) _ m=vimiim f 1 i -c — maximum \^—) 9X1 9X1 ^X N N Z E IT..(X M) Ax Ax' 1=1 J = 1 ,J for all Ax such that Ax1 t 0. 208 Proof. Construct the problem P P 1 1 N i maximize TT(X) = R(x) - c (x ) - E c (x) . . . .(a) (for a given a) satisfying conditions 1-2 in the neighbourhood * of each global solution x (not necessarily unique). Construct the corresponding problem Q Q N i maximize TT(X) = R(x) - E c (x) i=2 . . . .(b) subject to x* = x1A 1A where x is included in a global solution to problem (a). By (a)-(b) and Lemma 2-B, 3TT(X*A)Q 1 , IA. . f 1 W v r *A.P — - c. (x ) = maximum (-j—^ J E E TT..(X ) dx1 " AX i = 1 j=2 ,J for all Ax such that Ax1 #0 . . . .(c) *A 1A NA where x = (x ,«-«,x ) is a global solution to both problems (a) and (b). By Theorem 1 and Corollary 1-B and 1-C, 1 1A 92TTfX*A)Q 32TT(X*Z)Q p (x1A) = 9 nx^) = = . . . .(d) 1 3X12 3X1 2 where r *A *Z, {X ,'".X } 209 denotes the solution set to problem (b) . By (c) and (d), 1, 1 A. 1 , 1 A, . f 1 >2 N , * p (x ) - c (x ) = maximum ( J E Z TT..(X J Ax1 i=1 j = 1 IJ for all Ax such that Ax1 t 0 for any global solution x*A to problem (b). • 10. Theorem 2 Since the proof of Theorem 2 consists of several parts correspond ing to the statements (A-E) to be proved, it may be useful to precede the proof by a brief statement of the methodology that is common to these parts. As mentioned in Section 2.4.4.2 of Chapter 2, essentially Corollary 4 can be used to transform the comparative statics problem of determining the direction of change in equilibrium level of input 1, resulting from a change in the factor cost schedule c^(x^), to a problem determining the existence of an equilibrium for particular specifications of c^(x^). This statement can be elaborated upon as follows. From Corollary 4 (or, to be exact. Lemmas 2-3) we can deduce the following: , , 3Tr(xAf n . „ . . .... Sc^tx1*) 1, 1A, (a) — . = 0, for all I,IS equivalent to * p (x ), 3xr 3X1 P A (b) TT(X) concave in the neighbourhood of x is equivalent to > 3X1 in the neighbourhood of x , where the left hand statements in the equivalences a and b are the necessary and sufficient conditions for an interior local solution to the problem P at x . Therefore, 1, l. _ ~l 1 ,~T since a c(x)=wx (w exogenous) can always 1 1 1A be constructed such that w = p (x ), the slope of the derived demand schedule can be deduced from the answer to the following question: given particular properties of R(x) and c'(x) for i ± 1, and (a1) always, (b1) sometimes, or (c1) never p true that TT(X) is concave in the neighbourhood of x Depending on whether a', b' or c' is correct. 1, 1, _ 11 c (x ) = W X such that w = p (x ) is it < 0 * *r IA* ^ 9P (x ) > o or > 0 respectively. 211 Theorem 2. Suppose that conditions 1-6 are satisfied. Denote the the domain of p^x1) as X^, and denote a wage or rental rate that is exogenous to the firm as w'. Then the slope of the firm's derived demand schedule is related to certain properties of R(x) and c'(x) (i = 2, •••fN) as follows. (A) If R(x) is strictly concave,5 then 8p (x } ^ 0 and 11 11 11 ^x D p (x ) > p (x + e) for all (x ,x + e) in X , where e > 0. (B) If R(x) is concave, then 8P (x * 2 0 for all x1 in X . (C) If RUx) ^ AR(x) for all A > 1 and x > 0 but R(x) is not concave, then (1) ^P X ^ ^ 0 always for at least some x^ in X^ but N . 9p'(x') , (2) for some R(x) and Z c'(x) : . 1 U Tor i-2 9x 1 • VD some x in X (D) If R(Ax) = AR(x) for all (x,A) > 0 and c' E w'x' for i = 2, ••.,N, then 9p (x ] = 0 for all x1 in XD. 9x (E) If R(Ax) > AR(x) for all A > 1 and x > 0 and c1 E w'x' for i = 2, •••,N, then iE_!21_L > o and p^x1) < p^x1 +e) for all (xVx1 + e) in XD, where e > 0.6 Footnotes on the following page (5,6). 212 Proof. Part A (Introduction) Construct the problem P PA : maximize TT(X)A = R(x) -c^x1)^ - E c'(x). . . .(a-1) i=2 Given that p^x1) is constructed from the above R(x) and N i 11 E c (x) and various c (x ) (Definition 2), i=2 if p^x1) is defined for x1A (i.e., x1A e X^), 1 1 A A then there exists a c (x ) such that P has . . . .(a-2) . .. A _ , IA NA, a solution x = (x - ,«««,x ). Construct the corresponding problem Q Q N i maximize TT(X) = R(x) - E c (x) i=2 subject to x1 = x1A By (a-2)-(a-3) and Theorem 1-A, x is a global solution to problem (a-3) . . . . (a-4) (a-3) 5 The firm's total benefits function R(x), which is simply a total revenue function if the firm maximizes profits, is strictly concave if and only if (1) R(Ax) < AR(x) for all X > 1 and x > 0, and (2) all isoquants of R(x) are strictly convex for x ^ 0. Likewise, R(x) is concave if and only if (1) R(Xx) < XR(x) for all A > 1 and x > 0, and (2) all isoquants of R(x) are convex for x ^ 0. ^Note the asymmetry between statements C and E: p^(x1 >p^(x^+e) for decreasing returns to scale and fixed factor prices (i $ 1), whereas p^x1) < p^x1 + e) for increasing returns to scale and fixed factor prices (i i 1), where e > 0. 213 Replace cl(x1)A in PA with a cVx1)^ such that If 1A.B 1, 1A.A . c^x ) + c^x ) ... .(a-5) cj^x1)6 = 0 for all x1 (a-6) which results in the problem B B 1 1 B ^ i' P : maximize TT(X) = R(x) - c (x ) - .|2c (x).. . .(a-7) By (a-1)-(a-3), (a-5) and conditions 1-2, Uj(xA)B = 0 i = 1, --^N. . . . .(a-8) Given conditions 1-2, any x is a local solution to a P if and only if TTjU)*3 = 0 (i = 1,---,N) and TT(X)P . . . .(a-9) is concave at x and P ^ P TT(X) is concave at x (1) if [7T|j ] is negative P definite at x, and (2) if and only if [TT.. ] is ... .(a-10) ~ 7 negative semi-definite in the neighbourhood of x 7See Karlin (1959), p. 406. 214 p where [TT.. 1 denotes the Hessian matrix of IJ P N i Tr(x)r = R(x) - I c'(x) i = 1 at x. Part B (proof of Statements A and B) Since the negative of a convex function is concave, and the sum of a (strictly) concave function and a concave function is (strictly) 8 i concave, condition 4 (c.^ ^ 0 for all i, j, K and x) implies that TT(X) is (strictly) concave if R(x) is (strictly) ... .(b-1) concave. Given that TT(X) has a maximum over the convex feasible set of all x £ 0, •n[x) attains a unique local maximum over all x ^ 0 if TT(X) is strictly concave •nix) attains either a unique local maximum or a convex set of local maxima (hence every local maximum is a global maximum) over all x H if •nix) is concave. By (a-10) and (b-1), . . . .(b-2) (b-3) R(x) concave => maximum 1 2 N N Ax1 I I TT..(X)P Ax'Ax' < 0. . .(b-4) = 1 j=1 " Ax1 i 0 , ., for all x. 8Footnote on following page. (8) 215 By (b-4) and Lemma 3, R(x) concave => pj(x1) ^ 0 for all x1 e XD ... .(***) which is statement B of the Theorem. By (***) and Corollary 2-A and 2-C, if p^x1) = p^x1 + e) for R(x) concave, then p^x1) = pVx1 + Xe) for all 0 £ X S 1. By Theorem 1-B, if p^x1) = p^x1 + Xe) for all 0 £ X £ 1 where x1 is B included in a solution to a problem P (a-5 to a-7), then x1 + Xe for all 0 ^ X S 1 is included in a local g solution to P . By (b-3) and (b-5)-(b-6). TT(X) strictly concave => p^x1) 4 p^x1 + e) for any ,11 , VD (x ,x + e) e X . By (***), (b-7) and Corollary 2-B, (b-5) (b-6) (b-7) R(x) strictly concave => plfx1) £ 0 and ' f***i 11 11 11 D i p'(x') * p'(x + e) for all (x',x' + e) eXu q which is statement A of the Theorem. o See Lancaster (1R68) for a summary of most of the properties of concave functions and sets that are used here. ^ Corollary 2-B (p^x1) is single-valued for each y} eX^) implies that the result obtained by statement.^ ai^i (b-7); i.e., statement A, is independent of the assumption that c^fx ) = 0 for all x1. 216 Part C (proof of statement C) N . For a given R(x) - Z c (x) and {x} E a particular subset i=2 2 N x x of x that defines all possible factor proportions ( —, — ), we can x x construct a problem N : Z i = 2 c C 1 1 C i P : maximize TT(X) E R(X) - c (x ) - Z c'(x) ... .(c-1) where 11 1 , 1,c _ , .. 1 ... .(c-2) c. ,(x ) = 0 for all x TT(X)C ^ 0 for'all x e {x} . ... .(c-3) Assume that R(Xx) ^ XR(x) for all X > 1 and all x > 0 (c-4) By (c-3) and condition 4, ,c (c-4) => (a) TT(YX)" £ 0 for all y 2 1 and all x e {x} (b) TT(COC)C 2 TT(X)c for all 0 < a ^ 1 and all x > 0. By (c-5), (c-4) => {all x | TT(X)C > 0 and x 2 0} is closed and bounded. (c-5) (c-6) 217 Given that {all x | TT(X)C 2 0 and x 2 0} is non-empty: (c-6) and Weierstrass's Theorem imply that (c-4) => problem Pc (c-1) has a solution. . . . .(c-7) Given that this solution is interior: (c-2), (c-7). Corollary 4-A and Corollary 2-B imply that R(Ax) ^ AR(x) for all X > 1 and x > 0 ==> plfx1) ^ 0 ' f***i ID ' ' ' for some x e X which is part 1 of statement C of the Theorem. Given that an interior A IA NA A point x = (x , •••,x ) solves problem P (a-1) for an appropriate 1, 1,A c (x ) , VD 0, x e X ..... (c-8) by Definition 2, and TT(X)A is concave at xA . . . .(c-9) by (a-9)-(a-10). Since the sum of non-concave function and concave functions is not necessarily concave, TT(X)A concave at xA =t=> TT(X)B concave at xA ... .(c-10) A B By the definitions of TT(X) and TT(X) (a-1) and(a-7). 218 NN A A ' • NN A R • * Z Z 7T..(XA)A AX'AXJ = Z Z TT..(XA)B AX'AXJ i=l j=1 'J 1=1 j=1 'J for all Ax such that Ax"* = 0 By (a-10) and (c-11), (c-11) N . for some R(x) and Z c (x) satisfying (c-4) and i=2 condition 4: N N A R j maximum Z Z TT..(X ) Ax Ax' = maximum . . . .(c-12) Ax-Ax=1 i = 1 j=1 IJ Ax'AX=1,Axx^0 N N A R Z Z Tr.'.fx ) AX'AXJ > 0 . 1 = 1 j = 1 'J By (a-3)-(a-4) and Lemma 2-B, 32TT(XA)Q f 1 } 1 ^— = maximum -— ^ .. 3X1 Ax1 * 0 Ux1 J i=1 j=1 IJ Z Z TT..(XA)Q Ax' Ax' ... .(c-13) By (a-3)-(a-4), (c-8), (c-12)-(c-13) and Theorem 1-B, N . for some R(x) and Z c (x) satisfying (c-4) and i = 2 condition 4: ... .(c-14) 1/ Ii ^ n * 1 VD p.(x ) > 0 for some x e X which is part 2 of statement C of the Theorem. 219 Part D (proof of statement D) Suppose that, for problem P (a-1). R(Xx) = XR(x) for all X > 0 and x 2 0 ... .(d-1) c'(x) = w' x1 i = I, —fN . . . .(d-2) TT(XX)A = 0 for at least one x t 0, all X > 0 ... .(d-3) ^ A X ~ in TT(X)A 2 0 for all x t any Xx .' ... .(d-4) By (d-3) and (d-4), " A all Xx are global solutions to a problem P satisfying (d-1)-(d-4) .n By (d-5). Lemma 3 and Corollary 2-B, if R(Xx) = XR(x) for all X >0 and x 2 0 and ~~ 1,1, for all x1 e which is statement D of the Theorem, .(d-5) c'(x) = w'x1 for all i # then pj(x]) =0 ... .(***) Given (d-1) and (d-2), (d-3) and (d-4) are necessary for the satisfaction of condition 1 (hence are implied by the satisfaction of condition (1). If (d-3) or (d-4) is hot satisfied, then either x* = 0 or the problem is unbounded. ^Statement (d-5) is in effect Samuelson's substitution theorem (Samuelson, 1951). 220 Part E (proof of statement E) Suppose that R(Xx) > XR(x) c'(Xx) = Xc'(x) i = 2,---,N for all X > 0, x > 0 . 3 By the definition of TT(X) (a-7), (e-1) (e-1) => TT(XX)B > XTT(X)B for all X > 1 and all x . . . .(e-2) By (e-2). B (e-1) => TT(X) is not concave at any x. ... .(e-3) By (a-10), (c-9), (c-11) and (e-3), 2 (e-1) => maximum Axi N N . n E E Tr..(xA)y AX'AX1 SO (e-4) 1=1 j=1 By (c-8), (c-13), (e-4) and Theorem 1-B , (e-1) => pltx1) > 0 for all x1 e XD. ... .(e-5) By (e-5) and Corollary 2-A, given (e-1): if p^x1) = p^x1 + e) for a (xVx1 +e) eXD, then pVx1) = p1(xV+Xe) . . . .(e-6) for all 0 < X ^ 1. 221 g By the definition of TT(X) (a-7) and Theorem 1-B, p^x1) = p^x1 + Xe) for an (xVe) and B ... .(e-7) all 0 2 X ^ 1 => ir(x) has a local solution which is presumably interior. By (a-8)-(a-9), (e-3) and (e-6)-(e-7), (e-1) => p^x1) t p\x1 + e) for any (x1 + e) e . . . . (e-8) By (e-5) and (e-8), R(Xx) > XR(x) for all X > 1 and x > 0 and c'(x) = w'x' for i = 2,-«-,N => p](x1) > 0 and ... .(***) p^x1) * p^x^e) for all (xVx1 +e) e XD which is statement E of the Theorem. • 11. On the Hicks-Andrieu Formula for the Elasticity 12 of Derived Demand Given statement E of Theorem 2, we can easily demonstrate that a solution to the formula for elasticity of industry derived demand developed by Hicks (1963, pp. 241-6) and generalized by Andrieu (1974) is not necessarily consistent with the static maximization hypothesis. From the first order conditions for an interior maximum for competitive firms and assuming an industry production function 12 This section of the Appendix supplements section 2.3.3 of Chapter 2. 222 12 13 F(x ,x ) homogeneous of degree p, Andrieu develops the following formula for the industry elasticity of derived demand for input 1: , _ ax1 w1 e2k1 + a12 - Za12(1-kl)e2 dW X (1-k^ - Z(e2 + kla12) where 1, 2 d(F /F ) a = atx /x J / J- , ' (industry elasticity of factor 1 x /x 2 1 substitution) = 9D(p) . jp_ 1 ~ y (industry elasticity of product 9p demand) 2 2 2 e = 9S ^ 1 . ^HL- (industry elasticity of supply 9 (w ) x for input 2) 1 1 k, = ——— (factor share for input 1) 1 py z = (p - 1) - P/n (formula 15, p. 413). For p = 1, equation 1 reduces to the formula of Hicks. If T) -> +°° and e2 -*• +°°, then the numerator and denominator of equation 1 approach [k1 - Za12 (1-k1)]e2 and -Ze2, respectively, and Z + p -1 > 0 for p > 1. So 13 For p i- 1, these conditions are compatible in the presence of external economies or diseconomies of scale for the individual firm. 223 sign (X^) = sign [Z^ 2( 1 - k^) - k ] < 0 for n + 00, e2 -* + «>, p > 1 . . . . (2) given only Z > 0, a12 > 0 (convex isoquants for F) and 0<k1 < 1. Given perfectly elastic product demand schedules and supply schedules for input 2 at both the industry and firm level (so that changes in the level of output produced or input 2 employed do not lead to shifts in price schedules faced by individual firms, the industry derived demand schedule for input 1 would be equivalent to the derived demand 1 2 schedule for an individual firm facing the production function F(x ,x ) and identical price constraints. Therefore, the contrast between statement E of Theorem 2 (p!j > 0) and the more ambiguous statement 2 above implies that (a) a subset of the solutions '{(Xj, Oj2, n, e2, k^, p) } to formula 1 is inconsistent with the static maximiz ation hypothesis, and (b) various qualitative relations calculated by means of formula 1 will be more ambiguous than is warranted 14 by the static maximization hypothesis. 14 On the other hand, various qualitative relations implied by the static maximization hypothesis happen to be represented correctly by formula 1. By formula 1: X1 = 0 for p = 1, n -*• + °°, e2 + 00 (Hicks, 1963, pp. 373-4), which is equivalent to statement D of Theorem 4. By formula 1: > 0 for p < 1, o^2 > 0, n ->• +«>, e2 +00 and X1 £ 0 for p < 1, o-\2 < 0# n +oo/ e2 ->• +°°, which is in accordance with statements B and C, respectively, of Theorem 2. These conclusions are not surprising, since Hicks and Andrieu could not incorporate second order conditions for a producer problem P maximum into their formulas. 225 APPENDIX III QUANTITATIVE COMPARATIVE STATICS AND DERIVED DEMAND: DETAILS OF THE MODEL Page 1. Introduction 226 2. Restrictions Implied by the Maximization Hypothesis 227 2.1 Comparative Static Implications of the Maximization Hypothesis 228 2.2 Restrictions Corresponding to [IT..] Negative Definite... 233 3. Restrictions Implied by Additional Properties of [TT..