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Evaluating the aggregation biases in a production economy : a stochastic approach Fortin, Nicole M. 1988

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EVALUATING THE AGGREGATION BIASES IN A PRODUCTION ECONOMY: A STOCHASTIC APPROACH by NICOLE M. FORTIN M.Sc, Universite du Quebec a Trois-Rivieres, 1981 M.Sc, Universite de Montreal, 1978 B.Sc, Universite de Montreal, 1976 A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of Economics We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August 1988 © Nicole M. Fortin, 1988 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of ^ c gr^c r - v—Cc~s The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 DE-6G/81) ABSTRACT This dissertation • presents a theoretical framework to analyze and evaluate aggregation biases. These biases measure the information lost when macro relations evaluated in terms of aggregates do not capture all of the distributional properties of micro relations. The framework is developed in the context of producer theory, but it may be used to determine the biasedness of any representative agent model and to study general relationships between exact-aggregation macro parameters and their microfoundations. The model is based on a stochastic interpretation of the production characteristics which encompasses that of previous stochastic aggregation models (Houthakker, 1955; Hildenbrand, 1981; Stoker, 1984; Lewbel, 1986a). It admits the construction of "true" aggregate relations which can be compared to pre-specifed macro relations. Many of Theil's (1954, 1971) statistical results concerning the relations between micro and macro parameters then can be formalized at the population level and generalized to non-linear functions. A moments decomposition of the "true" aggregate relation makes it possible to identify the sources and causes of potential aggregation biases. Thus, the functional-form restrictions of exact-aggregation models (Gorman, 1968a; Blackorby and Schworm, 1984, 1988) are found to be neither necessary nor sufficient conditions for consistent aggregation, if the aggregates are taken to be the usual totals or averages. Traditionally, similarity among firms, either as a maintained hypothesis or as the long-run outcome of perfect competition, has proved to ensure exact aggregation. Here, economic diversification may also provide an alternative set of circumstances under which the aggregation biases may be minimized. In the case of an average-representative firm, the output aggregation bias is explicitly derived. Empirical analyses confirm that the magnitude of the bias increases as higher moment terms in the production characteristics increase in importance. Conditions under which exact-aggregation macro parameters possess stable microdefinitions are obtained; they explain Fisher's (1971) simulation results. Empirical results show that such macro parameters are relatively stable (within the estimated confidence intervals) when based on periods of relative economic stability. Finally, theoretical implications for macroeconomet-ric modeling and policy evaluation are explored. i i CONTENTS Abstract ii List of Tables vi List of Figures vii Acknowledgements viii Chapter 1. Introduction 1 1.1. Overview 3 1.2. The Nature of Aggregation Problems 7 Part I. Literature Review Chapter 2. Exact Aggregation over Firms (and Factors) outside Equilibrium 14 2.1. The Klein-Nataf Problem 14 2.2. Separability Conditions 16 Chapter 3. Exact Aggregation over Firms (and Factors) with Optimizing Behavior 2 0 3.1. Intersectoral Aggregation of a Fixed Factor Arbitrarily Distributed across Firms in the Presence of Efficient Factors - The Fisherian Ap-proach 21 3.2. Full Aggregation of a Subset of an Efficient Factor in the Presence of a Fixed Factor Arbitrarily Distributed across Firms 27 3.3. Full Aggregation with Hicksian Intrasectoral Aggregation - The Brown-Chang Approach 33 Chapter 4. Stochastic Aggregation Models 37 4.1. Intersectoral Aggregation Using Stochastic Capacity (or Efficiency) Dis-tribution - The Houthakker Approach 37 4.2. Intersectoral Aggregation Using a Parametric Distribution of Variables and Attributes 44 Part II. Aggregation over Firms in a Stochastic Environment: Theoretical Framework Chapter 5. Output Aggregation outside Equilibrium - Basic Framework 5 2 5.1. Definition of Full-Information Aggregates 54 5.2. Moments Decomposition of the Output Aggregate 57 5.3. Exact Aggregation Requirement and Resulting Biases 61 iii 5.4. The Average-Representative Firm Approach: A Completely Specified Case 63 5.5. Microfoundations of Exact-Aggregation Macro Parameters 65 5.6. Applications 70 5.6.1. Linear Aggregation 70 5.6.2. Quadratic Production Function 73 5.6.3. Generalized Cobb-Douglas Production Function 76 Chapter 6. Output Aggregation with Optimizing Behavior - The Problem of Profit Aggregation 78 6.1. Derivation of Profit Aggregates and Biases 79 6.2. The Case of the Average-Representative Firm 82 6.3. Stability of the Exact-Aggregation Macro Profit Parameters 84 6.4. Applications 85 6.4.1. Separable Quadratic Production Function 85 6.4.2. Generalized Cobb-Douglas Production Function 88 6.4.3. Generalized Leontief Profit Function 90 Chapter 7. Output Aggregation with Partial Optimization - The Problem of Cap-ital Aggregation 93 7.1. Profit-Aggregation Biases in the Presence of a Fixed Factor 93 7.2. The Case of a Firm of Average Capital 96 7.3. Exact-Aggregation Macro Profit Parameters that Are Capital-Depend-ent 97 7.4. Applications 99 7.4.1. Separable Quadratic Production Function 99 7.4.2. Gorman Type Forms 102 7.4.3. Normalized Quadratic Variable Profit Function 105 Part III. Empirical Analyses Chapter 8. Canadian Input-Output Data 109 8.1. Origin and Contents of Data Base 109 8.2. Tax Adjustment Process 109 8.3. Capital Rental Prices 112 8.4. Intrasectoral Factor Aggregation 120 iv Chapter 9. Bias Estimation for the Production Function 122 9.1. Variable Choices and Description 122 9.2. Average-Representative Firm's Bias 126 9.2.1. Linear Production Function 126 9.2.2. Quadratic Production Function 131 9.3. Exact-Aggregation Macro Parameters with Stable Microfoundations 133 Chapter 10. Conclusion 137 10.1. Modeling Implications 137 10.1.1. Consumption Function, Policy Changes and Aggregation Biases 138 10.2. Main Results 141 10.3. Future Research Strategies 146 References 151 v LIST OF T A B L E S Table 8.1 Data Items 111 8.2 McLeod, Young and Weir's 10 Industrial Bonds Yield Average 118 8.3 Volatility and Negativity of Alternative Capital Prices for Structures 120 Table 9.1 Means over Time 125 9.2 Means across Sectors 125 9.3 Correlations over Time 126 9.4 Correlations across Sectors 126 9.5 Average Representative Firm's Bias: Linear Production Function 129 9.6 Average Representative Firm's Bias : Quadratic Production Function 133 9.7 Macro Parameters with Microfoundations: (75-76-77) Time-Drift Coefficients — Linear Production Function 135 V I LIST O F FIGURES Figure 9.1 Joint Distributions of Ak and Kt (t=1970): Linear Production Function 130 Figure 9.2 Joint Distributions of Ai and Lt (t=1970): Linear Production Function 131 v i i A C K N O W L E D G E M E N T S I am indebted to my supervisory committee, Drs. Charles Blackorby, John G. Cragg and W. Erwin Diewert, for letting me pursue an otherwise risky topic. I also would like to express my appreciation to Dr. William Schworm for his early interest and encouragement. Dr. John Weymark deserves a special mention for persuading me to undertake and pursue a degree in economics, in the first place. A note of acknowledgement goes to Lawrence Ostensoe, whose preparatory work made the data base available. Finally, I extend my heartfelt gratitude to my parents. Without their moral and financial support, I would not have been able to complete this project. The support of my friends and fellow graduate students has been another indispensable ingredient to the successful conclusion of this endeavour. viii Chapter 1. Introduction The conditions for the exact aggregation of economic relations have been studied in a number of settings using different models of producer (Nataf, 1948; Fisher, 1965, 1969a, 1982; Blackorby and Schworm, 1984, 1988) and consumer behavior (Gorman, 1953; Theil 1954; Lau, 1982) and of economy-wide equilibrium conditions (Klein, 1946a, b; May, 1946, 1947). 1 In all circumstances, exact-aggregation conditions have been criticized for being overly stringent and impossible to satisfy empirically.' Surprisingly, the consequences of deviations from those stringent aggregation restrictions have attracted substantially less attention. This dissertation proposes to palliate this void by defining and quantifying the aggregation biases. 2 The study develops a theoretical framework to derive the biases and their properties, and undertakes some empirical analyses to evaluate their magnitude. The theoretical model focuses on the aggregation of production units (to be called firms). The empirical analyses use sectoral data from the Canadian economy in the period 1961-1980. In a production economy, exact aggregation requires that the relation using aggregate factors (the macro relation) describes the economy's production possibilities (Nataf, 1948) or, at least, the optimal level(s) of production (Fisher, 1965; Gorman, 1968a) as exactly as the combination of individual relations (the micro relations) do. 3 Theorists have derived necessary and sufficient conditions under which this exact-aggregation requirement is met. These conditions take the form of restrictions on the firms' technologies, restrictions that may carry through at the economy, level (Blackorby and Schworm, 1988), as well as restrictions on the aggregates (Nataf, 1948; Gorman, 1968a). Exact-aggregation theory generally imposes specific restrictions on the micro theory that make the macro theory consistent in terms of certains aggregates, such as totals and 1 See the surveys of Green (1964) and of Van Daal and Merkies (1984). 2 A statistical notion of aggregation bias in the macro parameters was introduced by Theil (1954, p.l22ff.; 1971, p.560) in the context of linear aggregation; it also was discussed in Allen (1959, p.700), Van Daal (1980) and Van Daal and Merkies (1984, p.l84ff.). 3 To avoid confusion, the terms "macro" and "aggregate" will be used in a more re-stricted sense than generally implied. The economic relation that runs in terms of aggre-gate variables will be called the "macro" relation, while the term "aggregate" relation will refer exclusively to a relation obtained by aggregating the micro relations, explicitly. 1 averages. 4 Typically, these aggregates have been assumed to provide adequate information about the micro relations. However, when aggregation is set in a stochastic environment, the usual aggregates may be considered -to provide only partial information about the distribution of the micro variables. Studying a similar aggregation problem, Stoker (1982) pointed out that a parallel can be established between the theory of aggregation and the theory of sufficient statistics. He showed that the usual aggregates are in fact sufficient statistics in special cases only. When they are not, the macro model will typically carry less information than the micro models and the micro relations will need to be severely restricted for there to be no information loss. The object of this study is to evaluate the information gap — distortion or loss — resulting from the use of macro relations that utilize aggregates that are not full-information, through the measurement of aggregation biases. These biases will measure only divergences related to the precise aggregative process modeled. In this sense, the analysis abstracts from the aggregation over factors or from any aggregative process imbedded in the micro variables. Also, it considers the aggregation of single micro relations into a single macro relation as opposed to the aggregation of systems of micro relations into systems of macro relations. Finally, it does not intend to address the problem of statistical aggregation biases due to error terms incorrectly specified as a result of the aggregative process (as in Grunfeld and Griliches, 1960; Aigner and Goldfeld, 1973). The identification of the aggregation biases, of their origin and of their relative mag-nitude is expected to make a useful contribution to the theory of aggregation and, more generally, to theories using a representative-agent approach. It may also have interesting implications for macroeconometric modeling, for industry analysis, and for any type of analysis using aggregate data. Considering macroeconometric modeling, Geweke (1985) has argued that the aggregation and expectations problems are of about the same or-der of magnitude. In relation to the econometric estimation of substitution possibilities among capital, labor, materials and fuel inputs, K o p p and Smith (1981) have reported that some results of non-association and false magnitude are due to aggregation errors. A s a contribution to the area of model choice and design, the comparison of biases resulting 4 Also, these have been chosen to be totals of firm-valued variables or sub-aggregates, as in Gorman (1968 a). 2 from different functional forms may provide criteria for the choice of functional forms that minimize aggregation biases. 1.1. Overview The dissertation is organized as follows. The second part of the introduction contains an overview of the different aggregative strategies; it positions the approach adopted here within the broader context of economic analysis. Chapters 2 and 3 review the main models of the exact-aggregation literature, concen-trating on producer theory. In particular, Nataf's (1948) solution to the aggregation of production functions (outside of equilibrium) and Gorman's (1968 a) solution to the aggre-gation of profit functions constitute the basic exact-aggregation results that will rexamined in light of the proposed stochastic framework. They notably require the additive separa-bility of the micro relations in the variables used as aggregates. In a stochastic framework, this condition will be shown to be neither necessary nor sufficient, if the aggregates are taken to be the usual totals or averages. Also, the Gorman conditions, as well as the more general aggregation conditions found in Blackorby and Schworm (1988), impose common-ality across firms of specific marginal components of the profit functions. It will be shown that such conditions can be relaxed in a stochastic framework. Chapter 4 summarizes the principal stochastic aggregation models. The models of its first section (Houthakker, 1955-56; Levhari, 1968; Johansen, 1972; Cornwall, 1973; Sato, 1975) assume fairly restricted forms for the production function of firms and use their capacity (or efficiency) distribution to describe a continuum of firms. Because of the restricted choice of functional forms both for the production function (fixed or variable proportions) and for the density functions (Pareto and Uniform), these models have limited empirical applicability. They, however, serve as forerunners to the present stochastic interpretation of the firms' production characteristics, where technological parameters are assumed also to belong to a probability distribution. The second section covers models (Stoker, 1984; Lewbel, 1986a, b) based on a paramet-ric distribution of variables and attributes. These models were developed in the context of consumer theory and are recast here in the producer's context. Insofar as they aim at reparametrizing the density function of variables and attributes in terms of its first 3 moments, they can be seen as precursors to the proposed moments decomposition of the "true" aggregate, developed in chapter 5. The applicability of these models, again, is reduced by the need to match empirical distributions with the model's stringent distribu-tional assumptions. The core of the proposed theoretical framework is found in chapter 5. A stochastic interpretation of the production characteristics is proposed first. Other stochastic aggre-gation models have considered either the technological coefficients (Houthakker, 1955-56; Johansen, 1972; Sato, 1975) or the variables (Stoker, 1984; Lewbel, 1986a, b) to be stochas-tic. Here, by contrast, both the technological parameters and the factor inputs are taken to be different for each firm and interpreted as stochastic variables. It is then possible to construct aggregate relations that are full-information in the sense of capturing all the distributional properties of the micro relations. Many of Theil 's (1954, 1971) results con-cerning the relations between micro and macro parameters can be generalized to non-linear functional forms. While Theil 's results are actually sample results, the present framework allows their population equivalents to be derived. A decomposition of the "true" aggregate production function in terms of moments of the characteristics is introduced. It makes it possible to identify the sources and causes of potential aggregation biases. This decomposition enables the functional form restrictions of exact-aggregation models and the distributional constraints of stochastic aggregation models to be considered. A general exact-aggregation requirement, based on the "true" aggregate relation, is proposed next. When this requirement is not met, the difference between the "true" aggregate relation and the macro relation results in an aggregation bias. The aggregation of production functions outside equilibrium is considered first as an illustrative example of the aggregation of economic relations. A representative agent's problem is studied thereafter. In such problem, one asks whether there exists an agent whose behavior, if replicated n times, reproduces macro behavior. Here, in addition, the problem of finding a representative firm whose technology is derived from the technology of non-identical firms is addressed. The case of a firm, whose technological parameters average those of the population of firms, is examined specifically. Alternatively, when the parameters describing macro behavior are not derived from micro 4 behavior but rather left to be estimated, the aggregation problem is posed in terms of the relationship between exact-aggregation macro parameters and their microfoundations. This relationship is studied and conditions under which the exact-aggregation macro parameters possess stable microdefinitions are determined. Applications to the aggregation of linear, quadratic and Generalized Cobb-Douglas production functions illustrate the concepts of moments decomposition, of output aggre-gation bias and of biases in the macro parameters. Most interesting are the stochastic interpretations of the linear aggregation problems studied by Theil (1954, 1971) and of Klein-Nataf solutions to the aggregation of non-linear forms. Chapter 6 applies the theoretical framework developed in chapter 5 to the aggrega-tion of profit functions, assuming price-taking and maximizing behavior from the part of producers. In terms of the stochastic framework, profit functions differientate themselves from production functions in that they run in terms of prices, which can be identical for all firms that is, non-stochastic. Conversely, prices may be assumed to differ across firms in order to capture a variety of competitive assumptions. This feature of the model allows the aggregation across industries (across different output prices), by contradistinction with previous exact-aggregation models. When perfect competition prevails and all prices are taken to be the same across firms, the components of the profit functions that comprise prices only cease to pose an aggregation problem. Under perfect competition, the aggre-gation problem is no longer an allocative problem; it reduces to the problem of deriving a macro technology from the micro technologies. This may be facilitated by increasing similarity across firms, which may be, in turn, a long-run result of perfect competition. Applications to a separable quadratic production function, to a Generalized Cobb-Douglas production function and to a Generalized Leontief profit function add concrete-ness to the profit-bias derivations. In particular, the aggregation of separable quadratic production functions corroborates Blackorby and Schworm's theorem (1988, Theorem I) and the Generalized Leontief profit function is shown to alleviate aggregation problems. Chapter 7 deals with what has been long-thought to be the most important stumbling block in the aggregation over firms: the problem of capital aggregation. In the Gorman (1968a) tradition, this chapter focuses on the aggregation of restricted profit functions. 5 The profit characteristics now include, in addition to technological parameters, output and factor prices, levels of capital inputs which are unlikely to be the same across firms. Thus, the dimensionality of the aggregation problem is increased considerably by comparison with the unrestricted case, even though it can be characterized in the same way. Also, the appeal to increasing similarity across firms brought about by long-run competition cannot be invoked. The more interesting insights of the chapter are given in the applications sections where the aggregative properties of the Gorman type forms are reevaluated. In a stochas-tic framework, the latter are found to be no longer necessary nor sufficient to ensure exact-aggregation, using the traditional totals and averages as aggregates. Commonality across firms of specific components of the profit functions and separability in the capi-tal inputs are shown to be too strong restrictions. The proposed weaker zero-covariance condition between appropriate components of the profit functions extends the linear in-dependence condition of Lau's (1982) fundamental theorem of exact-aggregation. It also carries an important implication. Whereas before, some degree of similarity among firms was thought to be an essential condition of exact-aggregation, now economic diversifica-tion, which entails small covarianCes between the profit functions' components, emerges as an alternative set of circumstances under which aggregation biases may be minimized. The aggregation of variable profit functions based on a separable quadratic production function and of normalized quadratic variable profit functions provide concrete example of these derivations. The motivation to study aggregation biases rather than exact-aggregation is supported by an underlying curiosity about the actual magnitude of aggregative errors. Thus, an essential part of the present study is the quantification of the aggregation biases. Whereas there may be a number of valuable options on how to proceed with this evaluation, the one adopted here was chosen on the basis of data availability and because of time constraints. It is by no means completely satisfactory; however, it has the merit of relying on "real world" data. Chapter 8 discusses the detailed content of the data base, along with tax-adjustment procedures. It also addresses the difficult issue of deriving reasonable rental prices for 6 capital. Studies of the volatility and negativity of alternative rental prices are included. The various choices of capital prices are found to generate wildly different intersectoral capital aggregates. Thus, three options are retained in the following bias analyses. Chapter 9 reports the empirical results. The output aggregation bias of an aver-age-representative firm for a simple production function of two inputs, capital and labor is calculated under the assumptions that the production function can be approximated by a linear, quadratic and separable quadratic forms. In the linear case, estimates of the time-drifts in the microdefinitions of exact-aggregation macro parameters and of the corresponding time-varying macro parameters are obtained. They allow the incidence of the distributional effects of the 75-76 oil shock on the stability of macro parameters to be studied. In the last chapter, a preliminary exploration of the modeling implications of potential aggregation biases for macroeconometric policy evaluation is undertaken. A simple linear consumption function is used to illustrate how a policy change will generally result in biases in the macro parameters. Chapter 10 concludes with a discussion of the main results and comments on the future steps of continuing research. 1.2. The Nature of Aggregation Problems In economic analysis, problems of aggregation across agents arise from the desire to describe the workings of the economic system either on the assumption that they mimic that of individuals or solely as a relation between aggregates. In both cases and for different reasons, there is a need for a legitimate link between the community's behavior and individual behavior behavior on one hand, and between the macro relations and their microfoundations on the other. Aggregation theory is concerned with the conditions under which such a link can be established. The first paradigm of economic behavior is the oldest and perhaps most, prevalent mode of economic reasoning. The use of reasoning by analogy was recognized by Jevons (1879) as an essential means of constructing economic relations: The general forms of the laws of Economics are the same in the case of individuals and nations; and, in reality, it is a law operating in the case of multitudes of 7 individuals which gives rise to the aggregate represented in the transactions of a nation. [...] It would not of necessity, happen that the law would be exactly the same in the case of aggregates and individuals, unless all those individuals were of the same character and position as regards wealth and habits; but there would be a more or less regular law to which the same kind of formulae would apply. (Jevons, 1879, p. 16-17) In this paradigm, the individual micro relation is the fundamental one, and it is used also to describe the operation of the whole economy. The need for rules that guarantee the legitimacy of the analogy is at the origin of the aggregation problem. Antonelli (1886), Gorman (1953), Nataf (1953), Eisenberg (1961) successively derived conditions under which individual utility functions can be aggregated into a community utility function, where the aggregation is carried out as a simple sum. This aggregation problem may be reinterpreted as the existence problem of a representative consumer, whose behavior, if replicated n times, conveys the community's behavior: A representative consumer exists if the market behavior of an aggregate of different consumers is as if it were the market behavior of a number of identical hypothetical consumers, each with the same level of income. (Muellbauer, 1976, p.979.). This notion of a representative consumer is paralleled in producer theory by the Kaldorian concept of a representative firm, where the firm is regarded as a small-scale replica of the economy. 