FORMULATION AND ESTIMATION , . _ Thesis for the Degree ‘of Ph,._'D. (-_ ‘ MICHIGAN STATE UNIVERSITY... ‘ ARSHAD ZAMAN ' v . ' 1970? ‘ L I B R A R Y Michigan State University %M y Major professor WM 0-169 ABSTRACT FORMULATION AND ESTIMATION OF A COMPLETE SYSTEM OF DEMAND EQUATIONS BY Arshad Zaman This dissertation constitutes an investigation into the principles underlying the formulation and estima— tion of complete sets of theoretically plausible demand equations. Upon an analysis of these principles, two alternative parametrizations of the convenient double— logarithmic system of demand equations are suggested. These two systems along with two other systems are esti— mated from five—series data on personal consumption expenditures in the United States. A comparison of the results reveals that the two double—logarithmic systems may indeed be serious competitors to the existing functional forms for demand equations. FORMULATION AND ESTIMATION OF A COMPLETE SYSTEM OF DEMAND EQUATIONS BY Arshad Zaman A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Economics 1970 /-/5’-'To To: Ammi, Abba, Asad, Arif, Amjad, and Ifti ii ACKNOWLEDGMENTS I am deeply indebted to Professor Jan Kmenta for having been a mentor to me during my course of study at Michigan State University. He first suggested the topic of this dissertation to me, agreed to serve as the chair- man of my dissertation committee, and guided me at each stage of its preparation. I shall forever suffer under the burden of his kindness and generosity. I am also indebted to my teachers, Professors Anthony Y. C. K00 and James B. Ramsey, who consented to serve on my dissertation committee. Their criticism of an earlier draft of this dissertation, and their willingness to discuss numerous issues at all times have contributed a great deal to the improvement of this thesis. My intellectual debt to Professor A. Goldberger's widely acclaimed review of consumer demand theory would be obvious to the most casual reader. Indeed, the first chapters of this thesis are to a large extent a mere paraphrasing of some of the ideas found in Professor Goldberger's paper. Although I have attempted to record explicit plagiarisms, I may at this point apologize for any passages where I have failed to give his paper due credit. I can only hope that I have not misinterpreted him in my numerous references to his work. During the course of preparation of this thesis I have had the opportunity of corresponding with Professors Anton P. Barten, Arthur S. Goldberger, and Henri Theil. They have all been extremely patient in answering to my queries, and have truly served to clarify my thinking on several issues. Professors A. P. Barten and Alan Powell were also kind enough to discuss my thesis with me while they were visiting East Lansing as guests of the Econo— metrics Workshop here at Michigan State University. To all of them I am sincerely grateful. Needless to say, I alone remain responsible for any remaining errors. iv TABLE OF CONTENTS Chapter Page 1. THE PURE THEORY OF CONSUMER'S DEMAND . . . . . . l 1.1 Introduction 1.2 Methodology and History The Axiomatic Foundations 1.4 Revealed Preference: A Digression 1.5 Theory of Consumer's Demand .6 Properties of Demand Functions 1.7 Aggregation Theorems 2. SEPARABLE PREFERENCES . . . . . . . . . . . . . 64 2.1 Introduction Definitions and Fundamental Results 2.3 Additive Preferences 2.4 Almost Additive Utility Neutral Want Association 2.6 Indirectly Additive Utility 3. EMPIRICAL MODELS OF CONSUMER DEMAND: STOCHASTIC SPECIFICATION, AND ESTIMATION . . . . . . 94 Introduction The Constant Elasticity of Demand System Rotterdam School of Demand Models Stone's Linear Expenditure System Utility Aspects of SLES General Linear Expenditure System 3.7 Other Models of Consumer Demand (GLES) IAALALI A ALIAS I . Chapter Page 4. A MODIFICATION OF THE DOUBLE- -LOGARITHMIC SYSTEM OF DEMAND EQUATIONS . . . . . . . . 161 4.1 Introduction 4.2 Two Theoretically Plausible Demand Models 4.3 Stochastic Specification and Estimation Data and Variables 4.5 Empirical Results A Comparison of the Models 4.7 Summary and Conclusions BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . 214 vi CHAPTER 1 THE PURE THEORY OF CONSUMER'S DEMAND 1.1 Introduction In this chapter we present the main results of the pure theory of consumer's demand under static certainty. This theory, as it is known today, represents a major triumph of axiomatic methods in the analysis of economic behavior. The current formulation, however, has evolved after a long period of extended debate over numerous and assorted issues. In section 1.2 we have attempted to provide a brief survey of the methodological and other issues that were the subject of this historical debate. The emphasis in this section, however, is more toward history. The theoretical issues have been examined in greater detail in sections 1.3 and 1.4. In section 1.3 we make use of hindsight to provide the current axiomatic formulation of consumer demand theory, while in section 1.4 we digress to examine briefly the theory of revealed preference which historically preceded the current formu— lations. Finally, in sections 1.5 and 1.6, the major results in the pure theory of consumer's demand are de— rived, and in section 1.7, the conditions under which aggregation over individuals and commodities may be car— ried out are formulated. 1.2 Methodology and History The purpose of demand analysis is to explain vari- ations in consumer expenditures by an analysis of cross— sectional or time—series data on consumption expenditures and prices. For the analysis to be meaningful, a model must be built in conformity with the a priori beliefs of the researcher. These a priori beliefs may be specified on an ad hoc basis, or be deduced from a set of axioms that are specified as the maintained hypotheses. The model should reflect these beliefs in terms of restric— tions on the functional form and the parameters of the demand equations. Both the ad hoc specification, and the axiomatically derived restrictions approach have been used in the literature on demand theory; and a wide vari— ety Of demand models have resulted from combining the two approaches in varying proportions. A broad distinction can, however, be made between two classes of models, which for lack of accepted terminology are labelled as "ad hoc models" and "axiomatic models." The ad hoc approach in demand theory is usually credited to the work of Gustav Cassel [1899], [1918], who was the first of the modern economists to revive the approach of Augustin Cournot [1838] and Leon Walras [1874] in viewing demand functions as empirical hypotheses. Un- like Cournot and Walras, however, Cassel expressed a positive distaste for "utility theory" which he considered unrealistic and full of error. Instead, he argued, that a theory of demand could be constructed independent of a utility substructure.l Cassel was not alone in viewing utility theory with some suspicion. Henry L. Moore, credited with being the founder of statistical demand analysis, was yet an— other major economist who espoused the ad hoc approach in demand theory. With the extensive statistical works of Moore [1914], [1922], [1925—26], the ad hoc tradition in demand analYSis was firmly established among empirical ‘ demand theorists. Models of demand were specified on an ad hoc basis and restrictions on parameters were imposed in a similar manner. From a methodological point of View it was unclear what the maintained hypotheses were, and 1The references above are from Stigler [1950], and W01d [1943-44]. The extent to which Cassel was successful in construction a demand theory independent of utility is a matter of debate. Wold [1943-44, Part III, pp. 77—90] has shown that Cassel actually imposed suffic1ent restric— tions on his demand functions to make them logically equi— valent to the Hicks-Allen [1934] indifference curve appriach. StiGler [1950] is of a similar Opinion. Houthakker [$96 ] however, has differed. Taking issue With Wold S [195 ] assertion that "revealed preference theory is a vagiant Of Cassel's approach, Houthakker declares that tiet gzégel weakness in Cassel's approach lies in the factft at'ons did not impose any restrictions on his demand unc i . hence it was difficult to provide an immediate interpre— tation as to what the estimates of the parameters really Signified. Despite these and other weaknesses the ad hoc approach to demand theory went unchallenged until recently.2 In order to evaluate the merits of the ad hoc approach it will be desirable to look at the more attractive and theoretically desirable axiomatic approach that is also available to the demand theorist. Indeed, an analysis of the axiomatic approach and its historical development pro— vides some insight into the reasons that led to the distaste for utility that had characterized the founders of the ad hoc tradition. The axiomatic models of demand have evolved from the classical theory of utility. The roots of this theory can be traced to late nineteenth century European writers. The prominent contributors to this theory were Gossen [1854], Jevons [1871], Walras [1874], Edgeworth [1881], Fisher [1892] and Pareto [1895].3 Unfortunately, the work a 2An example of a recent study in this tradition is the work by Houthakker and Taylor [1966] in which an exten- siVe model of consumer demand is built for the United States econOmy, and the "adding—up" restriction (see section 1.6) is imposed on an ad hoc basis. Since the number ofhcom—l modities is large—7665? eighty), this seems to be t e on y Viable approach. 3For a reference to these authors, and a disigggion of their works, I am relying primarily on Marshall éscu—I Bk. III, Ch. III), Hicks and Allen [1934], N. Eeoigwith Roegen [1936], Wold [1943—44], Wold ln aSSOCIa ::r [1954 Jureen [1953, pp. 81ff. and Notes), and Schumpe I Pp. 1054-1069]. of these writers was seldom unambiguous and was often marred by a pronounced lack of rigor. Moreover, the choice of the word "utility" and the style in which the theory was stated was responsible for a considerable amount of subsequent confusion both with regard to the dependence of utility theory on psychological laws, and also with respect to its philosophical foundations. It was this confusion of economic theory with psychology and philosophy, perhaps, that became the source of discontent with utility theory that many later economists exhibited.4 The situation was not long left unremedied. An- tonelli [1886] was the first to give a rigorous statement of the theory that had evolved up to that time. Other 4The initial confusion with regard to psychology arose in the context of the importance that the early theor— ists attached to the ability that goods possessed to ful- fill basic biological needs. This hedonistic aspect of the "utility" concept rendered it unacceptable to many. It was quite late in the history of utility theory that scholars realized that this was not a crucial assumption. This has been pointed out by Samuelson [1947a, p. 91]. Yet another a8pect of the same issue was the extended debate that arose over the confusion of Gossen's "law of satiable wants" with the Weber-Fechner "fundamental law of psycho-physics" (identical in formal structure to the Bernoulli-Laplace hypothesis about the marginal utility of income, proposed as a solution to the St. Petersburg Paradox). For a dis- cussion see Viner [1925, p. 37lff.], and Schumpeter [1954, p. 1058]. On the philosophical side the ethical and Utilitarian convictions of Gossen, Jevons, Bentham, Sidg- Wick and Edgeworth were the cause of numerous subsequent misunderstandings. On this see Schumpeter [1954, p. 1056]. The fact that utilitarianism had little to do with utility theory was emphatically pointed out by Marshall [1890, Bk. I, Chap. II, Sec. 1, n. 2; and Bk. I, Chap. II, Sec. 1, n. l]. writers were not far behind. The early twentieth century saw three other writers' successful efforts at a synthesis of the then current theory. Johnson [1913] gave a clear statement to the Jevons—Menger—Walras tradition of the posthumously so-called "cardinal" theory of utility; while both Pareto [1906] and Slutsky [1915] reviewed the exist— ing "ordinal" tradition and made significant contributions of their own.5 It was with these papers that two distinct traditions arose within utility theory. The orthodox "cardinal" tradition held on to the assumption that the consumer behaved as if he were maximizing a specific utility function, while the "ordinal" theorists were con— tent with the assumption that indifference curves were well-defined, so that the consumer could be maximizing a class of utility functions that were unique only up to monotonic transformations. The "ordinal" theory, derived independently by Pareto and Slutsky was extensively reviewed and given essentially its current content by Hicks and Allen [1934]. The primary contribution of the "ordinal" theorists was to demonstrate that most of the results derived by the cardinal theorists were accessible to them without 5The references to Antonelli's work are from the sources cited in footnote 2. For Pareto's work I am rely- ing primarily on the extensive discussion in Wold and Jureen [1953]. For the choice of authors to whom the "cardinal utility" tradition can be attributed I am re- lying on the verdict of Stigler [1950]. assuming the existence of a utility function. They assumed instead the existence of "indifference curves,‘ and using relatively elementary mathematical techniques derived most of the restrictions on demand functions that were known. In an attempt to consider the ramifications of the two existing approaches, Georgescu—Roegen [1936] took up the problem of "integrability" considered by Antonelli [1886], and discussed by Pareto [1906] and Slutsky [1915], and tried to derive its implications for the theory of demand proposed by Hicks and Allen. Georgescu-Roegen, in the same paper, made one of the earliest systematic investiga— tions of the axiomatic structure of the Hicks—Allen theory, which subsequently was a topic of considerable interest. On another front, Wold [1943-44], in an extensive paper used Volterra's [1906] formulation of the integrability conditions6 to synthesize the three approaches that he found in economic theory: (1) Pareto's theory of indif— ference maps, (2) The Hicks-Allen theory of marginal sub— stitution, and (3) Cassel's demand function approach.7 Although both Wold and Georgescu-Roegen provided useful analyses of the approaches found in utility theory, and 6For a discussion of the meaning and relevance of these mathematical conditions, see Wold and Jureen [1953]. 7The reference to Volterra [1906] is from Wold [1943-44, Part I, p. 86] and Wold and Jureen [1953, p. 90]. raised important issues, the problem of integrability8 continued to be a blemish on the Hicks-Allen theory. It remained for Samuelson [1950] to demonstrate that the problem of integrability was more of a mathematician's worry than a valid cause for concern among economists. Samuelson demonstrated that the integrability conditions are violated only when we are willing to attribute more than a fair share of inconsistencies to the behavior of the consumer. The investigation into the axiomatic structure of utility theory was carried further by Samuelson [1938a] through the "revealed preference" theory, which was claimed to have freed the theory of consumer behavior from the "vestigial" motions of utility. In effect, Samuelson laid down a set of axioms with regard to the preference behavior of the consumer, and showed (much like Hicks and Allen) that all of the results of the Hicks—Allen formulation were the consequences of his axioms. Following in his footsteps numerous writers experimented with an array of modified axiom sets that were capable of yielding the same results. Although the "revealed preference" theory was a source of great insight into utility theory, it was 81n simple terms the problem of integrability deals with the conditions under which the existence of "indifference curves" may be shown to imply the existence of a "utility function." subsequently revealed that Samuelson's theory was not radically different from the Hicks—Allen theory in logi- cal structure.9 It remained for Uzawa [1960] to provide a synthesis of demand theory by generalizing previous formulations and demonstrating rigorously that the exist— ence of demand functions satisfying certain regularity conditions was implied by, and in turn implied, the existence of a "preference relation" which possessed some specific properties. The consequence of these developments for the empirical demand theorist is that he "can adopt as his maintained hypothesis a definite" axiomatic structure which yields substantial restrictions on the demand functions. With an explicit statement of the maintained hypotheses, it is very clear that the estimates of the parameters of the "axiomatic model" are conditional on. Further, the empirical worker can put to test the maintained hypotheses themselves and thereby test if the empirical data confirm his a priori beliefs regarding the external world. 9Houthakker [1950] showed that the "revealed pre— ference" theory was logically equivalent to the indiffer— ence curve approach, while at the same time generalizing Samuelson's "fundamental hypothesis" so that it would imply integrability, and hence the existence of an ordinal utility function. Subsequently, Arrow [1959] showed that if the commodity space contains all its finite subsets, then the equivalence of Samuelson's "weak axiom" to Houth— akker's "strong axiom" is assured. Finally, Uzawa [1960] extended Arrow's general theorem and derived conditions for the demand function under which Samuelson's "weak axiom" is equivalent to Houthakker's "strong axiom." 10 It was stated that the "ordinal" theory yields mg§t_of the results of the "cardinal" theory without re- sorting to the more stringent axioms of the latter. This is not to suggest, however, that the cardinal theory is without merits. In fact, it is a "quasi-cardinal" approach that is found in the empirical analyses of demand.10 It is somewhat unfortunate that except for very small systems of demand equations (involving very high levels of aggre- gation), the "ordinal" theory does not provide enough restrictions to enable the empirical demand theorist to estimate all the cross-elasticities of demand, the own- price elasticities and the income elasticities. In these situations the pragmatic approach usually taken is to impose introspective a priori restrictions equating several (often all) cross-elasticities to zero. As mentioned above, the cardinal approach is better equipped to handle these problems. The pioneer work is of Klein and Rubin [1947], who proposed a specific functional form for a complete set of demand equations that came to be known as the "linear expenditure system." This system was shown by Samuelson [1948] and Geary [1950] to be implied by a specific utility function, which has come to be called the "Stone—Geary Function." Stone [1954a and 10The rest of the discussion in this section anticipates some of the future discussion, and has been included only to provide a very brief outline of current developments in the literature. 11 others), employed this function to estimate complete sets of demand equations. It was evident that Stone's system yielded substantial economies of parametrization, and re— sulted in estimates of all cross—elasticities without the imposition of artificial restrictions. Although it was well-known that if the utility function could be assumed to be of the Stone—Geary type, substantial economies of parametrization could be obtained, the general case of a quasi-cardinal utility function and its consequence for demand theory were not fully exploited. Frisch [1959] and Houthakker [1960a] were the first to analyze a specific type of utility function that resulted in powerful restrictions on the parameters of the demand system. This was the quasi-cardinal property of "additivity" (or Want—Independence in Frisch's terminology) that was in fact possessed also by the Stone-Geary function. In this sense the Frisch—Houthakker case of "separable" utility was a generalization of the Stone-Geary case. The general— ization from "additivity" to "almost additivity" was im— mediate. Barten [1964] gave rigorous content in terms of the Frishc—Houthakker theory to earlier ideas of Strotz [1957]. Based on these ideas, Houthakker [1960a], Barten [1964] and Theil [1965] proposed specific functional forms for the estimation of complete sets of demand equations. Finally, the work of Pearce [1961], [1964] merits attention. In the tradition of the ordinal theorists 12 Pearce has made a definite attempt to render the strictly ordinal theory as empirically fruitful as the quasi— cardinal variety. The motivation for such an effort may be derived from a critical look at the restrictive assump- tions that cardinal theorists must make. In particular, Pearce objected to the additivity assumption on grounds that it was not invariant with respect to monotonic trans— formations of the utility function.11 He proposed, in— stead, a new form of separability, "neutral want association,‘ which was shown to be invariant under mono— tonic transformations. From this assumption Pearce claims to have shown that substantial restrictions on the demand functions can be Obtained in a manner very similar to the cardinal theorists' models. In fact, Pearce [1964, p. 206] has claimed that all the computational advantages of additivity are available even if we make the much weaker assumption of neutral want association. In the review of empirical models below an analysis of Pearce's claim is made, and there is reason to believe that the case for neutral want association may have been overstated. llSubsequent discussion reveals that this may not have been a valid criticism of the quasi—cardinal approach. This is because the assumption of "strong separability," which is invariant with respect to an arbitrary non—linear transformation of the utility function, results in much the same restrictions on the demand functions as does the assumption of additivity. For further discussion see Uzawa [1964] and Chapter 2 below. 13 To summarize, then, the empirical demand theorist has the option of building an "ad hoc model" or an "axio— matic" model. It has been pointed out that the consensus is that the former approach is less desirable. Of the latter, there exist two distinct traditions. The "quasi— cardinal" tradition results in the most powerful restric— tions on the set of demand equations, although it does so at the cost of restrictive assumptions. The strictly "ordinal" approach is based on more palatable assumptions but does not afford the computational conveniences of the latter.12 1.3 The Axiomatic Foundations As we have noted in the previous section, the foundations of utility theory have only recently been investigated in a satisfactory manner. Prior to this investigation into the aximatic structure of utility, it had been the custom to accept the existence of a well— behaved utility function strictly on faith. Assuming that the consumer maximized his "utility function" which pre— sumably was well-defined and had certain desirable proper- ties, empirical workers conceeded to estimate elasticities 12The advantages of the cardinal models may have been overstated in this section. In subsequent sections it will be pointed out that the three acceptable cardinal models: Stone's Linear Expenditure System, Houthakker's Indirect Addilog Model, and the Rotterdam Model are all non-linear in parameters, so that estimation with restric— tions becomes non-trivial. 14 from available data on expenditures, prices and income. Critics of the approach were at a loss to comprehend the exact implications of such assumptions, and debate over their validity was often confused and universally incon— clusive. A need was felt, therefore, to decompose these assumptions into several primitive components in order both to promote understanding of the behavioral implica- tions of this axiom and to aid in a direct statistical verification, should it seem desirable. The effort is all but concluded with Uzawa's [1960] justly acclaimed synthesis of the investigation into the axiomatic structure of utility theory that began with Georgescu-Roegen [1936], and Wold [1943—44].13 The re- sults have been reassuring on at least two counts. First, it has been shown that a "well—behaved"l4 utility function does in fact exist under reasonable assumptions with re- gard to the properties and existence of consumer prefer— ences. Secondly, it has come to emerge that many of the basic results of demand theory can be obtained by using topological methods, directly from the properties of the preference relation, without any mention whatsoever of a l3Although Georgescu-Roegen [1936] did concern him- self with the axiomatic bases of utility theory, it seems that Wold [1943—44] was the first writer to be concerned with the conditions under which a real—valued, differen- tiable, order-preserving (utility) function can be shown to exist. 14 A definition of "well—behaved" will be given below. 15 15 The theorist thus has his option of utility function. using classical calculus methods, or of defining a pre— ference relation which satisfied certain additional con— ditions, and then use topological methods to derive the same restrictions on the demand equations. Since the economic content of the two approaches is practically the same, the classical calculus method is adopted in this paper. To do so, however, a brief discussion of a set of axioms that imply the existence of a "well—behaved" utility function is given below.16 Consider a single consumer who is confronted with a choice between a finite number of commodities which are labelled i=l,...,n. The quantity consumed of the ith commodity by the consumer is denoted by Xi' A "commodity bundle" will be defined as a n-dimensional vector, with xj as its jth component, and will be denoted by a non—sub- scripted lower case English alphabet, for example x or y. 15See, for example, the literature on revealed preference theory: Samuelson [1938a], [1938b], [1947a], Houthakker [1950], etc. In a more general context, these results are derived by Yokoyama [1953]. It might be noted that the term "topological methods" may be a misnomer. The term is used to describe the methods of higher mathe— matical analysis involving Real Analysis, Topology and Abstract Algebra, as opposed to "calculus methods" by which is meant the elementary calculus approach that is found for example in R. G. D. Allen [1935], and [1956]. 