(X*)] 234 3.1 Major Restrictions 236 3.1.1 Model with Output Exogenous 237 3.1.2 Model with Output and a Subset of Inputs Exogenous 242 3.1.3 Model with a Subset of Inputs Exogenous 248 3.2 Minor Restrictions 249 3.2.1 Knowledge of the Comparative Static Effects of a Change in R(x) 251 3.2.2 Several Special Properties of TT(X;CX) 253 4. A Minor Difficulty in Translating Between Local and Observed Comparative Static Effects 256 5. Restrictions as Confidence Intervals or Bayes Intervals 257 6. The Possibility of Additional Restrictions 260 7. Summary of Major Quantitative Restrictions 262 Appendix III. Quant itative Comparative Statics and Derived Demand: Details of the Model 226 1 Introduct ion In this Appendix we shall present a more detailed discussion of the methodology for quantitative comparative statics that was intro- . duced in Chapter 3. Proofs and a discussion of partial solutions to major computational problems will be presented in the next two appendices. The method of quantitative comparative statics can be schematized as obtaining global solutions to two nonlinear programming problems maximize z(tu) maximize z(f^) subject to Cf,:! tt = " ^'u subject to C^jl Z '^'>A [jT-,^] negative definite ^'j^ negative definite where (f^t'-'^ij^ i^O are endogenous 'variables and the scalar valued function z=z(^~") is the comparative static effect of interest. The restrictions C'Tr'.p U Z ~ L^l negative definite are the restrictions implied by the assumption of an interior solution to the firm's static maximization problem "maximize K{ x; oi)" (^(xjo*) is twice differentiable), the equations denote the relations between the Hessian matrix CTT^Cx*)! and a set of more readily observable parameters (0, and the restrictions denote the empirically derived restrictions (confidence-Bayes inter vals) for the parameters Here we shall discuss primarily a) the comparative static implications of the maximization hypothesis, b) various equations G(L^;J1, Q)r0 relating C^'j to more readily observable parameters and c) the interpretation of the solution values for z(^s) in the above problems when the restrictions (3° are formulated as confidence-Bayes intervals. 227 1. . Restrictions Implied by the Maximization Hypothesis It can be shown that the assumption of maximizing behavior Is essentially as realistic as the results of comparative static analysis, and that compara tive static methods usually are more appropriate than comparative dynamic techniques for the evaluation of community pasture programs. 1 Thus it is important to incorporate the restrictions implied by the maximization hypothesis. I.e. by the existence of an Interior static maximum, Into our methodology. However, In order to avoid placing arbitrary restrictions on the structure TT(X), we should model in this manner only those restrictions that correspond exactly to the comparative static Implications of the maximization hypothesis. The task of determining the precise comparative static Implications of the maximization hypothesis has been labelled the "integrability problem" In comparative statics (Silberberg, 1974a), and has been largely solved in the case of the dual approach to comparative statics (Epstein, 1978). In addi tion, necessary and sufficient conditions for consistency between the compet itive firm's factor demand schedules and the maximization hypothesis have been known since Hotel ling (1932). Nevertheless, the exact Implications of the maximization hypothesis for primal comparative statics apparently has not been demonstrated previously (even In the competitive case) for the problem N , , maximize TT(x;a) = R(x) - c^x^a) - E c (x ). P i=2 In this section we shall show that, for problem P, the usual set of primal restrictions [n ] 3x. = IJ 3a c1 olct symmetric and negative definite corresponds exactly to the implications of the maximization hypothesis for primal comparative statics. Thus the "integrability problem" is solved in this I . See Appendix 1 and Chapter 4. 228 special case.^ In addition, the restriction fnjjl negative definite Is ex pressed in a form that is more appropriate for our (primal) quantitative com parative statics model. ^.1 Comparative Static Implications of the Maximization Hypothesis Given that the primal problem P has an interior global solution x* where ir(x) is twice di fferentiable, the first order conditions for a maximum Imply that N I Z IT; ;(X*)3xJ» - C1 (x1*^) = 0 j=1 J IS" . 10 N I TT. .(x*)3xJ» =0 i = 2, , N i = 1 J 3a for an infinitesimal change da affecting the cost schedule c'fx1) for input 1, and the second order conditions imply that N N . Z Z TI . . (x*) dx dx <0 for all dx. (2.) 1 = 1 J-1 'J " Statement 2 is satisfied if and only if the Hessian matrix ["JJ] ot x* is either negative definite (implying that the strict inequality relation in 2 holds for all dx / 0) or negative semi-definite only (implying that the sum on the left hand side of 2 is equal to 0 for some dx i 0). In addition, Young's theorem imp Iies that [n|j(x*)] Is symmetric. (3) Statements (2) and (3) obviously exhaust the restrictions placed on [n ..(x*)] by the assumptions of an Interior maximum and twice differentiability of TT(X). "X . The "Integrability problem" in comparative statics has been described as a "major gap in the theory of comparative statics of maximization models" (Silberberg, 1974a, p. 171); but it Is easily solved for the general problem maximize T(x;a) subject to G(x;a) = 0 In the context of primal methods of comparative statics in essentially the same manner as for the special case maximize TT(x;a) For these reasons, a general discussion of the "integrability problem" in primal comparative statics Is included In Appendix 4. 229 For emphasis, the relation between statements t -3 and the restrictions on 3x* implied by the maximization hypothesis for problem P are presented here 3a as Proposition 1. Parts A and B of the Proposition are well known, and follow directly from the fact that a negative definite matrix has full rank and a matrix that Is only negative semi-defInlte does not have full rank. Thus, given that [TT . ^ (x*) ] (symmetric) is negative definite and that at least one comparative static effect 3x* exists for problem P, statement 1 and knowledge da of [n, .(x*)] and c1 (x**;a) are sufficient to define 3x* (which is unique). Given that [tjj(x*)] (symmetric) Is only negative semi-definite, statement 1 has multiple solutions for a particular [TI..(X*)] and c1 (x1*;**). However, 3a ,J ia by Part C of Proposition 1, 3x* is in fact undefined by primal comparative 3a static methods when [TTJJ(X*)] is only negative semi-definite and da defines 3 a shift in the firm's cost schedule for an input. The intuitive meaning of Proposition 1 may be clarified somewhat by the following argument. Given that [Ttjj(x*)] is only negative semi-definite, it can be shown that the derived demand schedule p'tx1) and the marginal cost schedule c|(xl;a) for any Input 1 have identical slopes at x1*. Thus, for the purpose of determining the comparative static effect of an infinitesimal change da (which depends only on the first and second order derivatives of TT(X) at x*), the situations shown in Figures f-A and R-B are equivalent to the 3 . Proposition 1-C can be proved essentially as follows (an alternative proof is presented in Appendix f-). The first order condition In the product market for an interior solution to problem P can be denoted as MR(y) - MC(y;a) = 0 (a) using obvious notation. The total differential of (a) yields (MR - MC )3y - MC = 0 ; (b) ^ y is a but [n..(x*)] only negative semi-definite Implies (by definition) that MR* - MC = 0 (c) Since (b) ana (c) are consistent only if MC = 0 , local (primal) comparative statics Is meaningless when [n j j(x*)] Is only negative semi-definlte. 230 Proposition 1. Suppose that conditions 1-3 are satisfied for a problem N maximize ir(x) = R(x) - c1 (xx ;<x) - E c'(x) (P) i=2 and that this problem has a unique global solution x*." Denote the set of comparative static effects of da for this problem as (3x*>. and denote the 3a system of total differentials of the first order conditions for a solution to this problem as U 3a cl o1" (1) where [n, . ] is defined as the Hessian matrix for IT(X) at x*, and c1 denotes the i J a exogenous shift in c|(x*;a) at xl*. Assume that is negative semi-definite and symmetric. Then (A) if [njj] 's negative definite: equations (1) have a unique solution 3x* ; 3a (B) if [TT J J 3 is not negative definite: equations (1) may have multiple solutions {dx}; but 3a (C) if [n. .] is not negative definite: 3x* is undefined ({3x*l is 'J 3a 3a empty), I.e. Sp'Cx1*) 3x*» - aVCx^a) 3x'» - sVCx1*,^) = 0 3x4 3a §7" 3a Sx^Soi by the first equation in (1) ; Sp'tx1*) - sVtx1*^) = 0 3x' 3X1-5 by equations 2,...., N in (1), [n ] negative semi-definite (and not 4. Assuming other global solutions in the neighborhood of x* rules out the possibility that [n .(x*)] is negative definite and does not alter state ments B and C. 'J negative definite); 231 so dx1* is undefined for a'cMx'^a) i 0. 8a 3x'3a Figured. A Discrete Analogue to [TT . j (x*) ] Only Negative Semi-definite 232 A. [TT.J(X*>] only negative semi-definite and x1* unique B. [n .(x*)] only negative semi-definite and xl* not unique {xl*B} C. A discrete analogue to [*jj(x*)] only negative semi-definite c^x1) 1 p^x1) XUA V»*B cjtx1) = p^x1) for all x1 X" the firm's marginal factor cost schedule for Input 1 the firm's derived demand schedule for input 1 the firm's solution set for Input 1 in case A the firm's solution set for input 1 in case B situation shown in ^-C. 233 In Figure "i-C, the solution set would be undefined after any downward shift in 5 the schedule.cl(x') for all x1 ; so dxx* is undefined when [n..(x*>] is only 1 3a ,J negative semi-definite and da defines a change in cj(xl) at x1*. ^ In sum, Preposition 1 implies that the set of comparative static effects {3x*} corresponds to the unique solution for (1) when the given [TT (x*)] is 3a ,J neaative definite, and that {3x*} Is empty when [n.,(x*)] is only negative 3a ,J semi-definite and c1 (xllf;a) + 0. Therefore, statement 1 plus the restrictions ia that [TT j j (x*) ] Is negative definite and symmetric correspond exactly to the 7 restrictions placed on {3x*} for problem P by the maximization hypothesis. 3a 2..3. . Restrictions corresponding to [n.^.] Negative Definite In specifying a system of equations that restricts fafj] to be negative definite, we utilize the following theorem: Theorem. A real symmetric matrix A is positive definite If and only if there exists a real lower triangular matrix H with positive diagonal 5 . If this relation In Figure 9-C extended throughout the negative orthant for x1, as Is In effect the case In local comparative statics, then the solution set {Ax1} also would be undefined for an upward shift in the schedule Aa cMx1) for al I x1. (o . This intuitive explanation of Proposition 1-C suggests that the un defined nature of {3x*} for [TT,.(X*)] only negative semi-definite Is fundamental 3a J to local comparative static methods rather than a peculiarity of primal methods. In other words, {3x*} (for problem P) Is undefined by any method whenever re-da strlctions employed In the method Imply that [n.,(x*)] Is only negative seml-deflnite. J 7 . For the general problem maximize TT(X;O) subject to G(x;a) = 0 , the exact implications of the maximization hypothesis for primal comparative statics are analogous to the above restrictions. See the discussion of "Integrability" in Appendix 4-. T * 234 elements such that A = HH . Since a negative definite matrix is simply the negative of a positive definite matrix, the following restrictions specify that the NXN real symmetric matrix [TTIJ} is negative definite: —TT , = h. , • n . ,+h5 7 • h . _+ +h. . • h. . all (i,j)' ij JJ ''2 J>2 ..J j,j sucn +na+ j < i \ (4-) hj j > 0 j=1,...,N where all hj j are also restricted to be real numbers. Restrictions (4-) com-prise N(N t 1) quadratic equalities and N bounds. 2 3 Restrictions implied by Additional Properties of [nfj(x*)] Given the maximization hypothesis, the comparative static effect 3x* for 3ct the firm's static problem N . maximize Tr(x;a) = R(x) - cl(xl;a) - £ c1 (x ) (P) i=2 Is defined by knowledge of the Hessian matrix [TT^(X*)] (negative definite and symmetric) and the exogenous shift c1 (x**;a) in the marginal factor cost IO schedule of Input 1. However, qualitative knowledge of the elements of [n..(x*)] and c1 (xl*;a) seldom determines 3x* qua IItatively, and direct quan-,J 1<x 3o" titative knowledge of the elements of [n j j(x*)] is in general very weak.1' %. This theorem can be inferred from Forsyth and MoIer (1967), pp. 27-29 and 114-115 plus Murdoch (1970), p. 232. 9. For a more general problem N j j maximize n(x;a) = R(x) - cl(xx;a) - I c (x ) 1=2 subject to g(x) = 0 , the implications of the maximization hypothesis are not as easily Incorporated into our quantitative comparative static methods. However, exclusion of such problems does not seem to limit our analysis significantly (see the discussion of constrained maximization in Appendix 4). 10, See Proposition 1 In the previous section. 11. See section 3.2.1. of Chapter 3. 235 Thus, even for the purpose of calculating qualitative restrictions on dx*, 3a there is need for a method of comparative statics that incorporates additional quantitative restrictions on [TC j (x*) ]. In this section, we shall show how [n.^.(x*)] is related to various poten tially observable and quantifiable properties p of the structure TT(X) of the firm's static maximization problem P. In contrast to the usual comparative sta tic approaches, which attempt to deduce knowledge of [n j j (x*) ]"*1 (and hence 3x*) from restrictions placed directly on [n..(x*)], we shall place restric-3a ,J tions directly on the inverse of matrices that are essentially submatrices of [n (x«)]. The vector of parameters p typically includes measures of the following types of properties of [tij (x*)]: (a) possibilities of factor substitution within any subset of inputs, (b) returns to an exogenous change in output when any subset of inputs is held constant and all other Inputs vary optimally In the static sense, and (c) changes in input levels corresponding to an exogenous change in output when any subset of inputs is held constant and all other inputs vary optimally in the static sense. A priori knowledge of a range of "reasonable" values for some of these paramet ers presumably is available in most cases. This knowledge would be derived from observation of physical processes, observation of firm behavior that ap proximates various short run comparative static effects, and from econometric estimation of physical processes and short run comparative static effects. By formulating these restrictions as cPnfidence intervals or as Bayes intervals, the corresponding feasible set for 3x* can also be Interpreted as a confidence-da 13 Bayes interval. 12. See section 3.3.2. Chapter 3. 13, See section 5. 236 However, the following elementary point should be emphasized: although we can easily formulate conditions that exhaust the comparative static implications 14 of the maximization hypothesis, we cannot formulate conditions that exhaust the relations between [n jj <x*)] and potentially observable data about the struc ture of the firm's problem P. Thus the relations between [TIJJ(X*)] and data that are presented here should be viewed only as a subset of all useful rela tions between comparative static effects and observable structure of the firm's maximization problem. 