1 An alternative popular notion of a representative firm is the one introduced by Marshall (1920, p.317), where the representative firm is seen as an average firm. The Marshallian representative firm is neither a struggling firm nor an extremely successful firm; it is just in between. It is the basis of Marshall's analysis of industries and has been credited for having established representativity modeling as an acceptable strategy (Van Daal and Merkies, 1988, p.611). In the present analysis, the problem of the representative firm will be addressed from the Marshallian point of view. If firms are non-identical, how well can a representative firm, which is a technologically average firm, describe industrial or national behavior? The conditions under which this representation is exact are considered also. However, throughout the present analysis, the aggregation question is approached with the aim of establishing by how much do macro results diverge from results obtained by consistently aggregating realistic micro relations. Traditionally, the focus has been 8 placed on obtaining exact-aggregation conditions which often turned out to be impossible to satisfy empirically. Under the second paradigm of economic behavior, the macro relation is seen as a relation purely between aggregates. The legitimacy of such a modeling strategy has been justified first by resorting to simple common-sense macro relations that fitted aggregate data well (Cobb and Douglas, 1928; Tinbergen, 1939) and that did not necessarily possess solid microfoundations. There are, however, two possibilities [...] first, to obtain by statistical methods, statements about averages or sums which have a significant relation to the [eco-nomic] system, or, otherwise, to describe a second system which consists of only a few variables, subject to relations which are suggested by experience. (Evans, 1934, p.37) Models of the pure sciences were given in example to justify moving away from reasoning by analogy (Evans, 1934; May, 1947). May cited Boyle's law from the theory of gases in physics as an example of a macro theory (the theory of mass behavior) which can be deduced from the micro theory (the laws of behavior of the particles) while being quite different from it. 5 "Because of the interrelations of the particles and the averaging of random movements, the macro laws had a distinct character." (May, 1947, p.59). Notice that the distinct character of the macro law is brought about by a theory of the interrelation of the microvariables. In economic analysis, it is this missing theory of the interrelations between economic agents that prevents the validation of truly distinct (or non-analogical) macro relations with solid microfoundations. In the aggregation of preferences, Arrow's (1951) impossibility theorem provided the first notable exception to this missing link in the form of a procedure that looks for a social ranking by imposing conditions on the interrelations between agents. More recently, Sah and Stiglitz (1986) have characterized economic systems under different "architectures" to study the nature of errors made by the system in the choice of alternative projects. In the earlier macroeconomic models (Hayek, 1933; Keynes, 1936; Hicks, 1937), es-5 Boyle's law, discovered in 1662 and also known as Mariotte's law, expresses the isothermal pressure-volume relation for a body of ideal gas. If the gas is kept at constant temperature, the pressure and volume are in inverse proportion, or have a constant product. This law is only approximately true. 9 pecially the business-cycle theories, the need for microfoundations was not paramount as these models were primarily descriptive. Relations solely between aggregates satisfied this end; the only potential problem was the justification of their genesis. Interpreters of Keynes (Van den Borre, 1958; Casarosa, 1981) have restated the Marshallian theory of the firm in terms of individualized aggregate supply and expected demand functions, thereby casting Keynes' aggregate supply and demand analysis in a representative agent framework and linking it to the first paradigm of aggregate behavior. Modern macroeconomic models do not restrict themselves to a positive description of the economic system but claim to provide policy guidance (Fischer, 1980; McCallum, 1980), to predict the outcome of economic disturbances (Bruno and Sachs, 1982) or simply to forecast G N P (Gordon, 1986). When these models do not adopt a representative agent view of macroeconomic modeling, the need for microfoundations stems from the need to describe aggregate responses to policy changes or market circumstances, which operate at the micro level. A legitimate link between the macro and micro theory would insure that these responses are modeled and estimated correctly. If macro relationships can be seen only as relations between aggregate variables with no consistent link to the micro relationships, how useful can they be in describing the aggregate outcomes of policy changes or market shocks that are transmitted via the micro relationships? The obvious incorrectness of policy recommendations based on unmicrofounded macro models has been illustrated in studies using slightly disaggregated models of the economy (Blinder and Gregory, 1984). In his discussion of appropriate aggregative strategies, May (1947) had already iden-tified the three main components of a theory of aggregation: the micro relations, the aggregates and the macro relations. He further suggested that the aggregation problem should be posed in terms of what macro laws are implied by the micro theory and certain aggregates taken as given. A second strategy exposed by Klein (1946a) takes as given the micro theory and certain properties which the macro theory must fulfill and tries to con-struct aggregates of the micro variables that are consistent with the micro theory and these properties. This approach resembles the one adopted by index-number theorists. However, index-number theorists first attempted to construct aggregates that fulfilled some internal consistency properties, such as Irving Fisher's (1922) weak factor reversal, before ensuring 10 their aggregative properties. More recently, "superlative" indices (Diewert, 1976, 1978) have aimed at preserving the properties of the economic relations in the sense of being exact for an aggregator function capable of providing a second-order local aproximation to an arbitrary twice-differentiable linearly homogeneous function. Finally, a third alternative outlined in Thei l (1954, p.5) takes as given a macro theory that runs in terms of certain aggregates and tries to find restrictions on the micro theory that makes it consistent with the macro theory. This last aggregative strategy and a variant of it, that does not take all aggregates as given, is the one used by the exact and stochastic aggregation models reviewed in the following chapters. The present aggregation bias study is designed to serve the validation of empirical models. Thus, it will take the aggregates as given — those being the ones available through statistical agencies — even though the proposed framework allows for virtually any type of aggregates to be considered. The micro theory will also be taken as given and, in particular, the important and dubious assumption about the independence of agents and the absence of externalities will be maintained. A s mentioned above, this unfortunate premise is a result of the unfinished state of theories of economic organization, such as theories of imperfect competition. As usual, the independence assumption may be made more tenable by assuming that the agents are sufficiently diverse and that their number is sufficiently large so that no single agent significantly influence the others. In the price space, this translates into perfect competition, which will be invoked occasionally. The proposed framework will accomodate the two paradigms of economic behavior presented above. If the representative agent approach is chosen, the aggregates will be pre-specified and the macro relation will be chosen to be similar to the micro relations, which are taken as given. Conversely, if the pure macro approach is adopted, the aggregates of the microvariables and the micro theory will be taken as given, and the macro function will be allowed to be any appropriate function of the aggregates. In addition to the three basic elements of aggregation theory, the study of aggregation biases will require the construction of a "true" aggregate of the micro relations. This "true" aggregate will then be compared to the macro function, defined either from the representative agent or the pure macro paradigm, and discrepancies will result in an aggregation bias. The construction of this "true" aggregate is discussed in detail in chapter 5, which constitutes the core of the 11 thesis. A review of current exact and stochastic aggregation models follows next. 12 Part I Literature Review 13 Chapter 2. Exact Aggregation over Firms (and Factors) outside Equilibrium Among all aggregation problems to be reviewed here, the aggregation outside equilib-rium is the one which uses the weakest competitive assumptions and leads to the strongest aggregation restrictions. Therefore it is studied first since other models consider these aggregation restrictions as a base or default case and/or try to relax them. 2.1. The Klein-Nataf Problem This special problem of aggregation outside equilibrium was raised by Klein (1946 a) and solved by Nataf (1948). Assuming each firm / (/ = 1,...,F) produces output j / using input quantities x^, ... , JN, according to x/ = / (a^ ; . . . , JN), is it possible to give a perfect representation of the production possibilities of the whole economy using only some functions of the inputs ipnirf,, • • • , n = 1,2,... , N ? In effect, Klein (1946a) was looking for conditions under which it is possible to find a macro production function G, defined as a function of aggregate inputs, ipi,... that would yield the aggregate level of output y defined by an output aggregator T. The problem is formally defined as the solution to: y=r(y\...,yF), = T(9l(xl...,x1N),...,gF(^,...,4!)), (2.1) Klein (1946b) thought that it would be preferable to derive such a macro production function regardless of the distribution of factors of production among firms since non-competitive practices, such as monopolies, may be prevalent in the real world. However, this assumption proved to render the aggregation possible only in very special cases and, thus became the center of a controversy (in Econometrica, 1946-47) on whether or not the institutional characteristics of production in the economy, including the equilibrium conditions, should be included in the description of its production possibilities. Klein (1946b) held the position that technological data should not be confused with institutional characteristics. In particular, he insisted that, if it were, it would then be impossible to trace the effects on the aggregates of a technological change in the production function 14 of one firm or industry. Meanwhile May (1946, 1947) and Pu (1946) argued that the technological possibilities available to the producers and the institutional relations among producers interact to determine the production possibilities of the economy. In the original Klein-Nataf problem, all inputs were regrouped in only two groups, allowing some intrasectoral aggregation. Let the set X of N inputs for each firm be partitioned into r mutually exclusive and exhaustive subsets [iVi,... , Nr], a partition called R. In the Klein-Nataf formulation the case where r — 2 was considered, while formulation (2.1) places each input in a separate class r = N (R =X). Since the original formulation does not lead to milder aggregation requirements than formulation (2.1), the latter is used for added clarity. It is, however, understood that the more general cases 2 < r < N include intrasectoral aggregation. Considering Klein's problem (2.1), Nataf (1948) showed that the existence of aggregate inputs, ip\,... , t/>N, satisfying the production function G required the existence of subaggregates, V»> ra = 1, 2,... , iV, for each firm, / = 1,2,... ,F, from which the aggregators and the production functions could be defined. Theorem 2.1. (Nataf, 1948) For consistent aggregation in the sense of relation (2.1) to hold, it is necessary and sufficient that all functions of the scheme be additively separable, as follows. The production functions gf, f — \, 2,... , F, need to be additively separable: (2.2) where the hf and i/>^  are monotonically increasing functions; the aggregators should satisfy: F n=l,2,...,N /=i and (2.3) F /=1 and the macro production function should be additive: y = V'ifci) + • • • + V W ( Z J V ) , (2.4) where xn = (a^ 'ni 15 Nataf's solution was found by integration under differentiability assumptions. Totally differentiating the macro production function G and the output aggregator T leads the following differential conditions, necessary for (2.1) to hold: dG dvbn 3T dq1 , , TTTT = JJTT' n=\,2,...,N, f=l,2,...,F, 2.5 These relationships imply that marginal rates of substitution among variables of the aggregators rpn must be the same as the marginal rates of substitution among the variables of the production function Examination of marginal rates of substitution for each of the partial derivatives, dG/dijr d^rjdl dr/d9k dgk/d4 dG/d^s' d^JdJJ dT/dg'! dg'/dx'/ 1 ' ) for all pairs r,s (r ^ s) and k:l [k ^ I) shows that these expressions depend only on the pairs concerned. 1 Furthermore comparing pairs for each expression shows that a proportionality factor can be used to relate the primitives to linear expressions. Treatments of this problem with weaker assumptions (continuity: Gorman, 1968b; absence of continuity and differentiability: Pokropp 1972, 1978) have not greatly increased the class of admissible functions. 2.2. Separability Conditions Whereas Nataf's aggregation conditions formally impose precise restrictions on the firm's production functions, on the aggregators and on the aggregate production function, it is the separability of the production functions that has retained most attention in further aggregative developments. Sono (1945, 1961) and Leontief (1947a, b) first formally related functional separability to the existence of input subaggregates. 2 The following Leontief-Sono conditions became the basis of many aggregation restrictions in different contexts. Dropping the firm index, assume that the production function y — g(x) = g{x\,... , x^) is continuously twice differentiate. T h e Leontief-Sono conditions for functional (weak) 1 See Green, 1964, p.37-38. 2 Although explicit references to the neighborhood of a specific point and the use of the adjective 'locally' are omitted, it is understood that the following concept of functional separability is a local one. It needs to apply at every point of the input space to become global. 16 separability of the production function g with respect to the partition R (r > 3) are satisfied if and only if: = -f*M*) = f o r a l l . . € N a n d k ^ ( 2 7 ) where the subscripts denote differentiation, gi{x) = dg(x)/dxi. The marginal rates of substitution between any two inputs z,- and Xj from any subset N3, s = 1,2, . . . ,r, have to be independent of elements outside the subset N3. Define the set of ordered triples that identify indices belonging to different subsets of the partition as: = {(i,j,k) Elxlxl: ie I3 A j <E It A k ISU It}, where / = { l , . . . , N} is the set of indices of X and, I3 and It are the sets of indices of N, and Nt. Additive (strong) separability requires that expression (2.7) vanishes for all i e T^. When the production function is additively separable, the marginal rates of substitution between two inputs Xi and Xj, respectively from different subsets Ns and Nt, do not depend upon quantities outside of these subsets. The relation between separability (weak and strong) of the production function and its functional form is given by the following theorem. Theorem 2.2. (Leontief, 1947b; Goldman and Uzawa, 1964) Let ips(x^) be functions of the elements of N3 only, x^ = (z,-), i(E N3, and assume that h{-) is a monotone-increasing function. a) Functional (weak) separability with respect to the partition R is necessary and sufficient for the production function to be of the form: g(x) = hlM^lM^),--- ,M^r))Y (2-8) b) Additive (strong) separability with respect to the partition R (r > Z) is necessary and sufficient for the production function to be of the form: g{x) = + Mx{2)) + •••+ Tpr{x{r])). (2.9) When the inputs i and j are separable from all other quantities zj. (Vfc), the production function is completely (strongly) separable as required by condition (2.2). For exact aggregation to hold, the marginal rate of substitution between any two pairs of inputs at every point of the input space has to be independent of all other inputs. Since the 17 approximation of given functions by separable ones is justified only if the given functions also approximately satisfy the Leontief-Sono conditions (Fisher, 1969a), it is not possible to extend the class of functions that can be aggregated in a practical sense. In empirical analysis, separability conditions sometimes are tested using the Al len-Uzawa elasticities of substitution (e.g. Hazilla and Koop, 1985). However, the use of Al len-Uzawa elasticities of substitution has been criticized (Blackorby and Russell, 1988), for lacking the basic properties that a multifactor elasticity of substitution should possess. A discussion of these shortcomings can be found in Blackorby and Russell (1988), where the authors advocate the use of the Morishima elasticity of substitution. The partial elasticity of substitution (AES) (Allen, 1938) between inputs z,- and Xj was defined first as: L>h=i 9hXh\9ij\ VXj\g\ i,j = 1,2,. N, (2.10) where \g\-is the determinant of the bordered Hessian matrix ( ° 9i 92 • 9N \ 91 9u 912 • • 9lN 9 = 92 921 922 • 92N V 9N 9N1 9N2 • • 9NN J (2.11) and \g~ij\ is the cofactor of in g. It was further expressed by Uzawa (1962) in terms of the cost function. Let the production function g be continuous, strictly quasi-concave and non-decreasing. Then the cost function C dual to g is defined by: C(y, p) = minx{p . x : y < g{x)}, (2.12) where p 2> O J V is a vector of positive prices. Assuming t h a t . C is twice differentiable, the A E S can be written as: _ C{y,p) Cy{y,p) i,j= 1,2,... ,N, (2.13) V Ci(y,p) q(y,P)' where Cj(y,p) = dC(y,p)/dp{ = x(y,p) by Hotelling's Lemma and C$ = d2C(y,p) /dpjdpj. The A E S measure the response of derived demand to an input price change, holding output and all other input prices fixed. Tests of separability using the A E S are based on the work of Berndt and Christensen (1973). Berndt and Christensen have argued that complete strong separability of the 18 production function at every point in the input space is a necessary and sufficient condition for the equality of all proper A E S <7,y. These conditions have been relaxed by Blackorby and Russell (1976). If the production function is also homothetic, it is necessary and sufficient that (Tik = Ojjfc for all k) G for some (s, t), for the production function to be strongly separable. A more informative measure of multifactor elasticity of substitution is the one proposed by Morishima (1967). The Morishima elasticity of substitution, which is the proper generalization of the original two-variable elasticity of substitution, is defined as: Mt,(y,p) = ~ a ' n | l ^ ' b ' r i l = ~'«(»->). « = U.-.ff, where e^ ,- and e„ are the cross and own price elasticitites. Blackorby and Russell (1981) have shown that the cost function is completely separable if and only if: M^y, p) = MM{y, p), V k) G T„ (2.15) where T3 - k) G I X / X I : i G Is A j E It A k<? /, U It for some t, s}. The potential advantage of using elasticities of substitution over a direct test of separability is that they may serve as measures of the degree of separability among a number of factors simultaneously. 19 Chapter 3. Exact Aggregation over Firms (and Factors) with Optimizing Behavior The so-called Fisherian approach to problems of aggregation over firms involves several variations, each pertaining to different restrictions on the mobility of factors. It was first initiated by Fisher (1965 a) who transposed the problem of aggregating capital of different vintages (Solow, 1964) into a problem of aggregation over firms. The approach was pursued further by a number of exact-aggregation theorists (Diamond, 1965; Stigum, 1967; Fisher, 1968a, 1969b, 1982, 1983; Gorman, 1968a; Blackorby and Schworm, 1982, 1983, 1984, 1988). This first set of problems tackles the question of capital aggregation, which was long considered to be the most important stumbling block in the aggregation over firms. The solution to this aggregation problem was found to require severe restrictions on the form of the firm's production functions and of the capital aggregates, as well as similarity conditions across firms. Later, the aggregation of efficient factors also was found (Fisher, 1982, Blackorby and Schworm, 1988) to require severe restrictions. This problem is reviewed in the second section of this chapter, which gives the conditions under which a partial aggregate of the efficient factors can be constructed in the presence of a fixed factor. The problem can be extended easily to one where all factors are taken as mobile. Such a question is explored in the third section, which examines the Brown-Chang approach. In contrast with Fisher's approach, the Brown-Chang model employs Hicks' (1938) Commodity Aggregation theorem to effect intrasectoral aggregation. An important underlying assumption of the Fisherian approach is the homogeneity of output. 1 When firms produce different outputs, Fisher (1982) proposed thinking of j / as dollars worth of output at fixed prices so that total revenue is maximized. Since prices generally vary over time and differently for heterogenous outputs, this approach by assumption has limited applicability when outputs are different. When the problem is considered in the dual profit function space, the homogeneity of outputs is analogous to the assumption of common output prices across firms or sectors. 1 That is, it assumes that outputs are non-differientiated products. 20 S.l. Intersectoral Aggregation of a Fixed Factor Arbitrarily Distributed across Firms in the Presence of Efficient Factors - The Fisherian Approach Assuming that M types of non-optimized inputs, to be called capital, and N types of optimized inputs, to be called labor, the firm's production function can be denned by: l/= gf(kf,lf), f=l,2,...,F, (3.1) where t/ is the level of output, kf is a vector of dimension M of fixed capital inputs, specific to the f-th firm, 1* > ON is a vector of optimized labor inputs, utilized by the f-th firm, and </ is assumed continously twice-differentiable. Under appropriate regularity conditions on the production function, the firm's dual variable profit function is given by: n'(p, w, */) = m a x ^ , / {pyf -w.l': ( j / , /') G )}, (3-2) where p 3> 0 is the output price, w Oyv is a vector of labor input prices, the same for each firm, and S^(kf) is the restricted (short run) technology set of firm / . As Diewert (1980, p.465a) suggested, this micro profit function can be written equivalently as the result of a two-stage maximization procedure: Ilf (p, w, kf) {Ylf{p,lf,kf) - w.lf : lf > 0N}, ^ (3.3) where Uf{p, lf, A/) = pr i / ( i , / / , kf) = m a x ^ {pr/ : /') G S'(A/)}. (3.4) Assuming further that output and labor are non-differientiated across firms, that is not firm-specific, their aggregates 1/ > 0^ and C > ON can be taken to be simple sums: F F y = T{y\...,yF) = Y,J a n d L = W\...,lF) = Y,lt- (3-5) f=i f=i A macro variable profit function then can be defined in terms of aggregate output and aggregate labor: Tl{p, w,k\... ,lf) = mzxyi£ {Py - w £ : (I/, C) € S{k\ if)}, (3.6) where the technology set of the economy is just the sum of the technology sets of the firms: S^k1,... , kf) = S/Li Sf(kf). If capital aggregation is feasible, the aggregate technology set can be described in terms of a capital aggregator K = ^(k1,... , if) > 0M, S(k\...,lf) = S(K). (3.7) 21 A s in the firms' case, macro profit maximization can be seen to occur in two-stage: n(p, w, kl,... ,kF) = m a x £ {Il*{p, £, k1,..., kF) - w£ : L > 0}, (3.8) where n*{plC,k1!...!kF) = maxy{py :{y,C)eS{k1,...,kF)}. (3.9) In its original formulation (Solow, 1964; Fisher, 1965 a), the problem of capital ag-gregation was set in the context of a single firm possessing capital of different vintages / (i.e. produced at time / ) . Given each single vintage production function, labor was al-located efficiently between the F production units using the different vintages of capital, the problem was then to describe total output as a function of total input of labor and of some "suitably weighted sum of surviving capital inputs of various vintages" (Solow, 1964). More formally, the "vintage" capital aggregation problem asks under what conditions is it possible to write maximized aggregate output 1/ as a function G of aggregate labor C and aggregate capital: ( F F 1 J/ = max ( 1 _,F \^9l(kf,lf) -Jl1'^ Z> lf>^\ V=i /=i ' (3.10) = g(k\...,\f ,t) = G\K,L\, where AT is a capital aggregate independent of labor, and where the marginal product of labor is the same for all firms: dgf/dl^Wi, f=l,2,...,F, i=l,2,...,N (3.11) that is, where labor is efficiently allocated across firms. Applying the Leontief conditions for the existence of a capital stock aggregator in con-junction with the efficiency condition (3.11), Fisher (1965a) derived non-trivial necessary and sufficient conditions for capital aggregation, but he was not able to secure a complete closed form characterization of the class of cases satisfying it. Considering the one-capital one-labor case (M = 1, N = 1), assume strictly diminishing returns to labor, that is < 0 (/ = 1,2,... ,F), where the subscripts denote differentiation. Then a necessary and suf-ficient condition for capital aggregation is that every firm's production function satisfy a partial differential equation that equates net normalized returns to labor across firms: I did = h(g(l), f=l,2,...,F, (3.12) km22 where the function h is the same for all firms. Stigum (1967) has generalized Fisher's results in the one-capital one-labor case to continuous production functions. In the one-capital multiple-labor case, Fisher singled out an interesting special case. If every micro production function can be written as: jrW) = AW),*'], (3.13) where hf is linearly homogeneous and $ is monotonic [i.e. exhibit exclusive capital-altering technical change, under capital generalized constant returns), a necessary and sufficient condition for capital aggregation is that the average product of each type of labor be the same for all firms. In the general case under constant returns to scale, Diamond (1965) and Fisher (1965 a) independently showed that capital aggregation requires that all firms have identical production functions except for a capital augmenting factor. Considering the "vintage" capital aggregation problem in the dual profit function setting, Gorman (1968 a) was more successful in finding closed form characterizations of the aggregation conditions. Gorman asked under what conditions on the technologies does there exist a macro profit function 77 and a fixed inputs aggregator ip such that: F n(p, w,kl,...,if) = ^ 2 n'(P, w, k?) = n[P, w,^1, ...,if)}. (3.H) / = I The original "vintage" capital aggregation problem has been reformulated by Diewert (1980, p.464) as follows: ( F F ) n(p, £,kl, ...,kF) = ma.xll l F ^ n / ( P , l ^ ) : ^ / / < i : , l f > 0 N \ , (3.15) V = i f=i } that is, the vector £ 3> 0^ of aggregate labor inputs is taken as given and is to be allo-cated across firms (or vintages) to maximized aggregate profit. Since Yl(p, t, kl,... , if) = Il*(p, L, k1,... , If) [(3.9) = (3.15)], Gorman's formulation encompasses the original prob-lem. 2 Theorem 3.1. (Gorman, 1968a) If and only if there exist capital subaggregates tp*, j = 1,2, ... ,M, for each firm f = 1,2, ..., F, such that the micro profit functions can be 2 For £ fixed, (3.8) is equivalent to (3.9). 23 written as: M nf{p!w,k/) = YJcJ{p,w)<pfj(kfj) + cf0(p,w)) f=l,2,...,F. (3.16) •;=1 The corresponding aggregates for each type of capital will be: F <Pj(k},...,kf) = J2<Pfj(li), J = h2,...,M, (3.17) /=i so that, letting K = (p(kl,... , kF) = (p\(k\,..., kf), ..., <pj(kj,... , k^)) denote the vector of capital aggregates, the variable profit function for the economy can be written as: F M YI nf(P> w>kf) = Yl ci(P> • • • » *T) + co(P, w)> ro 1tA /=i y=i y6-1*) = n[P, w, K], where c0(p,w) = J2/=i <?o(.P>w)-The specifications (3.16) for the micro profit functions imply that the capital aggrega-tors provide a measure of the quantities of the fixed factors in terms of their earning power (quasi-rents) at their efficiency price (marginal profitability) Cj(p,w). Since the marginal profitability of each type of capital is the same for each firm, efficiency in the labor market guarantees that any given distribution of the fixed factors is just as productive as another, that is efficient. Conditions (3.16) are necessary for consistent capital aggregation, under some regular-ity conditions on the technology sets (Gorman, 1968a). These regularity assumptions rule out increasing returns to scale as the technology sets are assumed convex. This does not seem too unreasonable, remembering that capital inputs are fixed in the short run. They require that capital aggregators <Pj(-) be continuous, which is an unfortunate property as it rules out lumpiness or discrete jumps in output due to small changes in investments. Notice also that the Gorman conditions (3.16), (3.17) and (3.18) are just Nataf's condi-tions (2.2), (2.3) and (2.4) applied to profit functions. A s a result of the integration of differential aggregation conditions (2.6), Nataf obtained affine functional forms, but further dropped the integration constant and overlooked the non-commonality across firms of the linear constants as he was looking for an output (profit) aggregator in terms of elementary 24 outputs. Since all profit functions (micro and aggregate) utilize the same elementary units (money), that part of the derivation is omitted when considering profit functions and func-tions of the non-aggregated variables enter the affine forms. Homogeneity in outputs and variable inputs and efficiency in the variable inputs' markets reduces the profit function aggregation problem to the aggregation of non-optimized inputs in a Klein-Nataf context. In practice, this type of capital aggregation is more likely to be justified within an industry where the fixed factors are not too lumpy and where perfect competition in the variable factors' market can be assumed, than throughout an economy as a whole. In the one-capital case, these conditions are not very restrictive, since in this case functions of the form c(p, w)<pf(kf) + <?0(p, W) can provide a second-order approximation to any twice-differentiable profit function H*(p, w, k/), as Diewert ( 1 9 8 0 , p .467) observed. However, when there are a number of capital goods, condition ( 3 . 1 8 ) requires that the capital aggregates be additively separable in the aggregate profit function which is more unrealistic from an empirical point of view. More recently, Blackorby and Schworm ( 1 9 8 4 ) have provided a primal interpretation of Gorman's conditions ( 3 . 