16The remainder Of this section relies heavily on Uzawa [1960], and the extensive discussion of Uzawa's paper, and other formulations by Pearce [1964]. 16 The set of all commodity bundles, C, will be called the "Commodity Space" and will be assumed to be the non- negative orthant of an n-dimensional Euclidean space. It is assumed that there exists a dichotomous "binary relation" P, defined on C, which is called the consumer's "preference ordering" if it satisfies the following additional requirements: Property 1: Irreflexivity. For any x in C, xPx. (where, P denotes the negation of P.) Property 2: Transitivity. For all x,y,z in C, If xPy and sz, Then xPz. Property 3; Monotonicity. For all x,y in C, If x>y, Then xPy (Definition: x>y if xi_>_yi for all i, and for some j, x.>y..) J J Prgperty 4: Convexity. For all x,y in C, If x¢y and xPy, then (l-A)x+xy P x for all 0u(y). The proof of the theorem is omitted due to the fact that no additional insight of economic relevance may be had from it. So far, the discussion has been of a general mathematical nature, with no explicit behavioral content. For the empirical analysis of consumer's demand, however, certain behavioral assumptions are needed. Before specify- ing these, however, we define the concept of an "attainable set of commodities": be constructed. Pearce [1964] has discussed why the "lexicographic ordering" does not seem to be of much empirical relevance. 20 Definition: The "attainable set of commodities" is defined to be a proper subset of the commodity space, C, and is given by {x: p'x = y, x in C} where p is the (nxl) vector of prices that the consumer must pay, and y is the prede- termined amount of total expenditure that the consumer can make, y being a scalar. The formal model of utility theory, then, rests on the following axioms: Axiom 1: (Existence of a preference ordering). It is as- sumed that the consumer possesses a dichotomous irreflexive, transitive, montonic, convex, and continuous "preference ordering" defined over the entire commodity space, C. Axiom 2: (Axiom of Choice). It is assumed that the con- sumer chooses that commodity bundle x, which is preferred to all other commodity bundles in the "attainable set" of commodity bundles. Although these two axioms are sufficient for an analysis of consumer's demand, and the principal results may all be obtained by using methods involving finite differences, it is usually assumed also that: Assumption: (Differentiability). The set of order- preserving utility functions that result from the prefer— ence ordering assumed to exist from Axiom l, are all at least thrice differentiable. The need for this assumption arises from the fact that Debreu's Theorem guarantees only the continuity of the utility functions, and makes no claims about differenti— ability. Having assumed differentiability, classical calculus techniques may be utilized in the analysis of the consumer's problem, and the derivation of demand 21 functions and restrictions upon them become straightfor- ward. On the basis of Axioms l and 2, and the assumption of differentiability, the consumer's problem may be formu— lated as: Problem: Given a set of prices {p1, p2,...Pn3, and a fixed level of total expenditure or income y, the consumer wishes to any utility function that preserves the preference ordering given by Axiom 1; subject to the "budget constraint" p'x=y. Once it is recognized that by virtue of the definition of a "monotonic" transformation, the choice of the specific utility function to be maximized is arbitrary, then the above statement of the problem may easily be seen to be identical in formal content to the classical statement of the consumer's problem. The advantage of the above approach lies, however, in identifying the set of primitive com- ponents of this assumption, and hence in increasing the insight into the behavioral implications of the utility assumption. Before proceeding to an analysis of the problem described above, and the derivation of the classical re- sults of demand theory, we shall digress to discuss very briefly some of the saliant issues in the theory of "re- vealed preference." In conclusion, we may mention here that two topics of current interest have been completely neglected in our discussion so far. These are the topics of dynamic utility maximization and of stochastic utility 22 and uncertainty. Our excuse for this omission is that de— spite their great significance these theories are yet in an embryonic state, and have, therefore, not been a source of many additional empirical tests. 1.4 Revealed Preference: A Digression No discussion of axiomatic utility theory would be complete without a mention of the revealed preference ap— proach, due to Samuelson [1938a, etc.], which historically preceded the simpler and equally general approach outlined in the preceding section. In fact, the general preference relation approach gained much from the discussions of re- vealed preference theory. In this section, therefore, we shall briefly survey the principle ideas underlying re— vealed preference theory, the related debate over transi— tivity, and the reasons why revealed preference theory may be considered to be a special case of the general theory outline in section 2. In the tradition of Gustav Cassel, and Henry L. Moore, Paul A. Samuelson attempted to construct a theory of consumer demand independent of the idea of a "utility" function. The resultant revealed preference approach started out with the assumption that given a set of market prices and income, the consumer selects a unique commodity bundle. Thus, a single—valued demand function, i Xi = h (pll°"lpnrY) (i=1,...,n) 22 and uncertainty. Our excuse for this omission is that de- spite their great significance these theories are yet in an embryonic state, and have, therefore, not been a source of many additional empirical tests. 1.4 Revealed Preference: A Digression No discussion of axiomatic utility theory would be complete without a mention of the revealed preference ap— proach, due to Samuelson [1938a, etc.], which historically preceded the simpler and equally general approach outlined in the preceding section. In fact, the general preference relation approach gained much from the discussions of re- vealed preference theory. In this section, therefore, we shall briefly survey the principle ideas underlying re— vealed preference theory, the related debate over transi— tivity, and the reasons why revealed preference theory may be considered to be a Special case of the general theory outline in section 2. In the tradition of Gustav Cassel, and Henry L. Moore, Paul A. Samuelson attempted to construct a theory of consumer demand independent of the idea of a "utility" function. The resultant revealed preference approach started out with the assumption that given a set of market prices and income, the consumer selects a unique commodity bundle. Thus, a single-valued demand function, _ i . Xi _ h (Plr'--rpn;y) (i=1,...,n) 23 was assumed to exist. Also, by nature of definition, the budget constraint, was assumed to hold. It becomes easy to show that from these conditions alone, we may demonstrate the validity of many of the principle results of demand theory to be derived in the next section. However, the two most import- ant results of demand theory: the symmetry and negative definiteness of the Slutsky matrix, are not deduceable from this assumption alone. To get at these crucial properties of the Slutsky matrix, an attempt was made to work backwards in order to infer the existence of some sort of a quasi-utility func— tion from the axioms of revealed preference.21 Pursuing this approach, it was found that the symmetry of the Slutsky matrix could not be proved from the assumption given above, alone. with regard to the negative definite- ness, however, the revealed preference approach led to some degree of success. To show that the Slutsky matrix was negative definite the revealed preference theorists introduced the so-called "weak axiom" of revealed prefer— ence: 21This interpretation is provided by Pearce [1964, p. 67]. This section has greatly benefited from the dis- cussion of revealed preference theory in Pearch [1964, pp. 24 The Weak Axiom: If a specific commodity bundle x0 is pur— chased at a given set of prices and income, when another commodity bundle x could also have been purchased; then the commodity bundle x1 will never be purchased when x0 may also be purchased at the prevailing income and prices. On the basis of this axiom, we may define a revealed pre— ference relation, R, which is antisymmetric, and is given by: xoRxl if and only if 2i pi(x$ - xi) 3 0, xofxl. Assuming only antisymmetry of R, it can be shown that the matrix of Slutsky terms is negative definite. However, the demonstration of symmetry of the Slutsky matrix re— quires assumptions with regard to the transitivity of R. Historically, the realization that it was necessary to assume the transitivity of the revealed preference re- lation, R, came after great difficulty. The first step was provided by Houthakker's [1950] "strong axiom" of revealed preference, which established the relation R*, where xOR*xS if and only if (s-l) commodity bundles can be found such that xoRxl, lex2,..., xs_leS. The "strong axiom" required that R* be antisymmetric. The motivation behind the introduction of the "strong axiom" are quite simple. The relation R* is transitive by definition. Combining this with the axiomatized antisymmetry of R*, Houthakker [1950] was able to show that a utility function can, in fact, be demonstrated by appealing to Debreu's Theorem, mentioned in the previous section, and noting 25 that R* satisfies all of the requirements of a "preference ordering" outlined in section 2. In fact, Uzawa [1960] showed that both the revealed preference relations R, and R* implied transitivity if the choice (demand) functions satisfied certain regularity conditions. With this demon- stration, it became clear that the relations R, and R* are in fact equivalent to the relation P introduced in section 2. To summarize, then, the revealed preference the— orists pioneered in an attempt to found a theory of con- sumer demand upon certain simple axioms derived from the observation of market behavior. It was found, however, that both the weak and the strong hypotheses depended upon the additional assumption of transitivity. Thus, the general "preference relation" approach of section 2, is seen to be equivalent to the revealed preference rela- tions approach. Indeed, the general approach gains con— siderably in simplicity, without any sacrifice in generality or rigor. 1.5 Theory of Consumer's Demand22 In the previous section it has been shown that the problem of the consumer is to 22This section relies heavily on Goldberger [1967]. 26 MaXimize: u = u(xl, X2""Xn) Subject to: zipixi = y where u = u(xl,...xu) is any utility function which pre- serves a "preference ordering," x. l is the quantity con- sumed of the ith commodity, with price, pi, and y is the predetermined value of total expenditure. In this version, the statement of the consumer's problem corresponds to the "ordinal" model of utility theory. The analysis of demand is usually conducted, how- ever, in the very closely related "cardinal" model, which may be given as follows: Maximize: u = u(xl, x2,...,xu) Subject to: Zipixi = y where the consumer is assumed to possess a specific utility function u(x), which it is assumed that he max- imizes subject to the budget constraint. It is immediate- ly obvious that the latter formulation is far more restrictive than the former. In fact, the validity of the second formulation as a relevant description of the process by which the empirical data under analysis is generated, is highly questionable. Fortunately, the methods of analyzing both formulations are quite parallel. In the following discussion, however, those results that are valid only under the "cardinal" model shall be specifi- cally pointed out. Needless to add, any results that are 27 true under the "ordinal" model are also true for the "cardinal" case, as the latter is a special case of the former. Recall that xi denotes the quantity consumed of the ith good (i=l,...,n); pi the price of the ith good (i=l,...,n); and y denotes the predetermined amount of total expenditure, or income. It is convenient to use a vector notation sometimes, and so the (nxl) vectors x and p are defined as: X1 p1 X2 p2 X = . , and p = Xn pn Given prices p, and income y, the consumer is assumed to select that commodity bundle x, which maximizes the value of an arbitrary (order-preserving) utility function u = u(xl,x2,...,xn) = u(x) subject to his budget constraint: or, p'x = y, in vector notation. 28 By virtue of the properties that consumer prefer- ences are assumed to have in Axiom l, and the assumption of differentiability, the utility function u=u(x) is real- valued, continuous, and thrice differentiable. To develop further notation, ui=ui(x) will denote the first partial derivative of u=u(x) with respect to xi; and u lj=u ij(x) will denote the second partial derivative of u=u(x) with respect to xi and xj. Note that the assumption of thrice differentiability implies that u .=u.., which shall always be assumed to be the case.23 Thijpaiillel matrix notation for these derivatives will be as follows:24 ul ' uln uX = . , U = . un . . . unn 23Alternatively, the existence of derivatives to the second order could have been assumed, with the addi— tional assumption that these second order derivatives were continuous. This slightly weaker assumption would, in fact, be sufficient to insure the equality of the cross partials. For a proof of this theorem in advanced calculus, see Cronin-Scanlon [1967, p. 90, Theorem 7]. The stronger assumption is made for simplicity, though the economic consequences of either assumption are practically identi- cal. 24The matrix development of utility theory, which has proved of great convenience is due to Barten [1964], [1966], and Theil [1965], and [1967]. 29 Two additional properties of the utility function that are a result of the properties that consumer preferences are assumed to have may be noted here. The monotonicity of preferences implies that all first partials of the utility function uj, (j=l,...,n) are strictly positive. Also, convexity of preferences imply that the matrix of second order partial derivatives U is negative definite every— where.25 This implies, incidentally, that ujj is strictly negative for all j=l,...,n; so that each good has diminish- ing marginal utility. Using the conventional Lagrange multiplier method for the solution of the constrained maximum problem, the following function is constructed: L(x,}\) = u(x) — Mp'x - y) where A is the Lagrangian multiplier. Maximizing L yields the solution to the constrained maximum problem. Differ- entiating with respect to x and A, and setting the deriva— tives equal to zero, we get the "first order conditions" (FOC) 25For a proof of the latter property, which, in— cidentally, is sufficient to assure a unique solution to the constrained maximization problem, see Lancaster [1968, p. 333]. 30 Or, in the more familiar non—matrix formulation, these are the (n+1) equations: u. = Api (i=l,2,...,n) (FOC) Zpixi = y (where the sum is from i=1, to n). These first order conditions are the foundations upon which a vast edifice of demand theory has been built. Borrowing the terminology of macroeconomic models, the set of (n+1) equations (FOC) may be construed as the set of "structural equations" of demand theory, while the set of demand equations and the equation expressing A as a func- tion of y and p, may be compared to its "reduced form." To solve for this "reduced form" we first note that the negative difiniteness of the Hessian, U, further guarantees that the set of equations (FOC) may be solved to yield x and A, as single-valued functions of p and y. These functions are denoted as x = h(p,y), and A = A(p,y) Or, in customary algebraic notation: x. = hj(pl,...,pn,y) (j=l,...,n) A = A(pl,...,pn,y) The first n equations expressing xj as a function of all prices and income are referred to as the "complete set of 31 demand equations." It is readily verified that the demand functions, hj(j=l,...,n) are invariant under an arbitrary non-linear monotonic transformation of the utility func- Bxi 3xi 553 , and 5;“ are also "monotonic invariant." These are the equations that tion. Hence, the partial derivatives are estimated by the empirical demand theorist from an analysis of expenditure and price data. Using the Barten-Theil notation, a matrix solution of the complete set of demand equations is possible. The matrix solution can then be easily interpreted in terms of the classical development in which Cramer's Rule was the workhorse for deriving various properties of the demand functions. To do so, some additional notation is required. Since the set of demand functions are also differentiable as a consequence of the differentiability of the utility function, we define the (nxl) income-slope vector xy, and the (nxn) price-slope matrix Xp, as follows: —8 F- 3 _— xl 8x1 . . . x1 3 3 8 Y Pl Pn x = . , and X = . . y C p O O axn an . . . an 3 3 3 Y __pl an Similarly, the A function is also differentiable, and the scalar Ay, and the (nxl) price—slope vector Ap, are de- fined as follows: 32 3A_ Spl Ay = g? , and Ap = E §A_ pn With this notation we may write down what Barten [1966] has called "the fundamental equation of the theory of consumer demand in terms of partial derivatives" as, (1)... Perhaps the best way to see the validity of equation (1) is to consider the sets of equations that result from a partial differentiation of the first order conditions, (FOC), with respect to income, y, and prices, p. Differ— entiating the set of equations (FOC) with respect to y, we get: 3x 3x 8x _1 2 n _ £31 _ u11 ay + u12 ay + ' ' ' + uln 8y Pi 3y ‘ 0 8x 3x 8x 2 n 8A _ unl 3y + un2 8y + ’ ’ ' unn 8y _ pn 3y 0 3x 8x 8x 1 2 n _ pl 53,—: + p2 T + . . + pn 59—- — l 33 This set of equations is represented by the (n+1) equations that result from equating the elements in the first column to the product of the two matrices on the left hand side of equation (1) to the corresponding elements on the right. In a similar way, the system of (n+1) equations resulting from equating the right hand side and the left hand side elements in the (j+l)th column of equation (1), are seen to be: 3x 8x 3;; u + u 2 + ‘ + u n 32 = 0 11 p:I 12 Epj in 35; - Pl 5p] 9x 8x 8x u. _.l + 11. _2+ . o . u. _n — p i = A 31 3p. 32 3p- in 3p- 3 3p- j J 3 3 8x 8x 8x 1 2 3A _ u —— + u —+ - . - + u — — p — — 0 n1 8 . n2 8 . nn 8 . n 8 . 1 p: p: p] p: 8x 8x2 an P1 ‘1an +92 E+...+pn8p_j =-Xj The (n+1)2 equations that constitute the matrix equation (1), can be solved for the (n+1)2 partial deri— vatives. The easiest procedure to adopt is to premultiply both sides of the equation by the inverse of the first matrix on the left hand side of equation (1),26 26The computation is straightforward. Baretn [1964] first adopted this procedure, and gave the formulae for the inverse. Hadley [1961, pp. 107-109] gives the relevant formulaes for the general case of inverting a partitioned matrix of the form above. 34 -l U p (p'U 1pm 1 — (U 1p) (U lpw U l __ I _l '1 P p' O (U p)' -l Carrying out the multiplication, we obtain the solution for the slopes, U_lp (p'U'lp)A.U'1—A(U‘lp)(U'lp)'-U‘1px' -1 A(U_lp)' + x' Reading off the blocks of equation (2), and writing in simple algebraic form we have " —l (3) . . . A = (22 uljp.p.) Y ij 1 3 where ulj denotes the (i,j)th element of the inverse of the Hessian matrix, U. 4 . . . ——&-= A 2. ij . f all ‘=1 ... n. ( ) 3y y( j u p3) or i , , 3X1 ij is jt (5) . . . 55; = Au —A Ay(£S u pS)(ztu pt) is Ay(>3S u ps)xj fOr i,j=l,2,ouo,nn (6) . . . 3%7 = -Ay(AZS ulsps + xi) (i=1,...,n) l 35 The equations (3)-(6) give the slopes of all the demand functions and the A function, in terms of prices and in- come. All of the properties that demand functions possess may be obtained from these equations. Before doing this, however, we shall write these equations in a form that is more usual. This is done by substituting equation (4) into (5) and (6). The revised version of these equations are: (7) . . . A = (2.2 uijp p.)‘1 y 1 j i 3 3xi ij 8 o o o — = A 2| u . I=l 0.0 n ( ) 3y y ( 3 P3) (1 I I ) axi ij A (Bxi)(3x.) (axi) (9) . . . 'a—I')" = All - T W— 43y — Xj F J Y (i,j=l,...,n) 8x. 3A - _ (_l) 1%] -_ (10) u I o apj —‘ B By + Xj \ay (j-l’u-o’n) We might note that equation (10) is called "Schultz's Relation," after Schultz [1938]. These equations have been used extensively by empirical demand theorists; and specially in the case of "additivy" (discussed below) they have proved to be a source of considerable simplification in computational procedures. In the next section we shall discuss the implications of these equations for demand theory. Before concluding this section, though, it ought to be pointed 35 The equations (3)-(6) give the slopes of all the demand functions and the A function, in terms of prices and in— come. All of the properties that demand functions possess may be obtained from these equations. Before doing this, however, we shall write these equations in a form that is more usual. This is done by substituting equation (4) into (5) and (6). The revised version of these equations are: = ij -1 (7) . . . Ay (ZiZj u pipj) Bxi i' (8) . . . 33,—: Ay (>33. u jpj) (i=1,...,n) Bxi ij A (Bxi)(8x.) (Bxi) ‘9"°'ap—.=M‘Ta‘r“‘lay‘xjw J y (i,j=l,...,n) 8x. i _ _ (_1) IE] -_ (10) a c 0 apj "' E 8y + Xj \ay (3-1’000’1‘1) We might note that equation (10) is called "Schultz's Relation," after Schultz [1938]. These equations have been used extensively by empirical demand theorists; and specially in the case of "additivy" (discussed below) they have proved to be a source of considerable simplification in computational procedures. In the next section we shall discuss the implications of these equations for demand theory. Before concluding this section, though, it ought to be pointed 36 out that equations (8) and (9) are a modified version of the Slutsky equation, or in the terminology of Hicks and Allen [1934], "the fundamental equation of value theory." This can be seen by denoting by Kij' the so—called "Slutsky term" A (3*) (“3) (ll) . . . Kij = Au " 5:; ~3-y— 5y— With this notation, equation (9) may be written as Hicks and Allen wrote it, (12) . . . where Kij’ is called the (Hicks—Allen) "substitution ef— fect" and the second term on the right is called the in— come effect" of a change in price on the quantity demanded. From equation (12), and from the fact that the demand functions (and their partial derivatives) are "monotonic invariant" it is readily seen that the Slutsky term, Kij’ is also invariant with respect to an arbitrary non-linear monotonic transformation of the utility function. Hicks and Allen attached great importance to equation (9) because they were able to show that Kij re— presented the change in the quantity purchased by the consumer of the ith good, due to a change in the price of the jth, if the consumer's income was changed so as to compensate for the price change in the sense of keeping utility unchanged. Thus, the consumer's response to a 37 price change could be decomposed into a (Hicks-Allen) "substitution effict" Kij’ and an "income effect" given by the term xj(§§i) Barten [1964], following Frisch [1959] and Houth— akker [1960], has further decomposed the Hicks-Allen "substitution effect" into a "specific substitution ef- fect" and a "general substitution effect." This has been done in response to the realization that there exists a difference between a "specific" substitution of one good for another‘in terms of the ability of the two goods to fulfill the same needs, and the "general" variety of sub— stitution of one good for another due to the change in real income that is accompanied by a price change. These ideas can be expressed more rigorously by an investiga— tion of the "indirect" utility function. The "indirect utility function" is obtained from the "direct utility function" u=u(x), by substituting the demand functions x=x(p,y) into the latter, to yield the value of utility that the consumer derives from selecting the optimal good bundle under prices p, and total expendi— ture y. The indirect utility function is denoted: u = u*(p,y) = u(x(p,y)) Differentiating with respect to y and using the chain rule for differentiation, we have 38 U H. where we denote the first partial of u with respect to xi by u., in the usual manner. Substituting for ui from i the first order conditions (FOC) we have 8x 8x *= __i_ *= _i_ uy Zi(Api) 3y , and uj Zi(Api) apj We now use two results which may be obtained by suitably differentiating the budget constraint in the first order conditions (FCC) and which are formally derived in the next section, 8x. 1 _ . 2i pi 5y— — l . . . (Engel aggregation) 8x. . i _ _ . . . (Cournot aggregation) and, Si pi 55; — xj Substituting these in the equation above, we have (13) . . . u; = A , and u; = eij (j=l,...,n) It is clear, then, that the Lagrangean multiplier, A, is the "marginal utility of income" so that the function A = A(p,y) gives the marginal utility of income as a func- tion of prices and income. These results may now be utilized to analyze an infinitesimal change in utility. Denoting the total differential of the utility function by du, we have 39 du = u* d + E. u? d . Y y 3 3 p3 A dy -AZj dePj using (13)- Similarly, the total differential of A= A(p,y), d , is an dA=Ad+z.——d. yy 3313ij Using Schultze's relation equation (10), we may write this as 8x, 8A dA=A d -z. [A—l+x.—]d. yy 3 By Jay p3 We may now derive two kinds of compensating variations in income. Setting du=0, we get the Hicks—Allen variety of a change in income that is required to offset a change dp in prices, in the sense of leaving utility unchanged, (dy)* = Z. xjdp On the other hand, in the analysis of Frisch—Houthakker decomposition27 of the response to price change, we are concerned with that change in income which will compensate the consumer for a price change dp, in the sense of leav- ing his marginal utility of income unchanged. This is given by equating dA = 0. 27The term is Goldberger's [1967]. 