3.1 Major Restrictions The restrictions on [njj(x*)] that are most Important in our method of quantitative comparative statics for a shift in a firm's factor supply schedule concern (a) possibilities of factor substitution within a particular subset of inputs, (b) returns to an exogenous change in output when a particular subset of Inputs is held constant and all other Inputs vary optimally in the static sense, and (c) changes In Input levels corresponding to an exogenous change in out put when a particular subset of inputs is held constant and all other inputs vary optimally In the static sense. The relations between [n^^(x*>] and these potentially observable properties of the firm's static maximization problem are detailed In Theorem 3 and Corollary 15, Ifc 5. Here we shall explain and elaborate upon these relations between [n (x*)] 14. See Proposition 1. 15. Theorem 3 and Corollary 5 owe much to Mundlak (1966, 1968), and in turn to Mosak (1938). ifc. Our quantitative comparative statics analysis could be extended easily to the case of a shift tn the firm's product demand schedule (see the re lated section of Appendix 40. 237 and properties (a)-(c) of the firm's static maximization problem. In contrast to the usual comparative static methods which place restrictions directly on the elements of [TT J J (X* ) ], these relations shall place restrictions on the in verse of matrices that are essentially submatrices of [TIJ.(X*)]. 3.1-1 Model with Output Exogenous Given the firm's static maximization problem N , maximize Tr(x;a) = R(x) - I c (x1;a') (P) i = 1 with solution x*, define the related problem where output is treated as exogen ous to the fi rm N maximize ir(x;a) = R(x) - E c'(x';a') 1 = 1 . <5> subject to R(x) = R(x*) Problem (5) can be expressed in Lagrange form as maximize Tf(x;a) - X(R(x) - R(x*)) (&) where the endogenous variables are (x,X) and the exogenous variables are (a, R). Suppose that the differentials of the interior first order conditions for (&) with respect to each of (a, R) yield a unique solution for all comparative static effects Ox**, 3X, 3x**, j)X).17 This assumption is equivalent to the 3a 3a 3R 3R" restriction that this system of differentials can be expressed in the form [A] [K] = I (7) 17. Since Proposition 1 can be generalized to the problem maximize (x;a) subject to G(x;a) = 0 (see the discussion of IntegrablIity in Appendix 4-), there is no loss In generality in assuming that 3x**, 3X, 3x**, 3X) Is uniquely defined for a given 3a 3a 3TT 3P! problem (fc). In other words, Ox**, 3x**) is uniquely defined if [TTj j(x*)] 3a ~W J is negative definite subject to constraint and Is undefined if fnj t(x*)] Is only negative semi-definite subject to constraint, and OX^ 3X) is also uniquely de-da 3TT fined or undefined (since a maximum or supremum is either uniquely defined or undefined for a given problem). 238 where the matrices [A], [K] and I are as defined in Theorem 3. [A] is the Hessian matrix [n..(x*)] bordered by marginal factor costs cj = (c'fx1*^1), cN<xN*;aN)>T!'J N [A] •~r— -C!T ! °. («.) [K] is a matrix of all the comparative static effects Ox**, 3A, dx**, 3X) 3a 9a 3R" 3R" for problem ((f), and I is an identity matrix. •I*? \%. The "revenue effect" 3x* ** is related to the corresponding output ~w~ effect 3x ** simp Iy as follows: 3x'** = 3x*** • 3R(y*) (by the chain rule) 9F dT 3R" 9y where y = F(x) and R(y) = R(F(x)). .Likewise, 3TT(x*) = 3TT(X*) • 3R(y*) and so "9? 3TT "~Ty 32TT(X*) = 3(3TT/3F) 3R(y*), which yields 3A = 3X • (3R(y*))2. §T 3R~ 3y W 9W 3y 19. Knowledge of comparative static effects in the presence of a con straint on expenditure for a particular subset of inputs could be easily incor porated into this approach. For example, consider the problem N maximize ir(x;a) = R(x) - I c (x ;a') S I i I subject to I c'(x1;a') = C i = 1 or equivalently S i i I -maximize TT(x;a) - X( E c (x ;a ) - C). 1=1 EEE Then it can be easily shown that the comparative static effects (3x , 3X , 3x_, 3a 3a 3TJ 3X ) for this problem are related to [K] as follows: 3U 1x11= cJ Kj j - cJ • K . i = 1,..,N j=1,..,S 3xiE = cJ" • K. = 3x'»* 1=1,..,N j=S + 1,..,N 3ajT jaj 'J "IST" |£ " -^aj" ^ • I.J * <J * 'Si • I.N * 1 J=,"-S •-^•KN*M =%T J-S+1....N 239 Theorem 3. Suppose that conditions 1-2 are satisfied for a problem P N maximize Tr(x;a) = R(x) - I c5 (x';a') i = 1 (1) 10 and assume that this problem has a unique global solution x* where the Hessian matrix for TT(X) is negative definite. Construct the related problem maximize TT(X;CX) subject to R(x) = R(x*) which can be expressed in Lagrange form as maximize Tf(x;a) - X(R(x) - R(x*)). Construct the symmetric matrix [A] (N + 1) x (N + 1) (2) ct i (N x N) (N X 1) c' i o (1 x N) X 1) where TT. . denotes the Hessian matrix for Tr(x;a) at x*, and (N xJN) c! = QcMx1*^1), , Zc™Un*;aN)). [A] necessarily has full rank, dx1 3x and denote its inverse as [Kj: [A]"1 = [K] always exists. (N + 1) x (N + 1) Then, (A) the comparative static effects for problem 2 are uniquely de fined as follows: a2cJ"(xJ*;aj) • K i,j = 1, , N 3x'** 3aJ 3xJ3aJ i = i,, , N 20, This theorem is easily generalized to the case c = c (x;a ) (1=1, ...,N), but the/fequations in the generalized theorem are somewhat more de tailed than here, and the generalized theorem will not be employed here. 3X =_92cJ(xJ«y) • K j = 1 N 3oJ 3xl3oP N + ''J 240 |g = ~KN + 1,N + 1 where K. . = element (I.j) of matrix [K], and K. . = K. . (i,j = » FJ » »J J F• 1,. ., N + 1);^ ' and (B) (a) The comparative static effects Ox*) for problem 1 are 3a unique, and (b) given that j £ K. , • 3cJ(xJ*;gJ) / -I,32 1=1 j-1 ' 3xJ 3x* for problem 1 is uniquely defined in terms of 3a 32c^ (x-|*;aJ ) and the elements of [K] corresponding to 3xJ 3aJ 3x** and 3x** for problem 2, as follows: 3aJ 3R 3x'* = 32cJ'(xJ*;qJ) • K. . + K. • 3R(x*) l,j = 1, , la? 3xT§c? 1 'J ' »N + 1 3aj 3R(x*) = jj 3c1 (xj*;^) • 3x'* j = 1, , Ha? 1 = 1 Sa^" 21. Thus 3x'»* =/ 32cJ(x'*;aJ) / 32c'(xj*;al) \3xJ** and 3aJ V 3xJ3aJ / 3x'3a' / 3a' 3X = 32cJ(xJ*;gJ) - 3xJ** d,j = 1, , N). 3aJ " 3xJ3aJ 3R N N 21. A sufficient condition for I I K. N . • 3cJ(xJ*;gJ) ¥ -1 (a) i=1 j=1 ' 3xT is that K, N + . ^ 0 (1 = 1, , N), which is equivalent to ruling out the possibility of inferior inputs Ox1** > 0 <=> K. _> 0). Condition (a) 3R ' would be violated only for a relatively few "appropriate" degrees of inferior ity; so condition (a) is not a serious restriction. 241 Thus equations (7) plus restrictions on the comparative static effects for problem (to) and on equilibrium marginal factor costs imply restrictions on the elements of the matrix [TT - j (x*) ] for problem P. These relations that are ex-pressed in equations (9) can be summarized as follows: (a) Z TT,,, ax1** - ck 3X, = ck , j=1 ,kT^~ ^ kotk N Z Tt.. 3x'** - cJ = 0 all j ^ k k=1, ,N 1 = 1 "3o^" N (b) Z c 3xu* = 0 1 = 1 1 ""aa*" N (c) Z TT. . 9xJ** - c4 3X = 0 a I I j . 1-1 >J~W J W N (d) Z cj 3x'™ = 1 where all partial derivatives are evaluated at (x*,a). Given knowledge of equilibrium marginal factor costs and of N-1 elements of 3x** and N-1 elements of 3x**, alI elements of 3x** and 3x** are known (see "JoT 3R 3ak 3R (b) and (d)). In this case, the comparative static effect 3x* for problem P 3o* could be calculated directly from the relations 3x'* = 3x!** t 3xu* • 3R(x*) i = 1,...., N*'3 ( «J.) 3a 3ak 3R 3ak 3R(x*) = j c! • 3x'* (10i 3a* 1 = 1 3a* except under unusual circumstances. 33. Statement (<?.) Is essentially Theorem 7-1 of Sakai (1973) (Theorem 7-1 has an obvious typing error). 2f. As can be seen from <<?!)-( 16), 3x5* is not a simple weighted sum of the pure substitution effect 3x'** and scale effect 3x^ ** in contrast to the 3ak 3R Slutsky equation in consumer theory. For the unusual circumstances under which 3x* cannot be calculated from ( 3)-(16), see Theorem 3-B. 242 In addition, when knowledge of (c'.c^,, 3x**, 3x**) is not exact, the re-3o*" "SR-strictions on 3x* are presumably increased by Incorporating restrictions on 3a 3x** as well as on 3x** into the quantitative comparative statics model. In 30?" 3o*" this more general case, the quantitative comparative statics model includes the 35 conditions implied by the maximization hypothesis, the equations [A] [K] = I and restrictions on elements of [K]. These restrictions correspond to the i N "reasonable" ranqe of values for Ox**, 3x**, c1 i,.., ckl and also for 3X = 32TT(X*) < 0.2fc,a? 3TT W Nevertheless, in many situations knowledge of the comparative static sub stitution and scale effects when all- inputs are variable Ox**, 3x**) may be 3a SR almost as scarce as knowledge about the comparative static total effect 3x* 3a1 itself. Considerably more knowledge about substitution and scale effects may be available for cases where subsets of inputs are fixed for the firm. 3.1.2 Model with Output and a Subset of Inputs Exogenous For many situations where knowledge about Ox**, 3x**) is quite weak, a 3a ~W narrower range of "reasonable" values for substitution and scale effects when some Inputs are fixed may be readily available. Moreover, this knowledge of 25". See Proposition 1. 2t>. See Theorem 3-A. Since [K] is symmetric and knowledge of Ox**, N cj i,.., cNaN* *s presumably greater than knowledge of 3X per se, restrictions ^® ^ > 3a on 3X_ seldom would be specified. 3a T7. 3X = 32TT(X*) < 0 for [n,.(x*)] negative definite (since 3TT(X*) = 0 W J "ITT" by envelope theorem, and An(x*) < 0 for a finite AR by [TIJJ(X*)] negative AR definite). 243 substitution and scale effects when various subsets of inputs are fixed may imply strong restrictions on the comparative static effect dx* for problem P. 8a This statement can be elaborated upon as follows. Given the firm's problem P, define the related "short run" static maximiz ation problem N maximize Tr(x;a) = R(x) - E c (x';a') i = 1 subject to R(x) = R(x») xJ = xT* j=S + 1, , N where output and an arbitrary subset of Inputs are exogenous to the firm at the equilibrium levels for P. This problem can be expressed In Lagrange form as N maximize Tr(x;a) - X(R(x) - R(x*)) - E yJ(xJ - *5*~) (U) J-S + 1 where the endogenous variables are (x1,.., xs, A, y^ + \>*, y^) and the exogen ous variables are (a, R, x^ + 1,.., x^). Suppose that the differentials of the interior first order conditions for (12) with respect to each of (a, PJ yield a unique solution for the comparative static effects Ox**S, 3XS, 3x»*S, dXs).2% This assumption is equivalent to the 3a 3a 3T* 3R restriction that this system of differentials can be expressed in the form [* ] [L] = I (13) where the matrices [7A ],[L] and I are as defined In Corollary 5. [A,,] consists of (a) the principal submatrix tn|jA] of [njj(x*)] that is formed by deleting rows and columns (S + 1,.., N) from [n^tx*)] and (b) the subvector c!A =. (cl(X'**;ol),..„ c!<xs«;c£>)T on the borders of [n,,A], i.e. i l S U 2%- There is no loss In generality In assuming that Ox**S, 3y^> 3a 3a W 3X ) Is uniquely defined for a given problem (12) (see first footnote in sec-3T* tton 3.f-0. 214 Corollary 5. Construct the problems 1 and 2, and the (N + 1) x (N + 1) ma trices [A] and [K], as in Theorem 3. Partition the Hessian matrix Tt-. and marginal factor cost vector cj of [A] as follows: (N xJN) [TTjj] = (N xJN) A "ij (S x S) n..C _(T x S) where S + T » N. "TT A ! "ij | ciA-| (S x S) | (S x 1) ~7»~T i 1 o (1 x S) j (1 X 1) I B I TT . . c! = [ciA Ic'B] ' i ' i (IxN) (IxS) (IxT) (S + 1) x (S + 1) [7AJJ] necessarily has full rank, and denote its inverse as [L]: [AJJ]"*1 = [L] always exists. Construct the problem maximize ir(x;a) = R(x) - I c (x1 ;a ) i = 1 subject to xJ = xJ* j=S + 1,...., N where x* is the unique global solution to problem 1. Construct the related problem maximize Tr(x;a) (3) subject to R(x) = R(x*) xJ = xJ* j=S + 1,... which can be expressed in Lagrange form as N , . maximize Tr(x;a) - X(R(x) - R(x*)) - T, r (xJ - xJ*). (4) j=S+1 245 Then (A) the comparative static effects for problem 4 are uniquely defined as foI Iows: 3x'»»S = 32cJ(xJ*:gJ) - L . i,j=1, , S "loT 3xJ3oJ ''J 3x'«*S = L. - . 1 = 1, , S 3R 3XS « _32cJ(xJ*;gJ) • L- . . , j= 1, , S 3oT 3xJ3aJ b ''J # =-Ls+i,s+, where L . = element (i,j) of [L], and Li j = L. ,(I,j = 1, , S + 1); i ,J ' »J J»1 and c (B) (a) the comparative static effects 3x* for problem 3 are unique, and 3a S S (b) given that I I L. - , . • 3 c* (xJ* ;aj) 4 -1, Ss i=1 j=1 ''b 3xJ 3x*^> for problem 3 is uniquely defined in terms of 32c^(xJ*;gJ) and the elements of [L] corresponding to 3xJ 3oJ 3x**^ and 3x**^ for problem 4, as follows: ~3aT~ 3R 3x'*S = 32cJ(xJ»;aJ) • L. . + Lt c + , • 3R(x*)S i ,j=1, , S 3R(x»)S = I ac'cx'^a1) • 3x'»s j=1, , S. do? 1 = 1 Sx^ 3gJ S S I i • 3*?. Assuming that I I L , • 3cJ(xJ»;gJ) ^ -1 has Implications 1 = 1 j = 1 ''b 1 3xJ N N j , , analogous to those of assuming that I I K. M . . • 3cJ(xJ*;gJ) ¥ -1 (see i=1 j=1 d>T footnote to Theorem 3). 246 c|AT ] 0 [L] is a matrix of the comparative static effects (3x**S, 3XS, 3x*»S, 3Xs) and 3a 3a CTFT W I is an identity matrix. By (13)-(1f), knowledge of the elements of [L] and c|A places restrictions of [Tt,,(x*)J. Thus knowledqe of the comparative static effects ( 'J "To" W c 3X ) for problem (13) and of equilibrium marginal factor costs places restr1c-3R tions on the "long run" comparative static effect 3x* for problem P. 3a1" Moreover, the comparative static effect 3x* for problem P can almost al-ways be defined precisely in terms of a set of comparative static effects {(3x**^, 3x**s)} for an appropriate set of problems (14), and these relations 3a1 3TT are implicit in our standard quantitative comparative statics model. The com parative static effects Included in this set will differ In terms of the parti tion Into fixed and variable inputs and the choice of shift parameter a'. This important relation between 3x* and various sets {Qx»»s, 3x»»S)> can be demon-3a' 3a W strated as follows. Consider a comparative static effect 3x*^ for the problem So*"" N I I I maximize ir(x;a) = R(x) - E c (x ;a ) 1=1 ^ (15) subject to xJ = ~>J* J=S + 1,.., N . 3x*S is almost always uniquely defined by the relations analogous to (1S)-(W) So*" 30* Since [L] Is symmetric and knowledge of (3x**s, c1^,..,- c| s> presum-s ~3PT" la Sa ably- is greater than knowledge of 3X per se, restrictions on 3Ab seldom would 3a 3a be specified (see Corollary 5-A). 247 3x'»s = Zx]**s + 3x}»* • 3R(x«)S 1=1,.., S (It) 3aK 3ak 3R 3ak 3R(x»)S = Z c|3x'*S (H3ak 1 = 1 l1ST plus a qiven comparative static effect (3x**^, 3x**^) for the corresDondinq -3^ sir 3 I problem (14). In addition, the following relations are satisfied for problem (15): [n A] [P] = I (It) i j where [n|jA3 Is the principal submatrix of [nj.(x*>] obtained by deleting the rows and columns (S + 1,.., N) from [TIJJ(X*)], [P] is symmetric and P, . = 3x * / o| j. Thus any principal submatrix of (n,.(x*)] can be uniquely defined by (I?) and a suitable partitioning of x Into variable and fixed inputs, a suitable choice of shift parameters {a^}, and the corresponding given {3x*^}By repeating this procedure, [n..(x*)] and 3x* can be deter-mined. In sum, 3x* can almost always be defined precisely in terms of (3x»»S, 3x»*S ) for an appropriate set of problems (14). Moreover, the 3aJ 3R restri cti ons [n. Ax.*)] 3x = J 3a1" [r\ ] [L] = I 1 1 ola , [n..(x*>] negative definite 'J V U<i) (a) uniquely define the comparative static effects for problem P and a series of problems (14) for a given [n^ .(x*)] and a given series of partltionings into variable and fixed inputs, and (b) define 3x* in terms of [n (x*)] and a 3o7 !J 3l„ See Corollary 5-B. 32. See Corollary 6. 33. The symmetry of [P], i.e. 3x'»S / c! ; = Zx^*S / c' , for all I, j= TSX~ / Ja ~3a?~ / . ,a 1,.. S, implies several degrees of freedom in selecting a {or},{(3x** , 3x** )} 3aJ 3"R in order to define a given [n j jA]. 