1 6 ) on the firm's profit function that emphasizes the separability of the different types of capital goods. They describe the firm's profit function as the result of the optimization of M production functions, one for each type of capital, as if the firm comprised different plants each using a single type of capital. Under the assumption that the technology sets are bounded, the Cj(p, w)(p^(k/j) can be treated as the images of profit functions with dual technology sets Sj(cpj(kj)), j = 1,2, ... ,M and condition ( 3 . 1 6 ) can be set in the primal. Theorem 3.2. (Blackorby and Schworm, 1984) Assuming that the aggregators (p^(kj), j = 1,2, ...,M are unbounded above and that the technology sets S^(k/), f = 1,2, ...,F are bounded above, the micro profit functions Tl^(p, ui, kf) will satisfy ( 3 . 1 6 ) , if and only if there exist production functions hj, j = 1,2, ..., M, and f/0, where the hj are linearly homogeneous such that / M JVf ,/(*/,//) = m a x , / ( 1 ) ) . . . ; I / ( M ) ) / / ( 0 ) E M ^ ^ ' W I + W O ) ) : £ ^ ' ) = / / } ' ( 3 . 1 9 ) where l^(j) are the labor inputs associated with the capital inputs of type j; 25 the economy's production function will be: , M g(k\ ...,kF,t) = m a x £ ( 1 ) £ ( M ) , £ (O ) { E hjl<Pi(k}> •••,#'), A-/')] + ^ o ( £ ( 0 ) ) : M \ (3.20) w/iere v?j(-) J« defined by (3.17), £ ( / ) = Y^=\ ^[j) and F F Ao(i!(0)) =max / 1 ( 0 ) > > j F ( 0 ) jXVo^O)) : E = £(0)}- (321) l / = i / = i j This theorem has been stated equivalently in terms of the technology sets. The production function characterization, however, underlines better how restrictive the Gorman condi-tions are. Not only do the production functions of the firms have to exhibit some sort of additive separability with respect to capital types, but the firms have to use combinations of labor inputs and firm-valued quantities of capital goods according to the same linearly homogeneous production processes hj, for each type of capital, so that the marginal prod-uct of each type of capital is the same across firm. Under capital-generalized constant returns, the linear homogeneity requirement in the hj processes can be dropped. The only processes that are allowed to differ across firms are the ways in which firms value the different types of capital through the <p^- aggregators and, use labor inputs independently of capital through the bfQ production processes. Under constant returns to scale, these last firm-specific components are omitted. When the production functions are written as functions of the appropriate capital aggregate: 9f(kf,lf) = hf[<pf{kf),lf] and g{kl,..., kF, C) = G[(p(k1,..., kF), £], (3.22) restrictions (3.19), (3.20) and (3.21) are found to be equivalent to a set of differential conditions (Blackorby and Schworm, 1984, p.639). These differential conditions require that the Hessian matrix be non-singular and can be expressed in terms of the determinant of the Hessian matrix: \Hkjl{h/)\ = 0 and \Hk.,{G)\ = 0, j = 1,2,... , M (3.23) 26 where H^-i is the Hessian matrix with respect to 1*) of tf and G, respectively. In the macro case, these conditions (3.23) are necessary and sufficient for consistent aggregation to hold, and thus provide a simple test of whether a particular aggregate production function can allow capital aggregation. In the micro case, restrictions (3.23) are necessary for consistent aggregation but not sufficient, since they do not include any conditions that would make firms appropriately similar. Another interesting feature of restrictions (3.23) is that they imply for any (k[,... , k/M, l^) the existence of directions in the (kj, /-^-spaces, j = 1,2, . . . , M, in which the production function is linear. The required commonality across firms is that the direction of linearity at the different optimal points (k[,... , l/M, l?), f = 1,2, ... ,F, be the same for all firms. This is analogous to Gorman's (1953) result. S.2. Full Aggregation of a Subset of an Efficient Factor in the Presence of a Fixed Factor Arbitrarily Distributed across Firms This section discusses the full aggregation of an efficient factor in the presence of other efficient factors and of a fixed factor arbitrarily distributed across firms. The mobility of the factors to be aggregated implies aggregation conditions that are more flexible than the Gorman conditions. Under these circumstances, the firm's production function is defined by i / = / ( t f , t f , / / ) , f=l,2,...,F, (3.24) where j / is a measure of output, tf is a vector of dimension M of fixed capital inputs specific to the / - th firm, tf > OR is a vector of optimized inputs, such as materials, > Ojv is a vector of optimized inputs to be aggregated, such as labor. Intrasectoral aggregation of the labor inputs is now based on the separability of the production function, more specifically intrasectoral labor aggregators 4^(1^), f = 1,2,... , F, are assumed to exist for each firm, so that yf = / ( A / , * / , / ' ) = tf(tf,tf,<tf(//)) = tf(tf,tf,l'), (3-25) where V = dJ(tf) £ HI denotes the intrasectoral labor aggregates. Under appropriate regularity conditions on the production function, the firm's dual variable profit function is given by: q, w, kf) = m a x ^ y ( / {pj/ -• q. J - w lf : / < / ( t f , J, lf)}, (3.26) 27 where p 3> 0 is the output price, q 2> OR is a vector of material input prices, w 3> 0/v is a vector of labor input prices, the same for each firm. Under assumption (3.25) that the production function is separable in the labor aggregate, the firm's labor aggregated profit function can be written as: J7'(p, q,u>, kf) = max^y,/ {py> - q. J - w \ f : / < tfftf, a/, l')}, (3.27) where OJ S> 0 is the price of the labor aggregate. The micro variable profit function (3.26) can then be written as the result of a two-stage optimization procedure, involving the maximization of the labor aggregate: nf(p, q, w, kf) = max,/ {n*f(p, q,lf, kf) - C ^ l ' ) } , (3.28) where the non-labor profit function, to be called revenue function, is defined by: n*\p:q^,kl)=™xjj{pyI-q.J:yl<hf{kf!xf,\})}, (3.29) and where the cost of the labor aggregate is denned by: uV = Cf(w,lf) = rmnlf{wlf : <fJ (lf) > ]/}, f=l,2,...,F, (3.30) so that OJ = w»V/V. Th is first stage can be regarded as the construction of the labor aggregate in value terms. Assuming that the output and variable factors are non-differientiated, the intersectoral output aggregate I/, intersectoral labor aggregates ti and materials aggregates X S IR^ are defined as simple sums: F F f=i f=i F and X = <p(xl,..., TF) = J. / = i (3.31) The full labor aggregate is further define by either equalities: F F C=${Ci,...,CN) = ^ ( i \ . . . ) i F ) = E 1 / = E^(//)- (3-32) /=1 f=l 28 Two aggregation problems proposed by Fisher (1968 a) can be studied using this formu-lation: i) the aggregation over mobile factors and ii) the aggregation of a subset of the mobile factors, both in the presence of immobile ones. The general full-labor-aggregation problem asks under what conditions is it possible to write maximized aggregate output t/* as: / F F y* = meixxl x F l l i ... | (F j £ < / z ' ' ; / ) < X, J >0N, F 1 ^2lfi<Ci, lf{>0, i= 1,2,... , N \ (3.33) f=i  } = G(k1,...,k , X, i \ , . . . , CN) = G[k\...,k?,X,Z}l where materials and labor are optimally allocated across firms. Considering problem i), under the assumption that the labor aggregators <tf for each firm are homogeneous of degree one, so that the production functions can be taken to be labor-homothetic, Fisher (1968a, p. 397) found that a necessary and sufficient condition (Fl) for full labor aggregation is that the production functions all be members of the same family, that is the labor aggregators must be the same for all firms. When only a subset of the mobile factors are aggregated (problem ii)), this condition is both necessary and sufficient when each firm's production function is either constant returns, capital-generalized constant returns or labor-homothetic (F2) (Fisher, 1968 a, p. 406). More general results have been obtained by Blackorby and Schworm (1988) while investigating profit functions. Let the economy's labor aggregated variable profit function be defined by: F F n{p, q,u>, k1,...,^)^ max y z l ^ £ {Py - £ ?• ^  - u£ : y < £ ^ ,<t>! [lS)), / = i / = i F xyc7) < c,if>oN}. (3.34) As in the firms' case, the economy profit maximization problem can be written in terms 29 of the labor aggregate as occuring in two-stages: n(p, q,oj, k1,..., kF) = n(p, q,w,k1,...,kF) = maxC {II*{p, q, Z,k\ ... - C{w, £)}, (3.35) where the economy-wide non-labor profit function or revenue function is: F F n*(p, q, t,k\..., kF) = maxyji t F {pi) - ]T , . J : 1/ < £ ^  ^ ( O ) , (3-36) /=1 /=l and the economy-wide cost of the labor aggregate is: F F OJC = C{w, £ ) = min ( 1 j F { J ^ w / ' : ] L ^ ( F / ) ^ £>)• ( 3 3 7 ) / = i / = i In terms of profit functions, the full labor-aggregation problem asks under what conditions on the firm's production function does the following equality hold: F Yl{p,q,w,k\...,kF) = E n / ( P > 9,v,*f) = = n(p, q.w^1,...,^). f=l (3.38) Since the firms' technology sets can be written in terms of the labor aggregates, (3.38) can be restated in terms of the two-stage optimization process: n*(P,q,ik\...,kF) = Y,n*(p, F and C{w, t) = Y^C!{w, P). 7 (3-39) / = i The ensuing aggregation restrictions are given by the following theorem. Theorem 3.3. (Blackorby and Schworm, 1988) Under the assumptions to be called R ; that the firms' production functions are i) continuous, ii) increasing, concave and bounded above in (a/, 1^) for all kf > 0, iii) increasing in kf and iv) satisfy (3.25) with continuous labor aggregates <^(-), unbounded from above, full labor aggregation, in the sense of relation (3.38), will hold if and only if either a) Cf{w,\f) = c0(w)V + c{{w), f=l,2,...,F (3.40) or, b) n*f{p,q,\f,k/)=n0{p,q)lf + n{{p,q,lcf), f=\,2,...,F (3.41) 30 or, c) Cf{w,\f) = < £ ( « ; , 1 ' ) + 4(«0 ( 3 - 4 2 ) w h e r e ] = a i (3.43) and /?* /(p,7,l/,tf) = ^ ( p , ? , l / ) + T i ( p / ^ * / ) (3-44) « * e r e = ,'(!') (3.45) orerf where l = Sf(lf) for f=l,...,F. (3.46) In order for there to exist a full labor aggregate, all firms' profit functions must be similar in either of three ways. Let V = A?(p, q, w, k?) be the firm's demand of labor aggregate. In the case a), the firm's demand function is assumed to vary, given a fixed wage and holding the total demand of aggregate labor of other firms constant, with changes in (p,q,kf), so that there exists directions of linearity in the demand function space. In this case, the profit functions are required to be similar only in their valuation of the labor aggregate. The marginal cost of aggregate labor co(w) is the same for all firms and the optimization in (3.28) implies that the marginal revenue of aggregate labor is also equal to this common wage index. The affine form of the cost function in the labor aggregate is derived as a requirement for the sum of functions to be a function of the sum of the arguments, as in (3.32) and (3.39). In the case 6), the labor demand of every firm is assumed to be independent of capital. The non-labor profit functions will then be additive in aggregate labor and capital. Affinity in aggregate labor will allow the economywide adding up and the same marginal revenue TIQ(P, q) for each firm will ensure sufficient similarity. As in case o), optimization in (3.28) will imply that the marginal cost of aggregate labor is equal to the common price index. In the case c), the assumptions of cases a) and 6) do not hold, the labor demand functions are independent of capital but prices and wages are interdependent. In cases a) and b), similarity among firms of either the marginal cost or the marginal revenue, respectively, was sufficient to ensure exact-aggregation, given the special structure of the corresponding functions. Case c) imposes equally stringent structures on both the cost and the revenue functions, but requires that the marginal cost and revenue with respect to the firms' labor aggregate be similar firms. 31 These aggregation conditions can be further described in terms of the production functions and labor aggregators that are dual to the non-labor profit functions and labor cost functions in Theorem 3.3. Theorem 3.4. (Blackorby and Schworm, 1988) Assuming R, full labor aggregation, in the sense of relation (3.38) ; will hold if and only there exist functions hf and (fJ that satisfy (3.25) and either a) 4/{if) = Ml1) (3-47) where <j>o is linearly homogeneous for f = I, 2,... , F, or,b) / /(A / ,3 / ,1') = max / / {ho{J0,\f) + : 4 + A = ^} (3.48) where ho is linearly homogeneous and <fJ is homothetic for f = 1 ,2 , . . . , F, or, c) satisfies case a) and hf satisfies (3.48) but ho need not be linearly homogeneous. In case a), only restrictions on the aggregators are required: the labor aggregators have to be linearly homogeneous and identical across firms; this result is identical to Fisher's (F l ) . In case b), only restrictions on the production functions / / are imposed. The production functions need to be additive in aggregate labor and capital spaces with the same linearly homogeneous function of aggregate capital. In the case c), there are restrictions on both the production functions and the labor aggregators. These results have been extended (Blackorby and Schworm, 1988) to solve other problems proposed by Fisher (1968b, 1982), such as the existence of a joint capital-labor aggregate and the simultaneous aggregation of capital and labor. S.3. The Brown-Chang Approach As problems of intrasectoral factor aggregation can subdivided in two distinct branches according to whether a Hicksian price proportionality or a separability approach is used, full aggregation problems also can be distinguished by their choice of intrasectoral aggregation approach. The approach of Brown and Chang (1976) relies on Hicksian aggregation while 32 the solutions of Fisher (1968 a, 1982) and Blackorby and Schworm (1988) are based on separability requirements. 3 The Brown-Chang (1976) analysis considers a full factor aggregation problem in a simple general equilibrium model, where prices are treated as endogeneous and where constant returns to scale prevail. By making factor prices endogenous, the model tries to correct the dissociation of Gorman's (1953) restrictions on functional form from any effects on price, and attempts to link to economywide forces the prices of factors to be aggregated in a Hicksian context. Since both factor proportions and factor prices are endogenous in a general equilibrium context, restrictions on one set of variables must affect the other. By relying on very strong (and highly unrealistic) assumptions and by specifying very restrictive sector interactions, the Brown-Chang approach leads to relatively mild aggregation conditions. It is interesting as a simplified ideal case. The Brown-Chang model assumes perfect competition and neoclassical technologies. The model also requires that the rate of profit r be exogenous (r to be determined by the rate of time preference or by fiscal or monetary policy). Brown and Chang first consider a general equilibrium model in which there are two capital producing sectors, both of which use labor and the two capital items to produce their outputs of capital. Note that all factors are assumed mobile (or optimized), so that here capital goods could alternatively be seen as intermediate products. Let the two production functions be described by: / = * £ , / > ) , / = 1 , 2 , (3.49) where j / is a capital output associated with the / - th type of capital input, and kf2 are the quantities of two distinct capital inputs, is a measure of non-differientiated labor inputs and gf is assumed homogeneous of degree one. The full-aggregation problem asks under what conditions can (3.49) lead to an aggregate production function G: 2 V = X V ! / = G{K,t), (3.50) f=i where K = <p(Klt K2), with Km = £ / = i k/m {m = 1,2), so that capital is aggregated both intrasectorally and intersectorally, and £ = S/=i &• 3 Hicks' Commodity Aggregation Theorem states that when the prices of a group of goods change in the same proportion, at the optimum this group of goods can be considered as a single good. 33 To perform full aggregation, Brown and Chang combine general equilibrium conditions with the conditions of Hicksian aggregation. Two sets of equilibrium conditions are derived from the general equilibrium requirement. Let the firm's profit function be given by: Tlf{p,w,q) = m a x s W { F y + qi k[ + q2 kf2 + : tf < /(tf, / ')}, f = 1,2, (3.51) where the nominal wage rate is taken as numeraire, / (/ = 1, 2) are the capital output prices in terms of the numeraire, and q > 02 is a vector of capital rental prices. Then, the zero profit condition reads: M = 1 + iA + ?A, /=1,2. (3.52) Assuming depreciation away, the gross rental rates become: qm = pmr (m — 1,2) and the marginal productivity equilibrium conditions are: dgf 1 , dgf pmr , 57 = 7 a n d M. T / ' m = M - (3'53) If the Hicksian aggregation conditions are satisfied, there will exist q*Q and qo such that: m=l % j ^ l q ° which implies that: ~ d r ~ - ~ d r ~ = °- (3-55) Solving (3.52) for / in terms of r, taking the ratio of the two expression and differentiating with respect to r, condition (3.55) becomes: djp'/p2) _ l / 1yjL_ 2 ^ dr {p2)2rA\P I1 P P (3.56) fyl k\ \ fy2 &\ k2 k\ . ,2 where A . - fr) - fr) - f-l(r)2. Since the term in parentheses represents the difference in wage shares in the two sectors, this leads to the principal Brown-Chang result: intrasectoral (mobile) capital aggregation is assured if the labor shares are the same in the two sectors, for all rates of interest. This condition can be applied to models with joint products and can be generalized to models with many primary factors. The equal-labor share condition amounts to equal 34 capital/labor cost ratios in each industry and has been compared to Marx's case of equal organic composition of capital. When considering a more complex model (many-sector, many-capital, equal-depreci-ation rates), the equal-labor share condition will imply the satisfaction of the Hicksian aggregation condition and permit intrasectoral aggregation over all capital inputs. In the new two-factor model, the equal-labor share condition is equivalent to equal capital intensities in values terms as well as in physical terms. Note that the reduction of the system to a two-factor model is an essential feature for its solution. The Gorman conditions for intersectoral aggregation are then .satisfied and all capital goods can be aggregated into a single capital good sector. When labor shares in all sectors are equal for all interest rates, the economy reduces to a single aggregate sector. This result could be seen as a special case of Fisher's (1968a, p.397) aggregation of mobile factors. Under constant returns to scale and in the presence of fixed capital inputs, Fisher showed that a condition for subaggregation is that every firm employs all mobile factors (to be aggregated) in the same proportion. When there are no fixed factors, this condition implies the equal-labor share condition in the Brown-Chang model. The merit of the Brown-Chang analysis in very special aggregative circumstances is to bring to the fore the interdependence between factor proportions and factor prices. In this type of constant returns general equilibrium model, if Gorman's similar factor-ratio conditions for intersectoral aggregation are satisfied, the condition for Hicksian in-trasectoral aggregation will be satisfied simultaneously (Zarembka, 1976). When the pro-duction functions happen to be such that their expansion paths are parallel straight lines through their respective origins, the optimal factor ratios will be the same for all firms. Since all firms face the same factor prices, labor shares clearly must be the same in all sectors, thus the Hicksian commodity aggregation condition is satisfied. However, the con-verse will not hold: the conditions for composite aggregation do not guarantee that the Gorman conditions will hold. The practical implications of Brown and Chang's result have been highlighted by Fisher's (1971) experimental discovery. Using simulation experiments, Fisher (1971) sug-gested that the relative constancy of labor's share was responsable for the success of aggre-gate Cobb-Douglas production functions in predicting wages and not the other way around. 35 Brown and Chang have shown in their theoretical model that when the equal-labor share condition holds, then the aggregate production function can be Cobb-Douglas of the sectoral forms is also Cobb-Douglas; more generally, the economywide production func-tion can be explicitly represented by any of the sectoral production functions. In effect, Fisher's experiment used Cobb-Douglas production functions at the firm level simulating Brown-Chang's result. This result will be corroborated in the present proposed stochastic framework. 4 See chapter 5. 36 Chapter 4. Stochastic Aggregation Models Earlier discussants of aggregation problems recognized that the "pattern of distri-bution, i.e. a definite way in which inputs must be distributed both as among different individual firms and as among different types of inputs" (Pu, 1946, p.300) was a key ele-ment in the formulation of a unique macro production function. These "patterns" may be formalized mathematically by probability distributions, which add a distinguishing feature to the following aggregation models. By contrast with preceding exact-aggregation mod-els, which use totals as output and inputs aggregates, the stochastic aggregation models construct aggregates of output and input variables on the assumption that the micro vari-ables are probabilistically distributed across firms. Most often, the stochastic variables are assumed to belong to a known probability distribution. Thus, these models obtain exact aggregation by imposing functional-form restrictions both on the production functions and on the probability densities. 4-1. Intersectoral Aggregation Using Stochastic Capacity (or Efficiency) Distribution In a path-breaking paper, Houthakker (1955-56) introduced the idea of describing a continuum of firms using a probability distribution. He assumed that the micro production functions were subject to fixed proportions between output and variable inputs while fixed factors limited capacity output. In the special case where capacity output follows a Pareto density function, Houthakker showed that the resulting macro production function is Cobb-Douglas. Houthakker's model addresses the aggregation of variable inputs. It has been recast by Sato (1975) to include capital aggregation, a problem which it solves unsatisfactorily since it would imply infinite production capacity (Sato, 1975, p.27). Under the assumption of fixed factor proportion, capacity output is described by: (4.1) where j/4iax is capacity output, which is limited by the fixed factors, and l2 are the variable inputs, 6^  and lf2 are the fixed efficiency coefficients. 37 Let w\ and w2 be the prices of and Z ,^ the same for all firms, and let the price of output be the numeraire. It will be profitable for the firm to operate as long as its quasi-rents are non-negative: yf - - w2l[ = - tuiftj - w2y2) > 0. (4.2) The zero quasi-rent condition + = 1 is rewritten as: ff2 = (l — u>il/1)/w2 • Under the assumptions that the firms are small and numerous, and have different production possibilities, Houthakker postulated that there exists a continuum of fixed coefficients (that give rise to the random variables B\ and B2 )• In particular, he considered the special case where the density of capacity output (which he calls the input-output distribution) is a generalized Pareto: tth, h) = PBhB2( bi> M = 6b^-lb^-\ (4.3) where S, fii and (32 are constants. The validity of expression (4.3) as a proper output capac-ity distribution may be questioned (Hildenbrand, 1981). Without appropriate restrictions on the domain of integration, Houthakker's generalized Pareto input-output distribution may have an infinite mean. 1 Then, integrating over all profitable values of the efficiency coefficients, aggregate output can be written as: y = l l ^ W i ^ W M & i - (4-4) Jo Jo Similarly, the aggregates of the variable inputs will be given by: £ 1 = / U ' 1 / W2 b1<f>{b1,b2)db2db1 and Z2= / 1 / W2 b24>(bl,b2)db2db1. Jo Jo Jo Jo (4.5) Inserting the Pareto density function (4.3) into the aggregates (4.4) and (4.5) gives: y = W ± ± ^ m + l i h + l ) i (4.6) £ l = a /»! + ! P2B(fil + l,h + iy ^ d C2 = —L—B[ftl + 1 J 2 + 1), 1 For example, Chetty and Heckman (1986) required that the parameters of the Pareto distribution be greater than 3. 38 where B(-) is the Beta function. So that, aggregate output may be rewritten as: V-(R .Lfl , n r^ (^ l + 1,^2 + 1)1 gj+gS+T jgpj^+T p g r f e n : y - w + h + i ) [ ^ 1 + i ^ 2 + i j ^ £ 2 ( 4 7 ) which is a Cobb-Douglas in the variable inputs' aggregates. Notice that the aggregates depend on the price level and are, in this sense, optimized aggregates. Houthakker's model has been extended by other authors. Levhari (1968) allowed the elasticity of substitution (<r) to be greater than 0 but less then 1 and derived a macro C E S production function using Houthakker's procedure. Cornwall (1973) applied Houthakker's aggregation model to the corresponding dual profit function. He thus demonstrated that this aggregation procedure could be carried out for less simple models, even though an easily analyzable form may not always be obtainable. Johansen (1972) and Sato (1975) have developed the procedure to a fuller extent, proposing somewhat more general frameworks. Johansen (1972) followed Houthakker in using a Leontief fixed coefficients model. But , he justified the "fixedness" as being the ex post outcome of ex ante micro production func-tions that include all possible designs. Once the design is chosen however, no substitution between variable inputs can be made; it is a putty-clay technology. 2 Let the ex ante firm's production function be: yLa* = 9f(lLm,x,lLm,2,lS)l f=l,2,...F, (4.8) where j / m a x is maximum output (or capacity of the firm), and tfmax2 a r e ^ n e amounts of the first and second inputs when the firm operated at full capacity and kf is the amount of capital invested. Ex post, the technology has been chosen for the values, y£iax> 4ax l> ' m a x 2 a n d tf» which are no longer variable. The short-run production possibilities are then described by: jf^mnVyf^ o r ff = f/.yf> where j{ = - ? p d ; * = 1,2 , . J / m a x J / m a x <• ' ' 0 < j/ < j / m a x . 2 A generalization where input coefficients vary with the degree of capacity utilization in such a way that marginal inputs requirements are non-decreasing is also considered (Johansen, 1972, p. 46). 3 9 In the case where the coefficients fr( and l/2 are countable, Johansen derived a short-run macro production function as the result of an optimization problem which maximizes aggregate output (defined as a total) subject to the availability of the variable inputs. When considering a continuum of firms, Johansen assumed that there exists a capacity distribution PB^.B^I, b2) that represents all the production possibilities. The potential sources creating dispersion in the capacity distribution are discussed extensively (chap. 3.1). Aggregate output and inputs are then written as: y = J L PBi,B2{f>i,b2)db1db2 = G , } V ^ ] (4.10) Ci =11 bipBl,B2{bi, h ) dbi db2 = w*)> i = !> 2 -J ^ 0 ( ^ , 1 0 2 ) where the region of integration is the region of utilization determined, as in Houthakker's, by the zero quasi-rent line: U(wi, w2) — { ( i ( ,^ ) : b*l > 0, lf2 > 0, wib{ + w2\J2 < 1} and where G{w\, u>2) and Hi(wi,w2) (i = 1,2) are the supply and factor demand functions in terms of relative prices. It is possible to invert the demand functions, «/i = I T ^ i t x , ^ ) , w2 = H ? { Z X , Z 2 ) , (4.11) since the Jacobian of the tranformation is positive. Then, a general expression for the short-run macro function can be obtained as: y = G*(J:1!C2) = G[H-11{C1,I:2),H21{L1)C2)}. (4.12) Johansen's analysis thus theoretically provides an aggregative solution to a "wider variety" of forms for the macro function and the capacity distribution than Houthakker's. But the assumption about the form of the micro production functions is still overly restrictive. Also, the wider variety of capacity distribution functions (distribution along a right-angled curve, along an hyperbola, uniform distribution bounded by a right-angled curve, uniform over a rectangular region) that give explicit polynomial macro production functions do not have a significant empirical appeal. Sato's (1975, chap. 2) analysis is more interesting insofar as it disentangles the production function from the probability distribution that describes the continuum of 40 firms. The micro production functions are considered ex post, capital is explicitly modeled and the proportions between inputs do not need to be fixed. The input-output relation of the f-th firm is summarized by the production function: yf=g{afk/,^lf) = g^)o/kf> f=l,2,...F, (4.13) where 3 / is the firm's output, kf is the fixed capital input, P is the labor input, of and 0 are the efficiency parameters and g is the production function, common to all firms. It is assumed that g is a continuous, differentiate and linearly homogenous and that jjj is strictly concave. The units of measurement must be specified to ensure that: g(0)=0, 3(00) = 1; j'(O) = 1, ?