40 8 IAA (S ) j ** = Z. . . (dy) j y y + X] dpj X. 3 (3X3) 2. [ ——— + x.] d . J ¢y 3y 3 p3 Where ¢ denotes the inverse of the "income elasticity of the marginal utility of income" denoted By, and given by28 -l _ _ 3A ¢ :EA‘ (F)A Consider now the total differential dxi of the demand function for the ith good,29 (“1) (“1) dx. = ——— d + Z. ——— d . 1 3y y J 3p p: Substituting from equation (9), we have (a) [. MPH—13’") (“)1 “1‘ W— 9”sz ‘E V 3y ‘X: F dpj 28This is the terminology of Barten and Theil, Our is closely related to what Frisch [1932], [1959] called the "money flexibility" which in our terminology would be equal to E . Houthakker [1960] called ¢y the "income flexibility." 29The argument could have been carried generally for all goods by considering the (nxl) vector of differen— tials of the demand functions, dx. For clarity, however, we have chosen a single good. The extension to matrices is obvious and can be found in Goldberger [1967, p. 17]. 41 With this formulation of the differential of the ith demand flunction and the two concepts of compensation, we may easily analyze the Hicks-Allen and the Frisch-Houthakker decomposition of the response to an infinitesimal price change of the jth commodity dpj. In the Hicks—Allen case, consider a change in the demand for the ith good, (dxi)* in response to the change in the price of the jth good dpj, when the consumer is compensated in the utility sense, so that dy=(dy)*. We have then (8x1) (3x1) dx. * = ——— d + Z ——— d ( 1) 3y y 3 apj p3 and, d * = Z. x.d .. ( y) J J p: (”if So that dx. * = Z. ——— d . ( l) J apj p3, (axi)* ij (axi)(3x.) where 55; = Au — ¢y §§_ §§l = Kij and is the Slutsky term introduced before. Thus, Kij measures the response of the quantity demanded of the ith good to the change in the jth price when income is com— pensated in a manner so as to leave the level of utility unchanged. The total effect of a price change is thus decomposed into a "substitution effect" and an "income effect." Incidentally, goods may be classified in the 42 Hicks-Allen theory as substitutes, independent, or com— pLenents according as Kij is positive, zero, or negative. In the Frisch-Houthakker decomposition, however, we consider a change in jth price and analyze the effect on the demand for the ith good, when income is compensated so as to leave the marginal utility of income of the con— sumer unaffected. Thus, we have (axi) (Bxi) mi)” = '87 dY‘“ 2j ”E dpj 8x. nd d ** = z. [ (——l) + .] d . a ’ ( y) 3 ¢y 3y X] p: So that on substitution we obtain, 3X. ** (dx. ** = 2. (——£) d . l) J apj p3 3x. ** where, (§_£) = .. . 1 p3 J (“1) (no) 6") = — + — + x_ ..— Bpj ¢y 3y 3y 3 3y = Aulj Just as Kij was the Slutsky price slope, Fij is the Frisch30 price slope, and measures the response in the demand for the ith good due to a change in the jth price, when income 30After Frisch [1959, p. 184] who introduced this type of marginal utility of income compensated price re- sponses in terms of elasticities. The term Frisch slope, is used by Goldberger [1967, p. 18]. 43 is compensated in such a manner so as to leave the marginal utility of income of the consumer unchanged. The total effect of a price change has now been decomposed into a specific substitution effect, a general substitution ef- fect, and an income effect. On a suggestion by Houthakker [1960, p. 248] goods may be classified as substitutes, independent, and complements as the specific substitution effect, F is positive, zero or negative. ij’ A schematic representation due to Goldberger [1967, p. 19] may serve to illustrate the two decompositions of equation (9): (axi) ij (axi)(3x.) (axi) = Au - _ _ X- Specific General :gfzit Substitution Substitution L";2§:2:-J Effect Effect L—_Hi:ks:Allen Substitution Effect __I We might note here, that while the Hicks-Allen decompo— sition is unaffected by monotonic transformations of the utility function, the further decomposition of the sub- stitution effect in the Frisch-Houthakker fashion is af— fected by arbitrary non-linear monotonic transformations of the utility function. 44 1.6 Properties of Demand Functions The first order conditions and the equations (7)- (10) that were derived from them, have been used in the previous section to decompose the consumer's response to a price change. More fundamentally, however, these re— sults can be used to derive important restrictions on a complete set of demand equations of the consumer for whom Axioms (l) and (2) and the assumption of differentiability applies. Indeed, as Samuelson [1947, p. 97] puts it, "utility analysis is meaningful only to the extent that it places hypothetical restrictions upon these demand functions." Prior to a formal derivation of these results we shall introduce some concepts that have historically been used in the analysis of demand. Recall that the complete set of demand equations is given by, xi = xi(pl,p2,...,pn,y) (i=l,2,...,n) and the related marginal utility of income function by, A = A (P11P2;---rpnry)- Although the analysis of demand can easily be conducted with reference to the conventional mathematical concept of "slope" of the demand function, or the partial derivative; it has historically been analyzed with reference to "elas— ticities" or logarithmic partial derivatives. We define, 45 then, the (Cournot) "price elasticity of demand" (for good i with respect to the jth price) as, 3(log x.) p. (3x. = 1 = _i_l) eij 8(log pj) xl apj If i=j, then eii is referred to as the "own—price elas— ticity" and otherwise as the "cross—price elasticity." In a similar fashion we may define the "price elasticity of the marginal utility of income" as, 8(log A) eAj = 8(1og pj) Where these (Cournot) price elasticities measure the per- centage uncompensated change in the demand for ith good with respect to the percentage change in the price of the jth good, the alternative (Slutsky) "price elasticity of demand" is defined as, (pi i' A Bxi 3x. = (u n - (—(—)(4) 1] A 8y 8y (axi) (Bxi) = ——— + . ——— apj Xaay as before. From the analysis of the previous section, it is easily seen that sij measures the utility—maintaining income-compensated percentage variation in the demand for 46 thezith good with respect to the percentage change in the jth price . Also, we define the (Engel) "income elasticity of demand" as, 3(log xi) .1. (axi) = x . i = 8(109 y) 1 3y where, Ei measures the percentage change in demand with respect to a percentage change in income. Similarly, we define the "income elasticity of the marginal utility of income" as, E = _a.<. 2912).. = y. _a_ A 3(109 Y) A By ' We may note, once again, that EA is the celebrated "money flexibility" parameter used by Frisch [1932], [1959], and measures the percentage change in the marginal utility of income associated with a percentage change in income. In addition, we define the "budget share" of the ith good, W. l, as Note, that purely by definition, the following relation (called "Slutsky's relation") connects the Cournot and Slutsky price elasticities: Slutsky's Relation: s.. = e.. + w. E.. ij ij j l 47 Finally, we define the "elasticity of substitution" be- tween goods i and j, as With these concepts in hand we are ready to derive all of the known restrictions on a complete set of demand equations. The proof of these results may be obtained from the first order conditions, (FOC), directly, or from their solution in terms of the slopes given by equations (7)-(10). The procedure adopted is to give the most simple proof of these results. As a matter of notation, these restrictions are numbered from (R1) to (R7). Recall, the first order conditions, (FOC), u. = Api (i=1,...,n) (FOC) . . . The last of these (n+1) equations that must be fulfilled by a complete set of demand equations is (R1) . . . 2i pixi = y (Adding-up restriction) By differentiating partially the (n+1) equation of (FCC), called the "budget equation," and by multiplying and divid- ing by y/xi, we get (R2) . . . Zi wiEi = l (Engel aggregation) I 'fl’Y; ‘ 48 Alternatively, we may obtain the same result by multiply— ing equation (8) by pi and summing over i, and substitut- ing for Ay from equation (7). The third restriction, "Cournot aggregation" is similarly obtained by differen— tiating partially with respect to pj, the same "budget equation" and multiplying and dividing appropriately to convert slopes into elasticities and collecting terms, (R3) . . . E. wie.. = -w. (j=l,...,n) (Cournot aggregation) Alternatively, this result can be derived by multiplying (9) by pi, summing over i, substituting for (R2), multiply- ing and dividing appropriately, and noting that the inverse of the symmetric matrix U, is itself symmetric. The fourth restriction is the symmetry of the Slutsky terms, Kij’ and can most easily be proved by not- ing that U-1 is a symmetric matrix, so that ul3=ujl. This gives, ij A (axi)(3x,) Kij = Au "W “a? ”lay .. 8x. 8x. - ._ (—a) (—l) Ay y 3y = K.. 31 Recalling the fact that the Slutsky elasticity is given by, p. = —l K X. S.. .. l] l l] h I 49 we may state the symmetry condition in elasticity form: (R4) . . . wisij = szji (i,j=l,...,n) (Symmetry Relation) The restrictions (R2) through (R4) are the main content of the classical theory of consumer demand. They constitute independent restrictions on a complete set of demand equations and are responsible for the considerable economy of parametrization which results when it may be assumed that the consumer behaves in a manner which satisfies Axioms (l), (2). Additional restrictions that are not all independent of these may also be derived from these conditions. The "homogeneity condition" may be obtained by observing that if prices and income were multiplied by the same arbitrary constant, the first order conditions, (FOC), remain unaffected by proportional changes in income and prices. Stated in a more rigorous way we have (R5) . . . z. e.. = -E. (i=l,...,n) (Homogeneity) which states in elasticity form the condition that demand functions are all homogeneous of degree zero in income and prices. This suggests that a proof of the result (R4) may be obtained by applying Euler's theorem after estab- lishing the homogeneity of degree zero of demand functions. More rigorously the result may also be proved by multiply— ing equation (9) by pj, summing over j, and substituting 50 conditions (R2), (R3). This proof points out the fact that the "homogeneity condition" is not independent of (R1), (R2), and (R3), from which it can be derived. In this sense it imposes no additional constraint on the demand functions. A familiar result, that has been utilized by Pearce [1964] as a restriction on a set of demand functions is (R6) . . . 2i piKij = Zj ijij = 0. The validity of this result may most readily be seen by multiplying equation (9) by pi, summing over i, and using Engel aggregation to clear terms. Repeating the same pro- cedure for pj, the truth of (R9) can be established. A somewhat longer procedure adopted by Pearce to prove the same result may perhaps be more intuitively appealing. The term ZipiKij may be proved equal to zero by summing the two equations that may be obtained by (1) partially differentiating the budget equation with respect to pj, and (2) partially differentiating the budget equation with respect to y and multiplying by xj. The second term x Zj ijij can be shown to be zero by substituting for 55% from the Hicks-Allen fundamental equation of value theory into the homogeneity condition. It might be noted that a classical consequence of this condition is that not all 51 goods in the budget of the consumer can be complements in the Hicks—Allen sense.31 A final result, which is stated without proof,32 is that the (nxn) matrix of Slutsky terms, Kij’ is nega- tive definite. The reason for this is simply that due to our assumption of convexity of preferences, the Hessian matrix U is negative definite. We state the result form— ally, (R7) . . . K is negative definite, where we have denoted by K the matrix of Slutsky terms. We might point out that from equation (2) we have: K = AU"l — A AY(U—lp)(U—lp)' Alternatively; (R7) may be stated as, K11 K12 K <0’ >0, etc. K ln K 21 22 which is the familiar result that the principal minors of a negative devinite matrix alternate in sign, beginning with the first minor being negative. 31The fact that no such condition holds for substi— tutes points to an asymmetry in the Hicks—Allen theory that Houthakker [1960] has called a "minor blemish." It was in response to this aspect of the Hicks—Allen theory that the further Frisch—Houthakker decomposition of the total substitution effect was proposed. It should be noted that the definition of substitutes and complements on the basis of the sign of the specific substitution effect suf- fers from no such bias. 32 For a proof see Pearce [1964, pp. 54—7]. 1 I“- 52 The restrictions (Rl) to (R7) constitute all of the known restrictions on a complete set of demand equa- tions, and have been utilized with great effect by empiri- cal researchers in demand theory. We might point out that in this section we have stated all of these restrictions in Slutsky price elasticity form instead of the usual Cournot form. The only exceptions being the homogeneity condition (R5) and Cournot aggregation (R3). This has been done because it is the Slutsky elasticities that are usually estimated by researchers. The reason for this is immediately obvious when we consider the Symmetry condi- tion as it is often stated, in Cournot form: E. + (w.)-le.. = E. + (w.)_le.. i j ij j 1 31 The validity of this can be seen by direct substitution of the Slutsky relation in (R4). Due to the importance of the Slutsky elasticity in actual empirical practice, we note two further results that are the direct consequence of Cournot aggregation, the Homogeneity condition, Engel aggregation, and the Slutsky relation: (R5') . . . Zj sij = O (i=1,...,n) (Slutsky homogeneity) (R3') . . . Z = 0 (j=l,...,n) (Slutsky aggregation). .w.s.. i i ij 53 1.7 Aggregation Theorems In the previous sections a considerable amount of theory has been developed with regard to the single con— sumer's demand for basic goods. It has been shown that demand functions resulting from utility—maximizing behavior must obey certain restrictions. It is unfortunate, how— ever, that typically the data that are available are on a community or nation's expenditure on aggregated "expendi— ture categories" which subsume within them a large number of basic goods. It is meaningful to ask, then, what additional assumptions are required before we can apply the theory developed so far to the aggregate data that is available. In answer to this question, we give in this section the necessary and sufficient conditions under which, (1) basic-goods may be aggregated into "composite" goods in such a way so as to insure that the aggregate demand functions possess all of the desirable properties of the micro demand functions; (2) single-consumer demand functions may be aggregated into community demand func— tions with the latter possessing all of the properties of the former; and (3) individual utility functions may be aggregated to community ”behavior function" such that the latter gives rise to a community demand function which is both consistent with, and possesses all of the properties of, the individual demand functions. 54 Of these, the conditions pertaining to the aggre— gation over commodities is the most straightforward. To restate the problem, we seek a procedure whereby the vast number of basic commodities that enter the individual's utility functions may be condensed into a fewer number of composite commodities that represent sets of the elementary commodities. In doing so, we must obviously sacrifice some of the information that the micro data contain. For aggregation to be "consistent" however, we require that this sacrifice of information should not affect the re— sults obtained. More precisely, aggregation will be said to be "consistent" when a knowledge of the macro relations and the values of the independent macro variables leads to the same values of the dependent macro variables as would be obtained if all the micro relations and the values of all the independent micro variables were known.33 Specifically, in the context of utility theory, it is useful to think that the consumer maximizes utility in the following manner: he first allocates the total expend— iture between composite goods by reference to the price indices of these goods; and then allocates the expenditure on each category among its primitive components, the basic 33These ideas are expressed by H. A. J. Green [1964, pp. 3-5, 35] who provides an excellent survey of the literature on aggregation. As will be obvious, much of this section relies on the initial chapters of Green [1964]. 55 goods, by a reference to their prices.34 In this frame- work an aggregation procedure will be said to be "consist- ent" if there exists a quantity index and a price index for the composite commodity such that, (1) maximizing a utility function whose arguments are the quantity indices gives rise to aggregate demand functions "consistent" in the general sense defined above; and in addition (2) the product of the price and quantity indices for the composite commodities gives rise to the same value of expenditure on the composite commodity as would be obtained by summing the expenditure on each of the elementary commodites con— tained in the composite commodity. Before starting the formal result, it would be useful to give the following definition: Definition: A function u = u(xl,x2,...,xn) is said to be "weakly separable" if and only if the commodities (xl,...,xn) can be partitioned into groups G1, G2,..., and when 8(u./u.) -——%§—l— = 0, for all i,j 6 GS, k t GS, where ui and uj denote the first partial derivatives of u with respect to xi and xj respectively, as before. We are now in a position to state the basic theorem on the aggregation over commodities: 34This is the celebrated "utility-tree" concept proposed by Strotz [1957]. Strotz [1957, 1959] and Gorman [1959a] analyzed in detail the consequences of this two— stage maximization procedure using the concept of "func— tional separability" due to Leontieff [1943] and Sono [1961]. In fact, Strotz and Groman anticipated many of the useful results that were later established by Frisch [1959], Houthakker [1960], and Barten [1964]. These latter results are discussed in Cahpter 2. 56 Theorem: The necessary and sufficient condition for a “consistent" aggregation of goods is that the utility function be weakly-separable, and that each quantity index be homogeneous of degree one in its elementary commodities. For a proof of the theorem, the reader is referred to Green [1964]. A more complete proof is provided by Gorman [1959a] and Strotz [1959]. To gain an intuitive insight into this theorem we note two facts. The first condition requires weak separability of the utility function, which may be argued for on grounds of the assumption of the two- stage maximizing procedure for the individual consumer. Given weak separability, the only other condition is that there should exist the possibility of constructing a quantity index for the composite commodity such that a proportionate change in all of the quantities of the ele— mentary commodities gives rise to a change in the quantity index of the same proportion. We see, then, that it is 35This is Theorem 4 in Green [1964, p. 25], and is originally due to Gorman [1959a] and Strotz [1959]. Gor- man and Strotz offered three alternate conditions that were also necessary and sufficient for consistent aggre— gation. These were: (i) that there be only two composite commodities; or (ii) that the utility function be weakly- separable with respect to a partition of the goods into two groups, with one group consisting of a single commodity and the other group consisting of composite commodities; where it should be possible to construct quantity indices for the composite commodities which are homogeneous in their arguments; or finally, (iii) that the utility func— tion be strongly separable. It should be noted that the condition we have stated in the theorem is a generalized version of the celebrated "Composite Commodity Theorem" anticipated by Leontieff [1936] and established by Hicks [1939]. The Leontieff—Hicks case of price proportionality is a special case of this theorem. As a parallel degenerate case, with limited economic relevance, Green [1964, p. 25] cites the possibility of quantity proportionality among the elementary commodities. 35 57 not unreasonable to consider that the consumer maximizes a utility function whose arguments are quantity indices of composite commodities, subject to the aggregate budget constraint which limits the expenditure on all composite commodities to total expenditure. This procedure gives rise to demand functions for the composite commodities which possess all of the properties possessed by the demand function for the elementary commodities. These demand functions, however, are for the single-consumer. We seek now to find conditions under which these functions may be aggregated over individuals to give rise to community demand functions. In the case of aggregation over individuals, how- ever, there does not exist a procedure similar to the case of aggregation over goods. The temptation exists, never- theless, of attempting to find conditions under which util- ity functions may be aggregated over a community to give rise to a community "behavior function."36 Indeed, 36This approach trespasses on the domain of welfare economics. The "behavior function" has, however, only limited welfare implications; for it cannot be unequivo- cally asserted that social welfare has increased if the aggregate of utility increases. For a discussion of wel- fare issues see Pearce [1964, pp. 127-132] and Green [1964, pp. 55-7]. Our interest in the existence of a community behavior function is restricted to the possibility of an additive-type community behavior function which may result from similar individual utility function, thus resulting in the same economies of parametrization for the community demand functions as are available for the individual de— mand functions. Thus we are interested in the community behavior function only to the extent that it might result in all of the "cardinal" properties of micro demand func- tions in the macro demand functions. 58 Samuelson [1956] motivates his discussion of social indif- ference curves by their bearing on community demand curves. The temptation always is to define a social-welfare func- tion, somewhat like a "composite" utility function, so that by direct appeal to the results derived for the single-consumer theory, we may easily deduce the same re- sults for the community demand theory. The efficacy of this approach has been a source of some controversy, and in this dissertation a position will not be taken on the relevant issues. Instead, we shall take a somewhat neglect- ed path which leads surprisingly to conditions quite similar to the existence conditions for a community behavior func- tions. Without recourse to a community utility function, we might ask under what conditions will the aggregate de— mand functions possess the properties of the single-con- sumer demand functions. Specifically, we seek necessary and sufficient conditions under which community demand functions satisfy the restrictions (R1) to (R6) that individual demand functions were shown to possess, in section 1.4. To do so we develop some notation37 as follows: Let the subscript i=1,...,n refer to goods as usual; and the superscript k=1,...,K refer to individuals. Denote 37This line of attack was adopted by Roy [1952], whose notation we adopt, with slight modification. 59 the demand function for the ith commodity for the kth individual: k k k . xi = Xi(p1’°"'pn’ y ) for all i,k where yk refers to the kth individuals income. Defining community aggregates of demand for the ith good and com- munity income respectively in the natural fashion: we denote community demand functions: xi = xi(pl,..., pn, y) for all i. To facilitate the exposition we define the following parameters: xk k k k k___ i k_y k_3(1ogy)_ (3y) Finally, if we denote as usual the Cournot price elastic- ity, the Slutsky price elasticity, the Engel income elasticity, and the budget share of the ith commodity in the community demand function respectively as eij Ei’ wi; and their counterparts for the kth individual's demand function by e3 Sij’ BE, WE; then it is easy to ij' i verify that the following relations hold between the macro SDI I le elasticities and the micro elasticities: 60 _ k k .'. E. = z Bf bf dk for all i,k i k i i w = Z w wk ck for all i,k i k i i _ k k k -l - (wi c )(bi) With these definitions, we state the two main results: Theorem: The community demand function is homogeneous of degree zero if and only if dk = 1. Also, Theorem: The Slutsky term is symmetric in the community demand function if and only if ckdk = b? for all i. (dek = bk => 5.. = Z 8*. bk ) l ij k 13 i It can also be shown that if the community demand function is homogeneous of degree zero, and possesses symmetric Slutsky terms, then the Adding-up restriction, Engel Aggregation, Cournot Aggregation, and the condition (R6) ZipiKij = szjKij = O are fulfilled. As the proofs of these results are trivial we do not state them.38 An intuitive interpretation of these results is immediately available. The first theorem states that if the community demand functions are to be homogeneous of degree zero, as they must be, then it is necessary and 38Roy [1952] proves the first Theorem. 61 sufficient that the "elasticity of income distribution," to use Roy's term, be unity. Alternatively, we must. assume that the proportion of national income held by every individual is unaffected by changes in the national income. This is reasonable to expect, for the community demand curve is a function of the prices and community in- come, while the individual demand functions admit an extremely large number of income variables yk, k=1,...,K. The price that is paid for aggregating all of the income variables is, as we should expect, paid by the assumption that income distribution of the community remains un— changed. The requirement for the symmetry of Slutsky terms is far more stringent. In combination with the requirement of no change in income distribution, the condition of the second theorem is in fact equivalent to the restriction that W? = wi(i=l,...,n; and for all k); or, the restriction that every commodity has the same share of every consumer's budget. This is indeed an extremely stringent restriction for it implies that every consumer allocates his expendi- ture in identical proportions between the commodities in his budget: the actual quantities of each commodity purchased being dependent only on his income. This im- plies, incidentally, that the Engel curve for each com— modity is a straight line through the origin, and is identical for each individual. 