248 (3x**S, 3x**s) in terms of a subset of elements of I",.(x*)] and (c1,.., cN). -IT ~W ij 1 N Thus the relation between 3x* and a (Ox**S. 9x*^S)) is implicit in (19). laT da dR In addition, 3x* for problem P apparently can be uniquely defined in 9ctr terms of 3xf_ = 32TT(X*)S for some problems (U) plus a subset of {(3x**S. 3R W 3aJ 3x«*S)} specified above. In other words, knowledge of a 3X for a problem (12.) 3R~ IW can "substitute" for some knowledge of {(3x**^, 3x**^)} in the determination 3aJ ~dW~ 34-of 3x* for a problem P. dar In sum, restrictions on {(3x**S, 3x**S, 3XS)1 for various problems (1a) 3aJ 3^~ 3aJ and {aJ} plus the relations (I*?) may imply considerable restrictions on 3x* dar for problem P. Since knowledge of substitution and scale effects will be de fined primarily in terms of problems (12) with various subsets of fixed inputs, these restrictions derived from a model with output and a subset of inputs exo genous are a very important aspect of our quantitative comparative statics mode I. 3,1.3 Model with a Subset of Inputs Exogenous In addition, direct knowledge about the total effects of da1 when certain Knowledge of a 3X can "substitute" for some knowledge of {(3x**S, ~3£ 3aJ 3X«*J )} jn -f-hg determination of 3x* if pj(x**) is a function of (among other 3R - c <5a~r n things 3Xb. Since 3Xb = 3Tr(x»r for a problem (12) and pICx1*) = 32TT(X*)v for a W IE dW 3x' 1 problem Q corresponding to problem P, this relation between p}(x'*) and 3X w seems somewhat reasonable. Differentiating the first order condition p'-c} =0 with respect to a1 yields pftx1*) 3x2«- c?,3x1« - cl ,= 0; so 3XS influences 3x'* and hence must "substitute" for some knowledge of 3R <• c 3a1 {(3x»* , 3x*»b)> If 3Xb influences pUx1*). ~3V~ W IPT 1 249 subsets of inputs are fixed may be available. Such knowledge can be specified as restrictions on comparative static effects {3x*^} for various "short run" static problems of the form (15; N maximize Tr(x;a) = R(x) - I c'(x1 ;a') i = 1 subject to xJ = xJ* J=S + 1,.., N. These restrictions plus the following relations US) can be incorporated into our quantitative comparative statics model: [ir,jA] [P] = I where fn|j^3 Is "^he principal submatrix of [TIJJ(X*)] obtained by deleting the rows and columns (S + 1,.., N) from [rt..(x*)], [P] is symmetric and P.. = J ' J 3x * / c*. ;. In this manner, knowledge about a "reasonable" range of values 1ST I JaJ for 3x*^ corresponding to any problem (15) places restrictions on 3x* for prob-dbT~ 3a7 lem P. 3.3. Minor Restrictions Other forms of knowledge about the structure of the firm's static maximi zation problem P may be available and useful In defining "reasonable" limits on the comparative static effect 3x* for problem P. These additional forms of W knowledge are of at least two types. First, there may be knowledge of the com parative static effect of a change in the demand schedule for the firm's output or in the firm's production function. Including the corresponding restrictions in our standard quantitative comparative statics model seems likely to lead to a small reduction in the range of feasible values for 3x*. If such compara-3a1" tive static effects and its "short run" variations with fixed inputs are in cluded In our model, then our model incorporates knowledge of all types of 35. See Corollary 6. 3<D . On the standard model, see Proposition 1-A, Theorem 3 and Corollaries 5-6. 250 Corollary 6. Construct problems 1 and 3 as above, and partition the (negative definite) Hessian matrix TT. . as above. Then the comparative (N xJN) static effects for problem 3 are uniquely defined as follows: 3x'*S * 92cJ(xJ*;g-i) « P i,j=1, ,S dor1 3xJ 9aJ 1 'J where P. . = element (i,j) of [n..^]"1 (which always exists), and 1 'J (S'x S) P, ; = P, : <» J - 1» S). 1 »J J »1 251 comparative static effects that can occur realistically at the level of the 37 single fi rm. Second, there may be specific knowledge about the functional form of the firm's static maximization problem P. The following examples are considered here: separability of ir(x;a) in x, linear homogeneity of TT(x;a) in a, fixed factor proportions for R(x), and homotheticity of TT(x;ot) in x. The first two properties, and presumably many other special properties of v(x;a), are easily incorporated into our quantitative comparative statics model. Such restric tions will be useful when (a) observation and/or theory suggests that such a property is closely approximated, or (b) sensitivity of comparative static results to such properties is an important Issue. In these circumstances, the Imposition of such properties or of limits on the "degree of deviation" from such properties can be useful in our quantitative comparative statics models. 3.2..1 Knowledge of the Comparative Static Effects of a Change in R(x) Knowledge of the comparative static effects of a shift In the firm's reven ue or benefits function also places restrictions on [njjJ- Define the firm's static maximization problem P as N maximize TT(x;ot) = R(x;a°) - I c (x ;a ) (P) i = 1 i.e. allow for the possibility of shift parameters In the firm's revenue or 39 benefits function as well as in the firm's factor cost schedules. Total 37. See section k. 3S For example, calculating the sensitivity of comparative static results to the property of separability may provide a rough estimate of errors due to Inappropriate aggregation of inputs in a quantitative comparative statics model (see section 3.2-3. of tV\\s Appendr* a-net section 3-i o-f A.ppen4 i x. 6"). 39. Here we are only interested in knowledge of 3x* as a means of obtain-9a0* ing knowledge about 3x*. For several brief remarks on the quantitative compar-3a1" at!ve statics of a shift in a firm's revenue or benefits schedule, I.e. on the case where our ultimate interest Is knowledge of 3x* rather than 3x* (1^0), see 3au 3a1 Appendix 4-. 252 differentiating the interior first order conditions for problem P with respect to a0 yields [71,.] 3x* = J 3a°" R. o ia where R. 0 = 32R(x* ;a°). Thus restrictions on 3x* and R. o (i=1,.., N) ,a 3x' 3au 3^ ,a Imply restrictions on [nj.(x*)]. However, restrictions on 3x* and its decompositions seem considerably less 3o7 important for our purposes than are the restrictions specified by Proposition 1-A, Theorem 3 and Corollaries 5-6. This statement can be elaborated upon as follows. Prior knowledge of 3x* per se generally appears to be quite weak. 3o7" 3ctu 3F 3a""" Moreover, the decomposition relating 3x* for problem P to (3x**, 3x**) for the prob I em N maximize Tf(x;a) = R(x;a°) - £ c (x1 ;ct') i = 1 V (20) subject to F(x) = F(x*) only leads to some of the restrictions on [TI.J(X*)] already specified by the relations presented in Proposition 1-A and Theorem 3. 3x** is simply the scale ~W effect already defined in Theorem 3, and 3x** = 0 is already implied by the 3a7~ restrictions [A][K] = I specified in Theorem 3. On the other hand. 40. Given that the firm produces a single output y according to the pro-duction function y = F(x) and that a0 does not enter F(x), R. o = (R|a°/R )c! where R(x;a°) = R(F(x);a°). If we further assume that R(y;a*T = P(y;a°)y; 1 then Ry = P(y*;a°) + Py(y*;a°)y* and Rycto = Pao(y*;ct0) + PyQo (y*;a° )y*. 4.1. In addition, this Implies that 3x* is uniquely defined in terms of W ([TI..], RlCto,.., o) given that [rt j , <x*) ] is negative definite. Of course, [n..t!x*)] Is not uniquely defined in Terms of Ox*, R „,.., RM^O)* J So0" 10t f3. Note also that exact knowledge of Ox** 3x**) plus the restriction [TI * * (x*) 3 negative definite can only define 3x* up to a positive scalar multiple J 3oT (see Theorem 4-C and the related discussion In Appendix 4). 253 c knowledge of the comparative static effect 3x* where various inputs are fixed 3o7~ may not be as weak as knowledge of 3x*, and may place additional restrictions doF on [TTJ J (x*)] Therefore, knowledge of the comparative static effect of a shift in the firm's revenue or benefits function and of such "short run" decompositions may help somewhat in determining "reasonable" upper and lower bounds on 3x*. 3a"1" 3.2.2. Several Special Properties of Tr(x;a) The following properties of the functional form of the firm's static maxi mization problem P are considered here: separability of TT(X;OI) in x, linear homogeneity of TT(X;OO in a, fixed factor proportions for R(x>, and homotheticity of TT(X;CX) in x. The first two properties are easily incorporated into our quantitative comparative statics model. In addition, limits on the "degree of deviation" from these properties are easily included in our model. However, it appears that the last two properties (especially homotheticity) cannot be in corporated into our model. A twice differentiable function R(x) (with non-zero first derivatives everywhere) Is defined as "weakly separable" with respect to the subset {1,.., m} of the firm's. N inputs when R(x) = f(y,xm + xN) for some (scalar-valued) functions f and y(x1,.., xm). This is equivalent to the "Leontief conditions" 3 (R./R.) = 0 for all i,j e {1,.., m} and k e {m + I,.., N} over all x.m>*S (21) 43. See Corollary 7-A in Appendix 4. 44. See Leontief (1947). In addition, R(x) = f(y(x1,.., x"1), z(xm+1> .., x*1)) is equivalent to (21) plus 3 (R./Rt) = 0 for all I, j e {m + 1,.., N} 3x*" J and k e (1,.., m) over all x. (2T) 45". For discussions of separability and aggregation, see Blackorby et al (1978) and Diewert (1977). 254 Thus (21) plus the assumption c1 = c'(x';o) (i = 1,.., N) are equivalent to the condition N Tr(x;a) = f(y(x\.., xm), x™ + xN) - CU1,.., xm;a) - I c'(x';a) i=m+1 m i • C(x\.., xm;a) = I c'(x' ;a) i = l which implies that inputs (1,.., m) can be correctly aggregated in specifying the firm's objective function TT(X;O.) as well as its revenue function R(x). moreover, given c' = c'(x1 ;ot) (I = 1,.., N), inputs (1,.., m) can be correctly aggregated In specifying TT(x;a) only if inputs (1,.., m) can be correctly aggre gated In specifying R(x). Therefore, at an interior solution to the problem N i i maximize TT(X;O) = R(x) - Z c (x ,a), i = 1 inputs (1,.., m) and associated comparative static effects can be treated cor rectly as an aggregate if and only if the following restrictions are satisfied TT j kcj - TTj kc j = 0 for all i ,j e {1,.., m} and k e {m + 1,.., N}. (22) Moreover, limits on the "degree of deviation" from the possibility of correct aggregation can be specified roughly in our model by restrictions of the form Pl ' <cj>2 1 *,kcj-*Jkc! < P2 • <cj>« 7°" < p1 < p137 V (23) QL2 < pz < puz In addition, on occasion we can reasonably assume that (in the neighbor hood of equilibrium) H-d>. Under reasonable conditions, approximation to Leontief conditions is equivalent to approximately correct aggregation (Fisher, 1969). This also seems to imply that R(x) approximates Leontief conditions when It approximates fixed factor proportions and is twice differentiable. Y1. If Leontief conditions are to be Incorporated directly Into the com parative static model (as opposed to being used simply as a "justification" for the construction of a model that includes aggregate inputs), then approxima tions of the form (23) rather than (22) apparently should be used. This state ment can be explained as follows: exact separability for a matrix [Ttj,(x*)] Implies that [n|*(x*)) Is negative seml-defInite only (see Proposition 2 In Appendix 5) and Focal comparative statics Is undefined In this case (by Propos ition 1). N . 255 TT(x;a) = ct°R(x) - E a'c'tx') (24) i = 1 I.e. R(x;a°) is linear homogeneous in a0 and c'(x';a* ) Is linear homogeneous i 4-"2 in a (i = 1,.., N). Then N t E 3x'» • aJ = 0 1=1,.., N {25) j=0 ~Ta? where a0 = R,<x*;a°) / R, 0<x*;a°) Ja > (2W a1 = c|(xU;a') / cjal (x'*;^ ) 1 = 1,.., N 49 Limits on the "degree of deviation" from linear homogeneity In a (25) can be incorporated Into our model In a manner similar to (23). However, the special property of fixed factor proportions for R(x) is not cjet Incorporated satisfactori ly into our comparative static model for the un constrained problem "maximize Tr(x;a)." As can be seen from (9 ), fixed propor tions between all factors is equivalent to the following: 3x'** = 0 and 9x'»* / 9xJ»» = x'» i,j=1,..,N. (?7) ""laJ 3PT / ~~~W xT*" The second statement in (37) Is equivalent to the condition that the "iso-profit lines" of ir(x) for different levels of R have identical shapes in a neighborhood of x*. This condition can be designated as "homotheticity of Tr(x)" at x*. However, imposing such homotheticity on our model may . re quire either exact knowledge of x* or exact knowledge of the third order partial derivatives of TT(X) at x*. Nevertheless, we can at least specify the following consequence of homotheticity of TT(X): 4?. Note that the effect of the communtty pasture programs on the firm's pasture supply schedule can be described more accurately as a parallel shift in the schedule rather than as an equI proportional change In the marginal factor cost of pasture at various activity levels (see section 3.5 Chapter '&'). T>ws the local effect of the community pasture programs cannot be described accurately In terms of (24). 49. The proof of this statement Is quite simple. Condition (24) Implies that factor demands are homogeneous of degree zero In a, i.e. x*(a) = x*(Xa) for all scalar X > 0. Then (25) follows directly from Euler's theorem. Equa tions (2fe) follow directly from the restrictions R(x;a°) = o°R(x) and c!(xl;a!) = a!cl(x') (1=1,.., N). 256 N t. N i, 50 Z TT (x«)3xk»» = I TT (x*)9xk*» i ,.i = 1N. (3?) k=i lk W k=i Jk air Thus, the following restrictions in our model are implied (but not equivalent to) the special property of fixed factor proportions of TT(X) at x*: N.N 3x'*» = 0 and I Tr (x*)3x*** = £ TT (x*)3xk*» C W> 3aJ k=1 ,k W k=1 Jk W i ,j=1, , N. 4- A Minor Difficulty in Translating between Local and Observed Comparative Static Effects Here we note that there can be difficulties in incorporating all knowledge of the form presented In section. 3 o-V ih"«s Appe^'ix into a local comparative statics model, but these difficuI ties'wiI I seldom pose serious problems in the use of our quantitative comparative statics model. The ambiguity relates to the problem of translating between local and global comparative static properties of TT(X;O). The comparative static effects included in the model presented here are formally defined in terms of local properties of TT(X;CI) and shift parameters (c1 ., etc.); whereas the counterparts lot of these effects that are "observed in reality" depend on more global proper ties of Tr(x;a). Moreover, local comparative static effects are linear homogen eous in the shift parameters. For example, 5^. Homotheticity of TT(x) IS equivalent to the statement that TT. (XX) Y JTT.(XX) for all x and all scalar X > 0, and a scalar yJ for each (l,j); :) = i »n MA*/ IUI aii A anu aii sv.aiai A * u, ano a scalar y J TOT eacn SO homotheticity and TT.(X*) = TT,(X*) = 0 imply that 3TT;(X*) s 3TT;(X*), which is J 31T J3tf statement (23). Since (2S) can be satisfied by a 3x** that does not preserve 3R initial factor proportions, (28) does not imply homotheticity of TT(X) at x*. 51. The assumption of fixed factor proportions "contradicts" the assump tion that TT(X) is continuous at x*. However, this "contradiction" is trivial: the statement 3x'** = e, where e is arbitrarily small, does not contradict the 3cJ assumption that TT(X) is twice differentiable. i.e. 8x* is linear homogeneous in c1 , for a given [n..