{oo) = 0. (4.14) For each value of $ , the (inter-labor-efficiency) capital aggregate or capacity density function is denned as: <p(P)= I ak{a,/3) da > 0, (4.15) Ja&A where ak represents the augmented capital stock which is comparable across firms and A is the domain of labor efficiency parameters af. A s in Houthakker's model, the efficiency parameters are the random variables: </>(/?) = PB{P), where pa(/3) is the probability density o f S . The labor market is assumed perfectly competitive, so that the marginal product of labor equals the wage rate: = °' ' = ? * ( # ) « " " ' ( 4 1 6 ) where h the inverse of jjf exists by virtue of its monotonicity. Thus output can be written in terms of the wage rate and efficiency parameter as follows: Vf = 9 h(^)]a/kf, w<0f. (4.17) Since the firm's quasi-rent is j / — wl^, it will be profitable to operate only if w < $. Then , integrating over all profitable values of labor efficiency, aggregate labor becomes: C(w)= ^l-h(^)pB{P)dp, (4.18) 41 where 8 is the supremum of the set of 3 values. Similarly, the output aggregate is given by: t/H= J g[h(^)]pB(B)dd. • (4.19) Note that 2/(0) is the total capacity of the industry or the economy; it will be taken to be the (complete) capital aggregate. Sato then supposed that there exists a macro function G that yields the aggregate level of output: 3MC K (4.20) y = G(K,3MC) = G where K is some capital aggregate, fixed in the short run, 3M is a constant and where G is linearly homogeneous by virtue of the linear homogeneity of the micro functions and G is strictly concave. Sato assumed that it is possible to appropriately measure K and 3M such that: G(0) = 0, G(oo) = 1, G'(0) = 1, G'(oo) = 0. (4.21) As before, the short-run equilibrium conditions require that: 8MZ K or £(«,)= * f f ( » ) / C , pM~\pMj"> (4.22) where H is the inverse of G'. Thus aggregate output can be written in terms of the wage rate and efficiency parameter: • y{w) = G[H( w K. (4.23) The existence conditions of an aggregate production function with capital and labor aggregates can be derived by combining (4.18) with (4.22) and (4.19) with (4.23). Theorem 4.1. Sato (1975) Assuming that the functions g and G are strictly concave, that they satisfy (4.14) and (4.21), and that h and H are the inverses of if and G1, respectively, then an aggregate production function with capital and labor aggregates, such as (4.20), exists if and only if and y(w)= f g~\h(^)}PB(3)d3 = G[H(^) K. (4.24) (4.25) 42 hold where (4.26) Sato has applied these conditions to a number of specific cases with variable success. The framework allows for the aggregation of variable (or fixed) proportions micro pro-duction functions into CES (or Cobb-Douglas) macro function with the Pareto efficiency distribution. The rectangular capacity distribution induces a macro production function of exponential form; the exponential efficiency distribution leads to macro production function of transcendental logarithmic form. But the aggregation of non-identical micro Cobb-Douglas or CES production functions did not lead to analyzable capacity density distributions. Despite its novel and seemingly general approach, Sato's procedure holds a number of drawbacks. First, its relies on a known (or analytically derived) efficiency distribution which is unlikely to be corroborated in practice. Since the form of the efficiency dis-tribution consistent with this aggregative scheme depends on the functional form of the production functions, whose validity must be assessed, problems of empirical plausibility are compounded. Sato attempted to validate the approach with empirical data. In partic-ular, he studied the empirical efficiency distributions of Japan's cotton spinning industry, of the Norwegian fish-food products and non-electrical machinery industries and of U.S. manufacturing industries. While overall he found strong correlations between efficiency and firm sizes, the efficiency distributions (especially that of labor) varied considerably across industries. In the U.S. case, 75-80 percent of the total of 300-odd industries ex-hibited bell-shaped or single-peaked distributions. However, when Sato tried to fit a Beta distribution to the data, he found that the approximation was at best passable in some instances and not too good in other cases (Sato, 1975, p. 205). Furthermore, the aggre-gate production function associated with the Beta type proved to be not easily analyzable (Sato, 1975, p.250). Secondly, the output and labor aggregates are also contingent on the functional spec-ification of production since they are based on the efficiency coefficients. The capital aggregate depends on short-run labor efficiency, as well as on capital efficiency, but its measurement may be facilitated by the fact that it is a full-utilization measure. Sato's 43 capital aggregate measures the total productive capacity of the industry or of the econ-omy, regardless of whether it operates at full-capacity or not. Sato (1975, p.XXIII) argues that this solves the problem of measurement of capital. However, this capital aggregate may have not direct connection with the conventional measures of capital services utilized in the micro production functions. The inflexibility of allowable functional forms for both the production and density func-tions may restrict the empirical validity of the Houthakker approach, but their simplicity facilitates theoretical developments. The model has been extended to a dynamic setting by Chetty and Heckman (1986). These authors studied a model where the firms use labor and capital in fixed proportion, but can acquire capital over time - albeit at an installation cost - and where firms maximize discounted expected revenue. They analyzed both the long-run steady state and short-run dynamic properties of the model. Under stationary expectations, the model produces a long-run macro function resembling a Cobb-Douglas technology that depends on investment prices and on the rate of interest. Seemingly, the aggregative properties of Leontief micro production functions into a macro Cobb-Douglas production function under independent Pareto efficiency distributions are carried through in this dynamic setting. Chetty and Heckman use the argument that this macro function could not result from a more conventional aggregative scheme to question the validity of representative firm models. However, the validity of such models should be assessed under less restrictive assumptions. The work of Stoker (1984), briefly summarized in the next section, discusses the validity of such models, in the case where all firms share the same functional parameters. 4.2. Intersectoral Aggregation Using a Parametric Distribution of Variables and Attributes More recent stochastic aggregation models (Stoker, 1984; Lewbel, 1986a, b) have been developed in the context of consumer theory. For sake of consistency, they will be recast here in the restricted terms of the aggregation of production functions. When modeling consumer behavior, it is convenient to use the same functional form for all individual micro functions and to assume that there exists a vector of attributes (observable or not) that capture all individual differences. The vector of attributes is choosen in such a way that the functional parameters of the model can be assumed to be the same for all individuals. 44 These models pose the aggregation problem in terms of the existence of a representative agent which, while describing macro behavior, also captures individual behavior. Lewbel (1986a) explicitly looked for an exact-aggregation macro model that describes the mean of the dependent variable in terms of the mean of the explanatory variables, by imposing restrictions on the functional form of the micro relations and on the distributions. Using similar sources of restrictions, Stoker (1984) investigated the uniqueness of a "true" aggregate model under the representative agent hypothesis, rather than that of a macro model. His analysis is thus more general and less bound to empirical practice than other aggregation frameworks. Extending the above modeling procedure to the producer's context, let the firm's production function be defined by: yl = g(J,a!,6), / = 1, 2,... , F , (4.27) where i/ is the firm's output, a/ is a JV-vector of inputs, af is a M-vector of attributes and 0 is the 5-vector of parameters of the function g, the same for all firms. 3 That is, g is written in augmented form. The attributes a/ capture the differences among firms in technology, managerial efficiency, scale economies, etc. and make the use of a common production function plausible. Lewbel (1986a) originally introduced a vector of prices (or other "macro" variables that are the same for all agents) into (4.27); this allows for optimized relations to be considered. Since it does not add to the analysis, here such a vector is left out to simplify the notation. The population of firms is assumed to be a random sample from a distribution with density PX,A(X> A l^ )> where 5 is an ^-vector of distributional parameters, 5 £ f]f C JRL. The domain of definition of PX,A{X, a|6) is ^(x,a) a f u u dimensional subset of M N + M . 4 Since aggregate data is most often available in terms of averages or totals, it is useful to -reparametrize the density function PX,A{X> A\$) m terms of the mean of the variables and 3 In the consumer's context, there is no distinction between inputs and attributes. Here, the distinction is introduced so that the description of micro production processes matches the characterization used in the aggregative scheme, by contrast with Antle's (1986) transposition. 4 A set n G J R N + M is full dimensional if it is not contained in a proper affine subspace of ]RN+M. 45 attributes /i(s,a)(^) = E,$(X, A). When such a reparametrization exists, via a mapping J one-to-one onto its range, 6* = J(6) = (»M(6),\{6)), (4.28) where /j.(XA)(5) is a full dimensional subset of 1RN+M, Stoker (1984) called the distribution mean full dimensional. The "true" aggregate output is defined as: / i„ = E , ( y ) = f g(x,a,9)pXiA(x,a\8)dxda= G*(8,9), (4.29) where E j ( y ) is assumed to exist and be finite, for all 8 € fl$ and f £ Q j . Assuming that the distribution is mean full dimensional, as in Lewbel (1986a, b), aggregate output can be rewritten as: liy= Gr*{vLM,\,0) =Es*[g(X,A,6)], (4.30) where 8* = ( / i ^ ) , A ) . Let the mean / i (z ,a ) D e the aggregate of the variables and attributes and let h(8*) be a function representing the characteristics of the representative firm. Then, Lewbel (1986a) linked the specification of a representative firm that is exact in aggregation to the existence of a function h(8*) such that: Hy = G * ( / i M ; A , 0 ) = G(»{Xia),h(8*),6), (4.31) where G is the production function of the representative firm and where h(8*) does not depend on fi'^a) • He found exact-aggregation restrictions that apply both to the micro functions and to the distribution of firms. They constitute particular examples in which the micro relations are allowed to differ in specific ways. For all r €E IR", let [r] = r\ + r2 +... rn denote the sum of the components of the vector r and let ((r)) = r\ • r2 • • • rn denote the product of the components of the vector r. Let BQ G IR be the parameter that differs across firms and B\ G 1R N, d2 G I R M the vectors of parameters that are the same for all firms. Theorem 4.2. (Lewbel, 1986b) Assuming i) that 0Q is uncorrelated with (((a/)' ' 1 (tfY2)) across firms, ii) that the absolute moments of order 1, B\ — \ and B2 — \ of PX,A{x, a\6*) exist 46 for all (3i e Qp1, (32 G fi/32, iii) that PX,A{x, a\S) = PX,A{x, a\/j,^,X) and that PX,A{x, a\l,X) exists, 5 let the micro functions be log-log, \nyf = ln/?£ + J f tW] + I/?2W],. or equivalently, t/ = ^{({J)h)){({jf2)), (4.32) then the macro function will also be log-log, ln/z„ = \nB0 + [/3iln//i] + lf32\n^a\, or equivalently, ny = 50((/4X))((Ma2)), (4.33) where B0 = E ( S 0 | / ? i ^ 2 ) - E ( ( ( ^ i ) ) ( ( ^ 2 ) ) | M ( L ; A ) = l ) , (4.34) if and only if the distribution PX,A{x, o.) is mean scaled, i.e. such that Pk1x,k2A{kix, k2a) = PX,A(x; a); where k\ and k2 are constants. Note that because of the multiplicative form of the functional form, the macro constant BQ can play a special normalizing role. An exact-aggregation result could not follow if all parameters were required to be the same at the macro and micro levels. This result, nevertheless, provides a rationale for using macro Cobb-Douglas production functions while assuming a representative firm model, up to a constant (cf. Chetty and Heckman, 1986). Log-linear macro production functions also may be used in valid representative firm models under analogous assumptions. Theorem 4.3. Lewbel (1986b) Assuming i) that /3fQ is uncorrelated with {(e^^+^f2)) across firms, ii) that r PX,A{x> a\S*) is absolutely integrable for r = [x, a) and ((e^^1 +(a^2)), for all /?i G f2/31, /?2 G Qp2, Hi) that PX,A{x, a\8) — PX,A{x, o\l^{x,a)iX) and that PX,A(X, o|l,X) exists, 6 let the micro functions be log-linear, W = Wfo+lPiJ} + lP2</l or equivalently, yf = ^ ((e^1^2)), (4.35) then the macro function will also be log-linear, Pi H ln / i y = ln5 0 + Ifiilixj + IfoVaj, or equivalently, ny = B0{(e^ +M<* » ; (4.36) where B0 = E(5 0|/3i,/? 2) • E{((exl3l+A02))\n[X:a) = l ) , (4.37) 5 See Lewbel, 1986b, p.8-9. 6 See Lewbel, 1986b, p.23-24. 47 if and only if the distribution PX,A[X, A) J S mean translated, i.e. such that P^^x^+Aiki + X, k2 + A) = PX,A(X, a), where k\ and k2 are constants. As the preceding capacity distribution models, Lewbel's aggregative scheme has the important disadvantage of being valid only for two specific functional forms, no matter how popular they are, and for two corresponding classes of distributions. His contribution resides in the definition of wider classes of distributions which could be more likely to have some empirical plausibility. In effect, mean scaling can be interpreted as a.generalization of proportional distribution movement (over time). Lewbel has investigated the empirical evidence on mean scaling. Using distributional data on personal income and manufacturing sales, he found that the hypothesis of mean scaling was weakly rejected. Real mean income and real mean sales have been rising over time, but their distributions remained relatively constant. Stoker's (1984) analysis is more general insofar as it is concerned with providing a theoretical foundation, termed complete aggregation structure, under which a unique cor-respondence between micro functions and aggregate functions can be established. The uniqueness of this correspondence would justify the empirical practice of treating param-eters estimated with aggregate data as though they were individual parameters, that is justify the use of representative agent models. Stoker formally defined the completeness of the aggregation structure S = {C,P}, where C = {g(x,a,9) : 9 £ Ug} is the class of micro functions and P = {PX,A{X, a\b~) 8 £ Qg} is the class of all possible densities, as a property which embodies the conditions for its identification. If two arbitrary "true" aggregate production functions are equal, then they must have the same underlying micro functions. If different micro relations could lead to the same "true" aggregate relation, it would not be possible to identify the aggregate relation, that is, to find its correspondence with the micro relations. Definition 4.1 (Stoker, 1984) An aggregation structure S = {C, P} is said to be complete if P is complete for C, that is, if and only if for all g(x, a,8), g(x, a,9) £ C such that (4.38) 48 for all 8 6 Q$, then g(x, a, 9) = g(x, a, 9) almost surely. (4.39) The precise form of the aggregate function depends on the parametrization 8, but whether the aggregation structure is complete (identifiable) or not, does not depend on 8. With this definition, Stoker made a parallel between aggregation theory and the statistical theory of unbiased estimation and hypothesis testing. He thus established the following main results, concerning broad classes of distribution and functional forms. Theorem 4.4. (a) Stoker, 1984; b) Lehmann and Scheffe, 1950, 1955) a) If P is a class of mean full dimensional densities, then P is complete for CL, the class of linear micro functions. b) If P is a class of the exponential family form, that is it can be written as PX,A{X, a\8) = c(6) h(x, a)exp(7r(8)'D(x, a)), (4.40) where n:Q$ —> IR^ is an L-vector function of 8, D:U^X^ —> IR^ is a one-to-one L-vector function of (x, a), and c(8) = ([ h(x,a)exp{7r(8)'D(x,a))dxda) , (4.41) and if the range of n contains an open set in I R i ; then P is complete for the class Cu of all measurable micro functions. The first result is consistent with previous exact-aggregation results. It also allows patho-logical cases to be singled out; if the distribution families obey a linear probability move-ment, even linear relations would not be complete in aggregation. The implications of the second result are not as straightforward. For example, it implies that the class of normal densities Pjv is complete under quadratic micro functions CQ. However, the corresponding "true" aggregate function is written in terms of the means and variances-covariances of the micro variables. The latter being generally unavailable in empirical analysis, Stoker's result cannot be applied to conventional quadratic macro functions that include only the averages. When the underlying micro functions are non-linear, "true" aggregate functions and macro functions using conventional aggregate data do not coincide. 49 The other empirical problem with representative agent models using common param-eters, such as Lewbel's and Stoker's, is the definition of an appropriate vector of attributes that validates the use of common parameters. However, this problem is probably less im-portant in consumer theory than in producer theory. Data on the consumers' attributes, such as demographic information, are obtained more easily than measures of managerial efficiency. Despite these limitations, these stochastic aggregation models fill an important deficiency of exact-aggregation models by studying the correspondence between micro and macro (or aggregate) parameters. 50 Part II Aggregation over Firms in a Stochastic Environment: Theoretical Framework 51 Chapter 5. Output Aggregation outside Equilibrium - Basic Framework This chapter develops the basic theoretical framework that constitutes the core of the present analysis. The framework is based on a probabilistic interpretation of the characteristics of production of a population of firms. With respect to this probabilistic description, the model parallels the stochastic aggregation models reviewed in the preceding chapter; there are however demarcating differences. First, theoretically the same functional form is used to describe the individual relations, but the functional parameters can differ across agents and thus allow for actual differences in functional forms between firms, as some parameters can take zero or limit values. This modeling strategy deviates from that of representative agent aggregation models (Stoker, 1984; Lewbel, 1986a) without excluding it. Secondly, whereas the Houthakker-Johansen-Sato model treated the parameters (the efficiency coefficients) as stochastic variables and the Stoker-Lewbel approach assumed that the variables were stochastic, here both the functional parameters and the input variables are assumed to belong to a joint probability distribution. The first section introduces this stochastic framework and derives aggregates that are full-information in the sense of capturing all the distributional properties of the micro relations. Previous stochastic aggregation models using a parametric distribution (Stoker, 1984, 1986a; Lewbel, 1986b) imposed specific restrictions on the parametric representation in order to make the "true" aggregate function comparable with the macro function. Here, the recourse to problematic restrictions, as well as the need to estimate parametrically or not the distribution of production characteristics, are avoided. Instead, a decomposition of the "true" aggregate production function in terms of moments of the characteristics is proposed. The decomposition in terms of micro characteristics can be obtained exactly only for polynomials. However, decompositions in terms of functions of the characteristics are also helpful; they allow the sources and causes of potential aggregation biases to be identified. This strategy makes the approach distribution-free in the sense that it applies identically to any distribution, by contrast with the other stochastic approaches. Changes in the size of the population may entail changes in the moments of the distribution of 52 characteristics and are thus be accounted for implicitly. A general exact-aggregation requirement, based on the "true" aggregate relation, is introduced next; when it is not satisfied, an aggregation bias will result. In particular, the output aggregation bias is defined as the difference between the level of output obtained from a given macro relation and the aggregate level of output resulting from the "true" aggregation of the micro relations. In this chapter; output aggregation outside equilibrium is presented first as an illustrative example of the aggregation of economic relations. Output aggregation with optimizing behavior is better modeled in the dual profit function space and is taken on in subsequent chapters. Whether in a optimizing setting or not, the aggregation of economic relations is treated alike-. In the competitive case, the framework simply needs to apply to the optimized relations (e.g. the optimal supply and demand functions). A s a special case of a completely specified macro function, a representative agent's problem is considered thereafter. In the case of an average-representative firm, the output aggregation bias is explicitly derived, exactly for polynomials and approximately for non-polynomial functional forms. When the macro function is not completely specified, that is, when its parameters are left to be estimated, the exact-aggregation requirement establishes the connection between the exact-aggregation macro parameters and their microfounda-tions. This relationship shows that exact-aggregation macro parameters will have identical microfoundations over time, only under certain conditions. When these conditions are not satisfied, the exact-aggregation macro parameters will be unstable if their microdefinitions remain the same over time. Hence, time-drift coefficients are introduced to measure devi-ations in the microfoundations, which in turn are used to evaluate the potential biases in the exact-aggregation macro parameters. Applications to linear aggregation, to quadratic and Generalized Cobb-Douglas pro-duction functions further illustrate these concepts. In particular, many of Theil's (1954, 1971) statistical results concerning the relations between micro and macro parameters are formalized at the population level. The non-linear examples show the limitations of separability conditions and suggest alternatives. 53 5.1. Definition of Full-Information Aggregates Let the industry or the economy be composed of a set of firms, each with a production technology described by a firm-specific production function g*: IR+ —• IR + : 2 / = < r V ) , f=l,2,...,F, (5.1) where j / denotes the level of output, J is a TV-vector of inputs. Throughout most of this section, the time subscripts have been omitted for clarity, however it should be kept in mind that this framework applies to a time-series of cross-section data. The framework, which is designed to facilitate the construction of a "true" aggregate production function, has to impose more structure of the micro production functions than exact-aggregation models do. All micro functions have to belong to the same class of parametric functional form, that is, a finite parametrization of the class of micro technologies is assumed to exist. This is seen as being not unduly restrictive, especially in relation to empirical analysis. If the technologies g? are known, if their number F is finite and if they can be described using a finite number of parameters S', then it is always possible to represent such production technologies using the same parametric functional form, y? = g(xf,9f), by including the technological parameters as arguments of the function. For example, the worst case would take the sum of firm-specific production functions to be the common function. Let 8 = (81,... , 8F), let a = (a\,..., ct1^, ..., a f , . . . , aFF), where c/v •• • , o/^ are the parameters of gf and is the number of parameters of g*. Write gftf) = g{xf,af) in augmented form. Let g{£,8,a.) = 52f=18f g{£,otf), g:JRF+N+s -+ IR, where £ is a vector of dimension N and 5 = Ylf=i & • Then g[J, 9?) = g* (^O f ° r & = {<&> a 0 G JRF+S where S = (0 ; . . . , lf,... ,0) and af = (0,... , a{,... ,afgf, ... ,0). Note that g represents a specific function and not a class of admissible functions. If the gf are unknown, the choice of a flexible functional form makes the assumption that the same type of production function prevails across the industry or the economy econometrically concurrent with current practice. For example, assuming that the firms' technologies can all be represented by a translog function (Christensen, Jorgenson and Lau, 1971, 1973), they can be written in terms of the same function g: 1RN+S —• IR that includes 54 the functional parameters, where 5, the number of parameters, is equal to l+iV+ ^ ( y V + l ) in the case of the translog: N N N Wry) = 4 + xxin^) + i E ^ ^ K M ^ ) 1=1 i=l } = 1 where at = o£-, (5.2) where ^ = {af0,a{,... , a N , a { v ... otfNN). The micro function g is assumed common to all firms, the parameters 9? capturing all firms' technological differences, and each firm's output can be written as: yf = exp[g{xf,0f)] = g{a?l0t) = g{zf), f=\,2,...,F, (5.3) where J = (z/,0^) G IR^"1"5 are called production characteristics. Let a/ £ Q z C IR+, where flx is the domain of definition of the factor inputs and let af G Qa C IR 5 , where fia is the domain of definition of the technological parameters. Then fij == (r2j;;na) is the domain of definition of the production characteristics J. The function g is called the augmented production function and is identical for all firms. The production characteristics (z1,...,^) G IR F x IR^"1"5 can be interpreted as realizations of a random vector Z G I R i V + s \ The random vector Z is distributed according to a joint density Pz{z) of factor inputs and functional parameters. 1 In general, the distribution of inputs is assumed to depend on the distribution of functional parameters. Contrary to common practice, the density pz{z) is not parametrized a priori; furthermore, there will be no need to estimate this density (parametrically or not). 2 Hildenbrand (1981) has expressed doubts about the empirical plausibility of using a simple parametric distribution to represent production activities. However, the more compelling argument against the use of a parametric representation is that it permits the "true" aggregate production function and its macro version to be compared only under very stringent assumptions, as explained below, are assumed to exist. 1 For convenience, the distribution Pz{z) is assumed to be continuous. However, a discrete distribution could be substituted into the following derivations. For a discussion of the discrete versus continuous descriptions, see Hildenbrand (1981), Stoker (1984). 2 See Tapia and Thompson (1978), Devroye and Gyorfi (1985) for the nonparametric estimation of density functions. 55 The first and other higher and central moments of the density pz{z) are assumed to exist. 3 The first order moment, or mean, is defined as = E(Z) = / z pz{z) dz, (5.4) where the integral of a vector is denned as a vector of integrals, so that fj.z = (fj,Zl, ... , VxN, •• •, Hes) e MN+S. The aggregates of the micro characteristics that will enter the macro function are also defined in terms of functions of the distribution of Z and are denoted by ip2(Z). This vector of aggregates will be of dimension N+S, ^Z{Z) = (yjZl(Zi), ... ,ibZN (ZN+S)) , if each characteristic's aggregate depends only on the corresponding characteristic. However, there may be instances (Nataf, 1948; Gorman, 1968a; Lau, 1982) where the characteristic's aggregates involve relations between the factor inputs and the functional parameters; then the dimension of yjz(Z) may be different from N+S. More commonly, the characteristic's aggregates will be defined as a function of the first order moment of Z, ipz(Z) = /i[E(Z)], or as the expected value of a function of the characteristics, ipz(Z) = E[A(Z)]. In empirical analysis, this type of definition conveniently leads to commonly used input aggregates: the totals and averages. For example, aggregate capital or aggregate labor would typically be defined as the number of firms times the expected value of capital or labor across firms and evaluated as the sum of firm-specific capital or labor: ipk{K) = F • ~E{K) and ipi{L) = F • E(L), where K and L would be the ran-dom variables whose realizations are the firms' levels of capital and labor. Note, however, that this notation does not exclude the possibility that different types of aggregates may be used for different inputs. The stochastic framework provides a theoretical interpretation for the conventional summing up practice. On the basis of the density of the characteristics, "true" output aggregates that capture all the distributional properties of the micro relations, can be defined similarly, provided that the integral is well-defined, in terms of totals as: y = F-E[g(Z)}= F • I g(z) pz(z) dz, (5.5) Jnz 3 The functional form of<<7 actually determines which precise central and non-central higher moments are needed in each case. 56 where g: 1RN+S —• IR, so that \j € IR; or, in terms of averages, as: M » = ! = E ( Y ) = / s(z) P i r ( * ) (5.6) where Y is a random variable with realizations (y1,... , yF) and fiy €E IR. In terms of observed values, these aggregates are the usual statistics, the totals and averages, used in non-stochastic exact-aggregation models, such as Fisher's (1965, 1969a, 1982): F F Y = ]Tj/ and F=- ? Xy. (5-7) /=i /=i Note that this approach presumes that products are non-differentiated (homogeneous). In cases of aggregation across an economy, constant dollars can serve as the common non-differentiated unit of account. 