62 Interestingly enough, these are the very conditions on Engel curves that Gorman [1953] has shown to be suf- ficient for the existence of an aggregate of utility func- tions, the community "behavior function" mentioned in the beginning of this section. We have come upon these con- ditions, however, in a more direct manner, and have avoided the relatively difficult proofs that are needed to establish the consistent aggregation of utility func— tions. For completeness, however, we state the theorems due to Gorman:39 Theorem: (Necessity) A necessary condition for consist— ent aggregation of individual utility functions is that all Engel curves are straight lines, parallel for each individual. Theorem: (Sufficiency) It is sufficient for consistent aggregation of utility functions that all Engel curves are parallel straight lines through their origin, for each individual. To summarize, then, aggregate demand functions possess all of the "ordinal" properties of the single- consumer, basic—good demand functions if it may be assumed that the community's income distribution remains unchanged, and also that each individual's Engel curves are straight lines through the origin, and are identical. We have also shown that if the last property of Engel curves may be assumed then a community behavior function may be 39These occur as Theorem 9 and Theorem 10 respec— tively in Green [1964, p. 47, p. 49] where a proof is also given. 63 constructed by aggregating over the individual's utility functions. Since the aggregation procedure assumes the summing of functions of individual utility functions, we may conclude that if all of the individual utility functions were separable with respect to the same partition, then the community behavior function is also separable with respect to the same partition. Thus, all of the "cardinal" properties resulting from additivity—type assumptions on the individual utility functions carry over to the community behavior function. This has the effect of imposing the same parametric restrictions on the aggregate demand functions. Thus, the community's demand for composite commodities may be treated identically to the single— consumer's demand for elementary commodities if we assume no change in income distribution, and linearity through origin of all Engel curves. CHAPTER 2 SEPARABLE PREFERENCES 2.1 Introduction The set of restrictions on a complete set of de- mand equations that have been derived in section 5 of the previous chapter have been used with great effect in the analysis of data on consumer demand. The major difficulty that arises in the estimation of a complete set of demand equations is the large number of free parameters to be estimated. For n commodities it is required that esti- mates of (n2+ n) elasticities be provided. A preliminary reduction in these can result by imposing the independent restrictions of Engel aggregation, (R2), Cournot aggrega— tion, (R3), and Symmetry, (R4), derived in section 1.5. This reduces the number Of free parameters from (n2 + n) to a substantially less, yet fairly large number of l/2(n2 + n - 2). For example, consider the moderate case of n=lO, for which the imposition of these classical re— strictions reduces the free parameters from 110 to 54, which is still large in comparison to the number of ob— servations that are usually available. 64 65 In the light of these facts a need has been felt to impose additional restrictions on the consumer's pre- ferences in order to further restrict the number of free parameters to be estimated. The next efficacious course, and one that has proved of great importance, has been to incorporate the quasi-cardinal assumption that the con- sumer's preferences are "separabel" in some manner. In this chapter, then, we present a brief survey of this theory of separable preferences. In the following section (Sec. 2.2) we define the alternative separability assump- tion and state briefly their implications for the Slutsky terms of the demand equation. Subsequent sections are devoted to a more detailed discussion of the implications for complete sets of demand equations of the assumptions of additive preferences, almost additive preferences and neutral want association, respectively. Finally, we discuss the case of indirectly additive utility in the last section, Sec. 2.6. 2.2 Definitions and Fundamental Results Before defining "separability" of consumer pre- ferences in a rigorous manner, we might examine intuitively the motivation behind such an assumption. Strotz [1957], [1959], and Gorman [1959a,b] expressed the belief that the total utility that the consumer derives is the sum of the utilities of the "branches" of utility, which may themselves 66 be split up into further branches. This View of utility as a "tree" reflected the belief that the set of all commodities in the consumer's budget could be partitioned (perhaps repeatedly) into groups of commodities in such a way so as to insure that price changes outside any particu- lar group failed to affect the marginal rates of substitu- tion between goods within the group. Intuitively, this implies that the consumer's allocation of expenditure between several goods in one group of commodities, say food, is unaffected by changes in the price of goods outside the food group. On a priori grounds, these assump- tions do not appear unreasonable for broad aggregates of goods. In order to give rigorous content to these ideas, we first define the concept of a "partition" of a set. In doing so, we assume familiarity with the defines of a "set," the operations of "union" and "intersection," the "empty set" and the definition of "mutually exclusive" sets. We have then: Definition: A Partition, P, of a set N, is a set of sets, P = {N1,N2,..., Nk}; where the Ni (i=1,...,k) are mfitually exclusive subsets of N, whose "union" is N, i.e., .U Ni = N. Now we assume that the n goods can be partitioned ihio groups, Gl’GZ""’ Gk' with the number of goods in group Gi being denoted by ni, for all i=1,...,k. Of course, the sum Xi ni = n. We introduce also the notation x1 to denote 67 an (nix 1) vector of goods in the ith group, Gi' Goods within a group will be identified by a lower subscript, thus x; denotes the jth good in x1. Denoting the utility functions, once again, by u(x), and the marginal utility of the ith good by ui, we have:40 Definition: u(x) is strongly separable with respect to a partition, P, if and only if 3(ui/u.) __§§__l_ = 0 V, IENS, jsNt (s#t) k k¢(NS U Nt) Definition: u(x) is weakly separable with respect to a partition, P, if and only if, 3(ui/u.) “‘"SE—L = 0 V i, jeNS; k¢N k S It is easy to see that both strong and weak separability are invariant with respect to an arbitrary non—linear monotonic transformation of the utility function. Definition: u(x) is Pearce-separable with respect to a partition, P, if and only if it is weakly separable with respect to P, and is strongly separable with respect to a pointwise partition of the elements of P. In other words, u(x) is Pearce separable with respect to P, if and only if, 3(ui/u.) 3x = O Vi,j€NS; k#i,j k Once again, it is readily seen that Pearce-separability is a "monotonic invariant" concept. 1This section is but a paraphrasing of Uzawa's [1964] paper which has so succintly related the various concepts of separability, in addition to providing a proof of the necessary and sufficient conditions on the Slutsky terms under "neutral want association." 68 The consequence of both weak and strong separa— bility, (and hence of Pearce-separability,) both in terms of the functional form of the utility function and of the Slutsky terms have been proved by Uzawa [1964]. As the proof is mathematically involved, and offers little in- tuitive understanding, we have omitted it, but we state the principal results derived by Uzawa. (In our state- ment we have actually combined two theorems into one.) Theorem: u(x) is strongly separable if and only if either (i) u(x) = F(ul (x 1)+ . . . + uS(XS)) or (ii) W)( %) V ieNS, jeNt(s#t) & for all x, k(x) being some function of x. where, for the second condition it is assumed that u(x) is strictly quasi-concave; and the theorem holds in both cases for s>2. Also, a similar theorem for weak separability: Theorem: u(x) is weak1y_separable if and only if either (i) u(x) = F(ul(xl), . . . , us(xs)) .. st (8X1) (11:1) . . or (ii) Kij = k (x) §§—- 3y VieNs, jeNt(s#t) & for all x, kSt (X) defined for s#t. where, once again, for the latter condition it is assumed that u(x) is strictly quasi-concave. 69 With the aid of these theorems, and the definition of Pearce—separability, it is readily seen that: Theorem: u(x) is Pearce—separable if and only if, 3(ui/u.) either (i) ————3— = o i,jsN; k7£i,j Bxk s or (ii) u(x) = F(fl(ull(xi)+...+ulnl(xl), ... , nI fS(uSl(xi)+...+usnS(xS ) ) n 5 st (Bxi)(3x.) or (iii) Kij = k (x) 8y— 5;; IENS, jENt; & for all x, kSt(x) defined for all s,t. where for (iii) u(x) is assumed to be strictly quasi- concave . 2.3 Additive Preferences The pioneering works in the theory of separable preferences are those of Schrotz [1957], [1959], Gorman [1959], Frisch [1959] and Houthakker [1960]. In particu- lar, Houthakker [1960] considered the case of "additive preferences," (or "strongly separable" utility, in the terminology of Sec. 2.2), i.e., the case where the con- sumer's preferences could be represented by (at least one) "41 utility function which was "directly additive. A 41Alternatively, this could be called "complete want-independence" in the terminology of Frisch [1959], who defined "want—independence" between two goods as the case where the marginal utility of one good does not depend on the quantity of the other good. 70 utility function will be said to be "directly additive" if and only if42 u = u(x 1n) = u(xl)+...+u(xn) lloo.’ Now, since the Slutsky equation and the complete set of demand equations are all invariant with respect to an arbitrary non-linear transformation of the utility function, we may proceed to analyze the case of "additive preferences" by considering that specific utility function which is directly additive. We must remember, however, that this additive (canonical) form of the utility function is chosen only for convenience, and all of our results must be checked independtly for monotonic invariance. To proceed, then, if utility is directly additive then the Hessian matrix of the utility function, U, is diagonal, so that the inverse, U—l, is also diagonal, with elements: (14) . . . u. — 42Note that "direct additivity" of the utility function is not the same as "strong separability" or its equivalent "additive preferences." The case of strongly separable utility" or "additive preferences" exists if and only if the class of monotonic transforms of any utility function representing the consumer's preferences contain at least one "directly additive" utility function. 71 The major effect of additivity is that all price elasti— cities may be derived from income elasticities and the (inverse of) the income elasticity of marginal utility of income, or in Frisch's term the "money flexibility" parameter. To prove this it is helpful to rewrite the solutions to the slopes of the demand equations and the marginal utility of income function that are given as equations (7)—(10), in terms of elasticities. These may be written as: -l .. _ 3(lo A) _A 13 (15) . . . ¢ — I:8 log y)] — y (ZiZj u pipj) ¢ .. —l 8(log x.) A ij _ i = (2. u p.) (i=1,...,n) (l6) . . . Ei — 3(109 y Xi j j 8(1og xi) (l7) . . . e.. = —3———-———- = n.. - ¢E.E.w. — E.w. 13 (log pj) 1] 1 J J l J Auijp. (i,j=l,...,n) where, n.. = ____.l_ = s.. + ¢E.E.w. 13 xi 1] l j j = 3(10 A) = _ + —1 .= (18) . . . eAj 3(109 pj) wj(Ej ¢ ) (j 1,...,n) Note now that, Z. n.. = Z. s.. + ¢(Z. w.E.)E. J l] J l] 3 3 J 1 where use has been made of Engel aggregation, (R2), and Slutsky homogeneity, (R5'). Also, in the case of direct additivity, 72 Z. n.. = n.. j 1] ii So that, we have for the price elasticity of demand under directly additive utility: 2 ._. ¢Ei — ¢Ei wi - Ei wi i—j (l7DA)... eij = - E.E.w. - E.w. i ' ¢ 1 3 J l J #3 Or, expressing (17DA) in terms of the more conventional expression for the Slutsky price elasticity: 2 . ¢Ei - ¢Ei wi i H (.l. S.. = 13 — OEiijj i¢j The power of these results is immediately obvious. From the general case of (n2 + n) unknown free coefficients that are required for the estimation of a complete set of demand equations, the additivity restriction reduces these to the n unknown income elasticities and the "money flexi— bility" parameter, for a total number of free parameters of (n+1). Although equation (l7DA) expresses concisely the major implication of direct additivity, we record for completeness some of the results stated by Houthakker [1960] in his original derivation. In elasticity form, the fol- lowing results may be derived by simple manipulation from (l7DA): (19) i = —T (i,jI‘k) which states the fact that under direct additivity, the ratio of Cournot price elasticities of two goods with respect to the price of a third good is equal to the ratio of the Engel elasticity of the two goods. This is equa— tion (1) in Houthakker [1960]. Also, Houthakker's equation (11) may be derived from equation (l7DA), or more easily from the expression for sij derived from (l7DA), by noting that Ki' is, by J x. definition, equal to (pi)sij' This is the result of the j fact that the Slutsky term, Kij’ is proportional to the product of the income slopes, under direct additivity: . (no) -- (2°) Kij = ' I; '3? 3y ”‘3 Note that this result is the slope form (as Opposed to the elasticity form) of our expression for sij'43 43It is sometimes asserted (mistakenly) that the relationship (20) is not "monotonic invariant." That this is not so, is readily seen by defining the "canonical money flexibility" as 3x. 8x. Since K.. and (u = Ki./(3—£) (—l)). 13 J y 3y the income slopes are monotonic invariant, so is U. Of Course, under strong separability, the "cononical money flexibility" is equal to the "money flexibility," A/Ay, and for the canonical (or additive) form of the utility function. Thus, while the "canonical money flexibility" is independent of the choice of the utility function, the "money flexibility" is not. Mil, 74 The result (20) is important from another view— point. Houthakker's Theorem 1 establishes the condition (20) as being necessary and sufficient for utility to be directly additive. We note the theorem formally without . . 44 giVing a proof: Theorem: (Houthakker, 1960a) The utility function is directly additive if and only if the Slutsky term, satisfies equation (20). Kij' To conclude, direct additivity is a forceful re- striction on a complete set of demand equations and results in the utmost economies of parametrization. These econo- mies are not costless, however. Houthakker [1960] has pointed out that direct additivity rules out specific substitution: uij = O for i#j. Theil [1967, p. 199] has proved that direct additivity rules out inferior goods. Also, direct additivity rules out complementary goods, as pointed out by Goldberger [1967, p. 31].45 As a result, direct additivity is a meaningful hypothesis only when 44A more rigorous statement of the theorem and a proof are given by Uzawa [1964, Theorem 4, p. 392]. In fact, Uzawa has shown that the utility function is "strong- ly separable" if and only if the Slutsky term is propor- tional to the product of the income slopes. Of course, the factor of proportionality in the latter case is no longer the same as in Houthakker's theorem. Incidentally, Uzawa [1964, n. 6, p. 392] seems to have confused "strong separability" with "direct additivity" and has come to the conclusion that Houthakker's statement of the theorem and his proof suggest more generality than there is. This does not appear to be the case. 45 p. 31]. These references are given by Goldberger [1967, 75 the data are aggregated to a very high degree.46 In the estimation of relatively large complete sets of demand equations, where goods are defined in a narrow sense, the assumption of additivity becomes questionable. For in this case, there is reason to suspect the presence of specific substitutability, complementarity and perhaps inferiority. In view of the severity of these restrictions, the empirical researcher may want to go only half way towards the additivity hypothesis. This can be done by a straight- forward generalization of direct additivity that is due to Strotz [1957]. This is the case of "block-independent preferences" (or, "block additivity") where it is assumed that the set of all commodities can be divided into G groups, (with ng being the number of commodities in the gth group) in such a manner that the marginal utility of any good depends only on the quantity of the goods in the group in which the commodity is placed. This results in the block-diagonality (as opposed to "diagonality") of the Hessian of the utility function, U. Alternatively, we may define a utility function, u, to be block—additive, if it is of the form: 46It might be pointed out that this repeatedly cited effect of "aggregation" is itself only "observed" and there seems to be little theoretical support for such an idea. u = u(xl, ... , x ) n = Z ug(xg) I g=l where, x9 denotes the (ngxl) vector of goods in the gth group. With this formulation, we note that Utilizing this as a restriction, along with symmetry and Engel aggregation, we get the number of free parameters to be estimated in the case of block-additive utility as47 G l + 1/2 X n (n +1). g=l g 9 To see this, note that the price elasticities are given in this case by: eij = Ggh nij - ¢Eiijj — Eiwj isg, 36h and 69h is Kronecker's delta. 2.4 Almost Additive Utility In View of the somewhat restrictive implications of the assumption of direct additivity, it was proposed in the last section that block-additivity be assumed as a half-way measure. Barten [1964] has generalized on these 47This formula is given by Theil [1967, p. 199]. 77 ideas and has proposed the assumption of "almost additiv— ity" which yields the direct additivity case and the block- additivity case as special cases within the almost addi- tivity hypothesis.48 The motivation behind this assumption lies in the observation that the price elasticities may all be estimated from the income elasticities and the money flexibility parameter, if the off-diagonal elements of the inverse of U are zero. Thus, if the off—diagonal elements could be approximated by some function of the diagonal elements of the Hessian, then there would be rea- son to believe that a possible relation could be found where cross price elasticities may be estimated from the own-price elasticities and the income elasticities. It is with this in mind that "almost additivity" assumes that the off diagonal elements of the Hessian are not zero, as was the case in additivity, but "small" in comparison to the geometric mean of the corresponding diagonal elements of the Hessian. Formally, we define a utility function to be "almost additive" if second partial derivatives of the utility function may be written as: 1/2 . ._ ij cij(uiiujj i,j—1,...,n 48The statement is not quite true for block- additivity, as will be obvious after a definition of "almost additivity" is given. 78 where the cij are fixed constants given by: C.. 13 C.. 13 In addition, the cij are symmetric in the sense that c.. = c.. i,j=l,...,n and, finally, the cij (for i#j) are "small" in the sense that the inverse of the Hessian U, has elements that may be "adequately" approximated by: -1 .__ (uii) l—J ij 1, -l/2 ) i#j c..(u..u.. 13 ii 33 We note that the case of direct additivity corresponds to cij = 0, for all i,j. Whereas a specialized block- additivity results if we set blocks of cij equal to zero. In empirical practice, it is the latter assumption that is frequently used.49 49Although almost additivity has been widely ac- claimed as an ingeneous and useful hypothesis, as it no doubt is, there remain some implications of the "smallness" of the cij that have been a source of some discomfort for the present writer. In particular, it seems that there is a need to derive the implications for the functional form of the utility function that must be implied by the second order partial differential equation: _ 1/2 u.. - c..(u..u..) 1] 13 ll 33 with the constants being "small." In empirical work, Barten [1964] assumed these constants to be of the order of less than 0.2. It is easily verified that it is I he. 79 Two consequences of almost additivity are immediate- ly obvious. First, the elements that are zero in U, are zero in U-l. This means that if it is suspected that two commodities have a zero specific substitution effect, then this prior information can be incorporated in the model by specifying the corresponding cij to be zero. This would result in the uij being zero, which is necessary and suf- ficient for the specific substitution effect to be zero; for non—zero prices and finite income. Secondly, the ratio of the non-zero off-diagonal elements to the geo- metric mean of the corresponding diagonal elements, is the same in U, and U—l.50 The consequences of "almost additivity" are seen in a way quite similar to direct additivity. Recall equa— tion (17) which states in general the solution for price elasticities: necessary and sufficient for utility to be directly addi- tive, that the c's be zero for (i#j). What is unclear is the implications of c's of the order of 0.1, or 0.2. Also, there seems to be a need for examining the order of the error in the approximation used for elements of the in- verse. 50These results are stated by Barten [1964, p. 4]. He states the second result with the qualification, "apart from sign" due to the way he formulated his definition of "almost additivity." The formulation in this paper avoids this by neglecting to append a minus sign before the uii’ as Barten does. Barten's reason for doing so are to insure that diagonal elements "represent decreasing marginal utilities." Barten [1964, p. 4]. His 9 = -ci.. ij 3 80 (17) ... eij = nij — ¢Eiijj - Eiwj (i,j=l,...,n) In the case of almost additivity we have, Ap. i ._. x. u.. l_3 i ii ij - AP- 3 -1/2 1*] It is easy to see, therefore, that the following relation holds between the n..(i#j), and the n..,n..:51 13 ii 33 1/2 = c.. (n ) n.. ..n..w. 13 13 ll 33 3/wi Also, the constraint that the specific substitution effects sum to money flexibility times the income elasticity holds for each good. Thus, if mi non-zero cij are specified for each row of U, then we have to estimate l+2n+2mi elastici— ties, and O. There are 1+n+1/22mi constraints; Engel aggregation, Zj nij = OEi, and symmetry of specific sub- stitution effects. This leaves n+1/22mi free parameters to be estimated.52 This is a substantial reduction in the parameters, and shows the effectiveness of almost additiv— ity as a more sophisticated additivity assumption. 51In Barten's [1964] actual work, a different for- mula was used as an approximation to this formula. This is discussed in the section on almost additivity in Chap- ter 3, below. 52Barten [1964, pp. 607]. 81 2.5 Neutral Want Association A hypothesis, quite similar to Barten's almost ad- ditivity hypothesis, is that of "neutral want association" proposed by Pearce [1961], [1964]. Once again the restric— tion is on the off-diagonal elements of the Hessian of the utility function, and it is assumed, as before, that com- modities can be divided into groups, but this time the off-diagonal elements of the Hessian are assumed to be proportional to the product of the marginal utilities of the row and column goods, the factor of proportionality being the same for goods in the same group. Unlike almost additivity, though, neutral want association does not place any restriction on the size of these factors of propor- tionality. Also, neutral want association introduces all of the first derivatives of the utility function into the specification of the Hessian. Both these factors have the effect of making it a non-trivial problem to solve for the inverse of the Hessian, U, and to analyze the effect on the Frisch-Houthakker specific substitution effect as we have done so far. We resort, therefore, to an alternative mode of analysis used by Uzawa [1964] in which the impli- cation of neutral want association is seen upon the separ— ability of the utility function, and results are derived for the Slutsky term under these assumptions. To proceed, we need a definition of neutral want association. 