(x*)]. Thus difficult-ies arise in translating between local and observed comparative static effects if and only if observed comparative static effects are not linear homogeneous in the (parallel) shift of factor supply schedules, etc. Since linear homo geneity will be a special case for observed comparative static effects, there can in principle be difficulties in incorporating all knowledge relevant to estimating comparative static effects into our quantitative comparative statics model. However, in practice this local comparative statics model should be able to assimilate most of our empirical knowledge concerning comparative static effects. In other words, one can seldom make sharp distinctions between (e.g.) Ax1** AwJ for different ("reasonable") levels of shift in an exogenous wage wJ. ' 5 Restrictions as Confidence Intervals or Bayes Intervals Here we note that the set of constraints PLi < p' < pU' i = 1,.., M (30) on the potentially observable parameters p in our model can be Interpreted either as confidence intervals or as Bayes intervals. Thus the corresponding feasible set for the matrix [fl|j(x*)] and the vector of comparative static 52.. The linear homogeneity property of local comparative statics will generalize to more global comparative statics if the local structure of Tr(x;ot) is invariant in a suitable subset of x (the equilibrium path). Thus the asser tion that one cannot significantly improve upon linear homogeneity in practice is essentially equivalent to the following: there is considerably more know ledge about the "average" t"jj] within this subset than of the differences between ["jjl over this subset. 53.. Note that linear homogeneity of comparative static effects does rule out a constant elasticity of comparative static effects. However, the assump tion of constant elasticity Is commonly employed in order to obtain unit-free measures and for other conveniences, and this does not seem to reflect a belief that variations between neighboring [nj .] can In effect be measured more accur ately than their average. 258 effects 3x* can be interpreted in a simitar manner, and the addition of dar constraints to (30) leads to a reduction in the size of the confidence-Bayes interval for dx*. do? Knowledge of the parameters p typically should be expressed in terms of frequency distributions rather than as point estimates. Such distributions can arise in at least four ways. First, the vector p may vary significantly across a group of firms, and we may wish to estimate the range in individual response dx* across these firms. Second, p may vary significantly across time for an 3a1" individual firm, and we may wish to estimate the range of comparative static effects 3x* that could be associated with such a range in p. Third, p may be W observed with error, and this error will generally be stochastic. Fourth, * p may not be directly observed (with or without error); but there will be a prior distribution summarizing our subjective degree of belief about the un known values of the elements of p. Knowledge of distributions of the first three types implies particular con fidence Intervals for p. In other words, from a particular set of observations {p'} and from assumptions about the distribution of p' and of errors in observa tion, we could construct on X% confidence interval eLI <p'<eul. If the assumptions about the distributions of pl and of errors in observation are correct, then there is an X% probability that a random observation of the population of (true) p' will be contained in this interval. Likewise, if p' is not observed, an X% Bayes interval 54-. There will be errors in observing p in a truly static situation and errors in inferring values for p from observations of dynamic situations. Pre sumably the latter type of error would be more common and more serious and would be systematic (hence not normally distributed with a mean of zero). 259 j erg can be constructed from our prior distribution for p . Assuming that are independently distributed, X% confidence-Bayes intervals for these individual parameters together form an X% joint confidence-Bayes interval for p (30). 5y The feasible set {(p,[n.j])} for our model is defined by (30) plus the maximization hypothesis ([n j j ] negative definite and symmetric) and the equiva lence relations between p and [n,.] ij [A] [K] = I, [A ] [L] = I ,-. (3D where all matrices are symmetric (by the symmetry of [ttjj]). Since any vec tors p that are contradictory or Inconsistent with the maximization hypothesis cannot belong to the true population of vectors p, the feasible set {p} for our model and the relations (3o) must define identical confidence-Bayes levels for the true population of vectors p. . Thus the feasible set {(p,[ijj])} for our model forms an Y$ confidence-Bayes level for the true joint population of (p,[n..j) and for the true non-joint populations of p and ["IJ]* • J J Since the set of feasible ["jjl defines en Y.% confidence interval and the value of zOx ) Is determined for a given [n. ], It follows that the feasible set of z(3x ) defines X% of the probability distribution of the true population 3or of z(3x ). Thus the range of feasible z(3x ) defines at least an X% confidence-Sot* 3a1 55. In practice, these constraints on often may be derived from a com bination of observations and subjective belief. 5i>f This assumption of Independence, which Is Implicit In (3o>, can be relaxed by defining constraints.that directly Itmlt more than one element of p at a time, e.g. pt-'J £ p' + pJ £pU|J or pLIJ <, p • pJ £PU,J. Moreover, vectors p that satisfy (3o) but are logically inconsistent (by placing contrad ictory restrictions on ["it)) will be excluded from the feasible set of our model by restrictions of ine form (31). ST. Since the Inverse of a matrix Is unique, (30 defines a one-to-one correspondence between feasible p and ["{j]* 5%. The one minor exception to this statement Is that negative semi-definite only} Is excluded from the model although this set Is consistent with the existence of a maximum; but it can be shown that comparative statics Is undefined for this set (see Propos.it->°« »X 260 Bayes interval for the true population of zQx ).^^° Furthermore, we have now justified the following argument for incorporating as many restrictions of the form (30) into our comparative statics model as is possible: as the number of restrictions of the form (3o) that define a given X5& confidence-Bayes interval is increased, the size of the corresponding confidence-Bayes interval for zOx*) is decreased (or at least does not in-3a1" crease). The Possibility of Additional Restrictions Earlier we formulated conditions that exhaust the comparative static im-plications of the maximization hypothesis. On the other hand, the structure [fljj(x*)] can be described in many ways, i.e. in terms of a large (and perhaps unlimited) number of overlapping parameters and properties. In addition, incorporating observations of additional parameters and properties of [T|j(x*)] into our analysis will lead to a reduction in the size of confidence-Bayes intervals for 3x*. Thus the restrictions described in the two previous sec-tions are only a subset of all potentially useful relations between comparative 59. Whereas p and [n.,] are uniquely related by (30, z and [n. .] are re lated by r-1 J J IJ 3a7 LP z = zOx ). "So7" Thus feasible values for z may be duplicated by elements of the true population of [n. J that are infeasible for our model. I.e. the range of feasible z(3x ) defines at least an Y.% confidence-Bayes level. For simplicity, and as a first approximation when X is large, we shall generally assume that this range forms an X% confidence-Bayes level. fcO. Note that our arguments do not Imply that the probability distribution of (p, ["j]], z) within this Y.% interval of the true population and within Its corresponding feasible set are equivalent. Indeed, the probability content of the true population of (p, [ttj»], z) seems likely to be much more concentrated around Its mean than Is the uniformly distributed population of feasible <P» f^jjJ' z* +na"'' 's Implicit In the model. L I. See -Proposition \. C>2t. See the previous section. 261 static effects and observable structure of the firm's maximization problem. Here we attempt to assess the importance of the restrictions described in 'sections 2. a«d 3 relative to the entire set of relations between comparative static effects and observable structure. We conclude that the most Important of the generally applicable relations may well have been specified In these sections. The discussion is highly speculative. First, consider the set of all special properties that can be imposed directly on Tr(x;a) and Incorporated Into our comparative static analysis, e.g. properties such as separability. Such properties can be useful in particular cases. However, these properties typically are observed either to hold or not to hold for a particular firm. Thus the specification of limits on the "degree of deviation" from these properties tends to be arbitrary rather than based on observation. In other words, any .particular special property is not generally appropriate for our quantitative comparative statics model. In addition, the set of properties that are always true for the Individual firm may be large. I.e. the set of parameters that can be correctly specified as varying numerically over all firms may be large. Nevertheless, the only properties of this type that have come to mind concern comparative static effects. In the remainder of this section, we note the following: (a) all types of changes In exogenous variables that are observable at the level of the Individual firm have been incorporated into our analysis, and (b) all types of comparative static effects that can be derived by the usual methods (primal, primal-dual, dual) for these changes In exogenous variables have been Incorpor ated Into our analysis. Consider the following problems N III N III maximize R(x;ct°) - I c1 (x1 ;a') minimize I c (x1 ;a ) 1=1 1=1 _ subject to R(x) = R C*>3. This Is true even when an "average" of firms Is to be modelled. For example, It Is not clear how to average one firm where TT(X;CI) Is separable and another firm where Tr(x;ct) is observed to be non-separable. N I 1 = 1 maximize R(x;a°) - I c'(x1;a') 262 S subject to I c1(x1 ;a') = C f = 1 with exogenous variables (a0,.., at*), (a1,.., a^, R) and (a0,.., a^, C) respec tively. An arbitrary subset of inputs can also be considered fixed for these problems. These problems appear to accommodate all types of changes in exogen ous variables that are observable at the level of the Individual firm, and relations defining [TT.J(X*)] in terms of comparative static effects of isolated changes In these variables have been incorporated into our model. ^'^^ The form of these relations between [tij .(x*)] and comparative static ef fects of isolated changes in exogenous variables for the above problems has been derived by primal methods rather than by primal-dual or dual methods. Further relations Involving comparative static changes in equilibrium x and X cannot be derived by primal methods; but it is not immediately obvious that relations between [n (x*)] and equivalent comparative static effects ex pressed in a different and more observable form cannot be derived by primal-dual or dual methods. Nevertheless, It appears . that primal-dual and dual methods do not lead to any additional comparative static properties that can be associated with [n|j(x*>]. 7 Summary of Major Quantitative Restrictions In this section, we summarize the most Important of the previously estab lished relations between (a) knowledge of various parameters of the producer problem N I 1 = 1 maximize ir(x;a) = R(x;a°) - Z c'(x1 ,-a1 ) fc>4. See section 3. A simultaneous change in two or more exogenous variables would be realistic If (e.g.) the factor supply or product demand schedules (or produc tion function) faced by one farm receiving community pasture are significantly affected by the activities of other farms receiving community pasture. The effects of simultaneous changes In exogenous variables can In principle be Incorporated easily Into our approach; but such modifications do not appear appropriate for the community pasture programs studied and In general will not be easy to quantify. fcfc. See the discussion of primal-dual and dual methods that Is included in Appendix 4. 263 (b) the maximization hypothesis, and (c> the comparative static effect 9x* for 3a7 this problem. These relations are presented In Figure 3. First, note that the (N + I)2 equations in the system [A] [K] = I can be reduced to (N + 2)(N + 1) equations without any loss of content, and the (S + I)2 2 equations in a system [A ] [L] = I can be reduced to (S + 2)(S + 1) equations 2. without any loss In content. For example, the restrictions implied by the (N + I)2 equations ln,j] j cjl _-,4--!--- [K] = [M] = I _ci I °_ in our model are expressed exactly by the (N + 2)(N + 1) right hand upper tri-2 angular equations of the system [M] = I: [o^T = [o^-] or equivalently Ms s - I» . *or a'' ('»j > such that i = 1,.., M I N + 1 and j > I. (3a) The argument for this statement can be sketched as follows. The matrix ,"[TtiiJ!c!~, c|T j o J (N+1)x(N+1) has full rank by the restriction that [n j j ] Is negative definite.^7 Thus the equations Mj j = Ij j (i = 1,.., N + 1) determi ne (K^ j ,.., KN + ^ j) for a I I j = 1,.., N + 1 and any feasible [TT|J3 - Therefore, the equations Mj N + ^ » 'i, N + 1 (1 = 1"" N + ]) deTertnIne (Ki, N + !»••» *N + 1, N + I5 ,n our model, and (given the value for N + i = KN + i ^ *he eaua+'ons Mi, N = Ij N (I = 1,.., N) determine (K^ N»*«» N}' e+c* Thus +ne stations between any feasible matrix [A] and any symmetric matrix [K] that are Implied (el\ See Theorem 3. ; (o%. The reader can verify that, tn contrast to the case of right hand upper triangular equations, necessary and sufficient conditions for determining [K] are not automatically satisfied In the cases of left hand upper triangular equations and of left or right hand lower triangular equations. 264 by the restriction [A] [K] = I can be incorporated into our model as the (N + 2)(N + 1) equations (32). Likewise, the restrictions implied by the 2 (S + 1)2 equations [L] = [N] = I "[*,/] i c!A" i 1 1 '.AT I Lci I ° in our model are expressed exactly as the (S + 2)(S + 1) right hand upper tri-2 angular equations of the system [N] = I: Nt : = I| : for all (i,J) such that i = 1,.., S + 1 and j > I. (33) ' »J ' »J •— As can be seen from Figure 3, the total number of equations and variables in the set of constraints increases exponentially with the number of inputs N and also with the number of decompositions. For N = 3, 37 quadratic equations and 45 variables are defined when each of the three possible decompositions, given one fixed input, is Included in the system. For N = 4, relations A-C involve 29 quadratic equations and 44 variables, and the set of all possible decompositions, given 1 to 2 fixed Inputs, adds 76 quadratic equations and 76 variables to the system. For N = 5, relations A-C define 36 quadratic equa tions and 62 variables, and the set of all possible decompositions with 1 to 3 fixed Inputs adds 235 quadratic equations and 235 variables to the system. Thus the size of the system of constraints Is particularly sensitive to the number of decompositions that are Included In the system. For N > 3, It is quite tedious to Incorporate all such decompositions Into the model. However, In general knowledge of the structure of the firm's static maximization problem will relate only (or primarily) to a subset of decompositions. 