4 A t this point, however, no assumptions about product prices are made. 5.2. Moments Decomposition of the Output Aggregate Stochastic aggregation models have used expressions such as expected output (5.6) and have attempted to solve them. In the Houthakker-Johansen-Sato model, the production function (or capacity output) was intertwined with the density function . 5 The integration problem then was solved simply by the choice of an appropriate density function. In the particular case where the density of capacity output follows a Pareto distribution, if all firms operate with fixed (or variable) factor input proportions, then the "true" aggregate (and macro) production function will be Cobb-Douglas (or C E S ) . In a different but related stochastic framework, Aitchison and Brown (1957) and Van D o o m (1975) have considered univariate log-log models, such as: ln(j /) = ln(tf) + 41n(a/) ; where j / is a dependent variable, a/ is an independent variable, b is a coefficient that does not vary across agents, ln(</) is a constant. By assuming that the distribution of X is log-normal, with the variance of ln(X) constant, these authors have explicitly aggregated log-log micro models into a log-log macro model. More recently, Maccini (1984) invoked similar assumptions to construct macroeconomic behavioral relationships of price and output decisions with solid microfoundations. He supposed that the variables 4 See section 8.1. 5 See section 4.1. 57 and parameters of the micro relations were distributed log-normally and independently of each other. Such distributional assumptions have been somewhat relaxed by Lewbel (1986b). Lewbel showed that "mean scaling" is a necessary and sufficient restriction on the distribution of X for the exact aggregation of log-log micro relations into a log-log macro relation. 6 Nevertheless, these developments suffer from the important disadvantage of being valid only for very specific functional forms and probability distributions. From the viewpoint of the present setting, the approach that attempts to restrict the family of allowable distributions of micro characteristics to permit the aggregation of given functional forms has limited value for yet another reason. Since the stochastic elements include the functional parameters, it seems more difficult to hold a priori assumptions on a precise distribution family. This task may be even more formidable if, given that the assumption of stochastic independence between the characteristics is likely to be empirically refuted, the density pz[z) ls undoubtably multivariate. More importantly, the empirical verification of such assumptions would require extensive micro data not available at the present time (Diewert, 1980). Lewbel's results, as well as Stoker's (1984, 1986a) developments, are based on a parametric representation of the distribution of characteristics which is restricted in order to allow the comparison of the "true" aggregate production function and of a macro function of the usual aggregates. In Stoker (1984) and Lewbel (1986a, b), the distribution is assumed to be mean full dimensional, that is, to be parametrized in terms of the mean of the variables. In Stoker (1986a), the parametrization has to be such that an invertible mapping between the parameters and the mean of the variables exists. 7 In essence, with these modeling strategies, the aggregation problem is partially transfered onto the parametric space of the characteristics' distribution and embedded in the chief assumptions. The precise circumstances under which aggregation biases will be shown to exist, are when the mean of the characteristics provides insufficient information about the micro relations (or their parametric representation) or, when such an invertible mapping 6 See section 4.2. 7 Let 8 parametrize the density pz(z) = pz{z\8), and let \iz = E(Z|5) = H(8). Then, Stoker (1986a, p.174) assumes that 8 = H~x(fiz) exists, implying that fiy = / g(z)pz(z\8) dz = G*(8) = G(# - 1 ( / /Z)) = G(fiz). The latter is equivalent (or even stronger than) to an exact-aggregation requirement, unless z is allowed to include squares or cross-products of the variables. 58 does not exist. In order to resolve expression (5.6), this paper proposes an alternative non-parametric approach to the description of the distribution of characteristics. That is, to write aggregate output in terms of first and other moments of the characteristics that can be evaluated em-pirically and that have an interesting intuitive appeal. In the search for exact-aggregation restrictions, this type of moments approach has the advantage of utilizing both the func-tional forms of the micro relations and the distributions of the micro variables. It thus provides more options for restrictions than non-stochastic exact-aggregation models and allows for a wider variety of functional forms than current applicable stochastic approaches. The present moments approach aims at relating the "true" aggregate production func-tion to a macro function defined in terms of the first moment (or mean) of the charac-teristics. However, since the micro relations g(z^) typically exhibit some non-linearities in the characteristics, the expression for expected output (5.6) may also comprise higher moments (m > 1) of the characteristics as well as other functional forms not directly re-lated to moments of the characteristics. These higher moments may take the form of a functional parameter multiplying an input variable, of cross-products of input variables or of powers of input variables. Exact-aggregation restrictions are sometimes designed to eliminate some of these higher moments. For example, the Leontief-Sono separability conditions disallow the sum of cross-products. If the augmented production function g(z^) is a polynomial, it is possible to express its expectation E[g(Z)] in terms of first and higher central moments of the characteristics. The former are often used to define the characteristics' aggregates. The latter provide a measure of the interdependence of the characteristics, which is akin to a population measure of "separability". If the characteristics are distributed independently, covariance terms vanish. Assuming for simplification that the random variable Z £ IR 3, let g{Z) be 3-linear: l g(z)= £ (z1yHz2)iHz3ys, (5.8) •l .'2 "'3 = 0 then l l E[g(Z)]= Yl E [ W ^ ) * W » ] = £ ^ , 3 , (5.9) • l > » 2 ' » 3 = 0 ' l . , 2 ' , 3 = ° 59 where E[(Zi)'i (Z2)l2 ( Z 3 ) ' 3 ] = A**^  »2 »3 is called the moment of order >i+i'2+»3 of the random vector (Z\,Z<i,Zz). Similarly, central moments are denoted by m » 1 »2 v3 : m , - l V s = E p i - MIOO)'1 (^ 2 - Moio)'2 (3» - Mooi)'3]- (5.10) In particular, the third central moment can be expressed as m m = E[(Zj - /iioo)(^2 - MoioH^s - Mooi)] = C(yr[Zu Z2, Zs) — M m ~ MiooMon — M010M101 — MooiMno + 2 M100M010M001 (5-H) = M m - Mioo^on - Moio^ioi - Mooimuo _ M100M010M001, since /j,011 — mon + MoioMooi> Mioi = "Hoi + M100M001 and M110 = ™no + MiooMoio-Thus, E[Z1Z2Z3] = n n i =E(^i)E ( Z 2)E(Z3)+E(Zi)moii + E(Z2)fnioi + E(Z8)miio + n»iii . (5.12) More generally, let ejt = (0,1^,0), k = 1,2,3, be the canonical basis of IR3; denote Mtji^ig by fi[ii, i2, iz] for clarity. Then any moment (>i+»2+»3 < 3) of Z 6 IR3 can be expressed as: 3 3 1 3 3 M[*I,*2,*3] = M [ E 4 e * ] = JJM + . ] P mit^Jkek} M [ E ( 4 ~ J*)^], (5.13) 3 3 3 3 where M[0,0,0] = 1, p\Y,3kek] = 0 if X^ijfcejt < 0 3 and m[Yljkek\ = 0 if < 13-fc=l A=l A=l fc=l The expectation of g(Z) then becomes: k=l /c=l ' (5-14) E[ 5 (Z)] =g[E(Z)]+ ( E ™[ij*c*]M[i(**-y*)e*] •i,'2.'3=° h'h-h=° = g[E(Z)}+M[g(Z)}. The right hand side terms, excluding ^[E(Z)], are called the higher moments of expected output M[<7(Z)]. Section (5.6.2) provides a more readable illustration of this expression. 8 8 Note that despite the similarly of expression (5.14) with Jensen's (1906) inequality on convex functions, the latter result or Lau's (1978) extension to production function homogeneous of degree k does not apply directly. The usual concavity in factor inputs of the production function applies to the first N arguments of g, - the factor inputs - , but does not extend to the S others, - the functional parameters. Because the augmented production function g includes the functional parameters of the production function </, its global curvature is undetermined. 60 Decomposition (5.14) generalizes to any A:-linear function and can be extended to any polynomial, writing the n-th powers of a variable as a n-product. Non-polynomial functional forms, that are continuously differentiable, could theoretically be approximated with chosen accuracy by polynomials, using Taylor series expansion for example, and the above decompostion could be applied to the approximation. It is, however, easier to take the expectation of the series expansion of the production function about the mean directly as will be shown in the average-representative firm's approach. 5.8. Exact-Aggregation Requirement and Resulting Biases Exact-aggregation models are typically set in a non-stochastic environment and have fewer options to impose aggregation restrictions as they consider restrictions on the micro relations only. The ensuing aggregation restrictions may thus be too stringent or insuffi-cient when transposed in a stochastic environment. Most exact-aggregation models look for restrictions on the functional form of the micro relations, the 9? in this framework, such that a macro production function G defined as a function of aggregate micro variables yields the aggregate level of output: y = F-ny=G{MZ)), (5.15) where ^ Z(Z) € IR/'4"5 is an aggregate of the micro characteristics, such as denned in (5.4), and G:JRN+S -> IR. If, more generally, rp,(Z) & IRM, then G: MM -* IR. If a specific macro function is assumed a priori, then the exact-aggregation require-ment (5.15) serves to test the correspondence between the macro relation G and its mi-crofoundations. If the macro function is not known, which is generally the case, it may be assumed that the augmented macro function can be described in the same way as the augmented micro functions are: G = g. Any micro technology can be represented by the augmented production function g; it is interpreted as a particular draw from the population of technological parameters. The above assumption then supposes that the choice of macro technology is simply another draw and should not be restrictive. It is more general then the representative agent view of macroeconomic modeling, since the behavioral parame-ters can differ across agents and at the macro level. In macroeconomic modeling (Lucas and Rapping, 1969; Sargent, 1978; Galeotti, 1984), the representative agent conveniently 61 models the behavior of a typical agent which, if replicated n times, will mimic aggregate behavior. Identical parameters for all agents are usually assumed. The exact-aggregation requirement (5.15) is expected to hold for all production pos-sibilities in the Klein-Nataf context (Klein, 1946a, b; Nataf, 1948), while it is expected to hold only at the optimum in the Fisherian approach (Fisher, 1965, 1969b); Blackorby and Schworm, 1984, 1988). In the latter case, the efficiency conditions may impose further restrictions on the functional parameters #A The exact-aggregation requirement and the ensuing aggregation restrictions are then defined in terms of the profit or cost functions (Gorman, 1968a; Blackorby and Schworm, 1984, 1988). The application of appropriate derivations to these functions follows in the next chapters. Some exact-aggregation models use an aggregation requirement that is different from (5.15). Notably, the Klein-Nataf exact aggregation problem defines the output aggregate as part of the solution. 9 Also other models (Gorman, 1953; VanDaal and Meerkies, 1984, chap. 2; Blackorby, Schworm and Fisher, 1986, sections 3-4) search for conditions under which the economy's production possibility frontier can be written in terms of input aggregates and may not be concerned with output aggregation. However, if the exact-aggregation requirement is defined by (5.15) and fails to be satisfied, a total output aggregation bias can be denned as the difference between the "true" aggregate production function and a macro production function of the characteristics' aggregates: Note that the augmented macro function may be fully specified, as in the average-repre-sentative firm approach below, or may not, as in most exact-aggregation models where the macro parameters may be unknown. If the macro function G and the characteristics' aggregates rpz(Z) were known exactly, the aggregation bias could be evaluated directly using expected output y = F E (F) . More generally, a moments form for the aggregation bias may be obtained using the moments form derivation or approximation of expected output. The exact-aggregation requirement (5.15) is equivalent to requiring that the bias (5.16) (5.16) 9 Cf. Chapter 2 62 be zero. Thus, the separability conditions of exact-aggregation models may be rexamined by considering the bias By expressed in moments form. These conditions entail the dis-appearance of covariance terms between factor inputs and reduce the aggregation bias. However, since the separability conditions do not require that the variance or higher mo-ment terms vanish, in a stochastic framework they no longer entail the annihilation of the aggregation bias. Furthermore, they are no longer necessary conditions for exact aggrega-tion since the stochastic independence of factor inputs would be a sufficient condition for the annihilation of the covariance components of the aggregation bias. 1 0 5-4- The Average-Representative Firm Approach: a Completely Specified Case A representative agent's problem is considered next: Is there a representative agent whose behavior can be derived from the micro behavior of non-identical agents? The latter is described probabilistically by the density pz(z). The representative agent is assumed to represent macro behavior but its behavior is derived from the micro behavioral parameters. In the average-representative firm approach, the augmented macro function is assumed identical to the augmented micro functions, G = g and the characteristics' aggregates are chosen to be their first moment; the characteristics' aggregates vbz(Z) = (fix,fig) provide then a complete description of the macro function. More specifically, this average-repre-sentative firm's technology, which is also the macro technology, is an average of all firms technologies, = fJ-e- It corresponds to some extent to Marshallian concept of a representative firm: "a representative firm is in a sense an average firm" (Marshall, 1920, p.318). This case is of special interest because the parameters' aggregates (the macro pa-rameters) are then functions of the micro parameters only. This feature seems appealing since it corresponds more to an intuitive notion of a representative agent than a weighted average scheme in which the weights depend on the micro variables. 1 1 It also implies macro parameters that are constant over time if the micro parameters are, which is a desirable property. However, as the following theoretical and empirical results show, the average-representative firm bias may be quite important. 1 0 See the quadratic production function example of section 5.6.2. 1 1 See the linear aggregation example of section 5.6.1. 63 The exact-aggregation requirement and the aggregation bias are stated more ade-quately in terms of averages: fiv = g(ji,)' or V[g{Z)] =g[E(Z)} (5.17) implying B„y = ny - g{fiz) or B„y = V[g{Z)} - g[E{Z)}. When g is linearly homogeneous in its arguments, the total output aggregation bias and the average output aggregation bias coincide. If g is a polynomial, the average output aggregation bias is given by the higher moments of expected output: BH =M[g(Z)}. (5.18) If g possesses a non-polynomial functional form but is continuously differentiable, the following approximation to the aggregation bias may be used. Let the covariance matrix CpXF denote the second-order central moment of the dis-tribution of z, C = E[(Z-ii,)(Z-»,)']=•[ {z- fiz){z- fiz)'Pz(z) dz, (5.19) Juz where the integral of a matrix is defined as a matrix of integrals. Let H be the Hessian matrix of second order derivatives of g{z) with respect to z., Keller (1980) has shown, using a second order Taylor approximation to the function g(z), that the aggregation bias in average output can be approximated by: B„y « itr(/?C7), (5.20) where tr is trace operator and H is the Hessian matrix evaluated at iiz. 1 2 Given the skewness of presumed distribution of firms across the economy, a better approximation would include third order terms. These bias results imply that unless the micro relations are such as to entail the disappearance of the higher moments of expected output, the average-representative firm's approach does not have exact microfoundations. Theorem 5.1 If the macro function is chosen to be G = g and the characteristics' aggregates are chosen to be the first moments, ipz(Z) = \iz, then the aggregation bias will be 1 2 Keller studied a general representative agent's problem. 64 null if and only if the sum of higher moments (m > l) of the characteristics in the average output aggregation bias vanish. Proof: Express the production function g evaluated at z in terms of its Taylor series expansion: y = g{z) = g{z) + (z - z)'Vg+ \(z - z)'H(z - 5) + • • • , (5.21) where Vg is a vector of first order derivatives of g(z) and H is the Hessian matrix, both evaluated at z, and assume that the matrix of third order derivatives is bounded. Substituting (5.21) into (5.6) gives: M » = / [g(^ + (z-^Vg+^(z-z)'H(z-z) + ---}pz(z)dz r r (5-22) = 9(z)+Vg (z- z)Vz{z)dz+\%vH (z - z)'(z - z)Vz(z) dz+ • • •, since the trace operator is associative. The average output aggregation bias is then given by 5 ^ ( 3 = My - = (M* - z)'Vg+ l t r ^ [ E ( Z 2 ) - 2zV , + 1 z] +••• (5.23) Clearly at z — ybz(Z) — fj,z, the first order term vanishes, the bias is then identically equal to the sum of higher order terms, o Note that this assumption may be more or less restrictive than requiring that the micro relations be linear. Since linearity in factor inputs a/ is generally equivalent to 2-linearity in the production characteristics it does not guarantee the elimination of all covariance terms. On the other hand, the elimination of higher order terms may result from properties of the distribution of Z allowing a non-linear production function. 5.5. Microfoundations of Exact-Aggregation Macro Parameters By contrast with the average-representative firm approach which specifies both macro factor inputs and macro parameters, exact-aggregation models generally specify aggregates for the factor inputs only. For example, the macro factor inputs may be taken to be V'i(^) = F • nx = X, evaluated as the sum of the inputs. The macro parameters are required to satisfy some aggregation restrictions that often follow from restrictions on the micro 65 relations, but they are otherwise are left unspecified: = 0R, where 9R £ IR 5 denotes an unknown vector of functional parameters, some of which are restricted. Of course, if an exact-aggregation restriction requires that a functional parameter be the same across firms, the parameter becomes completely specified. In the present setting, if the macro parameters are left unspecified, exact aggregation will require that the macro parameters 0M lead to no aggregation bias or equivalently that they belong to the zero-bias S-manifold described by the following equation: 1 3 G{X,9M)-y = 0. (5.24) As pointed out by Richmond (1976) in a slightly different context, the aggregation problem may be seen as an identification problem. Thus, the choice of a particular vector from this manifold may be helped by imposing the restriction that the exact-aggregation macro parameters be identical over time. This restriction may be thought as being ad hoc, but is common enough to justify its investigation. An alternative line of research could assume time-varying macro parameters. If the macro parameters are assumed constant over time, as they most often are, the relevant aggregation issue is whether such macro parameters possess stable microfounda-tions. Time-subscripts and indexes are now introduced. If the macro inputs are taken to be the sum of micro inputs, the description of the zero-bias 5-manifold may be rewritten as: G{Ft-ixx{t),eM) - Fr(iy(t) = 0. (5.25) Assuming that G is continously differentiable and that d[G(Ff [ix(t),9M) -Ft- ny(.t)] / <9l{Ff f^x(t)i FfHy{t)) is non-singular, then by the implicit function theorem 9M can, locally, be solved as a function of (Ft • nx{t), Ff (iy(t)). 1 4 Let the solutions to (5.25) be called 8M(Ft- fix(t), Ft- fj,y(t)). Then the following result generalizes one of Theil's remarks (1971, p. 570). If both the inputs and output aggregates vary over time, a particular solution to (5.25) will vary over time, unless the variations in the output aggregate are proportional to the variations in output resulting from changes in input aggregates. Note that changes 1 3 In general, in an n-dimensional space, a geometrical structure of dimension m < n is called an m-dimensional manifold. A surface is then a 2-manifold and a curve is the degenerate case of a 'unifold'. 1 4 Cf. Mas-Colell (1985), p.21. 66 in the parameters' aggregates, ^e{t), - the structural model-change effects (Stoker, 1985) - will appear as variations in the output aggregate Ft- fJ-y(t). Theorem 5.2 The exact-aggregation macro parameters 9'^(Ft-nx(t), Ft-ny(t)) =9M(IJ.X, fiy, Ft, t) will be constant over time if the inputs and output aggregates are constant over time Ft • /xz(t) = px and Ft • Py(t) = py for all t = 1,2,T, where px and py are scalars. Proof: Follows from the definition of exact-aggregation macro parameters, o Theorem 5.3 a) If G is homogeneous of degree A, in the factor inputs z,- separately, the exact-aggregation macro parameters 9M(Ft • fix(t),Ft • ixy(t)) will be constant over time if and only if the following proportionality requirements are met: Ft; (nXl(t),... ,(J,xN(t)) = {ktPxi:-- ,kmPxN) and Ft-ny(t) = Ktfry, t-l,2,...T, (5.26) where Kt = and where \ix and fiy are constant over time. b) If G is strictly monotone and homogeneous of degree A in all factor inputs, the exact-aggregation macro parameters 0M(Ff-fix(i), Ft-fJ-y(i)) will be constant over time if and only if the following proportionality requirements are met: Ft • ( M I X {t),... , fJ-xN{t)) = {hpxv ••• , kPxN) and Ft • fj.y{t) = kfpy, t= 1,2,... T, (5.27) (all factor inputs need to move in the same proportion over time), where px and py are constant over time. Proof: a) For the exact-aggregation macro parameters to be constant over time, the zero-bias 5-manifold (5.25) has to lead to the same solutions over time. Assuming two time periods t = 0,1, let px(0) and ny(0) be the inputs and output aggregates at time 0 and express the inputs and output aggregates of period 1 as a proportion of the aggregates of period 0: Fx • ( ^ ( l ) , . . . ,»XN{1)) = F0 • (fciiM.JO),... , W-jvtO)) (5.28) and F1-fty(l) = htF0-ns(0). (5.29) 67 In period 1, the zero-bias 5-manifold is now described by: G[knF0 • ^ ( 0 ) , ...,kmF0- »ZN{0),9M) -hiFo- /i„(0) = 0. (5.30) Since G is homogeneous of degree A, in each z,, equation (5.30) can be rewritten as : « i G^Fo • nT{0), 9M) FQ- M „ ( 0 ) = 0, (5.31) where ACI = F I ^ i • I n period 0, the zero-bias 5-manifold is described by: G(iVMz(O) ,0 M ) - F0 • /*„(0) = 0. (5.32) Clearly, the period 1 5-manifold will be identical to the period 0 5-manifold if and only if hi = KI. b) Sufficiency is proven using the same demonstration as in a) with the corresponding proportions (5.27) for inputs and output aggregates between period 0 and period 1. To prove necessity, assume that 6M are solutions to the period 0 S-manifold. Obtain the expression for the output aggregate in period 0 in terms of the macro function from (5.32), substitute it into the expression for the output aggregate in period 1 as a proportion of the output aggregate in period 0 (5.29) and get: Fi • /xy(l) = ^ G ( F 0 • Mz(0),0M) . (5.33) If 9M is also a solution to period 1 S-manifold, then using (5.33), G^Fi • Mx(l), 0M) ~ h1 G(F0 • /xx(0), 6M) = 0. (5.34) Since G is homogeneous of degree A in the factor inputs, equation (5.34) can be rewritten as: G[Fi • M*(1),0M) - G[h\/XF0 • Mx(O),0M) = 0. (5.35) Since G is strictly monotone, (5.35) implies Fi • [ix(l) = h\^Fo • (iX(0). o Under constant returns to scale, it suffices that the inputs and output aggregates vary proportionally over time for the macro parameters to be stable. This result explains the conclusions of some of Fisher's simulation experiments. Simulating the aggregation of Cobb-Douglas production functions, Fisher (1971) found that a macro Cobb-Douglas 68 production function, where labor is the variable input, would fit input-output data well if labor's share of total output happens to be roughly constant over time. Since the predictions retained were generated by estimates imposing constant returns, Theorem 5.3b applies. Exact aggregation implies that the macro parameters be a function not only of the micro parameters but also of the micro variables. 1 5 Thus, unless the population of inputs changes in proportion with its productivity, exact aggregation also implies that the macro parameters will be unstable over time if their microfoundations are defined identically over time. Changes in allocative efficiency or technological changes are incompatible with macro parameters having exact microfoundations that are consistent over time. In Stoker's terminology, the former changes in the nx(t) are referred to as distribution effects (1982) whereas, the latter changes in the fj.$(t) to appear as changes in the fiy(t) are called structural model-change (Stoker, 1985) or behavioral (Stoker, 1986b) effects . If the time-consistency of the microdefinitions of the macro parameters were not an issue, it would be possible to find solutions to the zero-bias S-manifold (5.25) that are con-stant over time, but whose microdefinitions are time-varying. Let k{i) = (ki(i),... , &jv(0) and h(t) be some functions of time such that: Fr nz{k{t)) = px and Fr Hy(h(t)) = py, t = 1, 2,... T, (5.36) at each point in time, then the solution 9M (jlx, py) is constant over time. The time-drifts k(t) and h(t) provide measures of the bias in the parameters in terms of deviation's from stable microfoundations. The above type of solution 9M(jIZ!jZy) is likely to be found in empirical analysis. Typically, macroeconometric models are of the form: yt = g{xt,0M) + et, t=\,2,...,T (5.37) where (y~\,..., yr, x\,..., X~T) are observed values of output and inputs aggregates, and et is an error term. The estimates of the macro parameters are calculated from the movement of the joint distribution of output and input aggregates over time: 6M = e{n,... ,yT,x1:... ,xT). (5.38) 1 5 See the linear aggregation example in section 5.6.1. 69 Because 0 is obtained by curve fitting (5.37), which is the empirical counterpart of the exact aggregation requirement (5.15), the latter will be satisfed almost exactly and, notwithstanding the fitting error, the empirical aggregation bias will appear as unstable microfoundations as in scheme (5.36). Thus, currently estimated macro parameters will generally not lead to a total output aggregation bias, but their microfoundations will not be stable over time and they will be sensitive to policy changes as shown below. 