82 Definition: (Pearce, 1961) Two goods i and j are defined to be neutrally want associated to a third good, k, if and only if, 3(ui/u.) Bxk where, ui, etc. denote the first partial derivatives of the utility function, as usual. To get an intuitive idea of the implication of neutral want association, note that what the definition amounts to is simply the assumption that the marginal rate of substitution between goods i and j is not affected by the amount consumed of the kth good. This is a straightforward generalization of the additivity assumption that the marginal utility of one good was un- affected by the quantity consumed of another good. Of course, the additivity relation was binary and hence sym- metric. This is obviously not true of the neutral want association relation. In the previous sections we were able to demon- strate the effect of the hypotheses of additivity and almost additivity directly on the inverse of the Hessian matrix, and hence on the Slutsky elasticities. Although we shall not be able to carry out this procedure in the case of neutral want association, we note, however, the effect on the Hessian matrix of the assumption of neutral want association. The elements of the Hessian, U, are esaily seen to be: 83 uii 1:3 lj 2 ieN , jeN ; for all s,t. A astpipj s t where the aSt are constants, Ns and Nt are the sth and t—th commodity groups, A is the marginal utility of income, and use has been made of the first order conditions. Thus, very much like Barten's almost additivity hypothesis, the neutral want association hypothesis also assumes that the off—diagonal elements of the Hessian matrix are func— tions of the known prices, the marginal utility of income, and a constant which remains the same for each commodity group. The comparison between almost additivity and neut- ral want association, however, cannot be carried on much further. The reason is that Barten formulates his hypothe- sis with the sole purpose of getting manageable and well- behaved off—diagonal elements in the inverse of U, while Pearce makes no such attempt. Thus, the methods of analy— sis, utilizing the Frisch-Houthakker decomposition, are no longer available to us for the purpose of analyzing the effects of neutral want association on economies of para- metrization. The alternative approach in analyzing the conse— quences of neutral want association is due to Uzawa [1964]. Consider the entire set of commodites grouped into k groups: G1, G2, ..., Gk' with x1 denoting the (nix 1) vector of 84 goods in the i-th group, and ni being the number of goods in the i—th group. We have, then, Theorem: (Uzawa, 1964) 3 (n../u.) l X = o, i,jeGS, k#i,j. k if and only if: t Bxi 3x. = ks (x) ——— ——1 ieGs, jth; x, kSt(x) defined for all s,t. and for all In other words, if goods may be partitioned in such a way that all pairs of goods in any group are neutrally want associated with any third good, then the income compensated (Slutsky) price elasticity between two goods is proportion- al to their income elasticities. The constant of propor- tionality, however, is the same for each pair of groups to which the respective commodities belong. Under neutral want association, then, the number of unrestricted para- meters to be estimated becomes 1/Zg(g+1)+n, where g is the number of groups, and n the number of commodities, [Pearce, 1964, p. 214]. This is indeed a substantial reduction in the number of parameters. Unfortunately, no empirical results seem to be available on the application of neutral want association to the estimation of demand functions, in the form outlined above. Instead, it seems, Pearce [1964] used additional assumptions in the implementation of neutral want 85 association to British data. In his estimation procedure, Pearce [1964, pp. 213ff.] adopted the following assump- tions, which appear to be quite restrictive: Assumption 1: The expenditure on any group of commodities bears a fixed proportion to total expendi— ture. Assumption 2: The income slope of the demand functions is assumed constant with respect to variations in prices and expenditures. The first assumption is equivalent to assuming that each commodity group may be treated independently, in the sense that price changes outside the group do not affect the allocation of expenditures within any group. If enforced exactly, rather than as an approximation, this would be equivalent to arbitrarily equating all the cross— elasticities outside each group to zero. Such an assump- tion would cast severe doubts on the results. The second assumption seems equally restrictive in its implications. Thus, a valid test of the plausibility of the neutral want association hypothesis remains to be carried out. 2.6 Indirectly Additive Utility A final theoretical restriction on the form of the utility function which results in economies of parametri- zation is the restriction of "indirect additivity," pro- posed by Houthakker [1960] along with its counterpart, direct additivity, mentioned above. Unfortunately, the (consequences of indirect additivity are difficult to 86 reconcile with ordinarily held beliefs about the behavior of the consumer. Also, preliminary empirical results on a specific indirectly additive utility model of demand have offered evidence of serious statistical limitations of the model. For completeness, however, we discuss the consequences of indirect additivity briefly. It has been noted in Chapter 1 that by substitut- ing for the demand functions into the direct utility function, we can get the indirect utility function: C II U*(plr ~00 I Pnr Y) U(Xl(pIY)I 00- I Xn(pIY)) where, for notational convenience we denote by p, as usual, the (nxl) price vector. Now, each demand function is homogeneous of degree zero, so that proportional in— creases in prices and income leaves the quantities demanded of each good unchanged. This means, then, that the in- direct utility function is also homogeneous of degree zero in prices and income. Thus, the indirect utility function can be written as: u = u*(y/pl, ... , y/pn). This is the canonical form of the indirect utility func- tion, and all discussions of the functional form refer to this function rather than the one above. 87 Rigorously, then, a utility function is said to be "indirectly additive" if the corresponding indirect utility function can be written as: u = u*(p,y) = 2i u*i(y/pi) The consequences of indirectly additive utility are best analyzed by the use of an identity due to Rene Roy [1943]. To derive this identity recall equation (13) of Chapter 1: (13) . . . u; = A, and u? = —ij (j=l,...,n) where u;, and u; denote the partial derivatives of the indirect utility function u* with respect to income y, and the jth price pj, respectively. As a direct consequence of (13) we have Roy's Identity: x. = - (i=1,...,n) Using Roy's identity and equation (13) it is relatively easy to derive all of the expressions for price and income slopes of the demand and the marginal utility of income functions. Note that from (13), we have A = u* Y YY a . and — = 11* u = 11* =1 0 o o n pj 173 31/ (3 ’ ’ ) where ugh denote second partials of u* with respect to the variables a, b. The income lepe is obtained by 88 differentiating Roy's identity; where for all i=1,...,n we have axl (u; uiy - u: u§y) F = — *2 (i=1,...,n) u -l = - * *. + x * b 13 (uy) (qu i uyy) y ( ) _ _ -1 3A 3A by (13) Similarly, by differentiating with respect to the jth price, we derive the price slope of the demand functions: For all i,j=l,...,n axi (u; uij - u: u§j Fz' *2 j u Y -1 8A =_* 'k _ (uy) (uij + xi apj) by (13) -1 3X: 3A =_* * ._ _ (uy) {uij + xi( Aay xjay)} by (10) -l —1 8x. = -A u#. + A A x.x. + x.——1 by (13) 13 y i j 13y Collecting results, we may write in concise form all of the solutions to the slopes in terms of the indirect utility function quite generally as: 3A 71 —— A = * ( ) 8y y uyy 89 3x i _ —1 3A 3A ._ 8x. 3x. _i — _ -l * -1 + ( ) (91) apj — A uij + A Ay xi Xj xi §§l (i,j=l,...,n) 3A . (101 ——— = u*. = u* =1 ... n ) apj y3 3y (3 ' ') These relations in fact can be viewed as a sort of "dual" to the equations (7)-(10), of Chapter 1.53 The consequences of indirect additivity are now obvious. If the utility function is indirectly additive then ugj = 0 for i¢j. This means that the price elastici- ties of the demand equations are given by: . = #. + _ . + . . ' '= ... eij nlJ ¢ wJ EJ wJ (i,j l, ,n) where, p. u*. 11*. = — _1 _ll 1] xi A and O is the money flexibility parameter. Note that nij = O, i¢j is the consequence of indirect additivity, so that the Cournot price elasticity of demand depends 53Several comments are in order. The duality re- lations linking direct and indirect additivity are explored by Samuelson [1965]. Further discussion of this aspect is at the end of this section where the case of "simultaneous additivity" is briefly touched upon. These relations are derived elegantly by Goldberger [1967, pp. 85-86] using matrix notation, similar to the Barten-Theil notation for the directly additive case. 90 only on the good whose price is changing and not on the good whose quantity is affected. This is the classical consequence of indirect additivity and was expressed by Houthakker [1960a] as: (i,ji‘k) .|... or, in our elasticity notation, quite simple as, Houthakker [1960] has given formulas for the price elasticities of demand that are related to our ex- pression for eij given above.54 We derive these formulas by noting, as Goldberger [1967] does, the effect of the Cournot aggregation condition on our results. Multiply- ing through by wi and summing over i; we have for j=l,...,n: - w = Z. w e.. 3 l 13 -l = Z w n* + (¢ + E.) w. 2. w. i i ij j j i i F3? p.p. U... _ = Z. (_l__l)__l + (¢ 1 + E.) W. 1 Y 3 J 54Goldberger [1967, pp. 87-88] has derived the correct version of the original Houthakker formula, which is in error. Our derivation is somewhat different from either Houthakker or Goldberger, but the substantive re- sults are the same. 91 So that, in general, we have, p.p. u#. -l _ i J 11 _ ._ Now in the case of indirect additivity this reduces to: —1 p2. ut. (¢ + Ej) wj = ( 3/y)( 33/A) - wj This in turn implies that —1 + E. = - n#. - 1 (¢ j) 3] or, n*. = - (¢'1 + E.) - 1 33 3 We may write, therefore, the following equations that give the price elasticities of demand under indirect additivity: 2 (pj/y)(ujj/A) - wj ..... (i#j) e..= ij 2 u* u* (Pi/y>( ii/A) - wi - (pi/xi>( ii/A) (i=j) where, the first of these quations is cited by Houthakker [1960] as equation (22). The own price elasticity is derived, however, in a somewhat different form. For sym— metry, we state the cross elasticity formulas corresponding 55 to Houthakker's own-price formulas, also. This can be done by substituting for nii in the expression for eij’ to 55This corresponds to Houthakker [1960a] equation (23), which was corrected by Goldberger [1967] and appears as equation (4.36) in the latter paper. 92 give an alternative expression for the price elasticities under indirect additivity: 1 (¢' + Ej) w. (iej) 3 e.. 13 1 (¢‘ + Ej)(wj - 1) - 1 (i=j) where, the latter formula is equation (4.36) of Goldberger [1967]. We see, then, that the case of indirect additivity imposes substantial restrictions on the number of free parameters to be estimated, for all price elasticities are estimable from income elasticities and the money flexi- bility parameter. Unfortunately, there is no direct motivation for assuming indirect additivity comparable to the direct additivity case. The result is that the in- direct additivity hypothesis has to be introduced solely on the grounds of resulting computational convenience. The empirical consequence of equal Cournot price elastici- ties of two goods with respect to a third price, seems dubious for obvious reasons. Thus, the indirect additivity hypothesis has not much to recommend itself to the empiri- cal worker.56 56In practice, indirect additivity has been imposed only within the context of the "indirect addilog" utility function proposed by Houthakker [1960b]. As Goldberger [1967, p. 92] has noted, Houthakker arrived at this func- tion by attempting to force the "constant elasticity of demand" system to satisfy the budget constraint. In doing so, however, the indirect addilog function became nonlinear in parameters, so that the most attractive feature of the 93 In conclusion, we might mention the case of "simultaneous additivity" mentioned by Houthakker [1960a] and explored by Samuelson [1965]. A utility function is defined to be "simultaneously" additive if both the direct and the indirect utility functions are additive. It was proved (incorrectly, as it turned out) by Houthakker [1960a] that simultaneous additivity implied unitary income elasticities. Also, Samuelson [1965] proved (once again, incorrectly) that simultaneous additivity implied unitary price elasticities. The exception to these results was given by Hicks [1969], who showed that these results were not true for the following utility function: n Hicks exception: u = u(x , ... , x ) + Z a.(1og x.)57 l r r+l J Samuelson [1969] has corrected the two theorems, and has shown that Houthakkers theorem [1960, Th. 3] holds for n=3, with the Hicks exception being the only exception. Also, Samuelson's theorem holds, except that at most one good may not have unitary price elasticity. constant elasticity of demand system — ease of estimation - was lost. 57We give the generalized Hicks' exception, given by Samuelson [1969]. CHAPTER 3 EMPIRICAL MODELS OF CONSUMER DEMAND: STOCHASTIC SPECIFICATION, AND ESTIMATION 3.1 Introduction In Chapter 1 it was shown that a "utility- maximizing" consumer possesses a complete set of demand equations which obey a set of restrictions on their par- tial derivatives. In addition, if it may be assumed that the consumers in a given community or nation possess iden- tical preference patterns, with linear Engel curves possessing zero intercepts, then it was shown that the community's demand for aggregates of goods is also a func- tion of all prices and national income. Further, this complete set of demand functions for the community possesses all of the prOperties possessed by the individual's demand functions. Finally, it was pointed out that under the same assumptions about Engel curves, a community "behavior function" can be constructed. The community's demand functions may then be derived alternatively by viewing the community as a single consumer attempting to maximize the community behavior function subject to the community bud- get constraint. Thus it was established that the 94 95 community's complete set of demand curves possess all of the properties that the individual demand curves were shown to possess. Under the assumptions above, it has been shown that the community's demand for aggregates of goods is a function of all prices and national income, with con- straints on the partial derivatives of the demand func- tions. Unfortunately, this is not sufficient to determine a unique functional form for the demand equations. The choice for functional form must be guided, however, by the condition that the demand equations must obey the restric- tions derived above. Historically, this point was often ignored by empirical researchers, and it is only in the past decade that a great deal of attention has been di- rected at the search for a "complete system of theoreti- cally plausible demand functions." In this chapter we discuss the three leading "theoretically plausible" functional forms that have been proposed in the literature. In addition, we discuss the "Constant Elasticity of Demand System" which is of his— torical importance, but is not theoretically adequate. 3.2 The Constant Elasticity of Demand System The "Constant Elasticity of Demand System" is based upon the double-logarithmic specification of the functional form of the demand equations, and has been the 96 most widely used specification in the estimation of demand functions.58 Despite the fact that it was well-known that the function was inconsistent with utility theory, the ease of estimation and direct interpretation of the parameters led empirical researchers to utilize this form and justify their actions by arguing that the specification was a good first-order local approximation. The "Constant Elasticity of Demand System" (CEDS) may be specified in its non-stochastic form as follows: log xi = ai + Ei (log y) + Zj eij (log pj) (i,j=l,...,n) where, as before, Xi is the quantity of the ith good, pi is its price, y is the total expenditure or "income," Ei is the income (Engel) elasticity, and eij is the price (Cournot) elasticity of demand. In addition, it is assumed that Ei and eij are constant with respect to variations in prices and income. The ai are constants which may be in- terpreted as trend terms. This specification will be referred to as the "Cournot specification" of the CEDS, as distinguished from the "Slutsky specification" which may be derived from the above by using the Slutsky Relation: log xi = ai + Ei (log y - ijjlog pj) + Zj sij (log pj) (i,j=l,...,n) 588chultz (1938, p. 83, n. 46) cites a July, 1929 paper by Leontieff in which this form was used. 97 where wj is the j—th commodity's budget share, wj E p,x./y, J J 59 The simplic- and sij is the price (Slutsky) elasticity. ity of the system is apparent. All coefficients are directly interpretable as elasticities. Finally, if an additive disturbace term is appended to the form, estima- tion becomes a trivial problem. The CEDS is, unfortunately, inconsistent with utility theory. This was demonstrated by Wold and Jureen [1953, pp. 105-7], for example, in the context of integra- bility conditions. They showed, to be precise, that the double-logarithmic demand function does not satisfy the integrability conditions, and hence is inconsistent with any utility-maximizing process, unless it is assumed that the indifference curves are of an empirically implausible form. A simpler and intuitively appealing argument can be made in fact against the plausibility of any system of demand equations in which elasticities are assumed con— stant. The argument rests upon the fact that if the income elasticity is constant then unless it is equal to unity, there exists a finite income for which the commodity will either not be bought at all, or be bought solely. Thus, if income elasticities are constant then they must all be 59The distinction between "Cournot" and "Slutsky" specifications will be made throughout, and is dependent upon which price elasticity appears in the functional form. The latter specification was used by Stone (1954a). 98 unity. This may be proved rigorously as follows: We have for all i=1,...,n. By definition: log wi 5 log pi + log xi - log y Taking total differentials, and assuming prices constant, d(log wi) _ d(log xi) - d(log y) (Ei — l) d(log y) where Ei denotes the income elasticity as before, and use has been made of the fact that the ratio of total logarith- mic differentials is equal to the partial logarithmic derivative if prices are assumed unchanged. Rewriting, we have: v7-.—i=(1~3i-1)‘:,—Y Now assume, in all generality, that at income y0 and prices p0 the budget share of the ith commodity was WE. Using the above relation we can calculate the incomes for which the budget share of the ith commodity becomes unity and zero. To do so we note that: o . u—wn . dwi < (l-wi) iff (Ei-l)dy < wo y i o . o and dwi > -wi iff (Ei-l)dy > —y 99 We have, therefore, the condition that: 0 0) in M the estimates depend on which equation is omitted." (This Statement by Pollak and Wales is also cited in an unpub- lished paper by Murray Brown and Dale Heien, on page 18). (Additional evidence for the uniqueness of estimates, i.e., 'fihe irrelevance of the choice of equation to be deleted, are provided by Powell [1969] in the context of the gen- eral (not Stone's) linear expenditure system. Powell has C(Dnclusively demonstrated that in any linear expenditure Siestem of demand equations, the choice of equation to be dEileted is arbitrary. 126 the subsistence bundles, Ci’ over time, irrespective of prices and income is an implausible specification. In addition, it might be pointed out that this kind of an assumption for the marginal budget shares, bi’ is even more questionable, since the bi cannot exceed unity. In- stead, Pollak and Wales propose "habit formation" models with regard to cit’ while assuming bit to be constant over time. In addition, the chief contribution of Pollak and Wales has been to develop a dynamic version of SLES, in which considerable attention is given to the specifica- tion of the structure of the disturbance terms. With their stochastic specifications, they develop a more satisfactory procedure for dealing with the implied sin- gularity of the covariance matrix of the disturbance terms. The Pollak-Wales linear expenditure system (PWLES) is derived from the non—stochastic specification of SLES as follows. Consider the non-stochastic SLES: pitxit " + b (Yt ' Z pitcit i k pktckt) where the t-subscript on the ci is appended to incorporate the belief that the subsistence bundles, Ci' change over time. The crucial assumption of the PWLES is that the subsistence bundle is a random variable, satisfying cer- tain stochastic assumptions. Thus, PWLES specifies the exact nature of the disturbance terms appearing in each demand equation. More rigorously, PWLES assumes that: 127 = * I = . = eit eit + uit (1 1,...,n, t 1,...,T) where the ui are random variables, assumed to possess t the following properties: (i) E(uit) = 0 (For all i,t) (ii) E(u2 ) = 02 (For all i t) it i ’ (iii) E(uitujt) = 0 (for all i#j, t) (iv) E(uitujt')= O (for all i,j, t#t') Substituting the equation for eit into the non-stochastic equation for SLES, given above, we get: * _ * pitxit pitcit + bi(yt Z2k pktckt ) + pitvit b (2 Where' pituit ' i k pktukt)' pitvit The implications for the distribution of the vit are easily derived from the specification of the moments of 111 If t. the uit are assumed to be multivariate normal, in addition, the vit are also distributed as multivariate normal, with mean zero. The covariance between vit from two different time periods is zero. However, under assumption (iv) the variance of the disturbances vit are independent of income and quantities but are inversely related tc>prices. Since prices fluctuate, the covariance matrix of the disturbances may not be assumed constant over time. Pollak and Wales 128 consider (ii) to be implausible, and suggest an alterna- tive specification: 2 2 2 2 ,. it) ‘ Oi E(Xi ) = c. .. , (ii) E(u t 1 xit where, (ii)' expresses the belief that the variance of the disturbance terms in the demand equations are larger for higher levels of consumption. Finally, to complete the specification of PWLES, the nature of variations in ait over time need to be specified. Pollak and Wales suggest the following alter- native hypotheses: . o * —_ * . —_ HypotheSis 1. cit k1 + ki t (i 1,...,n) - . * = + * HypotheSis 2. cil kl k Xil—l where the first hypothesis is of the Stone variety, while the second is the "linear habit formation" hypothesis which subsumes under itself the cases of constant cit (over time), and the "proportional" habit formation hypothesis, which results when ki=0 in Hypothesis 2. Using these specifications, Pollak and Wales de- rived maximum likelihood estimators for the coefficients, and estimated demand for four commodities for the United States, using data from 1948-1965. They also estimated these models for prewar (1930-1941) data, and found signifi- cant differences in the underlying utility functions for 129 the prewar and the postwar periods. Unfortunately, their estimates suggest that for postwar data the stochastic specification of variances of disturbances proportional to quantities is valid only for the "proportional" habit formation model under Hypothesis 2. However, the dynamic specification and the estimation technique were shown to affect the results of estimation significantly. We dis- cuss these results in some detail here because the model to be proposed in Chapter 3 is also estimated using the same data. 3.5 Utility Aspects of SLES Samuelson [1948] and Geary [1949] demonstrated that Stone's linear expenditure system could be derived from a constrained maximization of the so-called "Stone- Geary" utility function, which is given as: u = u(xl,...,xn) = Xi bi log(xi - Ci) where the function is defined for (xi - ci)>0, and for all i=1,...,n, O : bi i l, and ci 1 0. In addition, Zibi=l. Maximizing the Stone-Geary utility function subject to the budget constraint, gives rise to SLES: pixi = pici + bi(Y ' kakck)’ Thus, the a priori specification of SLES has been shown to be equivalent to the assumption that the consumers possess 130 a definite pattern of preferences, which they attempt to maximize subject to their budget constraint. In addition, the utility basis of SLES is a source of additional in- sight into the behavioral implications of SLES. The first point to be noted is that the Stone- Geary utility function is directly additive. Thus, with- out further analysis, the results of section 1.6 ensure that goods represented by SLES must be Hicks-Allen sub- stitutes. Also, these goods cannot be inferior, nor can they be complements, nor specific substitutes in the Frisch-Houthakker sense. The absence of specific sub- stitutability, inferiority, and complementarity suggest that SLES is a plausible specification only for broad aggregates of expenditure categories. In addition, it is easy to show that for SLES the implied Engel curves are linear, and the own-price elasticities are less than unity in absolute value. The fact that Engel curves are linear is obvious from the constancy of the marginal budget shares bi' which are defined as 3X1 . bi = pi (§§—) (i=1,...,n) Thus, given a set of prices, the consgancy of bi ensures that the slope of the Engel curves, (8;E)’ is also constant. The restriction on own-price elasticities is apparent 131 from the formulas derived in the last section. Recall that the Cournot own-price elastiCities are given by: C1 Ci . eii = "(l - X_) - bi ('X—i') (i=1,...,n) Ci . = - 1 + (1 - bi) £1» (i=1,...,n) Now, since 0 §.bi : 1 (i=1,...,n), the own-price elastici- ties, eii' lie between zero and minus unity. An examina- tion of the similar expression for the Slutsky own-price elasticities will reveal that they are subject to the same upper and lower bounds. These results imply, therefore, that SLES is a valid specification only for goods whose demand is "price-inelastic" in the usual sense. Further, the linearity of the Engel curves implies that the model should be applied only to those samples for which the sample variance of income is relatively small.72 The linearity of the Engel curves also result in an interesting analysis of the behavior of the budget shares as income varies. These results due to Goldberger [1967, pp. 53 ff.] may be derived as follows. Dividing the expenditure equation under SLES, by income, y, we get wi = (pici)/y + bi(y — Zk pkck)/y (i=ll°°°ln) 72These points are made by Stone [1965, p. 275], and Goldberger [1967, pp. 6lff.]. 