265 (First page of Figure 3) (A) first order conditions for a maximum . , 3x IK,-.] 11 3a1 N quadratic equations (B) second order conditions for a maximum -fTtjj] = [H] [H] N(N + 1) quadratic equations H. . > 0 I.J (j = 1. ---.N) N bounds (C) long run decomposition (see Theorem 3) I iT I ' I [K] = I (N+1) x (N + 1) (N+2KN+1) . 2 independent quadratic equations iL , i . iU c. < c. i c. i i i 3x dx 3F 3F K. L < c'. | K. . < K. . i.J '.J K. L < R • K. . < K. U '.J y LI '.J K. .L < -R2 • K. . < K. .U U y ".J R L £ R £ RU y y y (i = 1, ••••N) (i.i = 1, •••,N) (i = 1, •••,N and j = N+1) (i.j = N+1) 2(N + 1) bounds 2 bounds (D) decompositions, given fixed inputs (see Corollary 5); for each decomposition with N-S fixed inputs: I I T , A. 1 iA IWiJ 1 I Ci 3x iAT Ci | i**S [L] = I (S+1) x (S + 1) (S + 2)(S+1) 2 independent quadratic equations 3CT i**S L. .L < c! : • L. . < L. ^ '.J |0) i.J i,j 3x _ : L 7 < R • L. . < L. . 3F '-J y '.J I.J (i.j = 1,-",S) (i = 1, •••,N and j = S+1) 2(S+1) bounds FIGURE 3 Summary of Major Constraints for the Quantitative Comparative Statics Model 70 The mark " — " is placed above any symbol that refers to a constant rather than an endogenous variable in the model. "^For definitions of the symbols used here, see Theorem 3 and Corollary 5. R E 3 where y = F(x) and R(y) = R(F(x)). V Y 266 (Second page of Figure 3) (D) (continued) 2(S + 1)2 bounds (E) non -decompositions (output exogenous), given fixed inputs (see Corollary 6); for each non-decomposition with N-S fixed inputs: Iijj^l (P) - I | (S+2HS + D jncjependent quadratic equations (SxS) (SxS) i*S . : P. > < c! : • P. . < P. U (i.j = 1, ...,S) 3AJ '.) JOJ i.j i.j totals: (N+2MN+1) N(N+1) * N quadratic equations (N+2)(N+1) N(N + 1) + + 2N + 1 variables (S+2)(S+1) additional quadratic equations and variables for each decomposition 2 non-decomposition (C,D,E) with N-S fixed inputs FIGURE 3 Summary of Major Constraints for the Quantitative Comparative Statics Model1G-(Footnotes fcfl andTO are the same as the previous page) 267 APPENDIX IV QUANTITATIVE COMPARATIVE STATICS AND DERIVED DEMAND: PROOFS Page 1. On Integrability in Comparative Statics 268 2. Lemma 4 272 3. Proposition 1 6 4. Constrained Maximization , 282 5. Theorem 3 285 6. Corollary 5 294 7. Corollary 6 302 8. Quantitative Comparative Statics of a Shift in a Firm's Product Demand Schedule 305 9. Primal-Dual and Dual Methods of Quantitative Comparative Statics 319 268 APPENDIX IV QUANTITATIVE COMPARATIVE STATICS AND DERIVED DEMAND: PROOFS 1. On Integrability in Comparative Statics The problem of determining the precise conditions corresponding to the comparative static implications of the maximization hypothesis has been called one of the major remaining challenges in the theory of compara tive statics (Silberberg, 1971a), arid has been largely solved in the context of generalized duality theory (Epstein, 1978). Here we shall sketch a solution to this integrability problem in terms of primal methods of comparative statics and point out relations between the primal and dual approaches. Consider the general primal problem maximize Tr(x;a) . . . .(P) subject to g(x;a) = 0. with an interior solution x*(a) > 0. The only conditions that are placed on the Hessian [TT..(X*)] by the assumption of an interior solution and twice differentiability are negative semi-definiteness subject to constraint and symmetry. However, it is well known that the comparative static 9 x effect -r— is determined uniquely by [TT..(X*)] and g (x*) whenever 269 [TT..(X*)] is negative definite subject to constraint. Moreover, it can be shown that the condition [TT..(X*)] only negative semi-definite subject to constraint contradicts the notion of local comparative statics, i.e., the equations of the total differential are consistent with the implications of this condition only in the meaningless case where TT. (X*) - Xg. (x*) = 0 for all i.1 Thus the conditions of symmetry and negative definiteness subject to constraint for [TT..(X*)] correspond exactly to the comparative static implications of the maximization hypothesis, i.e., are necessary and d X sufficient conditions for a solution — to the total differential of first d a order conditions for P to be consistent with the existence of a maximum at 2 3 x*(a) when Tr(x;a) is twice differentiate. ' Thus we have the following conditions for economic integrability: (a) given a primal Tr(x;a) that is twice differentiate, integrability occurs if and only if [TT..(X*)] is negative definite subject to constraint, and (b) given a dual TT(CO that is twice differentiate, integrability occurs if [TT ] is positive definite (when TT(X;a) = R(x) - Z ai<J>(x)). OLOt j 1 N i i i For the unconstrained case where Tr(x;a) = R(x)- E c (x ;a), see the proof of Proposition 1-C. Proposition 1-C is easily generalized to problem P. 2 The importance of negative definiteness for integrability in the primal does not seem to have been noticed. In his survey article, Hurwicz (1971) states that semi-definiteness and symmetry of "indirect" (primal) as well as "direct" (dual) demand functions implies economic integrability. 3 Whereas primal methods appear to attain integrability in the general problem P, dual methods appear to be unsatisfactory for the general case where a enters the constraint function g as well as TT (Epstein, 1978). 270 These two conditions can be related roughly as follows. First, [TT..(X*)] exists and only negative semi-definite IJ n ... .(c) subject to constraint => x*(a) undefined; so the dual Tr(a) also is undefined. Second, the implications for the primal of the assumption [TT ] only positive semi-definite can easily be established in the competitive case N . . maximize TT(X;W) = R(x) -Z w x i=1 The total differentials of the first order conditions can be expressed in matrix notation as [yx*)] = 1 ; • • • -(d) (N x N) (NxN) so by Hotelling's Lemma [TT..(X*)] [TT ] = - I ... .(e) IJ WW only assuming that [TT..(X*)] and [Trww] are defined. Thus, by (d) and (e), [TTww1 only positive semi-definite => [TT.J(X*)] is undefined, i.e., R(x) is not twice differentiable.5 H For example, a competitive equilibrium is indeterminate under constant returns to scale. 5(On the following page). 271 Footnote 5 Thus knowledge of [Trww(p,w)] can be used to define [TT..(X*(p,w))], or vjce versa, if and only if Tr(x;p,w) is strictly concave at x*(p,w) or (equivalently) [Trww(p,w)] 1 exists. Analogous simple approaches also lead directly to the relation between second-order approximations for the production function/cost function and direct utility/indirect utility cases, except that here the implications of linear homogeneity in prices are avoided by considering the matrix system of comparative static equations that is defined by variations in the exogenous prices plus the exogenous output or income. 272 2. Lemma 4 4. Suppose that conditions 1-3 are satisfied for a problem P P 11 i maximize TT(X) = R(x) - c (x ;ct)- Z c (x) i=2 N * 1',2' and that this problem has a unique global solution x . ' Denote the set of comparative static effects of d a for this problem as * 8 x 8 a and denote the set of comparative static effects of dx1 for the corresponding problem Q Q N i maximize TT(X) = R(x) - Z c (x) i=2 1 I* subject to x = x as dx* > } - { <• 9xi J I (these sets are not necessarily non-empty). Then *A *Z dx , • • *,dx where ^ *A ^ *Z dx ,««',dx is the set of solutions for the problem N N 82Tf(x*)w . . maximize Z Z • : dx dx' 1 = 1 j=1 3x 3x Footnotes 1'& 2'on following page. 273 subject to dx1 = 1 ; and (D, fax 1 / . *A 9x *Z dx \ 1 * 9 x where is uniquely defined by the equation d 01 11* 1* n 1 1 * 1* 9p'(x' ) . 9x' _ 92c'(x' ; a) m dx1 9x1 9a 9xl2 9a 1 1 * 9 c (x ; a _ Q 9X1 9a * (assuming that -j — \ is non-empty, and d2c\x'i*; a) 9X1 9 a + 0 ) 1 * Assuming other global solutions in the neighbourhood of x does not change our results substantively. 2 The analysis is essentially unaffected by relaxing condition 3, i.e., by assuming c1 = c^(x;a). Statement A still holds for any problem Q. Given 32c1(x;a) 9x'sa for all i ^ 1 or equivalently c^x^) = c1A(x) + c1B(x1;a), statement B holds for the problem Q "maximize TT(X)^ = R(x) IA ^ i 1 —T* 9x1 -c (x) - Z c (x) subject to x = x ," where -r Is uniquely i=2 3a defined by 92TT(X*)Q . 9X1* _ S'c^lx1'^) . 9X1* _ 92c1B(x1*;g) _ Q dx1 2 da dx1 2 da 9X1 3a 274 Proof. Define the problem P N Z i=2 P 11 • maximize TT(X) = R(x) - c (x ;2) - Z c'(x) . . . .(a) where by assumption * problem (a) has a unique solution x . ... .(b) Denote the corresponding problem Q as Q N i maximize TT(X) = R(x) - Z c (x) i=2 (c) 1 1 * subject to x = x By Theorem 1-A x* is also a unique solution for (c); so by Lemma 2 8 x * 9 x * each for problem (c), and only these ———, is a 9xx 9xi solution to the problem (***) r. .-. i .±.\Q9x 9x maximize Z Z TT..(X*) i = 1 j=1 ,J 9xi -f^. 1* 9 x where = 1, which is statement A of the Lemma. By 9x^ Corollary 4, i 1 * 1 1 * p'(x' ) - c\{xl ; a) = 0 . ... .(d) Assuming that conditions 1-3 hold at the solution(s) to a P both before and after da. 275 1, 1*, 3x1* 1,1*, ax1* 1 , 1* , , , P1(X ]Ja—-CUlx ;a)^r-"C1a(x ;a)=0 .-..(e) by total differentiating (d). By Corollary 2-B, 1 1 * p.(x ) is uniquely defined. . . . .(f) By assumption 1 1 * 1 1 * c^(x ;a) and cia(x »2) are uniquely defined . . . .(g) By (f)-(g), 3x1* •* is uniquely defined by (e) for problem (a). . . .'.(h) By Theorem 1-A, ,1* 3d 1 3x if dx =3 for problems (c) and (a), respectively. * * 1 * * then {i^-dx1} = {22L_. £*_} ={|*_} 3x^ 3X1 3a da By (h)-(i), * * 1 * 1* {4*—} = {•^-x— • 4*-—} where -I*-— is uniquely determined 3a „i3a. 3a -1/ 0 x by the equation 1, 1*, 3X1* 1 . 1* .3X1* 1.1*. n pi(x ^Th--Q\\{x ;a)Wa--cia{x ;a) = 0 which Is statement B of the Lemma Q . (i) 3. Proposition 1 Proposition 1. Suppose that conditions 1-3 are satisfied for a problem P P 1 1 N i maximize TT(X) = R(x)-c (x ;a) - Zc (x) i=2 and that this problem has a unique global solution x Denote the set of comparative static effects of da for this problem as dx* {-5 }, and denote the system of total differentials of the first a a order conditions for a solution to this problem as 1 [Trij ] 9T = 'la (1) P P * where [TT.. ] is defined as the Hessian matrix for TT(X) at x . p Assume that [ir.. ] is negative semi-definite. (A) If [IT.. ] is negative definite, then equations (1) have a * 9 x unique solution -r— . o a (B) If [TT.. ] is not negative definite, then equations (1) may have multiple solutions {-5—} . a a 3 * Assuming other global solutions in the neighbourhood of x rules out the possibility that TT(X)P is negative definite at x*, and does not alter statements B and C. P 9 x (C) However, if [TT.. ] is not negative definite, then 7^- is * 9 x undefined ( {•»— } is empty) : 11* 1* 211* 1* 211* 9p (x' ) 9x _ 9 c (x ;g) 9x _ 9 c (x ;g) _ ^1 9a . 1 2 9a ^ 1^ 9x x 1 z 9x19a by the first equation in (1); . 1, 1* .21.1* . 9p (x ) 9 c (x ;g) _ . 1 12 9x' 9x' p by equations 2, • • •,N in (1), [TT.. ] negative semi-definite 'J 1 * 9 x (and not negative definite) and Lemma 3; so ^ ^ is un defined for 9 c (x ;g) ^ Q+,5 9x19 g * fx' g x \n order to obtain local comparative statics results {-5-^- } for da (where x + dx* is in the neighbourhood of x*), we must assume that „2 1, 1* , ,2 1. 1* . 9 c (x ;a) jfe 0 . If 9 C (X ;A) E 0 , but 9xx9g 9xx9g 2 1 1 9 c (x • g) 1 1 * $ 0 for some x # x leads to a change in global * * solution Ax , then Ax is finite and our methods no longer apply: (in general) equations 1 (and Lemma 4) are correct only for an infinitesimal change in global solution dx . 5' These results are essentially unaffected by relaxing condition 3, i.e., by assuming c^ = cVx;a). 278 Proof. By definition. P * P [TT.. ] negative definite <=> x [m. ] x < 0 for all x ^ 0 . . .(a) P P where [TT.. ] = N x N Hessian matrix for TT(X) at a unique solution * x for a problem P. By (a). P P [TT.. ] negative definite => [TT.. ] x i- 0 for any x i- 0 ... .(b) i.e., [TT.J ] has full rank N. Since a square matrix [A] has an inverse if and only if [A] has full rank, (b) implies that P P 9 x [TT.. ] negative definite => [TT.. 1 — = IJ 3 IJ 9 a r 1 <i 1a 0 0 (***) has a unique solution 9x 9 a which is statement A of the Proposition. By definition. -1 0 0 [A] = 0 -1 0 g is negative semi-definite only. . . . .(c) 0 0 0 By (c) / r h x -x 1 3 satisfied by all (x , x ). ... .(d) [A] 0 0 is 3 X J 0 6 | A - XI | = (-1-A)2(-A) = 0 has roots X = -1, -1, 0, which implies that [A] is negative semi-definite and is not negative definite. 279 By (c)-(d). [TT.. ] negative semi-definite only ^=> [7Tij 1 9d = 'la . . . .(e) has no solution 9x da We can show that given that [A] is an N x N matrix and C, X are N' x 1 vectors: [A]X = C has a unique solution x if and only if [A] has rank N,7 (f) [A] negative semi-definite only => [A]X =0 for at least one X $ 0. (g) By (e)-(g). if [TT.. ] is negative semi-definite only, the system ' 1 [7rij ] da- = 'la 9 x may have multiple solutions {-^—} which is statement B of the Proposition. By definition. 8 See Murdoch (1970), p. 112. [A] negative semi-definite only implies that there exists a scalar A* = 0 and a vector x* * 0 such that [A] x* = A*x* {see Madley, 1961, p. 256); so [A]x =0 for an x £ 0 280 P P [TT.. ] negative semi-definite only => X'[TT.. ]x = 0 for some x t 0 X'[TT..P]X ^ 0 ij for all x. .(h) By (h) and Lemma 4, P 8 x*1 P 8 x* 9 [TT.. 1 negative semi-definite only => [TT.. ] ~ = 0. . . .(i) > 3a 3a Assuming that conditions 1-3 hold before and after da. N j* .\ yx*)P!f--c]a(x1*;a) = 0 • • • j=i * N j* 1 *;M*)P ITT" = 0 • = 2, • • *,N. . . . .(k) j = l 'J da By (i) and (k). N i* P * P 9 x [TTJ. ] negative semi-definite only => E Tij.(x ) ^ ^ = 0 . . . .(I) 9 P 1*i By (h), maximum Z'[TT.. ]z = 0 where z is N x 1, and {z } includes z1 = 1. P 1 1 1* Therefore, given TT(X) E TT(X) -c (X ;a), the {z*:z = 1} is the solution set for the problem "maximize Z'[TT..^]Z subject to z' = 1"; so maximum 0 1 1 * Z'[IT.J ]z =c11(x ;a) given z1 = 1. Thus, by Lemma 4-A, *i * *, * 1*^ lTr. Qj = cLfx1*;^; so £*— [TT..P] ^- = 0 by Lemma 4-B 3xx IJ 3X1 11 9a ,J 9a and by TT(X)P = TT(X)^ - c^x^a). 281 1 I * which contradicts (j) for cla(x ;a) t 0. By Theorem 1-A and TT(X) = TT(XP - c (x ;ct), j_1 "J 3 a . = 1 Ij dx1 . . . .(m) By Lemma 1, N * Q 8xj* _ 92TT(X*)Q £ 7L.(x ) —= 1—o - ... .inj 1-^' 3x1 a? By Theorem 1-B, 82TT(X*)Q 3x — 2 P,(x ) .(o) By (j) and (l)-(o), P 1 1 * if [TT.. ] is negative semi-definite only and c1a(x ;a) £ 0, * * 1 * 8 x 1 1 * 8x 1 1 * 8 x then -r is undefined: p,(x cni(x ;a)5—— 8a 8a 11 da 1 i * - cla(x ;a) = 0 1 1 * 1 1 * by (j), whereas p^x ) - c^fx ;a) = 0 by the assumptions p tTTj. ] negative semi-definite only, (k) and Lemma 3 which is statement C of the Proposition. D 282 4. Comments on Constrained Maximization The problem P considered in our comparative static model is of the form "maximize TT(X;OO" rather than of the form maximize TT(X; a) j ... .