5.6. Applications The following three subsections contain applications of the proposed stochastic frame-work. These applications provide explicit expressions of the full-information aggregates and of their moments decomposition. For the average-representative firm, biases in mo-ments form are obtained under various stochastic restrictions. Explicit examples of the microdefinitions of exact-aggregation macro parameters, as well as implicit forms of the time-drifts coefficients are presented. Also, the Klein-Nataf exact-aggregation solutions are given to show how demanding they are. Each of the three sections uses a different functional form - linear, quadratic and Generalized Cobb-Douglas - for the production function, thereby illustrating the com-plications brought by increasing levels of complexity in functional form. The section on linear aggregation rexamines some of Theil's results in the simplifying light of the pro-posed stochastic framework. In the section discussing the quadratic production function, separability restrictions are imposed to show how they reduce the output aggregation bias of the average-representative firm. 5.7. Linear Aggregation A number of linear aggregation problems studied by Theil (1954, 1971) can be cast in this framework where they are easily interpreted. For illustration, a simple linear production function of two inputs is used. Let the production function be: yf = g(kf, 1^ a',/?') = aW + f=l,2,...,F, (5.39) where j / denotes the level of output, kf and V~ are respectively capital and labor inputs, a-70 and 8? are the functional parameters. Let Y,K,L, A, 8 be the random variables whose re-alizations are (y1,... , y^^k1,... , kF), (I1,..., lF), (a1,... ,aF), (81,... ,3F), respectively. In a first analysis of linear aggregation, Theil (1954, 1962) studied an average repre-sentative agent's problem. Theil asked under what conditions was the following equality satisfied: y= g(k,T,a,0), (5-40) which corresponds to G = g and yjz(Z) = \iz in the notation (5.15). Given (5.39), the average output aggregate is: E{Y) = E[G{K,L,A,B)\ = E(AK+BL) = ~E(AK)+V{BL), (5.41) The linearity of the functional form plays a crucial role in allowing the inputs to be separated in the evaluation of this expectation. The expectation of a product of random variables can always be decomposed as follows: E{AK) = V(A)E{K) + Cov(A, K), (5.42) where Cov(A,K) is the covariance of A and K and vanishes when A and K are distributed independently. Using this decomposition, the output aggregate can be expressed in terms of the characteristics' first and central moments: E(7) = E(A)'E(K) +COY(A,K) + E(8)~E(L) + COV{8,L), (5.43) and the average output aggregation bias is given by: = Cav{A, K) + Cov(S ; L). (5.44) Proposition 5.4 A sufficient condition for exact output aggregation, in terms of averages as in relation (5.17) [or, in terms of totals as in relation (5.15)/, to hold with G = g, where g comprises only first order terms in factor inputs and parameters and rpz(Z) = (fiz,fj.g) [or ipz(Z) = (Ffj.z,fj.g) j is that the covariance between x and 0 be null. (Their variance can be non-zero and no conditions need to be imposed on zero order terms.) Clearly, when A = a and 8=0 are the same for all firms, their distribution is independent of the input variables and, E(i?) = 67 and E(S) = 8. The output aggregate is then written as E (Y) = E(>l)E(Jf) +E(B)E(L) = aE(K) + /3E(L), (5.45) 71 which is estimated as y = ak + pT (5.46) satisfying (5.40). This condition is trivially verified, in this model, if the firms have the same technologies or equivalently if both capital and labor are efficiently allocated across firms. Another solution proposed by Zellner (1969) uses a random-coefficient model. Zellner assumed that the coefficients are distributed randomly across firms around a mean: A = a + 8a, 8=P + 8p, (5.47) where 8a and 8^ are random variables with E(<5a) = E(5g) = 0 . If the coefficients are assumed to be distributed independently from the inputs, the output aggregate becomes: E(r) = E(6T+ 8a)E{K) + E(/3 + fy)E(L) (5.48) = aE(tf) + jSE(L), yielding Zellner's exact aggregation result. Conversely, if this assumption is relaxed, for example the variance of the random terms could depend on the magnitude of the factor inputs, E[82} = a2 (K)2 and E[<5|] = a^(L)2, then the output aggregate would be: E ( F ) = a"E{K) + #E(L) + Cov (6 a , K) + Cov(fy, L) (5.49) and an aggregation bias remains. Theil (1968, 1971) further developed the random coefficient model into the so-called convergence approach. Summing (5.39) over firms and dividing through by F, he obtained: which can now be interpreted as the evaluation of, ^ w - ^ + l r 1 ^ - <5-51> Theil showed that if the factor inputs are assumed non-stochastic, (5.50) can be estimated consistently. He underlined that this assumption is "not innocent at all" (Theil, 1971, p.572), since it implies the stochastic independence of the factor inputs and the way in 72 which production reacts to the factors (the coefficients). Without this assumption, the coefficients of kt and 4 in (5.50) are obviously not constant except when the corresponding factor inputs move proportionally. More importantly, since the association between the macro coefficients and the micro coefficients and variables in (5.50) is arbitrary, this approach suggests that the micro relations can lead to a number of exact macro relations. Let the factor inputs aggregates be defined as the .usual averages, tpx(Xt) = Hz(t), but let the functional parameters' aggregates be unknown and denoted by a M and B M, so that i>z{Zt) = (/^(t),m(t),cxM, 0M). For G = g, the aggregation bias is given by: BIMy(t) = -E(AKt) + E(8Lt)-aM»k(t)-3M»l(t), (5.52) and the exact-aggregation macro coefficients are described by the surface: E(Kt)a M + E{Lt)B M - [E{AKt) + E{BLt)} = 0. (5.53) Theil's solution a M = E{AKt)lE(Kt) and B M = E(BLt)/E{Lt) obviously belongs to this surface. A more general family of solutions is: M h[E{AKt)]+82[E{BLt)\ (1 - ft)[E(AKt)} + (1 - S2)[E(BLt)} A = E W ~ — A N D " = m —' (5-54) where 0 < Si < 1 and 0 < S < 1 are scalars, Theil's solution being ft = 1 and ft> = 0. The macro parameters a M and 3M will vary over time as E(Kt), E(AKt), E(Lt) and E(5Lt) vary. Assuming that there exist some macro parameters a and 8 that are stable over time, their microfoundations can be written as „ ft(i)[E(i?^)] +82(t)[E(BLt)} ~ (1 - [ E ( ^ A i ) ] + (l-*2(t))[E(3Lt)] a = m) ^= m ; (5.55) for all t= 1,2,... , T, where the coefficients ft(t) and 82(t) measure the time-drifts in the microfoundations of the assumed" macro parameters. Clearly, if the population of inputs and output is stable over time, the ft(i) and 82(t) will be identical for all t = 1,2,... , T. Finally, "perfect aggregation" results can be obtained by using a definition of the characteristics' aggregates specific to the functional form aggregated, ip{Z) = (E(Gi-Xi), 73 ~E(@NXN), 9f,...,0N), a s proposed by Nataf (1948). Choosing the characteristics' aggregates to be: F F k* = j^^2afkf a n d pf = j-t2pflI> (5-56) / = i 7 / = i then Theil (1962) rewrote equation (5.41) as a macro function, y= G(k*,l*,6,i) =8k* + ^ l*, (5.57) without aggregation bias. Since the construction of these aggregates requires perfect knowledge of the micro relations, it has limited applicability Example 5.7.1 Quadratic Production Function The next level of complexity in functional form is represented by a quadratic function. It provides a more elaborate example of the decomposition of expected output in terms of first and central moments of the characteristics. The effects on the average output aggregation bias of restrictions on the distribution of the characteristics are illustrated. Separability conditions are imposed on the production function to show how they reduce the average output aggregation bias without annihilating it completely. Let a/ be an iV-dimensional vector of factor inputs and define the quadratic production function as: N N N ,=i ,=i j=i where o/- = a{:, for all i, j = g(a/,o/) f f f=l,2,...,F, (5.58) where j / denotes the level of output and where af = (o/0,a^,... ,oJN, o/n,... a^NN) are the functional parameters. Let Y be the random variable, X E IR^, A E ]R1+N+ 2 N ( N + 1 ) be the random vectors whose realizations are (y1,... , yF), (x1,... , x^), ( a 1 , . . . ,aF), respectively. 74 The average output aggregate can then be written as: N. N N E(Y) = E(A0) + EE(^1.') + E E WiiXiXj) i=i 1=1 y=i N N N = E(A0) + E [WMXi) + Cav(Ai, X{)] + E E [E(Ay)E(*,)E(^) i=l i=l ;=1 + V(Aij)Cov(Xi, Xj) + E(X , ) C o v ( ^ , Xj) + V{Xj)Cov(Aij, X,) + C o v ( % X0 Xj) . (5.59) In an average representative firm context, assuming that G = g and tyZ{Z) = the resulting average output aggregation bias is given by the higher moments: N N N B»y = E Cov(^ > * » 0 + E E [E(^)COV(X,- XJ) + E(X , )Cov( .% Xj) i=i i=i j=i (5.60) + V{Xj)Cov{Aij, X,) + Cov{Aij, X^ Xj) Proposition 5.5 A necessary and sufficient condition for exact output aggregation, in terms of averages in the sense of relation (5.17), to hold with G = g, where g is quadratic in factor inputs as in (5.58), and ^>Z{Z) = JJ,z is that the bias (5.60) be null. If the assumption that the factor inputs and their functional parameters are all dis-tributed independently across firms is maintained, the aggregation bias of the average representative firm reduces to: N B»y =E E( y I*)* 2W, ( 5- 6 1) n=l where <J2(Jf,) is the variance of Xi across firms. Under the assumption that the functional parameters A are distributed independently from the factor inputs (e.g. if the firms have the same technologies), the aggregation bias becomes : N N B»y = EEE(^)Cov(*'-'^)- (5-62) i=i j=i This bias can vanish in a number of circumstances since some of the E(Aij) could possibly be negative. In particular, the requirement that the expectations of the coefficients of second order terms in factor inputs be null (i.e their distribution be symmetric) will be sufficient and is less demanding then additive separability. 75 In the Klein-Nataf context, if additive separability conditions are imposed on the quadratic, it becomes: / ^ 0 + a ^ + 4(^]. (5.63) t=i Following Nataf's solution, if the characteristics' aggregates are defined as: MZ) = [F--E{A0),F •E[A1X1 + A11X21},...,F -V\ANXN + ANNX2N}), (5.64) a perfect aggregation result follows. Rather, if the input aggregates are defined as first moments ipx(X) = fj,x, the additive separability conditions do not lead exact-aggregation results in this stochastic environment. The output aggregation of (5.63) results in the average output aggregation bias: N N B»y = X) C o v(^ * .0 + X E M ^ V ) + Cov (An, (Xi)2)]. (5.65) i=l i=l Assuming perfect competition in all factor inputs, the efficiency conditions require that the marginal products of the factor inputs be equalized across firms, that is: d G -—. = oc[ + 2afJx = Pi, i=l,2,...,N, f=\,2,...,F, (5.66) which implies that Var(>?, + 2A#Xi) = 0, i=l,2,...,N. (5.67) Unfortunately, this restriction does not have an immediate interpretation, by contrast with the linear case, in terms of the aggregation bias. However, if the firms have the same technologies up to a translating factor, A{ = 67,-, An = 67,y, i = 1,2,... , N, then the perfect competition condition (5.67) is equivalent to ^ V , ) = 0 and the bias (5.65) vanishes. In a competitive environment, if the firms possess the same technology, they will employ the same amounts of inputs; they are technologically and allocatively equivalent up to a firm-specific factor O J Q . Aggregation in a competitive environment is studied more generally in the dual profit function space, where the above condition is sufficient but not necessary. 1 6 1 6 See chapters 6 and 7. 76 5.8. Generalized Cobb-Douglas Production Function The Generalized Cobb-Douglas exhibits a simpler form of non-linearities in the param-eters. Yet the second order approximation to the average output aggregation bias shows how important the bias can be. Let the production function be a Generalized Cobb-Douglas: yf = g^,lf,ccf,6f) = <*!,P! > 0, f=l,2,...,F, (5.68) where j / denotes the level of output, kf and V are respectively capital and labor inputs, oJ and 0f are the functional parameters. Let Y,K,L, A,B be the random variables whose realizations are (y1,... ,yF),(kl,... ,kF),(lll... ,lF),(a1,... ,aF),(dl,... ,0F), respectively. The average output aggregate can be written as: E ( F ) = ~E{KAL8) (5.69) = E ( ^ ) V{L8)+Cov{K\LB). Without further assumption, it will generally not be possible to express the output aggre-gate in terms of the characteristics' first moments. A trivial solution assumes A = B = 1, the functional parameters are then distributed independently from the inputs, with ~E(A) = E(S) = 1. If the output is stochastically distributed over firms, the factor inputs can be assumed independent; the covariance term vanishes and there in no aggregation bias. More generally, a second order approximation to the average-representative firm's output aggregation bias, using (5.20), is given by: + 2 [ ( / i O ^ M ^ " 1 + ^ l n ( M A ) ( w ) ' x «- 1 ( / i ;)^3Cov ( A : ; A) + 2na\n(N)^kr--l(mr0CoY(K, B)+n0(w - 1 ) ^ * ) " ° ( A * J ) " " ~ M (5.70) + 2/x /9ln(A**)(A**)' ia (M/)^- 1Cov(I ; A) + 2 [(//*)"« ( A ^ - 1 + ^ln(/i ( )(M*) M a (w)M / 3 _ 1]Cov(Z,, B) + l n ( M f t ) 2 ( M ) M a {mf^l + 2(Ai*)"aln(A**)(A*/)^ln(A«/)Cov(>l,S) + l n ( W ) 2 ( A ^ ( A W ) ^ -By contrast with the average-representative firm problem, the parameters of the macro function may be left to be estimated from the regression: E{Yt) = V{KflftM), t=l,...,T. (5.71) 77 The Generalized Cobb-Douglas can be seen as a linear function in the logarithms of the factor inputs, thus an implicit form for the exact-aggregation macro parameters aM and 8M is given by the surface: [ E ( ^ ) ] a M - [ E r f M - [ E ( ^ f ) ] = 0 . (5.72) A general family of solution for aM and 8M can be written as: aM = 8- I •> / * / J and 8M = 1 - 8) • 1 } ' / J . (5.73) ln[E(iQ)] ln[E(Li)] The macro parameters a M and 8M will vary over time as E(KfLf), E(if t) and E(Z () do. Assuming that there exist some macro parameters & and 8 that are stable over time, their microfoundations can be written as: aM_5(t] l n[E (#Lf ) ] M _ ( , \n[E(KfL?)} a ~ 5 { t ) ln[E(*)] * ln[E(Lt)] ' ^ for all t = 1,2,... , T, where the function 8(t) measure the time-drift of the microfounda-tions of the assumed macro parameters. Finally, a perfect aggregation result can be obtained under the following conditions. Since the Generalized Cobb-Douglas is additively separable under a logarithmic transfor-mation, yf = exp(o/ln(A/) + 8f\n(lf)), (5.75) Klein-Nataf's perfect aggregation result will apply to an output aggregate defined as F y* = e x p ( F - E [ l n ? ( £ ) ] ) or y* = exp ( I n / ) . (5.76) If the characteristics' aggregates are chosen to be: t K* = exp(V - ^ E ^ l n t f ] ) or k* = exp(V 1 ^ o / l n A / ) F and t* = exp(V ^ ^ E f S l n L j j or Z* = expjy - 1 ^ / 3 ' l n / ' ) , (5.77) the output aggregate y* can be written in terms of the characteristics'aggregates: y* = g[k*,l*,6n) = {k*)s{l*y, (5.78) without aggregation bias. However, since such aggregates are generally not available, the scheme has limited applicability. 78 Chapter 6. Output Aggregation with Optimizing Behavior - The Problem of Profit Aggregation This chapter applies the stochastic framework developed in the previous chapter to the aggregation of profit functions. The most important exact-aggregation results in producer theory (Gorman, 1968a; Blackorby and Schworm, 1988) assume some optimizing behavior and economy-wide equilibrium conditions and thus are given in terms of profit functions. The general problem of profit aggregation is studied first, the problem of capital aggregation or restricted-profit aggregation will be examined in turn. The exact-aggregation results, mentioned above, pertained mostly to capital aggregation problems or to aggregation problems under partial optimization. They will be reevaluated in terms of the present stochastic framework in the next chapter. This chapter pays particular attention to the implications of perfect competition on the aggregation conditions. In the problem of profit aggregation, all factors are taken as mobile. Price-taking behavior on the part of the producer is assumed, but economy-wide equilibrium conditions may or may not prevail. In order for the realizations of the firms' profit characteristics to be independent, it is also assumed that there is no joint production and that each firm's production decisions are taken independently. Different prices for each firm's inputs and output may be assumed. This situation could describe an otherwise perfectly com-petitive economy where royalties, subsidies and indirect taxes introduce price wedges in between firms. Alternatively, the prices could be assumed to differ across firms because of transportation costs as in Geweke (1985). A framework that supports different prices for the inputs and output allows different competitive assumptions to be analyzed. Uniformity of prices among firms can be assumed in those markets that are competitive. For example, the classic problem of capital aggrega-tion (Solow, 1964; Fisher, 1965) could be treated as a special case, where the prices of the mobile factors are common to all firms but where the price of capital differs across firms. In the empirical analysis of aggregation across an economy, the observed prices would typi-cally take the form of prices indices derived from the ratio of current to constant values, the latter becoming the quantities. In this instance, allowing for different output and inputs prices will capture the heterogeneity of products and factors across industries. 79 The profit aggregation derivations are applied to a separable quadratic production function, a Generalized Cobb-Douglas production function and a Generalized Leontief profit function in section 6.4 below. 6.1. Derivation of Profit Aggregates and Biases Let the firm's production function be described as in (5.1) by: . J = = gtf,Of), f=l,2,...,F, (6.1) where yf denotes the level of output, a/ » 0/v is a vector of inputs, 9* is a S-vector of functional parameters that characterize the firm-specific production technologies. At first no economy-wide competitive assumptions are made and each firm is assumed to face its own prices. Regularity conditions are now imposed on the production function in order to make use of duality theory. The assumptions D on the production function </ are: </ is continuous from above, increasing and concave with respect to 2/ for all fixed 9? (Diewert, 1973, 1982). Under these conditions, the firm's dual profit function is given by: W{pJ, J) = m a x , j {j/yf - vJ . a/ : j / < /(a/)} * ' (6.2) = p / y / (p / , « / ) — « / • xf(pf, vJ), where pf > 0 is the output price, vJ 3> Ojv is a vector of input prices and, applying Hotel-ling's Lemma (Hotelling, 1932), yf(pf, vJ) = dU.f(pf, vJ)jdpf is the firm's supply function and x{{pf, wf) = dU.f{pf, v/)/dv/n, n = 1,2,... , N, are the firm's demand functions. Under assumptions D on the technology, the profit function U.f(pf, vJ) is non-decreasing in pf, non-increasing in wf', closed, convex and linearly homogeneous in [pf ,vJ). Assuming cost minimizing behavior from the part of the producer, the profit function (6.2) can be rewritten as: nV, *S) = ™*yf W - <?V, yf)} (6-3) where the cost function C^(u/, yf) is defined as: Cf{u/,yf) = min j { « / . * / : yf < /(a/)} = vJ . xf (u/, yf). (6.4) Applying Shephard's Lemma (Shephard, 1953), x £ ( u / , yf) = dCf(wt', yf)/ du/n, n = 1,2,... , N, is the firm's conditional factor demand for input n. 80 Using the augmented production function g, the profit function (6.2) can be written equivalently as: n(t/) = n(/, v/, 9f) = m a x , j {pfyf - « / . / : / < ff(a/, 0')} (6.5) = f/yipt, vJ, 9f) - « / . x(j/, « / , #'), where i / = (;/, u/, 0^ ) € I R 1 + J V + S are the firm's profit characteristics, y(ps, vJ, 9s) and xn(pf, vJ, 9s), n = 1, 2,... , N, are respectively the augmented supply and demand functions. Assuming cost minimizing behavior, the augmented profit function (6.5) can also be defined in terms of the cost function as follows: n(p>, vJ, 9s) EE max^ {pfyf- C{vJ, yf, 9s)}. (6.6) The augmented cost function C(ws, i/, 0?) is defined as: C(v/, - m i l V • V* < di^, 9s)} = uf • x(vJ', y*, 9s), (6.7) where xn{vJ, ys,9s), n = 1,2,... , N, is the augmented conditional factor demand function for input n. The profit characteristics (v1,..., vF) G 1RF x I R 1 + J V + S are interpreted as realizations of a random vector V = (P, W,Q) G I R 1 + A r + , S . The random vector V is distributed over all firms according to the density pv(v). Note that pv{v) is a joint density of prices and functional parameters. It is easily seen that competitive assumptions implying uniformity of prices across firms can simplify the density pv{v) dramatically. The aggregates of the profit characteristics are denoted by 0 „ ( V ) and are defined similarly to the production characteristics' aggregates (5.4), in terms of the first moments of the profit characteristics, given the density pv(v). Generally, it will be desirable to define characteristics' aggregates that depend on the corresponding characteristics: tpv(V) = (ipp(P), ipw(W), r/>fl(8)). Furthermore, in the profit function context, average prices, ipp(P) = fip G IR and ipw(W) = fj.w G IR^, are more reasonable aggregates than total prices. The full-information profit aggregates thus capture not only the distributional proper-ties of the micro relations but also the competitive conditions that prevail in the economy. They can be defined in terms of total profits: U = F- f U{v) pv{v) dv= F-^, (6.8) 81 where n„ C JR^N • IR 5 is the domain of definition of the profit characteristics and where IT: jn1+Af+,? —> IR, so that Y\ £ IR- In terms of average profits, as the first moment: / / , = ] ^ = E[n(p, iv,e)] = / n(«) P K ( « ) (6.9) where fi„ G IR. In terms of observed values, the total profit aggregate Y\ corresponds to the economy's profit aggregate used in exact-aggregation models (Gorman, 1968a; Blackorby and Schworm, 1988). Perfect competition is assumed in output and inputs markets when P = p and W = w, that is, when no firm can influence product or factor prices. Then, the expression for expected profits (6.9) reduces to: E[U{p,w,&)]= j Yl{p,w,9) p@{9)d9. (6.10) If the supply and demand functions are polynomials in the parameters 9?, this expression can be written as: E [lT(p, w, 0) ] = p Eg [y{p, w, 9) ] , - « ; . E6 [x(p, w, 9) ] = p (y (p,«; ,E[e]) -Mtf[y(p,«7,0)]) - w. (x(p, «/,E[e]) - Ms[x(p, 0)]), ( 6 ' U ) where M#[-] are the higher moments of the functions with respect to the parameters 0, as defined in (5.14). The macro profit function is artificially defined in terms of the following optimizing problem: Il[tp,(V)] E m a x j / , ; {pTJ-wX : V < G(X,vbe{@))} (6.12) where \j and X represent the aggregate levels of output and inputs and are generally pre-specified. The real problem is posed by the choice of a macro function G, with macro parameters ^ ( 0 ) , such that the pre-specified aggregates satisfy (6.12). The macro profit function is expressed equivalently as a function of the profit characteristics %IJV(V) in terms of macro supply and demand functions: n[Mv)]=MP)y*(Mv))-Mw)*x*{Mv))> (6-i3) where y*{ipv(V)) and x*(ij>v(V)) are the macro supply and demand functions. Exact profit aggregation requires that the level of profits resulting from the aggregation of the firms' profit functions, described by F times expected profits fiT, equals the level of 82 profits given by the macro profit function Il[ipv(V)]. The total profit-aggregation bias can then be defined, similarly to the total output aggregation bias (5.16), as the difference: Bw = F-f Tl{v) pv{v)dv-ny>v{V)}. (6.14) As in output aggregation, the macro profit function may not be fully specified. Whereas aggregate prices may be known, as in the case of perfect competition, in general the aggregate parameters ipg(@) are not known. 6.2. The Case of an Average-Representative Firm As in output aggregation, the case of an average-representative firm is of special interest. When macro behavior is assumed to be that of an average-representative firm, the macro parameters are chosen to be the average of the firms' micro parameters: ip$ (0) = 1*8. In this case, the augmented macro production function is chosen to be identical to the augmented micro production function: G = g. The macro supply and demand functions are then identical to the micro functions, y*(-) = y(-) and x*(-) = x(-). If aggregate prices are chosen to be average prices, that is allowing any competitive assumptions, the average profit-aggregation bias is given by: Bft = F • U(v) pv(v) dv - n[fj,p,i2w,fj,e] = ( E [ P • y ( V ) ] - / i p • y(nP,fiw,fie)) ~ (E[W.X(V)} - /iw • x(^ p, fi^ne)) • (6.15) Under perfect competition in the sense that P = p and W = w, the profit character-istics become ipV{V) = (p, w, lie) and the macro profit function is: n(p,w,^e) = p y(p,w,iie) ~ w*x(p,w,fie), (6.16) and the average profit-aggregation bias is: B„T = P (Ee[y{p,w,e)] - y ( p , ii;,/**)) - w • (E«[X(P, «>, ©) ] - x(p, . (6.17) Thus, if the supply and demand functions are polynomial in their functional parameters, the average profit-aggregation bias is given by: B^ =pMe[y{p,w,e)} - w.Me[x{p,w,Q)}. (6.18) 83 Even under perfect competition, the average-representative firm may yield a profit-aggre-gation bias if the supply and demand functions are non-linear in the functional parameters. In the long-run, however, this bias may disappear as the distribution of firms becomes a degenerate case. The pressure of competition in the longer term may have force the firms to become "identical" as posed by earlier analysis (Friedman, 1976) or "redundant" as postulated more recently (Makowski, 1980). 1 In these cases, the distribution of the firms' technological parameters may be restricted in such a way as to entail the disappearance of its highers moments. In particular, if all firms are similar and earn zero profits, the average-representative firm will also earn zero profits. The important long-run result for aggregation theory is how similar the firms are rather than what their levels of profits are. However, empirical evidence (Mueller, 1986) indicates that there are persistent differences in profitability among the larger U.S. firms in the long run, so that the above theoretical ideal may not be attainable. Theorem 6.1 If the macro function is chosen to be G = g and the profit characteristics to be the first moments, tpv(V) = fxv, then the profit-aggregation bias will be null if and only if expression (6.15) vanishes. In particular, perfect competition alone may not be sufficient to ensure exact aggregation in the context of an average-representative firm. Proof: By construction of B^^, as shown in (6.15) and (6.18). Many authors (May, 1946; Pu, 1946; Bliss, 1975; Diewert, 1980) contend that perfect competition in all markets eliminates all aggregation problems. The statement of Bliss that a group of producers may be treated as if they were a single producer subject to the sum of individual production sets, provided only that all producers face the same prices and maximize net profit at those prices. (Bliss, 1975, p.146) can hardly be contested. However, it does not provide answers to the more practical empirical questions of how to relate the parameters of the macro profit (or production) function to the parameters of the micro relations. It is essential to find how the "mongrel" profit function can be linked to the individual profit functions to further provide answers 1 Agent i is a redundant firm if, when i stops being a firm, all agents including i can do as well (Makowski, 1980, p.215). 84 how it reacts to micro impulses and disturbances. Even if perfect competition prevails, non-linearities in the parameters of profit functions may imply that other conditions need to be imposed on these parameters, such as commonality across firms (or non-stochasticity), in order to find an average-representative firm. 2 This commonality may, however, be attained in the long run. T h e choice of a more complex functional form G alone would not solve these problems; the use of particular aggregates, a la Klein-Nataf, would also be required. 6.3. Stability of the Macro Profit Parameters If the functional form of the profit function is assumed rather than deduced from profit maximization subject to a production constraint, the vector of parameters 8? may not be known a priori. For example, in empirical analysis, the parameters 6* are left to be estimated through supply and demand functions. In this case, where the macro parameters are not p re-specified, the exact-aggregation macro parameters 9M will belong to a zero-profit-bias 5-manifold described by the following equation: n[rPp{P), rpw( W), 9M] - n = 0, (6.19) where 5 is the dimension of the vector 9M. Thus, in each time period, a number of exact-aggregation macro profit parameters can be derived from the micro relations. However, if the macro profit parameters are required to be stable (the same) over time, they will be completely determined over 5 time periods. Introducing time subscripts, if aggregate prices are taken to be average prices over firms, equation (6.19) becomes: n[vP{t)lfiw{t),6M} -Ft-nT(t) = 0 . (6.20) If average prices and/or aggregate profits vary over time, the solutions 9M(fip(t), /j.w(t), Ft-Hn(t)) to (6.20) will also vary over time, unless the variations in the profit, input and output prices aggregates are all proportional. Theorem 6.2 The exact-aggregation parameters 9M(fj.p(t), nw(t), Ft • fiw{t)) of a macro profit function n(/j,p(i), fiw(t), 9M) will be constant over time if all prices and profit aggre-gates vary proportionally over time. 2 See the separable quadratic production function example of section 6.4.1. 