132 where wi denotes the (average) budget share of the ith good, as before. This may be written as w. = (l -d) w? + d b. 1 i i pici where, w? = ———————-, the "subsistence budget shares" 1 2 p c k k k and, a = (y - kakck)/y , is the ratio of supernumerary income to income. It is easy to see, then, that in SLES the budget shares are weighted averages of the "subsistence" budget shares and the "marginal" budget shares for each commodity. Further, the weights appearing in the expression for wi may be identified as the Frisch "money flexibility" function. To see this, substitute the SLES demand func- tions into the Stone-Geary utility function to get the indirect utility function associated with the Stone-Geary function: C. II u*(pl,....pn.y) 2i bilog bi + log(y - Xi pici) - 2i bi log pi Thus, the marginal utility of income is given by A = (Y ‘ Zk pkck)—l The inverse of the income elasticity of the marginal util- ity of income, which by definition is the "money flexibility," ¢, is easily seen to be 133 ¢ = _ (Y ‘ 2k pkck) z 3(log A) '1 y 3(109 y) Thus, the ratio of supernumerary income to income, apart from its sign, is identical to the a above. In view of this, we may write the final expression for the (average) budget shares under SLES wi = (1 - |¢|) w; + |¢| b. (i=1,...,n) Noting first that wi and w: sum to unity, and secondly that wi, wi, |¢|, and bi all lie between zero and unity; an interesting implication of the response of budget shares to changes in real income, becomes available. We see that the (average) budget shares under SLES are bounded from below by the "subsistence" budget shares, and from above by the marginal budget shares. Their proximity to the upper and lower bounds is determined by the value of |¢| for the particular levels of prices and income. As income rises, with prices constant (or vice versa), |¢I tends to its upper limit of unity, and the (average) budget shares, wi, tend towards the marginal budget shares, bi' Similarly, a fall in real income due to changes in income or prices, results in |¢| to fall towards its lower limit of zero, and the (average) budget shares approach the "sub- sistence" budget shares.73 73The fact that the Frisch "money flexibility" function, lies between zero and unity in absolute value, and is the ratio of supernumerary income to total income, 134 In addition to the above properties, SLES has also the desirable property of aggregating perfectly over both individuals and commodities. This should be obvious from the section of aggregation in the previous chapter where we stated the conditions on Engel curves which were neces- sary and sufficient for aggregation. Thus, the linearity of Engel curves plays a crucial role in the existence of desirable properties in the linear expenditure system. Curiosly enough, it is this linearity of the Engel curves which constitutes one of the serious restrictions of SLES too. Finally, we point out that a utility basis may also be provided for the Pollak-Wales modification of SLES, by a direct extension of the Stone-Geary function. Pollak and Wales [1969] have shown that their stochastic specifi- cations for their dynamic demand functions are obtainable from a constrained maximization of the following stochastic utility function: {:1 ll ... X ... u(XlI I nlvll IVn) 2i bi log(xi- ci- vi) where the vi are random variables with a specified dis- tribution. Maximizing this utility function subject to lends particular credance to Frisch's [1932], [1959], proposition that "money flexibility" be treated as an index of welfare. 135 the budget constraint, yields the PWLES, given in the previous section: _ _ * where, v: = pivi — bi(2kpkvk). Thus, much like SLES, the PWLES may also be justified on grounds of a specific utility function. Indeed, the Pollak-Wales approach towards a stochastic formulation of the consumer's utility function admits of somewhat more generality than the Rotterdam approach, which is the only other model whose stochastic underpinnings have been ex- plored in detail. As Pollak—Wales [1969] point out, the "marginal utility shcok model" utilized by Theil and Barten (discussed in section 3 above) assumes that the stochastic terms enter the utility function in a specific manner, as follows: n u = u(xl,...,xn) + E x v klk k where vk are random variables. 3.6 General Linear Expenditure Systems (GLES) Although Stone's linear expenditure system has attracted the most attention, other linear expenditure systems have also been investigated and estimated. These "general" linear expenditure systems were proposed by 136 Stone [1954b], and were in fact specialized by Stone to yield SLES. The procedure by which this was achieved was simply to impose the classical restrictions on the general linear expenditure system. In doing this, Stone demon- strated that GLES satisfied the classical restrictions implied by demand theory only if it had the form of SLES. An alternative approach has been to consider the GLES and impose (approximately) the classical restrictions on the demand function in estimation. This approach was adopted by Leser [1958], [1960], [1961], and subsequently by Powell [1965], [1966], and Powell, Hoa, and Wilson [1967]. Estimation of the linear expenditure system was analyzed definitively by Powell [1969]. As an introduction to these models it might be instructive to consider the method by which Stone [1954b] arrived at SLES from GLES.74 Stone [1954] considered the GLES, (GLES) ... pixi = ai + biy + chikpk (i=1,...,n) where the ai, bi’ and the CR are assumed to be constant. By direct differentiation and manipulation of terms it is easily verified that the income (Engel) elasticities, and the price (Cournot) elasticities are given by, 74Our exposition follows Frisch [1954] due to his simplified presentation. Stone [1954], and Goldberger [1967, pp. 50-52] also present the same material. 137 bi E1 = VT; (1:1,. 0 o ,n) Ci p- p. . . and, e.. = —L-J' — —l 6.. (l,J=l,ooo'n) 13 xipi pi 13 where, wi is the budget share as before, and 5ij is Kronecker delta. It is easy to see, therefore, that the Slutsky price elasticities, sij’ are given by w.bi c..p. p. .. = —‘l—— + "ii-‘1 - —l 6.. (i,j-1,...,n) 13 W1 xipi Pi 13 The effect of the classical restrictions may now be examined. Homogeneity requires that E. e.. + E. = 0 (i=1,...,n) Substituting from above, we see that for the GLES, this implies that Ox. pl 1 a. X: e.. +E- = -( l) = 0 (i=1,...,n) Thus, GLES satisfies the "homogeneity" condition if and only if all the intercepts, ai = 0. Similarly, the "adding-up" criterion, Xipixi=y, if fulfilled if and only if, Ziai = 0, Zibi = l, and chij = 0 (i=1,...,n) The symmetry condition, wisij = szji’ requires, WJbl + ClJEY - w];— 6ij = wlbj + 0313; - W121. ij 138 Multiplying through by y, cancelling the last terms on each side, and substituting for pixi and pjxj from GLES, we have after transfering all terms to the left hand side, bi(aj+ bjy + chjkpk) + pjcij - bj(ai+ biy + chikpk) picji = 0 or, (biaj - bjai) + 2k pk(bicjk - bjcik) + (pjcij - picji) = 0 Utilizing Kronecker deltas, we may write down equation (21) of Frisch [1954, p. 509]: (in our notation) (biaj - bjai) + 2k pk(bicjk- bjcik + Cikékj ’ Cjkdki) where this equation holds for all i,j=l,...,n. Since this equation holds for all prices, both the first term, and the coefficient of the price term are identically zero: (biaj - bjai) = O, (i,j—l’ooo’n) Cik C‘k . . and' W = 8—73—87 (1.3=1,....n) l 1 k] j The first condition implies that ai = biy (i=1,...,n) while the second is true only if, 139 cij = (éji - bi) xj (i,j-1,...,n) where i, §5(j=l,...,n), are (n+1) constants that charac- terize the demand equations. Introducing these into the original expenditure equations under the GLES, we get pixi = biy + Zk(6ki- bi)pkxk (i=1,...,n) (Zbi = 1) = pixi + bi(y - kakxk) (i-l,...,n) (Ebi = 1) 75 which is Stone's linear expenditure system, SLES, with ck replacing xk shown that Stone's LES is the only form of the GLES which in our previous notation. Thus, it has been satisfies the homogeneity condition, the adding-up criter- ion, and the symmetry conditions. Indeed, the derivation above points quite forcefully towards the power of the classical restrictions. The impact of this, however, is to make the model nonlinear in parameters, and hence rob the GLES of its single most attractive feature: linearity. The estima- tion of SLES is quite cumbersone, as we have seen, and has been satisfactorily analyzed only recently. Leser [1958], 75A single-subscripted ck refers to the subsistence consumption level of the k-th commodity, in conformity with our previous notation; and should not be confused with the double subscripted c.k which were arbitrary coefficients in the GLES. l 140 therefore, examined the GLES with the view of maintaining the convenient parametric linearity of the system at the expense of only an approximate enforcement of the classi— cal restrictions. In Leser's linear expenditure system, (LLES), the expenditure functions are taken to be of the GLES form, initially: pixi = biy + Zj aijpj (i=1,...,n) where, it is understood that only the "adding-up" criter- ion is to hold globally, while the homogeneity and symmetry conditions are to hold only at the sample means of expend- itures and prices. It is apparent that the adding-up criterion will be met for all values of the independent variables if and only if and E. b. = l. (The latter condition, incidentally, insures that Engel aggregation conditions are satisfied globally). To achieve further economy of parametrization, Lese intro- duces the Hicks—Allen "elasticity of substitution" be- tween goods i and j, aij’ defined as wj aij = sij (i,j=l,...,n) 141 where Sij' and wj are the Slutsky price elasticity and the budget share of the jth commodity, respectively. Using the Slutsky relation, it is easily verified that the Slutsky price elasticity for the GLES is given by sij = pjaij/vi - 6ij + wjbi/wi (i,j=l,...,n) where, vi denotes the expenditure, pixi, on the ith good. By a suitable rearrangement of terms, and substitution from the expression for aij, we have =—— * — — _ ' '= aij wixjaij bixj + xiéij (i,j 1,...,n) which is derived by Leser [1960, p. 105] and is cited by Powell [1969, p. 921, equation (A.3)]. The equation is understood to hold at sample means of budget shares and quantities, wi, and ii, respectively. Substituting this in the GLES, we have = _ _ — * _ _ pixi pixi + bi(y ijjxj) + Ejaij(wipjxj) (i=1,.. . ,n) Leser's linear expenditure system (LLES) is de- rived from this equation by noting the fact that the demand functions will be homogeneous (at the sample means) if and only if Zj wja:j = O (i=1,...,n) 142 Substituting for aii from this condition we have, for LLES (LLES) ... p.x = pixl l i + bi(y - Z.pjxj) J + E at. a: .2]— a: .§. j¢i 3( 1p3 3 3pl 1) which is given by Leser [1960, p. 105, equation 2]. Un- fortunately, LLES is still quite rich in parameters, so that for purposes of estimation, Leser arbitrarily equated all cross-elasticities of substitution between goods, a§j(i#j). With this assumption, it is readily seen that LLES has (2n+l) free parameters. Also, it has been shown that LLES satisfies the adding-up criterion globally, and the homogeneity and symmetry conditions at the sample means. Estimation of the model is discussed by Leser, but the definitive solution is provided by Powell [1969]. Powell's procedure takes account of the implied singularity Of the covariance matrix of the disturbances. A generalized Aitken [1935] type of estimator is derived by Powell with the use of the Moore—Penrose generalized inverse for a matrix of less than full rank. We discuss briefly both Leser's procedure and the one suggested by Powell. Having equated the cross elasticities of substitu- tion, Leser was confronted by a linear model in which one parameter, aij = a* (identical for all i#j), occurred in all of the equations. Leser [1960, pp. 107ff.] adopted the procedure of accepting those estimates of a* which 143 minimized certain arbitrary linear combinations of residual sums of squares from the n fitted equations. Thus, denot- ing v. 1 = pixi, Leser obtained estimates of the parameters in LLES, which minimized: A 2 s ‘ ZiAiEt(Vit ‘ Vit) where, Ai are arbitrary constants, and vi represent the estimated values of Vi' To derive actual estimates, Leser proposed the following two assumptions: (i) A. l l (for all i) 2 or, (ii) A. l l/Zt(v. - 0. ) it it which correspond respectively to the minimization of the total sum of squares, and to the maximization of the sum of the R25. Powell [1966, p. 665, n. 3] subsequently credits a referee for pointing out that only under cri- terion (i) are the estimates of the parameters linear. Utilizing this procedure, Leser obtained the estimates of a*. Estimates of the other parameters were obtained in a similar fashion in a second round. It was noted that the least squares procedure ensured that Zibi = l; and ziti = 0, where ti were coefficients of the time trend variable. Powell [1965] has proposed an alternative estima- tion procedure for LLES which takes also into account the fact that the classical restrictions imply that the covari- ance matrix of the disturbance terms is singular. Although 144 Powell considers the general LLES, (a:j unequal for i#j), the "restricted" LLES under consideration may be estimated by deleting one equation (the choice is arbitrary) and using a restricted (a:j = a*, all i#j) Zellner's [1962] two-step procedure. Recovering the coefficients of the deleted equation by parametric restrictions, this estima- tion procedure leads to efficient and unbiased estimates under appropriate assumptions. The details of the pro- cedure parallel the discussion in Chapter 4 for the model prOposed in this paper. On another front, Powell [1965], and elsewhere, proposed a modification of LLES. The motivation behind Powell's model lay in the dissatisfaction with Leser's procedure of arbitrarily equating the cross elasticities of substitution. Indeed, from a strictly theoretical point of view, Frisch [1959] has questioned the use of the elasticity of substitution as a parameter in demand models, due to its peculiar disadvantage of tending towards in- finity as budget shares become small. (Thus, Leser's [1960] estimate of 0.5 as the value of the elasticity of substitution seems questionable when the mean budget shares for the nine commodities under question are reported between 0.020 and 0.259 [Leser, 1960, p. 108]). Of course, the advantage of using the elasticity of substitu- tion as a parameter lies in its symmetry. 145 To achieve symmetry, however, Powell took the approach of utilizing results from the theory of additive preferences. Thus, if it may be assumed that the class of utility functions which give rise to the GLES have as a member an additive utility function, then the results of section 1.6 of the previous chapter may be used in achieving a more parsimonious parametrization of the GLES. This, in fact, was Powell's approach. To derive his "model of additive preferences" (PMAP) consider the GLES: pixi = biy + Zj aijpj where, the intercept (not used by Powell), and an addi- tive time trend term (used by Powell) are omitted for simplicity. With the above specification, it is easy to see that (all 1.3) An expression for aij may be derived as follows: Bxi = p1(Kij — iji/pi) (for i¢j) (where, Kij is the Slutsky term). = k bibj/p' - ijl (for i#j) 146 where, k = - A5 L Y and use has been made of the fact that under additivity A (3xi (dx.) K_.=—_._ ._)__l ij Ay y 3y Note, however, that the consequences of additivity are not imposed globally, but are instead assumed to hold only at the sample means of the prices and quantities. Also, the adding-up criterion is met only if Which implies, since prices are arbitrary, that Using this relation, we deduce: where, we have used the expression for aij above. Finally, we may substitute these expressions for aij into the original formulation of the expenditure equation under GLES, to get Powell's "model of Additive preferences" (PMAP): (PMAP) ... pixi = pixi + bi(y - ijjxj) +kb.2.b. .". - ._. l J 3(p3/p3 pl/pl) 147 In this version, an interpretation of PMAP becomes available. Given a set of income and prices the consumer is assumed to pruchase "typical" quantities of goods, §i(i=l,...,n). The consumer then allocates his "supernum— erary income," (y — szjgj)’ proportionally among goods in accordance with their respective marginal budget shares, bi‘ This is parallel to the behavioral interpretation given for SLES with the "subsistence bundles," ci, of SLES playing the role of the "typical bundles," ii, of PMAP. Unlike SLES, however, PMAP assumes that the con- sumer adjusts his allocation of his supernumerary income to account for substitution effects resulting from price changes. This price response is given by the third term in PMAP, which, incidentally, sums to zero across equa- tions, and thus ensures that the adding-up criterion is 76 met for all prices and incomes. Powell [1966, p. 663], further notes that if the two terms, (y - ijjij), and - pixi — kbiZjbj(pj/pj -pi/pi) ), are deflated by the ith price, pi, then (in terms of the purchasing power (pixi of the ith good) we would obtain, respectively, an index of "real" supernumerary income, and a quantity index for the ith good. 76This interpretation, and much of the subsequent discussion of PMAP, relies to a great extent on the exposi— tion of PMAP given by Goldberger [1967, pp. 95—101], alongwith Powell [1966, pp. 663ff.]. 148 From a statistical viewpoint, PMAP possesses con- siderably more attractive properties than SLES. Firstly, the introduction of sample mean values (observable) in— stead of the "minimum" or "subsistence" values, Ci’ reduces by n the number of parameters to be estimated for PMAP, as compared to SLES. Thus, the number of total free para- meters to be estimated under PMAP is seen to be just n. In other respects, however, PMAP retains the same assump- tions as SLES. In particular, the behavior of price and income elasticities with respect to variations in expendi— tures and prices are assumed to be identical under the two specifications. It might be noted, however, that PMAP assumes the parameter "k" to be constant. This implies that Frisch's "money flexibility" (or the inverse of the income elasticity of the marginal utility of income) when plotted against income yields a rectangular hyperbola, at mean values of prices and income. This is due to the equality of k with (—¢y) at sample means. SLES, on the other hand, defines (-¢y) to be identical to "supernumerary income," a crucial variable. Finally, it has been noted that PMAP imposes the classical restrictions only approxi- Inately. Thus, PMAP is not strictly consistent with 77 ‘utility-maximizing behavior. __¥ 77It ought to be pointed out, however, that Gold- berger [1967, pp. 99ff.] has shown that PMAP may be derived from a constrained maximization of the Stone-Geary utility function subject to the additional restrictions that ‘ Cxi - kbi/pi) = ci(i=l,...,n); where ci are parameters of 149 Evaluating PMAP with respect to ease of estimation, we see that PMAP is non-linear in parameters like SLES. Thus, the estimation of PMAP is carried out by iterative procedures described under SLES. The comments of that section apply to the estimation of PMAP. 3.7 Other Models of Consumer Demand In the previous sections we have explored the most widely utilized empirical models in demand theory. There remain, however, an infinite variety of models that may be derived from any utility function which satisfies the several properties necessary for qualifying as a utility function. In this section we consider three specific models of consumer demand that are a result of three fa- mous utility functions proposed in demand theory. These are the Quadratic utility demand models, the "direct addilog," and the "indirect addilog" utility models of demand. Although all three are consistent with utility maximizing behavior, and hence satisfy the classical constraints, yet on intuitive and empirical grounds these models leave much to be desired. Their main contribution lies in providing additional insight into theoretically plausible models of consumer demand.78 the Stone-Geary function identified under SLES as subsist- ence bundles. 78Our discussion relies on Goldberger [1967] and Houthakker [1960]. 150 Quadratic Utilipy: The "Quadratic utility function" is given by (QUF) ... u = u(xl,...,xn) = 2i aixi - 1/22i2j bijxixj where the ai and the bij are constants. Although the Quadratic utility function was considered by Allen and Bowley [1935], it received little attention subsequently in the analysis of consumer demand, with the notable ex- ception of Rotterdam models where it was discussed by Theil [1967, pp. 183-188, 228-229], who used it as an illustration. To facilitate eXposition, we adOpt the following notation: _a1- 1’11 1’12 ° ' bln_ a2 b21 b22 ° ° ' b2n a = B = . _?n_‘ bn1 bn2 ° . . bnn where a is an (nxl) vector, and B an (nxn) pos. definite matrix of coefficients. The Quadratic utility function may then be given in its matrix form by (QUF) ... u = u(x) = a'x — 1/2 x'Bx where, x, is the (nxl) vector of quantities defined in 151 Chapter 1. By the usual methods adopted before, it is easy to see that the demand functions are given by: ij ij -1 ij ij X . = Z I D b - Z . Z ' b ' D Z . z I ' a I b - - Z ‘ b . 1 3 a3 ( 1 3 p1p3) ( 1 3 3 p1 y) 3 p3 where, bl] refers to the (i,j)th element of B-l. (For details of the derivation, Goldberger [1967, p. 73ff.] may be consulted.) From this we see that the expenditure functions under the Quadratic utility specification are given by: .blj J -a-bljp- - y)p Z bljp- _ i3 - pixi — piZ.ajb (ZiZ J j l l J j -1 J pipj) (212 or in matrix notation: px = pB la - (p'B_lp)'l(p'B_la - y)fiB_lp where, p, denotes the (nxn) matrix whose off-diagonal ele- ments are zeros, and the ith diagonal element is pi. An ingenious, but unreasonable interpretation may now be offered with regard to the behavior of the consumer reflected by the expenditure functions given above. To do so we note that the marginal utility of income function may be derived to be: A = (p'B—lp)_l(p'B_la - y) So that the positivity of the marginal utility vector uX ensures that: y < p'B‘la. 152 1a may be thought of as maximum at- This means that p'B_ tainable income, or "bliss income." If actual income, y, were equal to the "bliss income" the consumer would buy the "bliss bundle" of goods, given by B_1a. Thus, an interpretation for the consumer's behavior may be offered as follows. The consumer is thought to "buy" the "bliss bundle" conceptually, but since his income is below the "bliss income" he "sells back" the goods, receiving a fixed proportion, pi(2jbljpj/2i2jbljpipj), of his "infra- bliss deficit" of, (Zizjajbijpi- y)> 0, in return [Gold- berger, 1967, pp. 74-75]. This interpretation is (admittedly) at best tenuous. A further discussion of the Quadratic utility model is omitted due to its limited relevance for empirical work. We note, however, the principle results. It is relatively easy to show that under the Quadratic utility function, the demand functions may give rise to negative quantities for some income-prices. Also, it is possible that some goods possess negative marginal utilities at low income levels while the opposite holds true for other goods. From the demand functions, we see that the marginal budget shares are constant for given prices. Thus, the Engel curves are linear. However, under the Quadratic utility model, the marginal budget shares are not bounded between zero and unity, (although they sum to unity as usual). This is because the elements 153 of B-lp may be negative. The possibilities of negative marginal budget shares also implies that inferior goods are permitted under the Quadratic utility function. An additional property of this function concerns the perverse behavior exhibited by the Frisch money flexibility func- tion. In the case of Quadratic utility, for a given set of prices, the money flexibility falls from zero to minus infinity as income rises from zero to bliss income. Thus, its inverse (and not itself) may serve as a welfare indi- cator in this case. Finally, we note that if it is specified that bij = 0, for i#j, then utility becomes additive, and the additive quadratic utility function gives rise to the so—called Gossen map (see Samuelson [1947a, p. 93] and Allen and Bowley [1935, p. 139]). From the point of View of estimation, the Quadra- tic utility model possesses little attraction. In fact, the demand functions that result are quite complex, and have not been estimated by anyone.79 Direct AddilogyUtility: The direct addilog utility function (DAUF) was pro— posed by Houthakker [1960], alongwith its operational counterpart, the indirect addilog utility function, (IAUF). The DAUF is given by: 79We have but paraphrased Goldberger [1967, pp. 73-80] in this section. 