(D subject to g(x;a) = 0 i=l,«««,M which is the general classical problem. Here we shall point out that it seems difficult to extend our method of comparative statics to such a problem. How ever, we shall also note that this does not appear to be a serious limitation of our approach. The main (or at least serious) difficulties in incorporating problem 1 into our approach stem from the following: the second order conditions for 9x* 9 A a solution to 1 do not require that a matrix relating (-5 , —) to shift * j * parameters Tra(x ;a) and gQ(x ;a) be negative definite or semi-definite. Problem 1 can be expressed in Lagrange form as M . . maximize L(x,A;a) = Tr(x;a) - E A^g'(x;a) . ... .(2) j=1 Total differentiating the first order conditions for 2 yields N i* * * 9 x * * .^LxixJ(x 'A ;a) To" = "Lxia(x 'X ;a) M ?>Aj i * + Z g (x ;ct) i =1, ---.N j = 1 9a 1 . . . .(3) N i* i * 9x' i * Z 9i(x ;a) To-'= ~ga(x ;a) i = 1, —,N. j=l 1 283 Equations 3 can be expressed in matrix form as 0 1 C I x (MxM) | (M x N) f G I L X | XX (N x M) I (N x N) 3 X 3 a G a (M x 1) (M x i) 3x 3a L . xa (N xi) (N xi) > (M +N) x 1 (M + N) .(4) using obvious notation. Denote the (M + N) x (M + N) matrix in 4 as [L]. This matrix cannot be negative definite (due to the MxM submatrix of O's). In addition, [L] has full rank by the usual second order conditions for a constrained maximum;1** so [L] cannot be negative semi-definite only. Thus we cannot specify [L] as negative definite, and instead must specify the more clumsy conditions that the determinant of [L] has the sign N of (-1) , the largest principal minor of [L] has a sign opposite to this, and successively smaller principal minors alternate in sign, down to the principal minor of order M + 1. However, apparently we can ignore problems P of the general classical form (1) without restricting our comparative static method in any * * serious way. The solution set x = x (a) for a problem 1 can also be obtained as the solution set for a suitably defined unconstrained problem maximize Tr(x;a)' ... .(5) 10See Intrilligator (1971), pp. 496-7. 281 * * 11 where TT(X (a);a)' = TT(X (a); a) for all a. The form of Tr(x;aV implied by the particular problem 1 may not be obvious. However, we shall be interested only in specifying the ~ * restrictions on -r that are implied by a subset of possible o Oi forms Tr(x;a) and G(x;a) =0 for problem 1, and many of these restrictions can be incorporated into a set of equations * d X 12 G(-~ , p) = 0. In this case, defining quantitative com-o ot parative statics in terms of an unconstrained maximization problem does not lead to a serious loss in generality. Assuming that TT(X (a),a) > 0 for all a, we can "simply" construct * * 7T(x;a)' such that TT(X (a),a)1 = TT(X (a);a) for all a and TT(X;O0' = 0 for all * * combinations (x,a) that do not satisfy the relation x = x (a). 12 In some respects, the comparative static effects of fixed factor proportions can be modelled more accurately in terms of (1) than in terms of an unconstrained maximization problem. We can incorporate some —but not all — of the comparative static implications of fixed factor proportions into d X * a set of equations G(-r—, p) = 0 (see Chapter 3). On the other hand, any a OL particular example —but not the general case —of fixed factor proportions 1 2 can easily be expressed as G(x) = 0 (e.g., x - 2x =0). 285 5. Theorem 3 Theorem 3. Suppose that conditions 1-2 are satisfied for a problem P N . . j maximize TT(X) E R(X) - E c (x ;a) . . . .(1) i=1 * and assume that this problem has a unique global solution x where the Hessian matrix for TT(X) is negative definite. Construct the related problem maximize TT(X) subject to R(x) = R(x ) • which can be expressed in Lagrange form as * maximize TT(X) - X(R(x) - R(x )). ... .(2) matrix [A] (N + 1) x (N x 1) where ij denotes the Hessian matrix for TT(X) at x , (N x N) 13 i _ i i This theorem is easily generalized to the case c = c (x;ct) (i = 1, •••,N); but the equations in the generalized theorem are somewhat more detailed than here, and the generalized theorem will not be employed in our research. Construct the symmetric TT.. 1 i c. •j ; 1 (N xN) ; (1 xN) i i C. i 1 ! 0 (N x i); (1 xi) V J 286 9 A i 1,1* 1, . N, N* N. c. r 9c (x ; a ) 9c (x ;a J \ and ' n E 1 9x1 '' 3xN > (N x 1) [A] necessarily has full rank, and denote its inverse as [K] [K] a (N + 1)x(N+1) [A] ^ = [K always exists, Then (A) the comparative static effects for problem 2 are uniquely defined as follows: 9x'** _ 92c'(x'*;a!) 3aj 9xJ 9aJ ''' K. . i,j = 1f-fN 9x ** K i = 1 • • • N i,N+1 9R 9X _ 32c'(x'*;a!) *~ " 9x'9a' N+1'' N+1,N+1 3aJ 9xJ 9a 9X = -K K i = 1 • • • N 9R where K. . = element (i,j) of matrix [K], and K. . = K. .(i,j = I, •• \N+1) ;and '/J * i»J J/1 9 x (B) (a) the comparative static effects ( 9 a ) t"or problem 1 are unique, and • +u • v r is 9c'(x'*;g') j. . 15 (b) given that Z E K. • ^ t - 1 , i=1 j = 1 1 9xJ !** 1 i i* i 2 i i* i i** 1*Thus 3x . = (3 C (X ;a)/ 3 C,(X- ;a)) 3x. and 3 a' 9xJ9a' h'Sct' 9 a - * 2chx]*J)...jx!!l (i,j = 1,..., 9a1 9x' 9a' 9R N) N N ar'fx'*^! 15A sufficient condition for E E K. • d 1 • ' J * -1 (a) i=1 j=1 9x' is that K. „, , S 0 (i = 1,"«,N), which is equivalent to ruling out the possibility I, N +1 287 * 3 x -r- for problem 1 is uniquely defined in terms of 3a1 32cj(xj*;a!) * and the elements of [K] corresponding to 3xJ3 aJ dx-rr— and —— for problem 2, as follows: 3aJ 9R 3x1* = oW*-J) . K +K jRjxJ, iJ=1,...,N 3a' 3x'9a' ''N+1 3a* * N i i* i i* 3R(x ) _ " 3c'(x' ;a') . 3x^_ . . ... N 9 a1 i=1 3x 3 a' 7\ Proof. Suppose that x is a unique interior global solution for the problem P N . . . maximize TT(X) = R(x) - Z c (x ;a) . ... .(a) i = 1 Construct the related problem maximize TT(X) . . . .(b) 5T subject to R(x) = R(x ) which can be expressed in Lagrange form as JT maximize Tr(x) - A(R(x) - R(x )) . ... .(c) Footnote 15 (continued) of inferior inputs —> o, i = N). Condition (a) would be violated only 3R for a relatively few "appropriate" degrees of inferiority; so condition (a) is not a serious restriction. 288 By (a)-(b), x is the solution set to problem (c). ... .(d) By (c)-(d), TTj(x ) - ARj(x ) = 0 i = 1, •• -,N • • . .(e) R(x*) = R(x*) which are the first order conditions for a solution to problem (c). Total differentiating (e). N * i * i * Z TT..(X ) dxJ - cj^i (x ;ot ) - Rj(x )dX N R , 'J - X Z R..(x*) dx' = 0 i = N . . . .(f) j=1 N * j _ Z R.(x ) dx - dR = 0 ... .(g) i=1 1 given (a). By (c), X d-rr(x*) 3R N i** Z 7T.(x*) ~ =0 ... .(h) i=l 1 3R by (d) and conditions 1-2. By (h), (f) reduces to By (a) and conditions 1-2, (g) can be rewritten as N i i* i i -Z c!(x' ;<x) dx' - dR = 0 i=1 1 Construct the symmetric matrix i c. i (NxN) ! (1xN) ci : o i (Nxi) ; (ixi) [A] (N+1)x(N+1) where TT.. 'J (NxN) = Hessian matrix for TT(X) at x* c. /. ,M - r 1/ 1* K ™ i N" (1 xN) = (c^x ;a ),•••, cN(x ;a )J N, N* N, By definition, [A] has less than full rank if and only if N . . there exists a vector v t 0 such that Z v' IT.. - c. j=1 ,J = 1, N , c J Z vJ c! j=1 but this statement implies that 290 N N there exists a vector v ^ 0 such that Z I TT.. VV = 0 1=1 i=1 ,J which contradicts the assumption that [TT...] is negative definite. Thus [TTJ.] negative definite => [A] has full rank. . . . .(***) By (i)-(l). [A] X = C ... .(m) (Nxi) (Nxi) where 1 NT X = (dx ,"«,dx ,-dA) for problem (c) (Nxi) T .(n) C = (c. i(x ;a )da ,"-,c N(x ;a )da ,dRJ (Nxi) la Na By assumption, [A] 1 exists, and [A] 1 E [K] is symmetric . . . .(o) by the symmetry of [A]. By (o), X = [K] C . ... .(p) By (n)-(p), for problem (c) 291 • ** 9x —, = c! j(x'*;a!) • K. . i,j = 1,---,N 9AJ JoJ U 9x'' 9R 9 X = K i = 1 • • • N i,N+1 ' . >n j = -c\](j[j*j) -KN+1J J = l,—.N . . . .(***) 3a 1A_ = -K 3R N+1, N+1 K. . = K. . i,j = 1, •••,N+1 where K.j = element (i,j) of [K], which is statement A of the Theorem. By Propositionl-A and the assumption that the Hessian matrix for * TT(X) ([ TT.J]) is negative definite at x , *. I 3 x. I for problem (a) contains only one 1 i 9a1 . . • .(q) vector j = 1,***,N. 9aJ By the implicit function theorem and [TT..] negative definite at x , * we can solve the first order conditions of problem (a) for x as a function of a: i* x (a) i = !,•••,N for problem (a). ... .(r) 292 _ * * Given R 5 R(x ), x is also the solution set for problem (c); * so we can also solve (e) of problem (c) for x as a function of (a,R): ;*;**_ _ * x = x (a,R) i=1,"«,N given RE R(x ) . . . .(s) for problem (c). Substituting (r) into (s), i* i** r N~* ^ x = x (a,R(x (a),«—(a))J i = !,•••,N (t) By (t) and the rule for the differential of a composite function, •* :** •** * 9x 9x ^ 9x 9R(x ) . Kl , , —j- = r— + ——— • —p - I,J = 1,"*,N , . . . .(u) 9aJ 9aJ 9 R 9 a1 where 9R(x ) 9 a I R.(x ) i* i=1 9 a' j = 1,'--,N .(v) By (u) and (***), i* * 9x _ j . , j*. L .K K . 9R(x ) j ; = 1 ... N rwi —y--Cja,lX ,0fJ Kij+Ki/N+1 ''J ,N- • .IWj By (v) and conditions 1-2, 293 9R(x*) 3a' N I i.i i % c.(x ; a ) 3x 9 a) j = 1, ,N . .(x) By (w)-(x), I (NxN) Ki,N+1 (Nxl) -f-c! | -i (Nxl) | (1x1) (N+1) x (N+1) r 1* 1 9x' 9a1 • • N* 9x™ 3a' 9R(x*) _ 9a' (N+1) x 1 Ki • K N,j| 0 j = 1, ,N • (y) where I E identity matrix and K. = N+1'st column of [K], (NxN) (N'x1) Denote the (N+1) x (N+1) matrix in (y) as [L]. By (y) and the definition of a determinant. [L] N N .... • ^ = -1 - E Z K. ... • c.fx' ;a!) . ... .U) 1=1 |=1 ''N+1 J Since (y) has a unique solution 9x 9R(x ) 1 {j arbitrary) if and on|y if [L]_1 exists <• 9a1 ' 3 a' or equivalently |[L] | 0, (q) and (y)-(z) imply that g X (a) r is unique for problem (a) (j = 1,"*,N) 9cxJ° N N • J(xj*-a!) t -1 (b) given that Z Z K. N+1 j1 ,orj F * 9 x r- for problem (a) is uniquely defined in terms of 9aJ cjaj(x' anC' t'16 e'ements °^ corresponding to * * £^r- and for problem (c) Cj = 1, ---,N), 9aJ 9R as follows: 3x'* _ i J*. J. . „ . „ . 9R(x ) j * * = c| ,(x' ;cr) • K, :+K: Ma1 . ^2L± j,j=1,. m*J- = Z cV*;^) j=1, 9aJ i=1 ' 9aJ which is statement B of the Theorem. D 6. Corollary 5 Corollary 5. Construct the problems 1 and 2, and the (N+1)x(N+1) matrices [A] and [K], as in Theorem 9. Partition the Hessian matrix [TT..] and marginal factor cost vector c! of [A] as (Nxj\|) ' (1xN) follows: (NxN) f A 1 TT.. 1 •J j B 1 TT.. •J (SxS) | (SxT) c 1 TT.. 1 •'J | D u.. "J (TxS) ! V 1 (TxT) (IxN) ( -A ! •Bl i 1 • c. 1 c. > 1 4 (IxS) (IxT) where S+T = N. Construct the following symmetric matrix A (SxS) • .A : c! ! (sxi) .A c. ! o (IxS) : (ixi) [An] . (S+1)x(S+1) [A^] necessarily has full rank, and denote its inverse as [L] [A^] = [L] always exists. Construct the problem maximize TT(X) N . . . R(x) - Z c'(x';a ) i=1 subject to x' = x' j = S+1, "vN where x is the unique global solution to problem 1 Construct the related problem maximize TT(X) subject to R(x) = R(x ) j = S+1, • • »,N j j* x' = x' which can be expressed in Lagrange form as N maximize TT(X) - A(R(x)-R(x )) - Z yfx'-x' ) j=S+1 Then 296 (A) the comparative static effects for problem 4 are uniquely defined as follows: i**S 3x 32J(xj*;gJ) . L 3aj dxha) l,J U = 1 3x' a 3R Li,S+1 i=i,-.s 3 XS 9aJ 3X1 3R ' Zxho} S + 1'j j = i, •••,s LS+1,S+1 where L. . = element (i,j) of [L], and L- . = L. .(i,j = 1, •• *,S+1); ' 11 ' i J I»' and *S 3 x (B) (a) the comparative static effects for problem 3 are 3 a unique, and (b) given that EEL. _ • 5—r i=1 j=1 ''s 1 3xJ 3cJ(xJ ;aJ) , , 16 * -1, 16 . S S 3c'fx'*-ah Assuming that E I L. - . ^.^'^-i has implications analogous to those of assuming that E E K. N • ix . ;a ' i - 1 i=1 j=1 ''N+1 3xJ (see footnote to Theorem 3). 297 *S 9x * 9a1 32J(xJ*;oJ) for problem 3 is uniquely defined in terms of and the elements of [L] corresponding to **c **c 9 x 9 x —•• and — for problem 4, as follows: 9a1 9R :*c 2 i i* i * 9x' ^ = 9 c'(x' ;aJ) L + L 9R(x )3 9aj " 9x'9a' ' ''' !'S + 1 ' 9 a! * <i i i* i i*S 9R(x )b _ | 9c'(x' ;a') , 9x' 3 j = 1,...,S 3a' i = 1 9x' 9a' Proof. Suppose that x is a unique interior solution for the problem P N . . . maximize TT(X) = R(x) - Z c (x ;a) . ... .(a) i=1 Construct the related problems maximize TT(X) subject to x' = x' j = S+1, • • «,N maximize TT(X) subject to x' = x' j = S+1, • • *,N ... .(c) w R(x) = R(x ) . 298 Problem (c) can be expressed in Lagrange form as N j=S+1 maximize TT(X) - A(R(x)-R(x ))- I yi[xi-x] ) • (d) By (a)-(c), x is the solution set for both problems (b) and (d) By (d)-(e). m(x*) - ARj(x*) = 0 R(x*) = R(x*) i = 1, •••,S • (e) .(0 x> = x' j = S+1, ••sN • (g) which are the first order conditions for a solution to (d). By argu ments identical to (f)-(n) inthe proof of Theorem 3, [A„] XS = CS (S+1)x(S+1) (Sxl) (Sxl) given (g) • (h) where (S+1)x(S+1) A TT.. 'J c i (SxS) (Sxl) jA c. i 0 (IxS) (1x1) TT.. = submatrix for inputs i = 1, •••,S of the Hessian (SxS) for TT(X) at x* * " " (0) continued on following page) 299 r _ , 1, 1* 1. S, S* S ' Cj = (c^x ;a ), "',cs[x ;a )) (Sx1) Si S ^ x = (dx , •••,dx,-dX) . . . .(i) (S+1)xl _ 1 1*11 s «;*«;<;_ T cS = (cjai(x ;a')da\-.^c|^(x ;a )da ,dR) . (S+1)xi By definition, [A^] has less than full rank if and only if there s i i exists a vector v £ 0 such that E v' TT.. - c. = 0 i = 1, •••,S j = l ,J ! v'ci = 0 ; but this statement implies that S S there exists a vector v £ 0 such that E E TT.. V'V' = 0 i = 1 j = 1 lj which contradicts the assumption that [TT..] (hence [TT.. ]) is negative definite. Thus [TT..] negative definite =*> [An] has full rank (***) Then 1 (S+1)x(S+1) [A^] = [L] is symmetric . . . .(j) 300 by [A^] symmetric. By (h) and (j). S S X* = [L] cr By (i)-(k), for problem (d) the comparative static effects are uniquely defined as follows: **c c **c c 8x 3 9 Xs 9x 3 9 A3 da ' da ' 9R ' 9R 9x i**S 9a/ c' fx' -ah • L. . cjaTx ,CC) i,j 9x' = L. 9R ,S + 1 i = 1,-,S 9 A" 9a! j = 1,-,S 9 A-9R •S + 1,S+1 where L. . = element (i,j) of [L], and L. . = L. .(i,j=1, • • «,S+1) i, J ' i J J»1 which is statement A of the Corollary. By (b) and (e). * TTj(x ) X* = x' i* i = 1, "^S j = S + 1, • • •, N . .(I) 301 which are the first order conditions for an interior solution to problem (b). By (i) and (I), differentiating the first S first order conditions with respect to a} (j = 1, •••,S) yields [TT ] 9x ij 9 aJ (SxS) (Sxl) •c|ai(xJ*;aJ) a} = a , -",0^ . . . .(m) where 9x 9aJ 9x 1*S 9x S*S 9 a' 9aJ [TT..] Since negative (NxN) r Al definite implies that [7Tij is negative definite, (SxS) 9x 9aJ for problem (b) contains one and only one vector 9x 9 a (j = 1, •••,S). .(n) by (m), [TTJ.] negative definite and Proposition 1-A. By arguments (NxN) identical to (v)-(z) in the proof of Theorem 3, 9x *S s s • •* • given that EEL. _ , • c!(xJ ;aJ) t - 1 , i=1 j = 1 ''S+1 ' 9a/ for problem (b) is uniquely defined in terms of cj^fx1 ;a}) and the elements of [L] corresponding to ... .(***) ( (***) continued on the following page) 302 —. and for problem (d) (j = 1,---,S), as 3a1 8R f0"0WS: i?"=cI|oJ(x,*;J) 'LU + Li,s+1 '^X^ • ' • -(***} i,j = 1, "^S 3R( X*) S - I cW^a') .^L j=l,-,S. 3a1 i=1 ' 9aJ Statements (n) and (III) are equivalent to statement B of the Corollary, rj 7. Corollary 6 Corollary 6. Construct problems 1 and 3 as above, and partition the (negative definite) Hessian matrix [TTJ.] as above. Then the (NxN) comparative static effects for problem 3 are uniquely defined as follows: 8x'*S 92cj(xj*;aJ) D . . , _ 8a where P. . = element (i,j) of lTrij J (which always exists), (SxS) -J and P. . = P. ; (i,j = 1, -".S). ' • J J •1 Proof. Construct the problems maximize TT(X) = R(x) maximize TT(X) = R(x) i ~T*~ subject to x' = x' where x* is a unique global solution to problem (a). Partition the * Hessian matrix of TT(X) at x as follows: A B TT.. • TT.. 'J ' 'J (SxS) (SxT) C D TT.. , TT.. 'J 1 'J (TxS) i(TxT) . . . .