85 Proof: Since the profit function is homogeneous of degree one in prices, Theorem 5.3b applies, o Exact aggregation implies that the macro profit parameters depend not only on the micro parameters but also on the input and output prices. Problems in the stability of exact-aggregation macro profit parameters may arise as factor and output prices exhibit different inflation rates in practice. For example, empirical evidence from the Canadian Input-Output data base indicates that, in the aggregate from 1961 to 1980, output prices have risen 2.9 fold, capital asset prices 3.1 fold while labor prices increased 4.8 fold. 3 If an approximate aggregation result is judged satisfactory, such divergences may not be important. Again, empirical analyses are needed to evaluate the order of magnitude of allowable divergences. Note that exact-aggregation models usually maintain the assumption of fixed prices (Fisher, 1982, p.617), under which the microdefinitions of the exact-aggregation macro pa-rameters would be stable. Stability over time could be obtained alternatively by allowing the microdefinitions of the macro profit parameters to vary over time. Then, the aggrega-tion biases in the macro profit parameters would take the form of time-varying coefficients that would measure drifts away from stable microdefinitions as in (5.30). 6.4- Applications To provide a more concrete illustration of the biases in profit aggregation, the examples of the previous chapter are reconsidered. The linear production function does not yield a interesting profit function, but the separable quadratic production function provides useful insights. The Generalized Cobb-Douglas production function leads to rather clumsy and uninsightful profit-bias derivations. However, the Generalized Leontief, since it is a polynomial that is 1-linear in its technological parameters, gives an example of a functional form for which the average-representative firm, under perfect competition, induces no aggregation bias. Example 6.4-1 Separable Quadratic Production Function The choice of a separable quadratic production function instead of a full quadratic 3 See chapter 8. 86 function is made to facilitate the exposition. Consider again the separable production function (5.57): = ' ,oJ) where j / denotes the level of output, a/ is a TV-dimensional vector of positive inputs and af = {afQ, a?v ... , a/N, cJxl,... cJNN) are the functional parameters. The firm seeks to maximize profits: n(pj, p1, af) = maxjj {pfy]f - pf . jf : 3 / < g(xf, af)}, (6.22) where j/y > 0 is the output price and pf = (p[, ... ,pfN) 3 > O J V is a vector of positive input prices. The firm's factor demands are: x j ( p j y , Q / ) = = - i 7 - - i , i=i,2,...,N-2aiiPy 2an (6.23) and its supply function is: i=i l^n(Py) 4 < (6.24) The profit function is then given by: i = i 4 a « P » (6.25) Let Py be the random variable, P £ JRN, A £ I R 1 + 2 J V be the random vectors whose realizations are (py)..., pF), (p 1 , . . . ; pF) and (a1,... ,aF), respectively. Then the expected profits are: E[n(Py,P,A)) = E ( P A ) - f 4(P'4^f' (6.26) Assuming that the augmented macro technology is identical to the augmented micro technologies, G = g, the augmented macro supply and demand functions will be identical to the micro supply and demand functions, y*(-) = y(*) and x*(-) = x(-), and a general macro profit function will be given by: n*[*n(PM,(PU*W] = 1>n{P,)**oM -1 (6-27) 87 and the total profit bias is BT = F- E ( i V o ) - 1>Py{Py)rl>a0[Ao) - £( F 1 E N i = l {Pi-pM [4>Pi(Pi)-fPpy{Py)4>ai{Ai)]: A^A^py{Py) (6.28) Assuming perfect competition in the output and inputs markets, Py — py and P, = p,-, i = 1,2,... , N, so that aggregate prices are taken to be the common prices and the profit bias becomes: N r -2 p . , = p y [ p • E(JIO) - iM*o)i - E (*• • E y ' 1 i = i 'Ai,1 Ipa^Au) (6.29) For the aggregation bias to vanish, if the macro parameters are required to depend only on the corresponding micro parameters, they will need to satisfy: 4 ^ o ) = P - E ( M ^ ~ = - E ^ (6.30) However, the last two equations can be satisfied only if: II H II in particular, if A{ = 67,- is non-stochastic. In this case, ' 1>ao{A0) = F • E(/lo), ^ , (^ ) = a,- and ^ a{An) = (F • E[-J-(6.31) (6.32) satisfy equations (6.30) and a perfect aggregation results follows. The firms' profit function are then: N R{Py, V, &f) = Py<*o ~E ~ li 4PJ, P i ^ i + Q i P y (6.33) The firms' technologies are allowed to differ in the contribution of unaccounted factors QQ and in the curvature parameters at. When considering the aggregation over mobile 4 This requirement ensures that if these macro parameters were left to be estimated, they could be assumed stable over time. 88 factors, Fisher (1982, Theorem 1, p.621) restricted his analysis to constant returns to scale and required that the firms' production functions be the same (in the case where factor aggregation is not considered). By extending the analysis to production functions that do not exhibit constant returns to scale, it is possible to relax this restriction. Here the equality across firms of the parameter af will suffice. This restriction implies the equality across firms of the marginal profit attributable to optimized inputs up to a multiplicative constant. Example 6.4.2 Generalized Cobb-Douglas Production Function As in (5.62), let's assume a Generalized Cobb-Douglas production function. The firm's cost minimizing problem is then C(r>, «>,/?') = m i i y , , {r*kf+v/lt:yf< (kf)J (lf/}, (6.34) where / and vJ are respectively the prices of capital and labor. The corresponding conditional demand functions are: pf k(r>, «/,t/,o/,/?') = and \(rf ,u/ ,yf ,af ,p}) = a1vJ ~pf~7\ a! uf' •J+pf (yf}af+pf -af 1 (6.35) yf+pf (yf}af+pf > so that the cost function becomes: C{rr,J,yf,aJ,l3f) = [ipf pf af+pf 0fl f n -a af+pf 1 Pf (rJ)af+pf (yj^af+pf (yf) af+pf (6.36) = c(rf,vJ,af,df){yfyf+0f. Using the cost function C defined by (6.36), the profit maximization problem of the firm, max ^ { pfyf - c{rf,wf,af,0f) (yf)"f+Pf}, can be solved for the supply function: and the profit function is given by: n{pf,rf,wf,ccf,0f) = pf pfjaf + 0f) c(rf, wf,af,0f)_ pf(af + Pf) _c(rf, wf,af,pf) af+pf af+pf 1-af-pf l-«f-Pf - c(rf,wf,*f,pf) pfjgf + Pf) c(rf,vJ,af,pf) (6.37) 1 l-af-pf (6.38) 89 Note that this expression will be well-defined only for a + 8 < 1, that is, when the technology exhibits decreasing returns to scale. Let P, R, W, A, B be the random variables whose realizations are (p1,... ,pF), (r1,... , rF), (it*1,..., wF), (a1,... ,aF) and (81,... ,8F) respectively, where the (a 1 , . . . , aF) and (01, ... ,0F) are such that oJ + 0* < 1, for all / = 1,... , F. The expression for expected profits is then: E[n(p;a, W,A,B)] = E P P{A + B) _c{R, W,A,B) A+B - E c(R, W,A,B) P{A + B) c{R, W,A,B) l 1-A-B (6.39) Even under perfect competition, the non-linearities in the parameters make it impos-sible to express the expected profits in terms of the parameters' first moments. Assuming that the macro technology is also Generalized Cobb-Douglas, the macro profit function can be written as: 77* vbp{P)^r{R),^w{W)^a{A),ij0{B) M?) MP)(MA) + M&)) c{MR),1>»(w),i,a{A),MB)) MP)(MA) + M*)) c{MR),Mw),MA),MB))\ i.-i>a{A)-4>p{B) (6.40) If perfect competition is assumed in the output and inputs markets, Py = py and Pi = p~i, i = 1,2,... , N, the profit bias of the average-representative firm becomes: Br = p E P(A + B) [c(r,w,A,B) E( c(r,w,A,B) A+B 1-A-B P(A + B) p{E(A) + -E(B)) c{r,w,E(A)MB)) E(A)+E(B) l - E ( j d ) - E ( S ) c{r,w,A,B) (6.41) + c ( r , « 7 , E ( ^ ) , E ( S ) ) p(E(A)+E(8)) l - E ( i l ) - E ( S ) [c{r,w,E(A),E(B)) Thus, unless the firms are identical, in the context of a representative firm, the aggregation of profit functions based on Generalized Cobb-Douglas production functions generally will result in a bias. 90 If the-macro parameters are not pre-specified, contrary to the average-representative firm's case, but rather left to be estimated, the following equation implictly describes the microdefinitions of the exact-aggregation macro parameters aM and 8M: p(aM + 8M) c(r, w,aM,8M) c(f,w,aM,8M) p{aM + 8M) l-a M_pM [c{r,w,A,B)\ A+B l-A-B + E c{r,w,A,B) c(r,w,aM,8M)_ P{A + B) [c(r,w,A,B) 1 (6.42) = 0 It is not possible to obtain an explicit form for these microdefinitions. The Generalized Cobb-Douglas production function thus proves to be a poor choice to illustrate the profit-bias derivations. The above derivations are presented simply because this functional form is a very popular choice. Example 6.4-8 Generalized Leontief Profit Function Since it is sometimes more convenient to estimate supply and demand functions rather than production functions, it is useful to derive the profit-aggregation biases starting with an assumed functional form for the profit function. Let's now suppose that the production constraint is unknown and that the functional form of the profit function is a Generalized Leontief (Diewert, 1973) : N N n(p / 0 , / ,^) = EE'4(^)^Py)^ w h e r e ^ = $0 (6-43) »=0 j=0 where }/0 is the output price, p? 3 > Ojv is a vector of input prices and 0s = (lf00,... , '^y) £ IRJ^^"1"1) are the parameters which characterize the technology. The Generalized Leontief is a flexible functional form in the sense of that it provides a good local approximation to an arbitrary twice differentiable profit function. It also has the advantage over other flexible functional forms of being linear in its parameters and polynomial in prices, facilitating the estimation problem as well as the present bias derivations. However in general, it will describe a technology exhibiting diminishing returns to scale. Though necessary conditions are less stringent, convexity may be imposed by requiring that f/-- < 0, for all i^ j, which however rules out input complementarity. 91 Let Po be the random variable, P and 0 be the random vectors whose realizations are (PQ, ... , pF), (p1,... ,pF) and [91,... ,0F) respectively. Then the expression for expected profits is: E N N n(Po,p;0)J = £ £ E [ % P , 5 P J . i=0 ;'=0 1 1 ~ 2" (6.44) Assuming that the macro technology also can be described by a Generalized Leontief, the macro profit function can be written as: T V N 1 n*[MPo),MP),Me)] = E E ^ - W ^ . - ) ) 1 ^ ^ ) ) 1 - (6-45) i=0 ;=0 If perfect competition is assumed in the output and inputs markets, Po = pb and P, = p~i, the profit bias of the average-representative firm will vanish: E n(po,p,Mo) - n*[Po,p,fie] =0- (6.46) In other words, the microdefinitions of the exact-aggregation macro parameters are: bft = E(7%),; ;y = i,...,/v. If perfect competition does not prevail and aggregate prices are taken to be average prices, the average-representative firm will be biased: N N r 1 1 i r ^ = E E E[7%P, 5P/] - E(7%)E(P,-)b(P;)2- . (6.47) t=0 j=0  J In contrast with the average-representative firm problem, the parameters' aggregates in (6.45) may not be pre-specified but left to be estimated. Then, the microdefinitions of the exact-aggregation macro parameters, 6^ f, i,j = \,...,N = b^f), if aggregate prices are taken to be average prices, will be given by substituting the b^ into (6.47) and equating it to zero: i f = (E(P , ) -2E(P J)4) f f j j E f e ^ ] (6.48) /=0 *=0 where YliLo X ^ L o ^ ik ~ 1> l> k = I, • •. , N. If prices vary over time, the macro parameters with stable microdefinitions will also vary over time, unless the variations in prices are proportional. 92 Assuming that there exist some b^, i,j = l,...,N (6,y = bp), that are stable over time, their microdefinitions can be written as: % = (E(j\r*E(py)-i) EE*) E[W^1 (6-49) ;=o k=o with X^Io ^f=o /^t'(*) — 1) ^ = 1; • • • ; ^ ) where the coefficients measure the time-drifts of the exact-aggregation macro parameters. It will be possible then to evaluate the importance of non-proportional increases in prices with respect to the stability of the macro parameters. Not only is the Generalized Leontief a functional form that allows an exact representa-tive agent, but it is possible to derive explicitly the microdefinitions of its exact-aggregation macro parameters. 93 Chapter 7. Output Aggregation with Partial Optimization - The Problem of Capital Aggregation Since aggregation problems traditionally were assumed to vanish under conditions of perfect competition, the presence of unmovable capital was thought to be the main obstacle to the aggregation of the firms' production functions. Thus, the aggregation of capital has been at the center of many exact-aggregation studies (Fisher, 1965, 1969b, 1982; Diamond, 1965; Stigum, 1967; Gorman, 1968a; Blackorby and Schorm, 1984,1988). The main result of these analyses will be reconsidered here under the proposed stochastic framework. The first sections are a simple technical extension of the previous profit-bias derivations to variable profit functions and analyze the impact of the introduction of a fixed factor at a more general level. Specific exact-aggregation restrictions are rexamined in the applications sections; these sections provide the more useful insights. 7.1. Profit-Aggregation Biases in the Presence of a Fixed Factor Consider again the production technologies described by: yf = </(kf,li) = 9{V,lf,9f), f=l,...,F, (7.1) where j / denotes the level of output, kf denotes a vector of dimension M of fixed capital inputs, is a vector of dimension N of positive labor inputs (or more generally, of optimized inputs), 6? is a vector of dimension S of technological parameters, / is a continuous production function and g is an augmented production function. Whereas the capital inputs are at all times assumed fixed, the labor inputs are opti-mized given price-taking behavior on the part of the firm. However, economy-wide equilib-rium conditions in the output and labor markets may or may not prevail, so that different prices may be assumed across firms. Under the usual regularity conditions on the produc-tion function </, the firm's dual variable (restricted) profit function is given by: IlV, *f\ * 0 = max^/W - v/ . / ' : j / < gf{kf> lf)} = pf/h/, vJ; kf) -«/. /VVi *0> where pf is a positive output price, vJ O^ y is a vector of labor input prices and where y'fy, t i / ; A/) and lft(j/, v/; kf), i= 1,... , TV, are the corresponding variable supply and demand functions, respectively. 94 This micro profit function can be written equivalently as the result of a two-stage optimization procedure: nV, ^\ *0 = m a x / / / - Cf(v/, / ; k/)}, (7.3) where (vJ, yf'; tf) is the variable cost function: Cf{vJ, yf; tf) = mmlf{vJ.lf : yf < / ( tf , /')} EE « / . \f{v/, yf; tf), (7.4) where f[(itf, / ; tf), i = 1,... , N, are the conditional variable labor demands for each of the N types of labor. In terms of the augmented production function g, the variable profit function (7.2) can be written as: n(i?>) = Yl{pf, v/,9f; tf) = m a x , , { / / - vJ . /' : / < g{tJ, lf,9f)} (7.5) = / y ( / , vJ, 9!; tf) - ttf . 1(/ , ttf, 0>; tf), where <tf = ( / , t t f , ^ ; tf) G I R i + ^ + M are the firm's variable-profit characteristics, y ( / , itf, 9f;kf) and {{{pf, vJ, 0f; tf), i = 1,... , N, are the augmented variable supply and de-mand functions, respectively. Assuming cost minimizing behavior, the augmented variable profit function (7.5) can be written equivalently as the result of a two-stage optimization procedure: n(pf,wf!9f;kf) = maxyf{pfyf~C(wf,yf,9f;k/)}, (7.6) where C (« / , t f , ^;tf) is the augmented variable cost function: C ( t t f,/y ; t f ) = m i n d l y . / ' : / < g{kf,lf,9f)} = vJ . l(ttf, / , 0!; tf), (7.7) where ! „ ( « / , / , Of; tf), n = 1,... , TV, are the conditional variable labor demands for each of the N types of types of labor. As before, the variable profit characteristics (tf 1,..., tfF) G IR F x I R I + W + S + M A R E interpreted as realizations of a random vector "V = (P, W, 9, K) G jRi+^+s+M . The random vector "V is distributed over all firms according to the joint density pv (tf) of prices, technological parameters and capital inputs. In comparison with the unrestricted case, the aggregation of restricted profit functions is more demanding since the dimension of the characteristics' density function has increased by M. 95 The aggregates of the variable profit characteristics are denoted by Vv("V) and may be defined similarly to previously defined aggregates in terms of the first moments of the density pu($). Generally, the characteristics' aggregates will depend on the cor-responding characteristics: ip#(y) = {$P(P), ybw(W), VJ$(@), ^ ( i f ) ) . However, some exact-aggregation solutions (Gorman, 1968a) have used other types of aggregates: ip#("V) = (v<p(n Mw), iWe,ff)). The full-information profit aggregates are defined now in terms of the density pi/($) and thus capture the technological properties, the competitive conditions and the levels of capitalization of the economy: U=F-f n(&)pv{0)d# = F-»T> (7.8) where Q# C I R ^ + A f + A f • ]RS is the domain of definition of the variable-profit characteristics and where II: ]R 1+ J V+ S+M —• ]R) s o that f] S IR. Average profits may also be defined as the first moment: ^ = H = E [n(P, W,Q,K)}= f n(0) pv(0) dt?, (7.9) F A ? where fiv G IR. Perfect competition in the output and labor markets, P — p and W = w, will reduce the expression for expected profits (7.9) to: V[n{p,w,e,K)}= j I Tl(p,w,6,k)p{@iK){6,k)d9dk, (7.10) JukJne where Qk C JR^f is the domain of definition of the capital inputs and Qg C 1RS is the domain of definition of the parameters. This expression is substantially more complex than the corresponding expression (6.10) in the unrestricted case. At the macro level, the problem is posed by the choice of an appropriate production function G, with macro parameters yjg(©), and of a capital aggregator yJk(K) such that: . n [ M H = ™*y,zipy-v'Z-V <G{Z,M®)\MK))} (7.11) where 1/ and £, represent the aggregate levels of output and labor inputs and are generally pre-specified. The macro variable-profit function is expressed equivalently as a function of the variable profit characteristics ip&CV) in terms of macro supply and demand functions: 96 where y*(V»o(V)) and are the macro supply and labor demand functions. The total variable-profit aggregation bias can then be defined, similarly to the total output aggregation bias (5.16), as the difference: Br = F-f n(tf) pv(0)<fc>-#[lMV)]. (7.13) This bias will be fully specified when aggregate prices, aggregate parameters and aggregate capital are all given. 7.2. The Case of a Firm with Average Capital Now, let the average-representative firm be a firm of average technology, G = g and 0^(0) = fig, using an average amount of capital vbk(K) = ^ ik. Under arbitrary competitive assumptions, given that aggregate prices are chosen to be average prices, the average variable-profit-aggregation bias is given by: B„=F-f U^)pv{-&)d^-n[^p,^w,^ltxk] = (E [ P - y("V ) ] - / x p - y ( / / p , / i ^ M « ; M * ) ) - (E[W.1(V)] - fj,w.l(ij.p, (iw, w, w ) ) . - (7-14) Under perfect competition in the output and labor markets, P = p and W — w, the macro variable-profit function can be written as: IJ{p,w,fxg;fj,k) = p y{p,w,fig;^k) ~ w»\{p,w,ne\^k), (7.15) and the variable-profit aggregation bias of the average-representative firm becomes: B^^E[U(p,w,e,K)] - n[p,iv,neiiik] = P {^e,k[y{p,«>,®; K)] - y{p,w,ne,Hkj) k[l{p,w,G,K)} - \{p, w,ne,Hk))• (7.16) If the supply and demand functions are polynomial in the technological parameters and in the capital inputs, the average variable-profit-aggregation bias is given by: = p MSik[y(p, w,@,K)] - wmM$ik [l(p, w,e,K)]. (7.17) The bias is expressed in terms of higher moments of a joint distribution of technological parameters and capital inputs. Since these higher moments can take the form of a parameter times a capital input variable, it is unlikely that, in the presence of a fixed factor, the variable-profit bias of the average-representative firm will vanish. 97 Theorem 7.1 If the macro function is chosen to be G = g and the variable-profit characteristics to be the first moments, V,t?("^) = Mi?, then the profit aggregation bias will be null if and only if expression (7.14) vanishes. Proof: By construction of B^ , as shown in (7.14) and (7.17). The examination of the biasedness of an average-representative firm of average capital conveys a surprising insight: the introduction of a fixed factor brings about an increase in the complexity of the aggregation problems, rather than a change in the nature of these problems. A firm's choice of capital inputs is, in the short run, analogous to its choice of technology; as phrased by Fisher: the firm's technology is "embodied in its capital stock" (Fisher, 1982, p.615). The complications are brought by the fact that the combination of capital inputs and technological parameters will lead almost unavoidably to non-linearities in the characteristics of the variable-profit function. Furthermore, the long-run similarity argument that would make the higher moments of the profit characteristics' density disappear can be used no longer. 7.8. Exact-Aggregation Macro Parameters that Are Capital-Dependent In the unrestricted case, when the parameters of the profit function were not pre-specified, the exact-aggregation macro parameters could be defined implictly in terms of prices and profits. Because profits are linearly homogeneous in prices, the conditions for the macro parameters' stability (requiring that prices and profits move proportionally over time) though quite demanding were not impossible. In particular, they entailed a model with no money illusion; if all prices doubled, the ex ante and ex post exact-aggregation macro profit parameters would be the same. In the restricted case, however, when the macro parameters are not pre-specified, the exact-aggregation macro variable-profit parameters 9M will belong to a zero-variable-profit-bias S-manifold described by the following equation: n[ibp(P),yjw{ W),9M- tpk(K)] - II = 0, (7.18) where S is the dimension of the vector ,9M. Here, the exact-aggregation macro parameters also depend on capital aggregates. 98 As before, the zero-variable-profit-bias 5-manifold (7.18) yields a number of exact-aggregation macro profit parameters. The choice of macro parameters may be restricted by requiring that these parameters be stable (the same) over time. Introducing time subscripts, if aggregate prices are taken to be average prices over firms, equation (7.18) becomes: n[(ip{t),nw{t),9M;MKt)] - Ft-^{t) = 0, (7.19) and can be resolved over S time periods, provided that capital aggregates are p re-specified. The solutions 9M(/ip(i), nw(t), i>k{Kt), Ft • /^(O) t o (7-19) will vary over time if average prices, aggregate profits and/or the capital aggregates vary over time, unless the variations in the profit aggregate are proportional to the variations in macro profits resulting from changes in prices and capital aggregates. Of course, if the capital aggregates do not vary over time, the exact-aggregation macro variable-profit parameters will have the same properties as the exact-aggregation macro profit parameters. The restricted case differentiates itself from the unrestricted case in an important way. Since the restricted profit function is not necessarily homogeneous of degree one in the capital inputs, no general conditions for the exact-aggregation variable-profit parameters to be stable over time, can be devised. These will depend on the specific functional form of the variable-profit function. The particular case of homogeneous (in prices and capital, separately) variable-profit functions (Diewert, 1973; Diewert and Ostensoe, 1987) is of interest. If profits are linearly homogeneous is capital inputs and, if prices and capital aggregates vary with a factor of proportionality ht, it will suffice that profits vary over time in the proportion h2 for the exact-aggregation macro variable-profit parameters to be stable over time. Theorem 7.2 The exact-aggregation parameters 9M(iip(i),i*w(t), tb^Kt), Ff MTTM) °f a macro variable profit function Il(np{t),iiw(t),9M;ipk{Kt))i that is homogeneous of degree A in prices and capital aggregates separately, will be constant over time if fip(t) = ht • pp, Vw{t) = ht • pw, i>k{Kt) = k- Mfc and Ft- ^ (t) = (htkYpir-Proof: Applying Theorem 5.3. This result also indicates out to preferred types of capital aggregates that may be used in order to obtain the stability of exact-aggregation macro parameters. In the context of 99 theorem 7.2, the usual linearly homogeneous aggregates, such as ipk{Kt) — Ff E(ifj), will be preferred. On the other hand, normalized aggregates, homogeneous of degree zero, such as rpk{Kt) = Ft • 'E{Kt)/kmax, will confer linear homogeneity to the variable-profit function, leading us back to the unrestricted case. In the aggregation of variable-profit functions, the choices of capital aggregates and of functional form for the profit function are crucial to maintain the stability of the macro parameters. Since no long-run stabilizing arguments can be invoked, theoretically, it will be precarious, in most cases. Again, empirical analyses are needed to quantify this alleged instability. 7.4- Applications The following applications sections examine previous exact-aggregation conditions and propose new ones with the help of distributional assumptions. A first concrete example starts with a separable quadratic production function; the exact-aggregation conditions and the bias for the dual variable-profit function are obtained. The second example analyzes the aggregative properties of Gorman-type profit functions, which are found to be neither necessary nor sufficient to ensure exact-aggregation, using the usual aggregates. Parallels with Lau's (1982) fundamental theorem of exact-aggregation are established. Finally, for the empiricist, a new flexible functional form for the restricted profit function is studied and found to be of little convenience to model representative agent's behavior. A counterpart to the Generalized Leontief unrestricted profit function is still to be developed. Example 7.4-1 Separable Quadratic Production Function Again, the separable quadratic production function provides a more concrete example of the conditions under which optimizing firms with non-linear technologies can be aggre-gated. Let's now assume that all firms' production technologies can be described by a separable quadratic production function utilizing capital (M = 1) and labor (N = 1) : m2 f= 1,-~,F, (7.20) where j / denotes the level of output, kf denotes the level of capital, fixed in the short run, 100 tf denotes the optimized labor inputs, all non-negative, and ctf = (a^a^a*, 0^ ,0^ ) are the functional parameters. The firm seeks to maximize variable profits: n(p$,y,a>;tf) = maxyf,if iPyl/ ~^if -t/ < 9{ltf,lf,af)}, (7.21) where p{ > 0 is the output price and v/t > 0 is the price of labor. The resulting labor demand function, l(^,a>;tf) = 4-4; (7-22) 2a'a a\ is actually independent of the level of capital since the production function is separable in capital and labor. The firm's supply function is then: y ( ^ , a / ; t f ) = y i + ^ (7.23) So that the profit function is given by: nV„ «fy; tf) = tfja{ + a j t f + *i(tf) 2] - {Jl~fjU?- (7-24) Note that under perfect competition, the separable quadratic production function leads to a profit function of the Gorman type form: nfo, mh at tf) = h W + ^(tf )a] + PA - ("'"r")2 ,7 _ 4a;upy (7.25) = c*(p„)A*(tf> o4o4) + C0(p„,«J/;Q!{,af,Q!^), where p^  and w; are the common output and labor prices faced by all firms. Let Y, K, L, Ai, Ak, A\, Akk, An, Py and Wi be the random (across firms) variables whose realizations are (y1,... , yF), (k1, ... , kF), (I1, ..., lF), {cx\, ... ,af), (al, ... ,aF), (aj, ... ,af), (a^, ... ,aFk), (a\, ... ,aF), (p1,... ,pfy) and (w1,... ,wF) respectively. Then, the expected variable profits are: V[n(Py, Wh A;K)]= E(PyAi) + E(PyAkK) + E ( P y ^ K 2 ) - E (W,-PyAu) 4AUPV (7.26) Assuming that the augmented macro technology is identical to the augmented micro technologies, G = g, the augmented macro supply and demand functions will be identical 101 to the micro supply and demand functions, y*(-) == y(-) and 1*(-) = l(-), and a general macro variable profit function will be: bl>w(Wi)-MPy)^u(Au)]2 (7-27) Then, perfect competition in the output and labor markets, Py = py and Wi = u>i, gives the following variable profit bias: B* = py([F - tbai{Ai)] + [F -E{AkK) - il>ak{Ak)il>k(K)] . + [F-E(AkkK2) - taJAriM*)2]) -^{P • E [ ^ ] - ±) (7.28) 2 V *\-An1 - ib~JA»\) 4 V ^Au1 ib„..(A,A 2 V LAB> - V* „ ( ^ « ) 7 4 V L V rbau(Aa)J' For this bias to vanish, Gorman (1968a) proposed to choose a capital aggregate, that follows Klein-Nataf's solution (5.58): ^(a,t) {A,K) = F-E [AkK + A^K2] (7.29) where ib^ak^(A,K) in terms of observed values is simply J2f=i hk(kf,a?k,a^). This type of capital aggregate has the important disadvantage of requiring perfect knowledge of the micro relations to be computed. However, if it is used in conjunction with the solution, 1>ai(Ai) = F-EiAi), rbai(Ai) = at and rba{Au) = (F • E[^-]V\ (7.30) for the parameters' aggregates, it will yield an exact-aggregation result. Using more traditional capital aggregates, an alternative solution that would involve parameters and capital aggregates that are defined only in terms of micro parameters and of capital, respectively, could be based on distributional assumptions. For example, a perhaps not too elegant solution would require: Cav{Ak,K) = 0, Coy(Akk,K) = 0 and E(4a) = 0. (7.31) Then ibk{K) = F • E{K), TpAk{*k) = E{Ak) and V>ltt(•***) = E(yJ«) will yield an exact-aggregation result, albeit without a quadratic capital term in the micro and macro func-tions. 1 Note that such distributional assumptions allow for an informational^ richer 1 In order to preserve the concavity of the production function, the o/^'s and a^'s have to be negative. Thus E (./?&) = 0 implies AM = 0. 