154 (DAUF) ... u = u(x1,...,xn) = 2i ai xi where it is assumed that ai > 0, and 0 < bi < l for all i=1,...,n. It has been shown by Houthakker [1960, p. 253] that only a partial solution to the demand functions is available. This is due to the fact that the first order conditions are difficult to solve explicitly for the x's in terms of the prices and income. To get a partial solution, the ratio of marginal utilities may be equated to price ratios to give: (1 — bi)log xi - (l - bj)log xj = log(aibi/ajbj) (for all i,j) Since the direct addilog model is non-operational, (unless extremely tedious methods of estimation are adopt- ed), we omit further discussion, but note briefly the few properties that are known about this model. Houthakker's [1960] results that the ratio of income elasticities under DAUF are constant, may be easily derived by taking a total differential of the equation above, assuming prices con- stant: (1 - bi)d log xi - (l - bj)d log xj = 0 (all i,j) which implies, after clearing terms and dividing by dlog y, 155 A special case of the DAUF results when all the bi are equal, say to b. It is easy to show, then, that the resulting demand functions are given by -1 (b—l) (p-/a-b x. = 1 l) y (i=1,...,n) -1 (b—l) Zj pj(pj/ajb) Hence, all Engel curves are straight lines through the origin, so that income elasticities are all unity. Pollak [1967, p. 3] shows that this special case of the direct addilog utility function is a monotonic trasformation of the constant elasticity of substitution utility function. This is easy to see if we set ai = Di, b = —p. Then 1/b - —1 v(xl,...,xn) = u = (Zipix. p) /p 1 which is the CES utility function. Finally, for the spe— cialized indirect addilog function, it can be shown that the money flexibility is given by: ¢=(b-l)l which is negative but independent of income. Indirect Addilog Utility: Finally, we have the operational case of "indirect addilog" utility function due also to Houthakker [1960]. Estimation of demand functions under this utility function 156 was carried out by Houthakker [1960], although the demand functions which result had already been explored by Somermeijer and Witt [1956]. Somermeijer [1961], Russel [1965], Parks [1969], and Yoshiahara [1969], have applied the model to data from Netherlands, U.S.A., Sweden, and Japan, respectively. The extreme difficulty in estima- tion, the lack of good empirical fits, and the admitted lack of intuitive justification render the indirect addilog case as somewhat of a curiosity among demand models. The indirect addilog utility function proposed by Houthakker [1960] is given as b. = * = l o ... (LAUF) ... u u (y,pl,...,pn) 2i ai(y/pi) (ai = (log y) - Ej wj(log pj) With this formulation, several parametrizations are pos- sible. First, it may be assumed that the income elastic- ity, Ei’ varies as a rectangular hyperbola with respect to variations in the budget shares, wi, i.e., let wiEi = bi' where bi (i=l,2,...,n) is a constant. This is the assumption that marginal budget shares, which are the slopes of the Engel curves, are constant with respect to variations in expenditures and prices. This assumption of linearity of each Engel curve is crucial for aggregation to go through, as we have seen in section 1.6, and is, in fact, utilized both by Stone, and by Theil and Barten in the formulation of their models. Denoting by bi' the respective marginal budget shares, we obtain a partial 169 parametrization of the model. If, in addition, we assume that the quantity (wisij) is constant with respect to variations in expenditures and prices, as the Rotterdam models do, we obtain the following specification, which may be called the "double-logarithmic Rotterdam elasticity specification" model (DOLRES) ... wi(log xi) = bi(log y) + Zj cij(log pj) where, Cij’ denotes the constant (wisij), by assumption. Notice, that the above specification is, in fact, identi- cal to the Rotterdam model, if the logarithms of the variables are replaced by changes in the logarithms of these same variables. Before proceeding to discuss further similarities between the DOLRES and the Rotterdam models, an alterna- tive specification may also be derived. Note that the Hicks-Allen "elasticity of substitution" between goods, d.., is related to the Slutsky price elasticity, s.., by 13 13 the following relation: dij 13' 3" Thus, the Rotterdam assumption that the cij are constant with respect to variations in expenditures and prices, reflects the belief that the elasticity of substitution between goods is determined solely by the budget shares 170 of the two goods, and is, in fact, given by the ratio of a constant and a product of the two budget shares. Form- ally: where, cij is constant. Although, with the exception of Leser's LES, demand models have not been formulated in terms of the elasticity of substitution, we see that this can easily be accomplished in this model. The constancy of the elasticity of substitution between factors has been hypothesized in the theory of production and has been used to great advantage. Incorporating the same assumption for consumer goods, we propose the following double-logarithmic (constant elasticity of substitution) model, (DOLCES): (DOLCES) ... wi(log Xi) = bi(log y) + Zj dij wiwj(log pj) where, dij is the (constant) elasticity of substitution between goods i and j.83 Having given two parametrizations of the tradi- tional double-logarithmic model, we consider the implica- tions of the classical restrictions on the parameters. These are easy to derive. Engel aggregation implies that 83It might be mentioned that a similar version Of the Rotterdam model may also be constructed if it is so desired. In fact, the theory of production is rich in hypotheses with regard to the elasticity of substitution, and it may, perhaps, be a useful source of alternative parametrization for demand models. 171 the marginal budget shares, bi’ sum to unity. The symmetry condition implies that the coefficients c.. = for the 1] Cji DOLRES model, and the dij = dji for the DOLCES model, for all i, j. For the DOLRES model, the homogeneity condition, implies that the cij sum to zero for each equation: Zj cij = 0; while, for the DOLCES model the condition is that Zj w.d.. = 0, for all i. For both models, the J 1] homogeneity condition may be incorporated into the func- tional form, by deflating all prices by one of the prices. Without loss of generality, we may deflate all prices by the nth price to obtain the following specifications for the two models: (DOLRES) ... wi(log x.) l bi(log y) + Zj cij(log pj- log pn) (DOLCES) ... wi(1og xi) bi(1og y) + Z. d.. w.w. lo .— lo 3 l] 1 j( g P3 9 Pn) where the summation over the index j is understood to be from 1 to (n-l). In this version, with the bi's restricted by Zibi = l, the DOLRES and DOLCES models satisfy all the classical restrictions other than the adding-up criterion, and, of course, the negative definiteness condition on the matrix of Slutsky terms. The DOLRES model, in particu- lar, is identical to the preliminary formulations of the 172 Rotterdam models if the logarithms are replaced by changes in logarithms. Thus, the parameters of the DOLRES models have exactly the same interpretations as the parameters of the Rotterdam models, and provide a useful basis of comparison. This similarity between DOLRES and the Rotterdam model is not surprising. The Rotterdam model uses an exact expansion of the logarithmic differential of the demand function and approximates the infinitesimal changes by finite changes, while the DOLRES model is de- rived by considering a particular solution to the set of partial differential equations defining the logarithmic partial derivatives which appear as coefficients in the Rotterdam model. Thus, in a sense, we have merely formu- lated the Rotterdam model in an absolute, rather than differential, version. It is this similarity between the two models which we exploit to the advantage of the DOLRES and the DOLCES models, in attempting to solve the problem of the non-additivity of the demand equations. Recall that a similar problem arose with respect to a preliminary version of the Rotterdam model, in which the adding-up criterion was also not met. In the Rotter- dam model the solution was found by replacing the log- change in real income by an index of the log-change in the volume of consumption. This procedure was justified [Theil, 1967, p. 224ff.] by noting that both the log- change in real income and the log-change in the volume of 173 consumption were local quadratic approximations to the true cost of living index, involving errors of third order in logarithms of changes in prices and income. Thus, the two approximations were considered interchangeable for empirical purposes. This would suggest, admittedly on intuitive grounds, that an index of the absolute level of the volume of consumption--similar to the log-change volume index--may provide an adequate approximation to the level of real income. Replacing the logarithmic real income term (log §), by a log-volume index of the level of con- sumption, x*= kwktlog th' we may rewrite the DOLRES and t and DOLCES models in their final non-stochastic form: (DOLRES) ... wit(log x. = * lt) b. x II C" >4 x- (DOLCES) ... wit(log x ) it i t +3: d.. witwjt(log p.t- log pnt) 113 3 where the index i ranges over commodities, and the index t refers to time. Note that in doing this the bi may no longer be interpreted as marginal budget shares. We may, however, still consider them as approximations to the marginal budget shares. 174 To gain additional insight into the nature of ap- proximation involved, we might consider the relationship between the absolute log-volume index utilized here, and the log-change volume index used in the formulation of the Rotterdam models. The Rotterdam volume index is given by: = * _ Dxt 2k wkt(log xkt log Xk,t—l) where, w* kt l/2(w + kt wk,t-l)° Note that the use of arithmetic means of the budget shares as weights in the construction of the Rotterdam index in— sures that the index meets the time-reversal test. This is in fact, the reason why the Rotterdam model, in its current formulation, utilizes the arithmetic means of the budget shares as the common multiple corresponding to the role of the current period budget shares used in this sec- tion in the derivation of DOLRES and DOLCES models. An additional feature of the Rotterdam index is its purely "statistical" prOperty of being zero for the case when xkt = xk,t-l for all k. In comparison, the index we propose is an "econo- mic" index, and is given by: * = x 2k wkt(log xkt). 175 Even if all quantities were to remain unchanged, on purely statistical grounds it would be possible for the budget shares wk to change due to changes in prices, thus mak- t ing the volume index proposed by us to fluctuate due to variations in prices even though the volume of consumption does not change. Clearly, this would be an undesirable prOperty for a volume index to possess. To insure that the log-volume index proposed above does not change if all quantities remain unchanged, we must invoke the homo— geneity condition of economic theory, which insures that if 211 quantities remain unchanged, then the budget shares of the respective commodities could not have changed either. The log—volume index proposed here bears an expli- cit relation to the log-change in volume index utilized in the Rotterdam models.84 Using the notation developed above, it may be easily verified that the following re- lationship exists between the two indices: Dx = x* - x - 2i l/2(log (x t t t-1 )( wi’t_l) itxi,t—l) “it" Thus, we see that the change in the log-volume index pro- posed in this section diverges from the log-change in the Rotterdam volume index by a weighted average of the geo— metric means of the volume of consumption in the current and the preceding period; the weights being the differences 841 am indebted to Professor Henri Theil for point- ing out this relation in correspondence. 176 in the budget shares of the respective commodities over the two periods. This weighted average term reflects the difference in the two indices due to the use of absolute budget shares in one, and the arithmetic means of the bud- get shares in the other index, and, in fact, may be con- sidered to be the degree of error involved in approximating the Rotterdam log-change index by changes in the log- volume index. Although the relative magnitude of the term, ) )] (w - w Zil/2[log (xi it i,t-l tXi,t-l is not immediately obvious, we may still justify the use of the log-volume index on two grounds. First, since the Rotterdam index is itself an approximation, the degree of divergence of the log-volume index from the Rotterdam in- dex is not directly relevant. Secondly, this difference arises due to the use of arithmetic means of budget shares instead of the budget shares themselves. Thus, the volume index proposed here does not seem an unreasonable approxi- mation to the true real income. 4.3 Stochastic Specification and Estimation The stochastic specification of complete sets of demand equations has not been given the degree of atten- tion that it merits. The notable exceptions to this state- ment are the Rotterdam models, for which the "marginal 177 utility shock model" (discussed in section 3.2) is deve10p- ed with the aim of incorporating a stochastic element in the utility maximizing behavior itself; and the Pollak- Wales versions of the LES (discussed in section 3.3) for which a similar attempt has been made. Although the for- mal similarity between the Rotterdam model and the DOLRES and the DOLCES models could, perhaps, have been exploited with the intent of developing a stochastic model aprallel to the marginal utility shock model, we have chosen not to do so. Instead, we adopt the most convenient stochas- tic specification available, and assume that the disturb- ance term enters additively into the demand equations. With this specification, the disturbance terms in each of the demand equations may be interpreted as the allocation discrepancy due to random factors. To facilitate exposition, we shall write the demand equations for both the DOLRES and the DOLCES models as: (i=1,...,n) = b- yt + 23. k. . Z. + u. (j=l'ooopn-l) (27) yit 1 ij jt it (t-l T) '"pooo, where the bi and kij are constants, yit is the t-th ob- servation on the i-th dependent variable, and yt (=Xiyit), and xit are the t-th observation on the non-stochastic independent variables, y, and Xi' respectively. The uit are interpreted to be the t-th value of the i-th disturb- ance terms, and are assumed to possess a specified 178 variance-covariance structure. It is readily seen that with a suitable interpretation of the parameters and variables, (27) gives rise to both the DOLRES and the DOLCES models. If the uit are to be interpreted as allocation discrepancies, as we have suggested, then the sum of these allocation discrepancies in any time period must be zero. This can also be seen by recalling that the bi sum of unity and the kij are symmetric, and in the case of DOLRES, the ki. sum to zero for each equation. 3 Then adding over the i, (27) yields: 21 uit = 2i Yit ’ yt ' zizj kij zjt = Zj(zjtzikij) = zj(zjtzikji) = 0 A similar proof for the DOLCES is readily constructed. Thus, it has been shown that the disturbances in the demand equations sum to zero, and hence are not mutually independent. This is the cause of the singularity of the covariance matrix of the disturbance terms which must be present in any model of consumer demand which seeks to allocate total expenditure into its various components. Adopting the simplest assumptions with regard to the moments of the distribution of the disturbances, 179 uit’ the complete model may be formulated as follows: (i=1,...,n) (28a) yit = b. yt + Zj kij zjt + uit Ej:l:::::g;l) (28b) Zibi = l; kij = kji (for all i,j); (28c) ziuit = 0 (for all t) (28d) E(uit)= 0 (for all i,t) (28e) E(uisujt)= 0 (for i#j, s#t) (28f) E(uitujt) = Oij (for all i,j,t) (289) E(uisuit) = 0 (for all s¢t). In this formulation the set of equations (28) have a con- venient matrix representation due to Zellner [1962]: [Y1 x1 0 ' ' ' 0 81 u1 yn 0 X2 - - - 0 82 u2 (20) . = . . . . + : Ynj 0 0 ‘ ‘ ‘ Xn 8n un where, for all i=1,...,n: 180 ?11—' Pyl z11 z21 . . znl_ I11; in y2 z12 z22 . ° ° zn2 u12 2 = 2 Em- YT le 221' ° ° ' 7‘an fir with yit’ yt, zij defined as in (28).85 Even more compact- ly, (29) may be written as: (30) y = X8 + u where, y is the (nTxl) vector of observations on all of the dependent variables, and X is the (nT x n2) matrix of observations on the non-stochastic explanatory variables, 8 is an (n2 x 1) vector of parameters, and u is an (nTxl) vector of disturbances with mean zero, and covariance matrix, E(uu') = X. With this notation, the model represented by the equations (28a—g) may be completely represented as: y = XB + u (31) R8 = r E(uu') = Z, and £1 uit = 0, for all t. 85Note that the ells of xi do not depend on i, so that xi = xj 181 where, R is an appropriate restrictions matrix which to- gether with the vector r expresses the restrictions (3.3.2b), and Z is the covariance matrix whose elements are defined by the equation (28e,f,g).86 The derivation of a best linear unbiased (restrict- ed) estimator, which is a maximum-likelihood estimator under the assumption of normality of the disturbances, is trivial in the case that Z_1 exists, and is a direct ex- tension of the simple restricted least squares estimator outlined for example in Goldberger [1964, pp. 256-258]. Unfortunately, the restriction that the budget discrepan- cies, sum to zero for each period imply the singular— uit' ity of the covariance matrix 2. Thus, neither the likelihood function exists, nor is the "generalized sum of squared deviations" [Goldberger, 1964, p. 233] (y - XB)’ 2‘1 (y — XS) 8 defined. In a pathbreaking paper, Powell [1969] considered the set of all restricted estimators of B,{E}, and showed that a specific estimator, say 80, of any independent sub— set of the parameters, possesses the property that the generalized sum of the squared deviations associated with this estimator (which is defined by construction) is equal 86The y and x of (31) should not be confused with the earlier notation in which y was income and x referred to quantities of goods! w 182 to the following generalized sum of squared deviations n n associated with 88 {B}: n + n S*= (y-XB) Z (y-XBM where 2+ is the unique Moore—Penrose generalized inverse of the covariance matrix 2. The Moore-Penrose inverse of a matrix of less than full rank is defined as 1 1 2+ = Q' (QQ')' (T'TA‘ T' where, T is any column basis for Z, and Q is a matrix which satisfies the condition: Z=TQ The Moore-Penrose generalized inverse, is a g-inverse in the sense that: The procedure suggested by Powell [1969] is computation- ally equivalent to deleting as many equations as are necessary to render the covariance matrix to be of full rank, then estimating the remaining parameters using the conventional restricted least squares estimator, and finally recovering the left out parameters by the use of the restrictions matrix R. The force of Powell's result, which derives from the uniqueness of the Moore-Penrose inverse, is that the estimates of the parameters so 183 obtained are numerically invariant with respect to the choice of the equation to be deleted. We omit the details of Powell's proof, but heuristically derive an alternative formulation of the same result.87 Instead of looking at the generalized sums of squared deviations, we may alternatively consider the following generalized "likelihood" function which may be defined when the disturbances are assumed to be multi- variate singular normal, with the covariance matrix, 2, of less than full rank: L* = -l/2n(1og 2n) - l/2(y-XB)' 2+ (y—XB) where, 2+, as before is the Moore-Penrose generalized in- verse. We might seek a restricted maximum "likelihood" estimator which maximizes L* subject to an arbitrary linear constraint on the parameters: Note that the "likelihood" function L* is well—defined in the sense of being single valued. Thus, the solution of the constrained maximization of the "likelihood" function will give rise to an estimator which is also unique and 87It ought to be mentioned that Powell [1969] did not present his results for the case of arbitrary linear restrictions, but considered instead, some specific re- strictions on the parameters. However, his results seem to hold for the general case which we have presented above. 184 well-defined. The derivation is strightforward. Construct the Lagrangean function: L = - l/2n(1og 2n) - l/2(y-XB) 2+ (y-XB) + 2 A'(RB - r) where, A, is the vector of Lagrange multipliers of dimen- sion (Jxl), where J is the rank of the restriction matrix R. To maximize L, take partial derivatives with respect to B, 5? = X'Z+y - (X'Z+X)B + R'A Equating to zero, we have: 1 8 = (X'Z+X)-1(X'Z+y) + (x'z+X)' R'A Utilizing the fact that R8 = r, we have from this equation by multiplying through by R, and rearranging, 1 = (R(x'z+x>’ R')’1(r - Rb). >JZ (x'z+X)'l(x'z+y) where b Substituting for A in the original equation we have the restricted maximum "likelihood” estimator which is equiva- lent to Powell's estimator: E = (X'Z+X)'l(x'z+y) 1 l + (x'z+X)" R'(R(X'Z+X)— R')-l(r-Rb) 185 Since 2+ is numerically invariant with respect to the choice of any column basis, we may for computational con- venience delete arbitrarily the last equation, and esti- mate the following reduced system of demand equations: (32) = b. yt + zj k.. z. + u. (i,j=l,...,n-1; yit i ij jt it t=1,...,T) with the restriction that kij = kji for all i,j=l,...,n-l. The parameters of the n-th demand equation may be recovered by the left out restrictions of symmetry, and the Engel. aggregation conditions. Powell's theorem insures that this procedure gives rise to the same numerical estimates of the parameters. To estimate the reduced system of equations (32) we note first that the disturbances are correlated across equations. This would suggest using Zellner's (asympto- tically) efficient two-stage Aitken procedure. However, as is well known, in the case where the eXplanatory vari- ables in each equation are the same, Zellenr's procedure (ZEF) reduces to the ordinary least squares procedure (OLS). To impose the symmetry restrictions, however, we must once again resort to the Zellner-Aitken procedure. The formulation (289) restricts the disturbances to be non-autoregressive. Since the data utilized in the pre- sent study are time series, some sort of autocorrelation may be present. If it is assumed that the disturbances 186 follow a first-order autoregressive (Markov) scheme, then a modification of the Zellner procedure due to Kmenta and Gilbert [1970] may be utilized. This is a four-stage pro- cedure in which the Zellner-Aitken two-stage residuals are utilized to obtain a weighted first-differencing of the relevant variables before estimating the parameters by the two-stage ZEF-again. Formally, we consider the case where (289) does not hold, but instead we have: I .— 1289 ) E(uitui,t-s) ‘ pi Oii which is the specification which results if the disturb- ances uit follow a first—order autoregressive scheme: uit = piui,t—1 + Vit with the vit distributed independently and with zero mean, and constant variance: Var (Vi = (1 -p§)cii. The de- t) mand model may then be written as: + Z. k.. z. + u. yit — i yt j ij jt it (33) uit = 0i ui,t-1 + Vit with vit satisfying the assumptions of the classical linear regression model. To estimate the DOLRES and DOLCES models, we use the direct extension of the Kmenta-Gilbert four- stage procedure, ZEF-ZEF. 187 First, OLS residuals are used to get consistent estimates of the covariance matrix of the system of (n-l) equations obtained by deleting the last equation. Second- ly, using these estimates we obtain the restricted least squares estimates of the parameters, where the restric- tions imposed are those of symmetry. Thirdly, assuming that the scheme of autocorrelation is a first-order Markov scheme, we used the residuals of the restricted ZEF esti— mates to estimate the autocorrelation parameters, pi. Finally, we obtain once again the two-stage restricted ZEF estimates of the parameters, after correcting for autoregression by lagging the variables in the usual man- ner. 88 4.4 Data and Variables The data utilized in the empirical section of this study are from The National Income and Product Ac— counts of the United States, 1929-1965: Statistical Tables. The basic sources are Tables 2.6 and 8.6. The former give constant dollar expenditure on some forty-six expenditure 88Some small sample results by Kmenta and Gilbert [1968] reveal that the use of ZEF residuals, instead of OLS residuals for the estimation of the 9. lead to considerably more efficient results in small samplés. Indeed, the most efficient procedures found by Kmenta and Gilbert [1968] is to estimate the p. jointly on the basis of ZEF residuals, (called JOINTEST procedure by them). This procedure was, in fact, considered by Parks [1967, p. 503, n.], but was rejected as unnecessary. 188 categories, and the latter lists implicit price deflators (1958:100) for the same categories. From this data we constructed four aggregates: Food, Clothing, Shelter, and Miscellaneous. These aggregates were used by Pollak and Wales [1969], who estimated several variants of SLES from the same series that we have used. Pollak and Wales, how- ever, used only the data from 1948-1965 for the major part of their study, although they made comparisons between pre- war and post-war data, and found significant differences in the two samples. In this study, we have used the en- tire series from 1929-1964, but have constructed our aggregates exactly as Pollak and Wales [1969] did. We have also used the additions to the time series that be- came available recently. Thus, we added to our 1929-1964 data, additional data covering the period 1965-1968, which are provided in the July, 1969 issue of the Survey of Current Business. In the construction of the four commodity aggre- gates, Pollak and Wales excluded all durable goods, transportation services, and gasoline and oil. The classifications of expenditures that were aggregated into the four commodities are listed below: (the numbers in the parentheses refer to the two tables cited above). I. Food: 1. Food and Beverages (15) 189 II. Clothing: 1. Clothing and shoes (21) 2. Shoe cleaning and repair (54) 3. Cleaning, dyeing, pressing, etc. (55) III. Shelter: Housing (35) Household operation services (39) Semidurable housefurnishings (29) Cleaning and polishing preparations, etc. (30) Other fuel and ice (31) WDWNH o o o o 0 IV. Miscellaneous: Tobacco products (27) Toilet articles and preparations (28) Nondurable toys and sport supplies (33) Barbershops, beauty parlors, and baths (56) Medical care services (57) . Admission to specified spectator amusements (61) . Drug preparations and sundries (32). \lmLfl-bWNI—J The price index for the aggregates were constructed by taking a weighted average of the implicit price deflators for each primary expenditure item. The weights used were the ratios of expenditures on the primitive items to the total expenditure on the aggregate. In the definition of variables, we differ somewhat from the procedure used by Pollak and Wales. First, un— like Pollak and Wales, we estimate community demand functions in the aggregate, instead of estimating per capita expenditure functions. Secondly, we define the quantity of the i-th commodity, (xi), as the constant dollar ex- penditure divided by the implicit price deflator for the commodity. Pollak and Wales, on the other hand, chose to consider the constant dollar expenditures as "quantities" 190 (xi). Both procedures seem perfectly acceptable to us, thought, it appears that our definition might be a more natural one. 4.5 Empirical Results In this section we present the estimates, based on the four commodity data for the United States, for the parameters of the DOLCES and the DOLRES models proposed in sections 4.2 and 4.3. For the purposes of comaprison, we have also estimated from the same data, the Rotterdam model and Leser's linear expenditure system. Before con- sidering each model in turn, we make a few remarks with regard to some statistical peculiarities of the data under consideration. In the estimation of SLES models from the same data, Pollak and Wales [1969] assumed the data to be free of serial correlation. It seemed to us that this was not a plausible specification for any time series. How- ever, our results lead us to believe that if autocorrela- tion is present, it does not seem to follow a first-order Markov scheme. This is because our corrections for first— order autocorrelation were not very effective for most of the models considered. Specifically, even after we trans- formed the data, the estimates obtained showed the Durbin- Watson statistic to be significantly indicative of the presence of first order autocorrelation. A cursory visual examination of residuals plotted against time failed to 191 reveal any quadratic or higher order pattern. This would suggest then that the Durbin-Watson statistic is inappro— priate for the data under consideration. In the presenta- tion of our results, however, we continue to give the Durbin-Watson statistic, although it should be realized that its use as a test statistic may be limited.89 As it was indicated in the previous sections, the DOLRES model is estimated from the following specifica- tion: wit(log X. It) = bi(Z k wktlog th) + Zj cij(log pjt - log pnt) + uit where, i=1,...,n-l; t=1,...,T; k=1,...,n. The parameters bi are approximations to the marginal bud- get shares, due to the fact that the real income term has been replaced by a volume index, although we shall continue to use the term "marginal budget shares" to describe them. The cij are the income compensated (Slutsky) price elas- ticities, weighted by the budget share of the i-th commodity. Alternatively, the cij may be interpreted as the Hicks-Allen elasticity of substitution between goods 1 and j, weighted by the inverse of the product of the 891t would have been extremely interesting to ex— amine the applicability of some specification error tests devised by Ramsey [ ] for single equation estimates. Further research in this direction will be followed in another study. 