(c) By the assumptions that (N'^N) is negative definite and symmetric, -1 A -1 [TT..] and [TT.. ] exist and are symmetric. . . . .(d) (NxN) 303 N . Z c (x1 ;a ) =1 N Z c (x1 ;a') = 1 (a) . . . .(b) j = S+1, • • «,N By (a)-(b), x is the solution set to (b) as well as (a). ... .(e) By conditions 1-2, the first order conditions for a solution to (b) are 304 TT.(X) = 0 i = 1, —,S • (f) x' = x' j = S+1, • • «,N • (g) By (c)-(e) and (g), total differentiating (f) yields dx [TT.. ] • C IJ dxS -(Sxl) (SxS) (Sxl) • (h) where C = (c|^(x1 ;a1 Ida1, • • ^c^^(x^ ;aS)da^) . By (d) and (h), for (b) i*S 9ai CiaJ(X Pi,j i,j = 1, "',S A -1 where P. . = element (i,j) of [TT.. ] , and P. . = P. . (i,j = 1, • • *,S) i /J 'J '/J J/1 which is the Corollary. • 305 8. A Theorem on the Quantitative Comparative Statics of a Shift in a Firm's Product Demand Schedule Relations between various potentially observable properties of a problem 0 N . . . maximize Tr(x;a) = R(x;a ) - £ c (x ;a) ... .(P) i=1 and * 8 x 0 —jr- , when da implies a shift in the product demand schedule 8a faced by the firm, are presented in Theorem 4 and Corollary7.17 These relations differ from those specified in Theorem 3 and Corollaries 5-6 in one particularly important respect, which can be explained as follows. When a° is a parameter in the product demand schedule, * —Q— can be decomposed as 8 a : * • ** * 8x _ 8x m 8F(x ) 8a° 8F 8a° i = 1, • • «,N ... .(1) ** 8 x where is the comparative static effect of dF for the problem 8F maximize TT(X) = P( F(x); a°) F(x) - Z c'(x';a) i subject to F(x) = F(x ) 17The proof of Corollary 7 is not presented here (Corollary 7 can be established in an abvious manner by the methods used in other proofs) 306 (y = F(x) denotes the firm's production function). In addition, m**L = (1/R)? isiiA .12^ ... .(2) 9a y i=1 9x' 3a ** where R(F(x) ;a°) =R(x;a°) for all (x,a°). For a given vector—, 9F equations 1-2 constitute a homogeneous system of N+1 equations in N+1 9x* 9 F(x*) unknowns ( —„ , n ). Thus, equations 1-2 can determine the 8 a 9 a" 9 x * unique —^ only up to a scalar multiple, i.e., only ratios .9 a" 1* i* N* i*-\ 9X1 , 9x' ... 9x™ , 9x' ^ 0 0 ' * * ' 0 ' 0 9 a 9 a 9 a 9 a can be uniquely defined by 1-2. A similar statement holds for the de-*S 9 x composition of —g— (when a is a parameter in the product demand 9 a ** **c 9 x 9 x schedule). Therefore, knowledge of——or {— } defined by all 9F 9F possible partitions of x into fixed and variable inputs (and knowledge of * ~ 1 1 * N N * 9x R , c^x ),»»»,C|Sj(x )) is insufficient to define the unique —Q—. The y 9 a additional restrictions due to the second order conditions for a maximum * 9 x only imply that the unique —^- is determined up to a positive scalar ,*• . 18 ^ multiple. 18 The proof of this statement can be sketched as follows. The first order conditions imply that the "correct" Hessian [TT..*] and comparative static 9 x* effect —Q satisfy a system of equations of the form tTT,.*3 = [K] . ... .(a) IJ 9aU Nxi Exact knowledge of (£*— , R.c (x1 ),•••, cj(x" )) implies only the following 9F y relation: 307 Thus restrictions must be placed on other parameters in order to 3 x * obtain both upper and lower bounds on —g- by our methods. In particular, Sx* dCL knowledge of —r- (i ± 0) and its various decompositions seems quite important 3 a in the quantitative comparative statics of changes in the firm's revenue 3 x * or benefits schedule (whereas, prior knowledge of —g- and its decompositions 3 a is relatively unimportant in the quantitative comparative statics of changes in factor supply schedules). Theorem 4. Suppose that conditions 1-3 are satisfied for a problem P 0 N i i maximize TT(X) =. R(y;a).- I c (x ) ... .(1) i=1 where y = F(x) is a scalar function, and assume that problem 1 [TT.. ] IJ (NxN) has a unique global solution x* where the Hessian [TTJJ ] is 1 9 negative definite. Construct the related problem ll » [7T..*]][Y ' ^K-] = [K] -(b) (18 continued) - » [TT..*]][Y * ^7 Y 'J 3aC where y is an arbitrary scalar. Given that E77j j * 3 's negative definite: ^ 0 t^jj*] 's negative definite if and only if y > 0. Thus, relation (b) plus the second order conditions has the solution set T dx* > . _-, 19The comparative static effect •3-XQ- is undefined when [TT..] is 3a ' only negative semi-definite (see Proposition 1). 308 0 N i i maximize TT(X) E R(y;a ) - E c (x ) i=1 .(2) subject to y = F(x*) LetR = 3R(y*;«°) y sy = 32R(y*;a°) y a^ •ao 8x.3ao lya° y ' Construct matrix [A] as in Theorem 3, so that [A] 1 = [K] always exists. Then { ^or Pr°b,em 1 corresponds to the single solution to 3a the system [V 7^0 = - Ria0 (NxN) da (Nxi) 20lf R(y;a°) E P(y;a°)y, then Ry = P(y*;a°) + Py(y*;a°)y* and Rya° = Pao(y*;+ Pya°^y*;a°^y** For tne more 9enera' case where the firm sells all y units at an identical price and also receives non-pecuniary benefits B(t) from the t'th unit of y, R(y;a°) = P(y;a°)y + /y B(t)dt. where Rjao = (R,aO.'",RNaO) (Nxl) 309 T (B) the comparative static effects for problem 2 are defined in terms of [A] as follows: j ** 9F y ''N+1 j ** [ K ] = [A]"1 => = 0 i = 1, • • •, N 9 a — = -Ry2 'KN+1,N+1- i = 1,.»,N 9F [K] = [A]-' =>^ = Ru. 0 '\ac 9 a where K. . = element (i,j) of [A] 1, and K. . = K. . i / J 1•J J'1 (i,j = 1, •••,N+1); and N N . (C) given that E E cj • K. N+1 *-1 , * 3X7j- is determined up to a scalar multiple by R , c! and 9 a -1 9 x** the elements of [A] corresponding to ——, i.e., the 9F 3x* 9F(x*) following system has as solution the {all y( Q~• 0—^ 9 a 9a (y an arbitrary scalar): 310 iiiL = R . K • iL 0 y i,N+1 „ 0 i = 1,»'fN 3 a 3 a 3F N Z i=1 3x 3aC i* Proof. Construct the problem N . . maximize TT(X) = R(y;a)y - Z c (x ) i=1 .(a) where y = F(x), or equivalently N j . maximize u(x) = R(x;a) - Z c (x ) i=1 .(b) where R(x;a) E R(F(x);a) .(c) Total differentiating the first order conditions for an interior solution to (b), N HX1* -Z TT..(X*)^— + R. (x ;a) = 0 i = 1,---,N. j = l IJ 3a a . .(d) Since a negative definite matrix has an inverse (see a-b in the proof of Proposition 1), [TT..] negative definite => equations (d) has a unique 'J I***} <NxN> solution 3 a 311 which is statement A of the Theorem. Construct the problem N . . maximize TT(X) = R(y;a) - Z c (x ) i = 1 . . . .(e) subject to y = F(x*) where x* is the unique global solution to problem (b). Problem (e) oan be expressed in Lagrange form as N . . maximize R(y;a) - Z c(x ) - A(F(x)- F(x*)). . . . .(f) i=1 By (b) and (e), x* is the solution set to problem (f) as well as (b) (g) By the manner in which a enters into the objective function for problem (f), —— = 0 for problem (f) . ... .(h) 9 a By (f)-(g). TT.(X*) - AF.(x*) =0 i = 1, --^N ... .(i) F(x) - Fix*] = 0 . Total differentiating these first order conditions (i). 312 N . ^ Z TT..(X*) dx + R. (x*;a)da- F.(x*)dA = 0 . , II la i i = 1,--,N (J) N Z F.(x*) dx1 - dF i=1 ' since X = 0 (see h in the proof of Theorem 3). By conditions 1-2, Ry(y*;a) F.(X*) - c:(x' ) = o = 1,-,N .(k) where (x*, y*) solves problem (a). By (c). Rja(x*;a) = R (y*;a)F.(x*) i = 1, —,N .(I) By (j)-(k). [A] X = C .(m) where TT.. ! •J l (1/R )c. y i (NxN) ! (1xN) d/Ry)c|; 0 (Nxi) ; (1x1) X = (dx1 . N H)x1 C E (-^ d a a [A] (N+1)x(N+1) .~T . . .(n) (N+1)x1 313 H | = X |C | if every element of a row or column of a matrix C is multiplied by a scalar X to give a new matrix H, and C ^ = (adjoint ofC)/|G|. By these facts and the definitions of [A] (n)and[A], [A] 1 E [K] exists if and only if [A] 1 E [K] exists K.. = R a K.. if i = N+1, j = N+1 'J V " ... .(o) K .. = R K.. if i = N+1 or j = N+1, i i j ij y ij ' K .. = K.. if i = 1, --^N, j = 1, ...,N . ij 'j By (h), (m)-(o), the existence of [A] 1 (see Theorem 3) and the symmetry of [A] 1 (by the symmetry of [A]), for problem (f) 9x' 9F i** N R • K. ... i = 1,—,N y i,N+1 9x Z R. (x*;a)K.. i = 1, • N a .=1 jal ij 9F •1^- = R Z R. (x ;a) K.. , . 9a y .=1 jav N+1,j where (by k-l) R. = (R /R )c! and K.. = K.. 7 ia ya y i ij ji ky N+1,N+1 (i,j = 1, ...,N+1). By the definitions of [A] and [K], N AK = I => E c! K.. j=1 ' 'i N E c! K. . j=1 J J'N+1 314 i = l.-'.N By (p)-(q). j ** = R.. K. Mil i = 1,---,N 3F y i,N+1 j ** AK = I => ~— = 0 3 a = - R 2 K. KI , y i,N+1 3F AK = I => l±- = R 3 a y a .(***) which is statement B of the Theorem. By arguments analogous to (r)-(s) in the proof of Theorem 3, ;* •** •** * 3x _ 3x + 3x # 3F(x ) 3a 3a ^ p- 3a i = 1, ...#N ... .(r) where N i* 1F[X ] = E F.(x*) — . ... .(s) 9a i=l 1 3a By (h) and (r), 315 3_2L 9 a 12L 9F 9F(x*) 9 a 1,-,N . . . .(t) By (k), (s)-(t) and statement B of the Theorem (***), 1* I (NxN) — R K. ., -y i,N+1 ! (Nxl) d/Ry)c; -i (IxN) ! (1x1) (N+1) x (N+1) 9x 9 a 9x N! 9 a 9F(x*) 9a (N+1)xl .(u) where I = identity matrix and K.( N+1 = N + I'st column of [A] Denote the (N+1)x(N+1) matrix in (u) as [L]. By (u) and the definition of a determinant. N N - i i* |[L]| = -1 - £ £ K. • c (xJ ) . ... .(v) i=1 j=1 'IN J ~ -1 I^I Since [L] exists if and only if |[L]| 4= 0, (w) implies that N N „ . .* given that £ £ K. N • c (xJ ) 4 -1 , i=1 j=1 ' > the solution (-ir*-, to the linear homogeneous da da system (u) is unique except for a scalar multiple ((***) which is statement C of the Theorem. D 316 Corollary 7 . Construct problems 1 and 2 as above, and partition the r TT i (negative definite) Hessian matrix ij and marginal factor (NxN) cost vector c! at x* as follows: (1xN) [TT..] (NxN) A TT.. 'J (SxS) B , TT.. •J | (SxT) C TT.. •J (TxS) 1 D 1 TT.. 1 'J | (TxT) [c . A .B i i i ] (1xN) (IxS) (IxT) where S+T = N and x* is the solution to problem 1. Construct the matrix [A^] as in Corollary 2, and denote its inverse as [L]:!^^]1^ [L] always exists. Construct the problem 0 N i i maximize TT(X) = R(y;a ) y - £ c (x ) i=1 subject to x' = x' . . . .(3) j = S+1, •••,N, Construct the related problem maximize TT(X) subject to y = F(x ) x' = x' j = S+1., • • «,N which can be expressed in Lagrange form as N maximize TT(X) - A(F(x) - FTx~*l) - Z YJ(XJ-XJ ) (4) j=S+1 Then *S d X (A) the comparative static effects •? for problem 3 are d Ot uniquely defined as follows: 3a ya y j = 1 3xJ " A-1 where P. . = element (i,j) of [IT.. ] (which always exists), '' J ' J and (i,j = 1, -.S), and ' i J /1 (B) the comparative static effects for problem 4 are uniquely defined as follows: i**S 3F y '-s+1 j **s [L] = [A11 ]_1 => = 0 i=1,-..,S. 3 a ^ = "R 2 -L 9P Ky LS + 1,S + 1 S [L] = [A,,]"1 => = R o 11 3a° ya where L.. = element (i,j) of [A^] 1, and L. . = L. . (i,j = 1, •••,S + 1); and given that E Z d 1 . J • L. t-1 , j = 1 i = 1 9x' ''S + 1 9x*S i —— is determined up to a scalar multiple by R , c. and 9x**S the elements of [L] corresponding to —— , i.e., the 9F following system has as solution *S * S {all y (•3-^n— , 8F(xn} ) } (Y an arbitrary scalar) : 9 a 9 a — = L • — i = 1 • S . 0 Li,S+1 .0 ' 3 a 9 a 9F - (1/R ) \ 9ci(xi*} - 9X' 0 " l" v' i 0 9 a y i=1 9x 3 a 319 9. On Primal-Dual and Dual Methods of Quantitative Comparative Statics The quantitative comparative statics method presented in Chapter 3 is developed directly from the primal problem 11 N i • maximize Tr(x;a) = R(x)-c (x ;a) - I c'(x'). . . . .(P) X i=2 In Chapter 2 t we noted that "primal-dual" and "dual" problems can be formulated from P and that many standard comparative static theorems can be derived more easily from these problems than from P per se. Here we shall consider the possibility of using primal-dual and dual methods as substitutes or c
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Comparative statics and the evaluation of agricultural development programs Coyle, Barry Thomas 1979-12-31
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Title | Comparative statics and the evaluation of agricultural development programs |
Creator |
Coyle, Barry Thomas |
Date | 1979 |
Date Issued | 2010-03-05T22:05:30Z |
Description | Perhaps the most important task of any economic analysis of agricultural policy is to estimate the effects of policy on various economic measures such as income and output. This is usually done by combining economic theory with data. However, the economic theory seldom is fully descriptive of the situation and the empirical knowledge generally is far from complete. Thus, even aside from difficulties in aggregating gains and losses over individuals, economic analyses of policies are often unsatisfactory. The major purpose of this thesis is to extend economic theory and methods so as to be more descriptive of various agricultural policy situations and to make more appropriate use of available empirical knowledge. This leads us to relax some assumptions in the standard theory of the firm that often seem inappropriate, and to propose a potentially more effective method of incorporating available empirical knowledge of farm structure into economic analysis of policy. In addition, we also attempt to verify the appropriateness of other theoretical constructs of fundamental importance. First, the static theory of the firm is extended to the case of variable factor prices, i.e., factor prices endogenous to the firm. Under these more general conditions, we establish (among other things) (1) the relation between measures of surplus in factor markets and of consumer plus producer surplus, and (2) relations between the slope of a firm's derived demands schedule and various properties of its production function. It is shown that (2) provides additional support for the well-known fact that traditional qualitative comparative static methods can seldom be useful in economic policy-making. Second, we introduce a method of "quantitative comparative statics" that in principle overcomes this defect of established comparative static analysis. This methodology incorporates the available degree of empirical knowledge of the firm's structure without imposing further specification of structure (in contrast to, e.g., the traditional linear and nonlinear programming models of the firm, where a full structure must be specified). This degree of knowledge and its relations to comparative static effects of interest can be expressed as a set of quadratic equalities and inequalities. Then the range of quantitative as well as qualitative effects of policy that are consistent with our degree of knowledge of farm structure and the assumption of static optimizing behavior can in principle be calculated by nonlinear programming methods. Third, we consider the issue of the appropriateness of constructs of static optimizing behavior in predicting farm response to policy. We demonstrate that, by estimating an equilibrium shadow price for an input rather than (e.g.) supply response, one can reduce the significance of many of the problems associated with studies of supply response via representative farm models and investigate this issue more clearly. In this manner, we derive empirical support for the use of the construct of static optimizing behavior in predicting the effects of agricultural policy. |
Subject |
Agricultural administration Statics |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Collection |
Retrospective Theses and Dissertations, 1919-2007 |
Series | UBC Retrospective Theses Digitization Project |
Date Available | 2010-03-05 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0094636 |
URI | http://hdl.handle.net/2429/21566 |
Degree |
Master of Science - MSc |
Program |
Agricultural Economics |
Affiliation |
Land and Food Systems, Faculty of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Unknown |
Aggregated Source Repository | DSpace |
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