192 respective budget shares. These parameters with the same interpretations, also appear in the Rotterdam model, which, it may be recalled, is specified as follows: w? Dx. i _ I 1t — bi Dx + Z. c.. Dpjt + u t t j 13 it )I where, w? it = l/2(wit+ ), Dxi = (log x. t it - log Xi wi,t—i ,t-l I _ _ _ .- and Dpjt — (log pjt log pnt) (log pj,t-1 log pn,t-1)° The parameters bi are once again approximations to the marginal budget shares, though they are slightly different from the bi in DOLRES, by Virtue of the fact that we have used an absolute index instead of a differential index (see pp. 3-10ff.). The two models may, therefore, be compared directly with respect to the values of the para— meter estimates of Cij’ and to some degree with respects to the estimates of bi' The estimates of DOLRES obtained by using a restricted ZEF-ZEF (Kmenta-Gilbert) procedure, are reported in Tables 1.1, 1.2, and 1.3. It should be noted that at each step of the esti— mation procedure, the marginal budget shares are all posi- tive and lie between zero and one, as we should expect. In addition, the income compensated own-price (Slutsky) elasticities are all negative, as theory would lead us to believe, and the matrix (estimated) of the coefficients cij is negative definite. The DOLRES model, therefore, 193 Table l.l.--OLS/ZEF Estimates of Unconstrained DOLRES Model. i J "Income" Food Clothes Shelter Food 0.1532 -0.4643 0.1643* 0.1014. R2=0.7328 (0.0138) (0.1479) (0.1501) (0.0699) Dw=o.1125 Clothes 0.3143 0.1373 -0.0894* 0.1544 R2=0.7796 (0.0048) (0.0508) (0.0516) (0.0240) DW=0.1809 2- Shelter 0.2176 0.2101 —0.1711 -0.4563 R —0.9487 (0.0056) (0.0594) (0.0603) (0.0281) DW=0.2064 *Indicates that coefficients are less than twice their standard errors, and hence are not significantly dif- ferent from zero. Note: Although parameter estimates are identical for OLS and ZEF, the estimates of standard errors differ. This table records OLS estimates of the standard errors. Table 1.2.--ZEF Estimates of Unconstrained DOLRES Model Correcting for First Order Autocorrelation. "Income" Food Clothes Shelter 0.2978 —0.3575 0.0965* -0.0315* R2=0.9350 (0.0269) (0.0413) (0.0526) (0.0790) DW=0.6058 Clothes 0.2707 0.1283 -0.1170 0.2022 R2=0.8231 (0.0112) (0.0221) (0.0285) (0.0446) DW=0.9145 0.2214 0.1411 —0.0367* —0.3463 R2=0.8421 (0.0114) (0.0230) (0.0296) (0.0488) DW=0.6100 Food Shelter *indicates coefficients not significantly dif- ferent from zero. 194 Table l.3.—-ZEF Estimates of Constrained DOLRES model (With Correction for Autocorrelation).ar "Income" Food Clothes Shelter Fo d 0 2300 -0 3737 0 1432 0 1123* R2=°°9266 0 ° ' ° ' DW=0.4953 Cl th 0 2659 -0 1693 0 0304* R2=°°76°3 0 es ° ' ° DW=0.5295 2 R =0.79l6 Shelter 0.2266 -0.2328 DW=O.2564 aThe missing entries may be recovered by symmetry. bAsymptotic F statistic for restrictions: F2'99 = 12.052 (Sig. Prob. < 0.005). *indicates coefficients not larger than twice the standard errors of the unconstrained estimates. See text for a discussion. fulfills the negative definiteness criterion (for this sample) which it should be recalled, was the only aypriori knowledge that we held but did not impose in estimation. Note, also, that the demand equation for the fourth commodity group "Miscellaneous" may easily be ob- tained from the Tables above, by virtue of the fact that we restrict the marginal budget shares to sum to unity, and the cij to be symmetric and sum to zero for each equation. In Table 1.3, it should be pointed out, that the standare errors of the parameters are not reported due to the unavailability of this feature in the statistical 195 computer programs available at M.S.U. As an alternative, we used the well-known result (see Goldberger, 1964, p. 257) that the standard errors of the unrestricted estima- tors is an upper bound to the standard errors of the restricted estimators. With the use of this result, we see that with the exception of two parameters (about which we may not say anything), the rest of the parameters in Table 1.3 are significantly different from zero. In Tables 3.1 and 3.2 we report, respectively, the unconstrained and constrained estimates of the Rotter- dam model. No correction for autocorrelation was attempted for the Rotterdam model, which uses a differenced version of the data. We assumed that the differencing procedure would reduce the presence of any autocorrelation in the original time series. (This was, in fact true of Swedish data employed by Parks [1969], who reported that although the data indicated the presence of high serial correlation, the Rotterdam model indicated values of the Durbin-Watson statistic which were not indicative of the presence of serial correlation in its residuals.) In the case of our data, however, two of the three equations yield values of the Durbin—Watson statistic which would indicate the pre— sence of first-order autocorrelation, if such a scheme was assumed to exist. We attribute this result, once again, to the peculiar nature of our data, for which the hypothesis of first-order autocorrelation may not be tenable. within the theoretically expected range. 196 For the Rotterdam model also, the estimates lie The estimates indicate, once again, the negative definiteness of the, estimated matrix of Slutsky terms. More interesting, how- ever, is the considerable similarity in numerical magnitude of the estimated cij DOLRES model. for the Rotterdam model and the This is, indeed, what we had expected. Although we have not tested the hypothesis of equivalence of the parameter estimates under the two models, a cursory comparison seems to suggest that this might, in fact, be the case. Note also the differences in the estimates of the bi' which we had also anticipated. Table 3.l.--Unconstrained OLS/ZEF Estimates of the Rotter- dam Model. ”Income" Food Clothes Shelter Food 0.4626 -0.3358 0.1978 0.1064 R2=0.9618 (0.0376) (0.0344) (0.0444) (0.0688) DW=0.8002 Clothes 0.2174 0.1299 -0.1638 0.1739 R2=0.8596 (0.0267) (0.0243) (0.0314) (0.0486) DW=1.8211 Shelter 0.1856 0.1340 -0.0619 -0.3120 R2=0.4732 (0.0372) (0.0339) (0.0437) (0.0676) DW=O.9556 197 Table 3.2.--Constrained ZEF Estimates of the Rotterdam Model.a "Income" Food Clothes Shelter R2=0.9599 Food 0.4491 -o.3075 0.1420 0.0979 Dw=0.7306 Clothes 0.1816 —0.2217 0.0435* EZZEIE§23 Shelter 0.2328 —0.1750 3;:813égé aAsymptotic F statistic for restrictions: F3 99=3.764. Corresponds to Sig. Probability of 0.013. ’ In Table 2.1, 2.2, and 2.3 we present the estimates of the parameters of the DOLCES model, which, it might be recalled, is specified by: wit(log X. 11:) = 1Di (2 k wktlog th) + Zj dij(1og pjt - log pnt) + uit where the bi are to be interpreted as identical to their counterparts in the DOLRES MODEL, and hence are approxi— mations to the marginal budget shares; and the dij repre- sent the (constant) Hicks-Allen elasticities of substitution in the DOLCES model. It should also be noted that the assumption of constancy of the elasticities of substitution, will affect the parameter estimates of the b-, 1 so that the estimates of the bi should be expected to 198 Table 2.l.--OLS/ZEF Estimates of Unconstrained DOLCES Model.a "Income" Food Clothes Shelter Food 0.1545 —3.4778 3.8905 0.7654* R2=0.7683 (0.0121) (0.7854) (1.9128) (0.5759) Dw:0.1254 Clothes 0.3143 2.3113 -3.9855 3.0470 R2=0.8175 (0.0040) (0.6270) (1.5654) (0.4151) DW=0.2109 Shelter 0.2266 1.8475 -3.0030 -5.1947 R2=0.9431 (0.0055) (0.5632) (1.4068) (0.3288) DW=0.1586 aAlthough OLS and ZEF yield identical estimates of the coefficients, they differ on estimates of the standard errors 0 standard errors. * In this Table we report OLS estimates of the indicates coefficient not significantly different from zero. Table 2.2.—-ZEF Estimates of Unconstrained DOLCES Model Correcting for First Order Autocorrelation. "Income" Food Clothes Shelter Food 0.2855 -2.l652 1.3156* -0.6238* R2=0.9296 (0.0330) (0.2573) (0.7564) (0.8084) Dw=0.7031 Clothes 0.1729 1.1809 —5.0622 1.8112 R2=0.9783 (0.0039) (0.1100) (0.3343) (0.3418) Dw=0.7436 Shelter 0.1736 0.9274 -0.5776* -3.6817 R2=0.8511 (0.0132) (0.1818) (0.5452) (0.4470) DW=1.2317 *Indicates coefficients not significantly different from zero. 199 Table 2.3.--Constrained ZEF Estimates of DOLCES Model (with Correction for Autocorrelation).a "Income" Food Clothes Shelter R2=0.9254 Food 0.2777 -2.1017 1.1731* 0.4770* DW=0.6552 Clothes 0.1738 -5.0854 1.1051 fi;:3:§;§§ Shelter 0.1806 -3.6738 fi;:2;§3$2 aAsymptotic F statistic for restrictions: F = 4.740 (Sig. p. = 0.004). 3,99 differ for the DOLRES and DOLCES models, although the economic meaning of the parameter is identical for the two models. In evaluating the parameter estimates, we might note that for the DOLCES model, as before, the marginal budget shares lie between zero an unity as should be ex— pected. Also, the Hicks-Allen own-elasticities-of- substitution, which are equivalent to the income compensated own (Slutsky) price elasticities, are all negative as should be expected. The magnitude of the estimates of the dij parameters may be compared to the estimates of cij in the Rotterdam model and the DOLRES model, only for specific values of the budget shares, by using the relationship c.. w.w.d... ij 1 j ij 200 Before making this comparison, we shall examine the esti- mates of Leser's LES, for which direct estimates of the cross-elasticities of substitution are obtained. We pre- sent these results in Table 4.1. Leser's linear expenditure system, LLES, is esti- mated by the following specification: pit(xit ' Xi) = bi(zk pkt(xkt ' §k1) + Z d z + c t + u jsi ij ijt i it where, xi = (l/T)Zt xit' and if we denote wi = (l/T)ZIt wit , zijt = (wipjtxj ‘ wjpitxi)° Here, the dij are directly comparable to the dij of the DOLCES model. In the DOLCES model, however, we imposed the homogeneity condition by deflating one of the prices by the last price. This is not the case for LLES. In the LLES, the homogeneity condition is imposed only at the sample means of the expenditures, prices, and budget shares. Thus, the own-elasticity of substitution can be obtained in LLES by the following relation: d.. = -(1/6 ) z fild.. ii i j#i j 13 201 .mHOHHm onmocmum “hoop OOHSU ammo mmwa mum soaps muowHOHmmmoo moumoaoGHs me n flxfl a Hmoumnm bemoan oulx mop mo some mHmEMm map wouoooo x3 mum£3 ...omB w AmB\HvI n ..o .HMGO©MAU mnu so mmauuoo moan .mmumfifiumo moo co woman mum oaomu memo ca Concomoum manoeoammmoo UmmeHumo mop mo mnounm Uncommon ones memm.ou3o Amoeo.ov Imam~.ov Adamo.eo loose.ov Amooo.oo .omez whom.oumm mmmo.o Hemm.OI mmmm.HI mHmN.H mbmo.o . wwmm.ofl30 Amaoa.ov Ammhv.mv thmm.av Ammmm.ov Awamo.ov Houaosm oanm.oumm mmam.o mem.m mmHN.wI omvm.m Hmhm.o mmoN.Hu30 Amowo.ov Ahmmm.mv Abmmm.ov onao.ov Awmmo.ov . I . . . . . mogpoHU mmmm olmm mvma OI Nmmm ml oamm m mmoa on deem o voom.on3Q Ammva.ov Aoomm.mv Avmmv.av onmh.av Ammmo.ov boom moom.oumm «mnna.on sommv.bl ommH.m Hmmv.h Haqm.o pawns .omflz Hmuamom moouoao poem maoosH p.p.mmq m.hmmoq mo mmumsauum mmm\mqo pmsaopumcoossuu.a.e dense 202 .mmumEHumm ouncemummucs one noncommunoo may no muonuo oumpowum mob mo some OHHuoEOmm mop mo moam> mop OOHBD swap mmma mum moans mumpmamnmm pmuoanumon mop mo mmumsflumm mmumoaocHr mm.m .mooo.o coop mmwa mo muflaflomoonm OOCMOHmacme o no: new .ome.ma u m “ommo menu CH we .mnpmsfimm mo mooHuOAHummu AHMOGHHV mop moflpmmu How owumflumum m vapoumemmm moan .m.v manna ca mm oooflmuoo mum HMGOUMHU may mo masoEmHm one .COHuopHumoom mo hpflowummam was NO anumEEMm mop moHMO>cH an omno>ouon on has mOHHuom accompaolmmo woes eeom.ouzo mmma.oumm omeo.o rossm.o mmem.o hopeosm emem.ouzo oomm.oumm Heom.ou .seem.o- reme~.o oeo~.o mospoeo memm.ouzo same.oumm omoa.o- .mmmo.e .Hooo.ou .oome.o- Hoem.o ooos ozone .omHz kuamnm monuoHU poem OEOOGH I 1' (III L1 p.p.mmq m.uomoq mo mopssspmm emu poseoupmsoouu.m.e menus 203 This is somewhat inconvenient for the purposes of direct comparison. Also, we might note that the DOLCES model imposes all of the classical restrictions globally, while LLES imposes only the adding-up restriction globally. Thus, the similarity between the DOLCES and LLES is not as great as for example between the DOLRES and the Rotterdam model. We estimated LLES by a direct parallel to the estimation procedure used for DOLRES and DOLCES models. The ZEF/ZEF procedure was extended to the case of restricted estimation, where the restrictions were those of symmetry of dij’ Once again, we did not correct for the presence of autocorrelation, due to the peculiar nature of the time series under consideration, despite the fact that the Durbin-Watson statistics indicated the presence of first-order autocorrelation, if such a specification was assumed to exist. Although the estimates of LLES may not be directly compared to the DOLCES model because of the fact that own-elasticities of substitution are absent in LLES, we may still observe that the estimates of the cross- elasticities of substitution show a marked difference for the two models. Within LLES itself, the elasticities of substitution between two goods show considerable varia- tions in their unconstrained estimates. Thus, a cursory examination of LLES in its unconstrained form reveals, 204 for our data, that the elasticities of substitution do not appear to be symmetric. We may test the linear hypothesis of symmetry by observing the value of the (asymptotic) F- statistic, which is, in our case, associated with a proba— bility value of less than 0.0005. Using the asymptotic F-statistic as a proxy for our small sample test, we would reject the hypothesis of symmetry conditional on Leser's specification of the functional form. Although we impose the restriction, despite its implausibility, we note that this result raises some doubts with regard to the empirical plausibility of LLES. Anotherconsequence of imposing re- strictions, which are implausible, may be seen in the values of the restricted estimates of the elasticities of substitution which are all considerably less than twice the geometric means of the standard errors of the corre- sponding unconstrained parameter estimates. Of course, this latter conclusion must be tempered by the fact that the geometric means of the standard errors are at best an approximation to the upper bound on the actual standard errors . 4.6 A Comparison of the Models In the previous section we have reported the re- sults of the estimation of the Rotterdam, DOLRES, and DOLCES models, and Leser's LES, on the basis of U.S. data. In this section we compare these models, both with regard 205 to differentiating between plausible and unplausible specifications, and with respect to comparing the values of the income and price elasticities implied by these models. Several criteria may be used in the comparison of consumer demand models. The most obvious criterion is the comparison of the R2 values for each equation, under.the alternative specifications. These values give the pro— portion of variation in the dependent variable about its mean, explained by the explanatory variables. Alterna- tively, Parks [1969] and others have looked upon (l-R2), as a measure of "badness of fit" in opposition to the conventional "goodness of fit" criterion. The problem with R2 comparisons lies in the upward bias of R2 as an estimator of the population squared multiple correlation coefficient. This has been pointed out by Barten [1962]. An alternative criterion, prOposed by Theil [1965], and Theil and Mnookin [1966], is the "average information inaccuracy of prediction" criterion, used with consider- able success by Theil himself, Goldberger and Gamalesos [1967], Parks [1969], and others. This is based upon looking at budget shares, as probabilities that a wit’ given dollar in the consumer's budget will be spent on the ith commodity at time t. Thus, the wit can be treated as prior probabilities, and their predicted values as 206 posterior probabilities, w . With this interpretation, it the "information inaccuracy of prediction" at time t is defined to be It = wit log (wit/wit) Thus, a measure of how well a model predicts over the sample period is the arithmetic mean of the information inaccuracy of prediction, or the "average information inaccuracy of prediction" T I = (l/T) Z wit log(wit/wit) Unfortunately, the estimation procedure does not ensure that the predicted values of budget shares will not be negative. This has, in fact, been the case for the models estimated with our data. The first reason for this is that we are not, in fact, predicting budget shares direct— ly, but the logarithms of quantities weighted by budget t shares. Thus, the estimates of the budget shares are second round estimates, which turn out to be negative in several cases. Yet another index of the plausibility of the model suggests itself from the paper by Parks [1969]. Parks' procedure has been to test the symmetry hypothesis for each model before imposing the symmetry restriction. Actually, the test of the symmetry hypothesis is conditional 207 on the validity of the functional specification. However, since our belief in the symmetry of the Slutsky terms may be stronger than our belief in the validity of the specif- ic functional form, we may consider the significance probability of the asymptotic F statistic for the test of the (linear) symmetry hypothesis itself as an indicator of the plausibility of the model. Roughly speaking, this view of the F statistic expresses the belief that if it were known that either the functional form or the symmetry of the Slutsky terms (or both) were not valid, then we would be willing to reject the validity of the functional form before we reject the symmetry hypothesis. In Table 5 we present the (l—R2), and the signifi— cance probability of the F statistic under the alternative specifications. On the least "badness of fit" criterion, the DOLCES model clearly dominates all the other models, with the exception of LLES on Shelter. The Rotterdam model seems to do better than the DOLRES model, with the exception of the estimates for the Shelter equation, for which the Rotterdam model has an extremely low R2. Leser's LES seems to outperform the Rotterdam model on all but the Food equation. However, this must be tempered by the realization that the estimates for LLES violate the nega- tive definiteness of the Slutsky matrix, implied by theory. This can readily be seen from Table 4.2, where the diagonal elements are not all negative. For this 208 reason we shall omit further discussion of LLES. It seems, then, that on the basis of the "badness of fit" criterion, DOLCES clearly dominates all models, while the Rotterdam model seems to outperform the DOLRES model. On the "con- cordance with the symmetry hypothesis" criterion, reflected by significance probabilities of the asymptotic F statistic for symmetry, the Rotterdam model ranks highest. The DOLCES model is also plausible, but the DOLRES model does not ap— pear to be a serious competitor. Table 5. —-A Comparison of the Models with Regard to Badness of Fit and Significance Probability of the Asymptotic F statistic for the Symmetry Hypothesis. (1-R2) a (Sig. P.) Food Clothing Shelter Rotterdam 0.0401 0.1770 0.6528 0.013 DOLRES 0.0734 0.2397 0.2084 <0.005 “ DOLCES 0.0745 0.0232 0.1920 0.004 LLES 0.2183 0.0440 0.0475 <0.0005 aA high Sig. P. indicates that the Symmetry hypothesis is not rejected, and hence indicates a better model. 209 The three models may also be compared with respect to the implied values of the price and income elasticities, and the elasticities of substitution. In Tables 6.1, 6.2, 6.3 and 6.4 we present respectively, the income (Engel) ealsticities, the income compensated (Slusky) price elas- ticities, the price (Cournot) elasticities, and the Hicks— Uzawa elasticities of substitution, evaluated at the sample mean values of the budget shares, for the alterna— tive models. In comparing the income elasticities, it should be recalled that for the Rotterdam model, the DOLRES and the DOLCES models, the "income" elasticity is, in fact, ap- proximated by a volume elasticity. This is not true for the Leser's LES. Apart from LLES, then, the estimates of the income elasticities show considerable uniformity in the DOLCES and DOLRES models, and even in the Rotterdam model. With the usual definitions, Food and Shelter come out as "necessities" under all three models, while Cloth- ing is a "luxury" under all three models. The DOLRES and DOLCES models would lead us to classify the Miscellaneous items as "luxury" while the Rotterdam model would not. Clearly, then, while the numerical values of the estimates are comparable, differences do exist across models. Tables 6.2, 6.3 and 6.4 provide estimates of the price elasticities and the elasticities of substitution. Leser's LES is easily seen to violate the condition that is. J Table 6.l.-—Income (Engel) Elasticities Evaluated at Sample 210 Means of the Budget Shares. . Miscel— Food Clothing Shelter laneous Rotterdam 0.1242 1.1202 0.7792 0.9774 DOLRES 0.5757 1.6403 0.7585 1.9871 DOLCES 0.6952 1.0721 0.6045 2.6344 LLES 0.8514 1.2436 1.0554 1.0240 Table 6.2.--The Slutsky Matrix Evaluated at Sample Means of Budget Shares. . Miscel- Food Clothing Shelter laneous Rotterdam -0.7697 0.3555 0.2451 0.1692 rd DOLRES -0.9355 0.3585 0.2811 0.2959 8 DOLCES -0.8396 0.1902 0.1425 -0.5069 m LLES —0.0503 -0.0015 0.4857 0.4339 g Rotterdam 0.8760 -l.3676 0.2683 0.2233 g DOLRES 0.8834 —l.0444 0.1875 —0.0265 0 DOLCES 0.4686 —0.8244 0.3302 -0.0256 8 LLES -0.0036 0.0392 -0.2912 -0.2556 3 Rotterdam 0.3277 0.1456 -0.5858 0.1125 :1 DOLRES 0.3759 0.1018 -0.7792 0.3016 ’2 DOLCES 0.1906 0.1791 -1.0976 -0.7279 m LLES 0.6495 -0.1580 0.1727 0.6641 . Rotterdam 0.4841 0.2592 0.2406 -0.9839 0 DOLRES 0.8464 —0.0308 0.6452 -l.4608 .fl DOLCES -l.4501 -0.0297 -1.5572 -3.0370 2 LLES 1.2412 -0.2967 1.4208 2.3653 W 211 Table 6.3.—-The Cournot Price Elasticity Matrix Evaluated at Sample Mean Values of Budget Shares. . Miscel- Food Clothing Shelter laneous Rotterdam —l.2188 0.1732 —0.0908 0.0122 c DOLRES -l.1655 0.2651 0.1091 0.2155 0 DOLCES —l.ll73 0.0775 -0.0652 -0.6040 ,9, LLES —0.3904 —o.1395 0.2314 0.3150 m Rotterdam 0.4284 —1.5492 —0.0663 0.0669 g DOLRES 0.2281 —1.3103 -0.3025 -0.2556 8 DOLCES 0.0403 —0.9982 0.0099 -0.1753 3 LLES -0.5004 —0.l624 —0.6627 -0.4292 3 Rotterdam 0.0164 0.0193 —0.8l86 0.0036 u DOLRES 0.0729 —0.0212 -1.0058 0.1957 '3 DOLCES -0.0509 0.0811 -l.2782 -0.8123 g LLES 0.2279 -0.3291 —0.l426 0.5167 Rotterdam 0.0936 0.1008 -0.0514 -l.1204 O DOLRES 0.0526 —0.3529 0.0515 -1.7383 m DOLCES -2.5025 -0.4568 -2.3442 -3.4049 E LLES 0.8322 -0.4627 1.1149 2.2223 the matrix of Slutsky terms be negative definite. This would cast serious doubt on the plausibility of the LLES specification. The values reported for the double— logarithmic models show considerable concordance as well as a few marked divergences. For all of the three double— logarithmic models, the signs of the income compensated own—price elasticities are negative. In fact, all three models seem plausible on theoretical grounds. For only three price (Slutsky) elasticities (in the Miscellaneous column) the three models differ in sign from each other. 212 Table 6.4.—-The Hicks-Uzawa Elasticity of Substitution Matrix Evaluated at Sample Mean Values-of Budget Shares. . Miscel- Food Clothing Shelter laneous Rotterdam 2.1927 0.8203 1.2177 '3 DOLRES 2.2113 0.9409 2.1187 0 DOLCES 1.1731 0.4770 -3.6299 F LLES -0.0091 1.6258 3.1071 3 Rotterdam 0.8982 1.2117 (g DOLRES 0.6277 -0.1899 '3 DOLCES 1.1051 -0.1833 0 LLES -0.9747 -1.8302 g 3 Rotterdam 0.8053 H DOLRES 2.1595 g DOLCES -5.2121 m LLES 4.7556 . Rotterdam g DOLRES TIDOLCES z LLES Thus, there seems to be a fiar degree of concordance among the three models in the classification of the commodity groups as "substitutes" and "complements" on the Hicks- Allen definition. variations between estimates, The Cournot elasticities show wider from one model to the other. Remarkably enough, however, all Cournot own-price elas- ticities are negative, lending credance to the belief that "demand curves are downward sloping." The values of the Hicks-Uzawa elasticity of substitution may be looked 213 as the curvature of the indifference curves. These values vary considerably across models, and is positive in all but four cases. 4.7 Summary and Conclusions To summarize, then, of the four models estimated, the Rotterdam model, the DOLRES and DOLCES models seem all to provide theoretically plausible estimates of the parameters; while Leser's LES seems to be deficient on this count. The Rotterdam model and the DOLCES model seem to be comparable in terms of desirable properties, while the DOLRES model seems to be in some conflict with the symmetry hypothesis with regard to the Slutsky terms. It would appear, then, that the constant elasticity of substitution hypothesis, which proved to be of so much value in the theory of production may hold some answers to the choice of an efficacious functional form for a complete set of demand equations. It would indeed by interesting to compare the performance of the Rotterdam model under its original specification, to the CES specifi- cation to which it can easily be transformed. For the non-differential double logarithmic case, the CES specifi- cation seems to do clearly better, and the question with regard to the differential double—logarithmic (Rotterdam) model seems quite open. BIBLIOGRAPHY BIBLIOGRAPHY (Articles actually referred to in the dissertation are preceded by an asterisk.) 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