102 micro structure than assumptions requiring commonality of functional parameters across firms, which would be insufficient here. Example 7.4-2 Gorman Type Forms The Gorman Polar Form, and its PIGL and P I G L O G extensions (Muellbauer, 1976), have become standard basic forms to investigate economic problems while maintaining exact-aggregation. This first solution to the interagent aggregation problem is a special affine form which was popularized by Gorman (1953), derived previously by Antonelli (1886) and concurrently by Nataf (1953). Extensions to the original form, such as implied by the Gorman (1968a) conditions, may be problematic in a stochastic framework, as shown below. Let's assume that there are only two types of capital (M = 2), equipment (e) and structures (s), that perfect competition prevails in the output and labor markets, P = p and W = VJ, and that the firm's augmented variable profit function is additively separable in the profits attributable to the two types of capital inputs and to the other factors of production, and that the profitability of the two types of capital can be factored into a function of prices and a function of the capital inputs: Yl{p,w,9f; J, if) = ct{p,iv,af)he((?,a?) + c3{p, w, 0f)hs(sf, iJ) + c0{p,iv,if), (7.32) where the components of the profit function are written in augmented form: ce(p,w,af) and cs(p,w,0f) represent the price-dependent (marginal) profitability of equipment and structures, respectively; he(ef, a/) and h3(sf, tf) represent the firm's valuation of the different types of capital and c0(p,vl,^) denotes the profitability of the other factors of production. The additional condition that the price-dependent profitability of the different types of capital be the same for all firms,-needs to be impose on (7.32) for it to satisfy the Gorman (1968a) conditions. Let A, A, E, B, B, S and V be the random variables whose realizations are (a1, ... ,aF), ( a 1 , . . . , aF), (e\ eF), (0\ ... ,0F), {b\ bF), (s\ . . . , / ) and (7 1,..., 1F). Then the 103 expresssion for the expected variable profits is: E{U{p,w,@;E,S)} = E[ce{p,w,A)he{E,A)} + E{cs{p, w, B)h3(S, B)} + V[c0{p, w,T)} = E[ce{p,w, A)]H[he(E, A)) +Cov[ce{p,w,A),he(E,A)] + E[c,(p, w, B)}E[hs{S, B)} + Cov[c,(p, w, B), hs{S, B)} + V[c0{p, w, Y)} (7.33) Assuming that the macro technology can be described in the same way as the micro technologies are, the augmented macro variable profit function can be written as: n*{p,w,i>e(©)]yje(E),vJ3(S)] = ce{p,w,yja{A)) he{yje(E),iPa{A)) (7.34) + cs{p,w,vbp(B)) hs{yj3{S),4>b{B)) + c0(p,w,^{T)), The variable profit bias then will be: Bn = F- \ V[ce(p-,in,A)]v[he{E,A)] + Cov[ce{p,w,A),he{E,A)] - ce(p,w,MA))^e{E),^a{A)) + F-(v[cs& + Cov\cs{p,w,B),hs{S,B)]^ - c3{p,w,^0{B))hs{vjs{S),vbb{B)) + F ••E[c0{p,w,T)} - c0(p,w,^{T)). Gorman's exact-aggregation conditions (Gorman, 1968 a, restated in Blackorby and Schworm, 1984, Theorem 1.) first comprise a set of distributional assumptions, A = 67 and B = j3, which make the covariance terms in (7.35) vanish. They also ensure, if the cr(-), r — e,s, functions are linear in the parameters, that the expected price-dependent profitabilities equal that of the average-representative firm: E [cr(p, VJ, E)] = cr(p, w, £) = cr(p, w, rp^{E)) = cr(p, w, f ) (7.36) if S = e = E(S) = V €(3) Z = a,P r=e,s. Assumptions that are less stringent distributionally would require that the covariance between the cr(-), r = e, s, functions and the hr(-), r = e,s, functions be null. The need for some sort of "independence" between the cr(-), r = e, s, functions and the hr(-), r = e,s, functions has been recognized previously by Lau (1982) who required that analogous functions be "linearly independent" (Lau, 1982, Theorem 2, p. 213). The above zero-covariance condition is however a weaker assumption, especially if an approximate result is judged satisfactory in empirical analysis. 104 In effect, the stochastic independence of the price-dependent profitability of the capital inputs and of the firms' valuation of these different types of capital seems unlikely as a result of the optimization process. However, the annihilation of the covariances could follow from sufficient diversification in the economy. If a firm (l) relies on a high level of structures relative to its level of equipment, because of the technological processes involved (e.g. mining), it may attribute a high valuation (earning power) to its structures (h] = high), despite a profitability that could be comparatively low (c] = low). Whereas a firm (2) that uses a relatively low level of structure (e.g. chemical industries) may value it correspondingly low (h2 = low) with an equally low profitability (c2 = low). In such circumstances, the covariance between the hs(-) and the c3(-) functions could be small. That is, the technologically constrained valuations could be independent of the relative marginal profitabilities, if firms utilize sufficiently diverse technologies. The extent to which firms can differ technologically within the same industry may be questionable. But this result widens consistent aggregation possibilities across an economy. A second set of conditions (Gorman, 1968a) construct capital aggregates on the basis of the firms' valuation of the different types of capital: rPhe{he(E,A)) = F-E{he(E,A)} (7.37) Tph3{ha(S,B)) = F-E{hs(S,B)\, where iphe(he(E, A)) and iph3(hs(S, B)) are equal to J2f=\ ^ (e^, of) and Ylf=i h3{^, iJ) in terms of observed values. Again, these types of aggregates have the important disadvantage of requiring perfect knowledge of the micro relations to be computed and possess some empirical value only when additional distributional information is available. 2 Studying consumer behavior, Jorgenson, Lau and Stoker (1982) have called similar aggregates, index functions, and have correctly interpreted them as statistics describing the population. The present stochastic approach has the advantage of suggesting which precise "statistics" are needed to aggregate exactly the specific micro relations that may be derived from economic theory. An alternative solution would use the usual first moments as aggregates, il>e(E) = F • E(E) and ip3(S) — F • E(5), and require that the hr(-), r = e, s, functions be affine in 2 The case where he(e?, a!) = J and h3(J, iJ) = / is, of course, an exception to this complication. . 105 E and S with Cov(i2, A) = Cov(E, A) = Co\(A, A) = 0 and Cov(5 ; B) = Cav(S, 8) = Cov(B, 8) = 0. Moreover, in a stochastic framework, it is no longer necessary that the profit function be additively separable in the two types of capital inputs. A mixed term, for example heia(ef, sf ,6*) = 8? ef d, could be added to (7.32) as long as Cov(E, S) = Cov(E, A) = Cov(5 ; A) = 0. For each individual firm, the relative quantities of equipment and structures are likely to be dependent as a result of the optimization process. However, if the economy is made up of diverse enough technologies, the covariance between the two types of capital could be small. For example, some industries have relatively high level of structures (mining, electric power, petroleum and coal industries), some industries have relatively high level of equipment (automotive, chemical, primary metals industries), whereas others may use comparable amounts of each type of capital (broadcasting, rubber and plastic, miscalleneous industries). 3 Since the proposed alternative solutions to Gorman's solution are not a subset of the latter, Gorman (1968a) conditions cannot be seen as necessary in a stochastic framework, and they are sufficient only if capital aggregates that depend on micro characteristics are available. In Jorgenson, Lau and Stoker (1982), models of aggregate consumer behav-ior with exact-aggregation were implemented using index functions (analogous to the h functions) based on individual cross-section data on expenditure shares and demographic characteristics. Exact aggregation thus seems to imposed a clear choice between either im-posing simple affine forms or gathering the information that makes it possible to aggregate non-linear forms. Example 7.4-8 Normalized Quadratic Variable Profit Function As in the unrestricted case, the variable-profit function may be estimated from supply and demand functions. For that purpose, the choice of a flexible functional form is appropriate to approximate arbitrary production technologies. The normalized quadratic profit function is a recently proposed functional form (Diewert and Ostensoe, 1987), which supports the general assumption of non-constant returns to scale. It is a generalization of Lau's (1976) normalized restricted profit function. Let's now assume that the firm's variable profit function can be described by a nor-3 From Canadian Input-Output Data, see chapter 8. 106 malized quadratic. Let j/0 > 0 denote the output price, tf = . . . , pfN) be a vector of positive prices for the optimized inputs, tf = (k[,..., k/M) be a vector of non-negative capital inputs and 0? is a vector of parameters. Then, the following equation describes the normalized quadratic variable profit function: T / M \ N N , N \ M M zPo v / = i 7 i=o j=o lKi >i=o 7 /=i h=i M N / AT X M J / N \ N +' E E <W*f + 7 ( E "W) E « + 3 ( E * « ) + E /=1 i=0 *1 Vi=0 7 /=1 i=0 i=0 (7.38) where o/ > 0 M and /tf > ON+I are pre-specified parameter vectors; o,y = a;i- with alt- = a a = 0 for i = 0,1,. . . , N\ 6^ . = &w with 6^ = bhi — 0 for A = 1 7 . . . , M; bll = 0. There are a total of (iV+i)( ^ + M + i ) + M ^ + 1 ) free parameters. If the parameters - 0, / = 0,2,... M and = 0, i — 0,1,... , N, the functional form describes a technology with constant returns to scale. In this case, the normalized quadratic is linearly homogeneous in pf and tf, separately. The question of testing and imposing convexity is discussed in Diewert and Ostensoe (1987). Let Po be the random variable, P, © and K be the random vectors whose realiza-tions are (p\,... , pF), (p1,..., pF), (01,...,9F) and (k1,... , kF), respectively. Then the expression for expected profits is: E[ll(Po,P,Q;K)} = / M \ N N -j \ 1 / N \ M M E i=i i=0 j'=0 _2Ki i=0 ;=i h=i N + EE E [<^ ] +E +E w(EBiP) 1=1 i=0 L 1 i=0 7 1=1 1=0 7 EE[c»p»-]' (7.39) i=0 Assuming perfect competition in the output and inputs markets,Po = po and P = p, assuming that the prespecified parameters are the same for all firms, A\ = 67;, I = 1,... , M 107 and B{ = 0i, i = 0,1,. . . ,N, then (7.39) simplifies to: E [n(p0, p, 6; i()] = -EE W ( E [ E ( ^ W ] + Cov(A t f , #,) i=0 j=0 ( = 1 N M Af E[fl f t]E ','=0 ' / = 1 h=l L l M N r -I / JV v. Af r „ + XX>' E[Q]E[K,] + Cov(Q,Ai) + (E^P'j E E W h f ] + C o v ( £ , , - ^ ) /=i i=o L -1 N=o ' /=i L -fn - K i \=o y i=0 (7.40) Under these conditions, if the macro technology is also described by a normalized quadratic, j , Af \ N N n*[po,p,M®Y,MK)} = ^ (E^^)) E E P - P A ; ( ^ ) AT i=0 N Af Af n E ^ E E w ( = 1 ' i=0 j=0 Af iV + E i=0 ( = 1 /l=l Af ^ * i v y ( = 1 «=0 (7.41) i = i V ^ ) < M * i ) J J 1=0 A T the total variable-profit bias would vanish if the following capital aggregates were available: K, v>KLHL(KH KH, = yJKLL(K,,K1) = E fbbQkl{B0lK,) = -E Bo (7.42) and if the following stochastic assumptions were satisfied: Cov(A,j ; K{) = Cov (%, ^ ) = Cov (CU> Ki) = Cov(B U Jj-) = 0. (7.43) Because of the normalization by the variable K\, this type of functional form poses a number of aggregation problems that cannot be bypassed using simple stochastic assump-tions. 108 Part III Empirical Analyses 1 0 9 Chapter 8. Canadian Input-Output Data 8.1. Origin and Contents of Data Base The data used in the empirical analysis was obtained from Statistics Canada's Input-Output Division. This data set is a modified version of the one used by Cas (1984) and Cas, Diewert and Ostensoe (1986). It provides disaggregated measures of inputs and output, along with a few other interesting variables, in both current and constant dollars (thousands). The input-output tables are given for 37 sectors of the Canadian Economy corresponding to a level of aggregation similar to the "M" level described in Statistics Canada's report (1984) and comparable to the U.S. SIC two-digit classification system. A description of the accounting framework, the classifications systems, selected definitions and details of the mathematical treatment of analytical uses can be found in Statistics Canada's reports (1969, 1984). Other particular problems, such as the problem of constructing constant prices series and the relationship between the input-output tables and the system of National Accounts, are discussed by Lai (1982). Table 8.1 enumerates the available data items and gives a list of the sectors considered. The data is annual and covers the period 1961 to 1980. Different deflators in terms of producers' prices were used for each sector, so that price data for each sector could be computed. The constant dollars (real) values are used as quantities. 8.2. Tax Adjustment Process In order to utilize prices that are closest to the prices faced by producers, the infor-mation available on subsidies, royalties and indirect taxes (data items 47-50) was used to refine the crude deflator prices. The subsidies (or royalties) were subtracted (added) from nominal output to account for the amount of output actually produced. Producers' prices were then obtained from the ratio of nominal to real values: ti= {V"+yfTi)> f=l>2,...,F, (8.1) where Vy = j/y • j / is the current dollars value of gross output of sector / , with j / being the constant dollars value and p{ the price deflator, and where T{ is a subsidy (negative 110 T A B L E 8.1 - Data Items The 57 data items comprise outputs intermediate inputs, which are the inputs They are: Outputs: 1- Gross Output 2- Intermediate Output 3- Final Demand Output Intermediate Inputs from: 1- Agriculture & Fishing 2- Forestry 3- Mines, Mineral Fuels, etc. 4- Food &z Beverages Industries 5- Tobacco Products Industries 6- Rubber & Plastics Industries 7- Leather Industries 8- Textile Industries ^ 9- Knitting Mills 10- Clothing Industries 11- Wood Industries 12- Furniture & Fixture Industries 13- Paper &c Allied Industries 14- Printing & Publishing 15- Primary Metal Industries 16- Metal Fabricating Industries 17- Machinery Industries 18- Transport Equipment Industries Primary Inputs: 38- Imports (competitive) 39- Imports (non-competitive) 40- Government Goods 41- Labor Inputs 42- Raw Inventories 43- Finish Inventories 44- Capital: Material &; Equipment inputs categories. The inputs subdivide into ling from the 37 sectors, and primary inputs. 4- Exports 5- Re-exports 19- Electrical Products Industries 20- Non-Metallic Mineral Products Industries 21- Petroleum & Coal Products Industries 22- Chemical &z Chemical Products Industries 23- Miscellaneous Manufacturing Industries 24- Construction Industry 25- Air Transportation 26- Rail Transportation 27- Water Transportation 28- Motor Transportation 29- Urban Transportation 30- Storage 31- Broadcasting 32- Telephone 33- Electric Power 34- Gas Distribution 35- Trade 36- Finance, Insurance & Real Estate 37- Commercial Services 45- Capital: Structures 46- Land 47- Commodity Indirect Taxes 48- Subsidies 49- Other Indirect Taxes 50- Royalties 51- Capital Consumption Allowance (M&E) 52- Capital Consumption Allowance (S) 111 value) or royalty in current dollars. The price f?y that the /-th producer faces is the net revenue, after subsidies or royalties, received per unit of output. The data on commodity indirect taxes (data item 47) is available by sector only. It was assumed to include the taxes paid by consuming industries on their intermediate inputs, exclusive of imports and government goods. These taxes usually amounted to 2-6% of intermediate inputs in the primary and services sectors (1-3, 20, 26, 27, 30, 32, 34-37). They were less or around 1% in the manufacturing industries (4-23, 33), and were much higher (7-17%) in the fuel intensive transportation industries (25, 28, 29) and in the construction industry (24). These figures supported the hypothesis that these indirect taxes could be distributed proportionally among all intermediate inputs, neglecting the heterogeneity of tax levels, without introducing significant distortions. Note that the non-uniformity of tax levels is itself a factor that generally distorts productive efficiency. The commodity taxes were added to the nominal value of intermediate inputs in proportion of their total input contribution. The adjusted intermediate input prices for the consuming industries were then obtained from the ratio: vl Sf - H i ! Vii «=1,2,..,,37, *- i ' /=l,2,-..,37, ( 8- 2 ) where V[. = pf • J{ denotes the nominal value of input i consumed by the sector / , with J{ being the real value of intermediate input i and = p'y the corresponding price deflator, and where T{ represents the nominal value of commodity indirect taxes. The price p( faced by /-th consuming industry includes the commodity taxes paid by this industry, as well as the subsidies (or royalties) received (paid) by the i-th producer. The other indirect taxes (data item 49) comprise mainly property taxes for most sectors. They averaged 2 — 6% of total fixed capital (including structures and land values) for a majority of industries (1-2, 4-8, 11, 13-20, 22, 30-31, 35-37). They were less than 1% for the power industries (21, 33) and most transportation industries (25-27, 29); a little higher at around 1-2% for the mining, telephone and gas distribution sectors (3, 32, 34). A few manufacturing industries (9-10, 12, 23) paid between 7% and 13% in property taxes; these could be assumed to be located closer to urban centers. However, they were two outliers the construction industry (24) and the motor transportation industry (28) for 112 which the values of the other indirect taxes mounted up to 36% and 49% of fixed capital, respectively. In these cases, the taxes were assumed to include other permits and licenses, and the property taxes in the following capital rental prices calculation were appropriately reduced to the average taxes paid in other heavy industries (11, 16-20) for the first case and in the transportation industries (25-27, 29) for the second. The remaining taxes were counted under a new indirect taxes headings and were zero for all other sectors. Even with this ad hoc adjustment, the present treatment of the property tax rate is much more sophisticated than most studies which use a unique tax rate across industries (Denny, Fuss, Waverman, 1979). 8.8. Capital Rental Prices The production functions used in many economic analyses, including the present, relate flows of inputs to flows of outputs. It is thus important to distinguish between capital stocks and services and to use the correct measure of capital prices and inputs. Since the following empirical analyses utilize an aggregate (over factors) measure of capital inputs, this distinction is critical even in the estimation of production functions. Following Jorgenson and Griliches (1967) seminal article, the problem of imputing appropriate user costs for durables has been a troublesome and controversial topic. In their classic paper, Jorgenson and Griliches called attention to the fact that the aggregation of capital stocks is different from the aggregation of capital services. The aggregation of investment goods should be based on asset prices while the aggregation of capital services should utilize service prices. The capital services price or user cost of capital is usually thought of as a rental price, p{, incurred for use of the capital services and should be distinguished from the purchase price of an asset, q{ (for simplication, the derivations are given first for only one capital good). There exist a number of variations to the derivation of rental price formulae, but they all use a supply oriented approach. Continuous time (Jorgenson, 1967; Hall, 1968; Hall and Jorgenson, 1967) and discrete time (Christensen and Jorgenson, 1969) derivations equate the asset price of capital to the discounted value of all future capital services, allowing for deterioration and obsolescence of the capital goods and solve for the rental price of capital. In discrete time, Christensen and Jorgenson (1969) obtained the 113 rental price formula: Jt = U-i + ^t,t-%-l,\> (8-3) where r( is the discount rate, S[ is the combined depreciation and obsolescence rate and qftt is the unknown but expected asset price at the end of period t. The rental price of capital in period t must cover the interest earned given asset prices in the previous period plus the depreciation incured given expected asset prices this period minus the expected capital gains. Diewert (1980) proposed a similar derivation of the discrete rental prices for capital services, based on a firm's profit maximization problem. He assumed that competition would force the "leasing" firm to earn the prevailing rate of interest on its leasing activities. The purchase price of capital q{ minus the rental price j/t received this period times the opportunity cost of holding these funds (l + r[) must equal the depreciated expected value of the capital good next period: Wt ~ Ail + 4) = (1 ~ Btrfw (8.4) where (j[tt+1 is the expected purchase price at time t of the capital good to be purchased at time t+ 1. The ensuing rental price is given by: In the limiting case of a nondurable good (Sj. = 1), Diewert's specification has the advantage of yielding a consistent rental price (p{ = q{), whereas the Christensen-Jorgenson formula gives an unclear result (p^  = (1+ r[) g{_1) (Diewert, 1980, fn. 57). The former specification also takes a forward outlook at capital gains, which seems more appropriate, even if it extends only to the next period. Thus, specification (8.5) will be adopted in the present paper. Both formulae (8.3) and (8.5) can easily be modified to incorporate provisions of the tax code (Jorgenson and Griliches, 1967; Diewert, 1980). In the present study, the unavailability of data on investment tax credits precluded the inclusion of corporate income tax provisions,. making the estimation of an after-tax rate of return impossible to compute. However, data on property taxes was available and was included in the rental 114 price calculations. The data comprise five types of capital goods, raw (data item 42) and finished (data item 43) inventories, material &; equipment (data item 44), structures (data item 45) and land (data item 46), to be called k(, i = 1, 2,... , 5, of which only structures and land have been assumed to bear property taxes. The property taxes were given in effective terms (nominal value of property taxes collected). They were distributed in between structures and land in proportion to their respective contribution to fixed capital. The property tax rates r £ . t , i= 4,5 were computed as, yf yf rpf k4t T f k5t (yf + V f ) Pf (yf + v f ) <• = f » d <,= l f * > , (8.6) where T^t is the nominal value of property taxes, v£ t and vf^t are the nominal values of structures and land stocks, respectively. These property tax rates were incorporated into the rental price formula as follows: The empirical implementation of rental price formulae, such as (8.3) and (8.5), presents special problems for a number of reasons: i) the formulae include variables that are subject to interpretation or ii) on which data is not readily available, and more importantly, iii) the formulae rely on an equal-profit condition between the "selling" and the "leasing" activitites of the firm, which may not be satisfied in the short run. With respect to the latter argument, the use of one period capital gains is likely to yield erratic and even negative rental rprices. As argued by Denison, Since capital gains are highly erratic from year to year, the weights [service prices] must also change erratically from year to year. It could hardly be argued that market prices of capital goods and land fluctuate annually so as to maintain proportionality between capital values and the sum of earnings and capital gains, nor could firms adjust the composition of of their real assets annually even it they could foresee the pattern of each year's capital gains and losses. (Denison, 1969, p.9) One of the variables that may be subject to interpretation is r{; it poses the problem of choosing an appropritate discount rate. As pointed out by Diewert: 115 Which r would be used? If the firm is a net borrower, then r should be the marginal cost of borrowing an additional dollar for one period, while if the firm is a net lender, then r should be the one-period interest rate it receives on its last loan. In practice, r is taken to be either (a) an exogenous bond rate that may or may not apply to the firm under consideration, or (6) an internal rate of return. (Diewert, 1980, p.476-477) From a theoretical point of view, if formula (8.5) is used, since capital is assumed instantaneously adjustable, it has been argued (Harper, Berndt and Wood, 1987) that the after-tax rate of return should be the same for all sectors of the economy and the first alternative should be preferred. If formula (8.3) is preferred, neither alternative appears to be correct on theoretical grounds. On the other hand, a number of empirical studies (Jorgenson and Siebert, 1972; Berndt, 1976; Hazilla and Koop, 1984) have investigated the implications of alternative rental prices. In particular, a recent study by Harper, Berndt and Wood (1987) has concentrated on the empirical consequences of using different discount rates. The authors have analyzed the effects of using five different rates of return on the composition, volatility and negativity of capital rental prices. The discount rates considered were: an internal nominal rate of return (Ml), using a Christensen-Jorgenson specification with perfectly anticipated capital gains, an internal "own" rate of return (M2), where the capital gains were averaged across assets, a constant (=3.5%) external "own" rate of return (M3), using again averaged asset appreciations, with perfect expectations, a internal nominal rate of return (M4) with smoothed capital gains and an external nominal rate of return (M5), chosen to be the Moody Baa bond rate. Note that averaging capital gains across assets (as in (M2) and (M3)) defies one of the main purposes of introducing capital gains in the services price formula, that is to reflect differences in "rates of capital gain or loss among capital goods" (Jorgenson and Griliches, 1967, p. 267). With respect to the three criteria above, they found significant differences between the five alternative capital rental price formulae. Different specifications of capital gains measures had dramatic effects on the volatility of rental prices. The authors expressed a subjective preference for the internal nominal smoothed option (M4), where they used three-year moving averages to obtain measures of capital gains. But they also suggested that the internal own (M2) and constant external (M3) were viable alternatives to the 116 standard internal nominal (Ml). They had to discard the external nominal (M5) specifi-cation because of surprisingly negative results related to the equalization of ex ante and ex post property income. Because of the very special specifications employed, the Harper, Berndt and Wood (1987) study may not be readily transferable, but it serves an appropriate warning. Ac-„ cordingly, a number of different specifications for the capital rental prices formulae were explored here. The option of using internal rates of return was discarded following the computation of before income tax rates of return using the Canadian Input-Output data base. These rates of return were calculated by equating capital income, Yl^=i P^kt^iv ^° nominal value added, Vyt — /^t> which w a s nominal gross output minus nominal variable costs, and by solving for r[. They presented an important number of anomalies, that could not be obviously corrected. The rail, water and urban transportation industries (26, 27, 29) showed many years of negative rates of return, while other industries (2, 15, 21, 24), including the petroleum & coal industries, had a few odd years of negative returns. The construction (24) and motor transportation (28) industries exhibited rates of return over 100%. It thus seemed impossible to apply a general corporate income tax correction. As such, the variability of these internal rates of return was extremely high and lead to unreasonable rental prices. Sector-specific data on corporate income tax or investment tax credits may have helped to correct these anomalies, but these data items were unfortunately unavailable. A more viable alternative that can also be justified theoretically, as mentioned above, is the use of an external bond rate. Imitating other studies (Grunfeld, 1960; Miller and Modigliani, 1966; Evans, 1967; Holland and Myers, 1979), a corporate bond rate (from McLeod, Young and Weir, 1961-1980) was chosen to be the same discount rate across sectors. Table 8.2 reports averaged yearly values of monthly bonds rate for each of the twenty years under study. As exemplified in the Harper, Berndt and Wood (1987) paper, the specification of the capital gains is a crucial problem that greatly influences the volatility of rental prices. Three alternatives have generally been proposed to estimate the producer's expectations of future prices: i) the use of perfect expectations, f£.[ t i + 1j = ^t+i (Jorgenson and Griliches, 117 T A B L E 8.2 - McLeod, Young and Weir's 10 Industrial Bonds Yield Average 8.40 1976 10.58 8.31 1977 9.72 8.42 1978 9.96 10.01 1979 10.74 10.73 1980 13.11 1967; Fraumeni and Jorgenson, 1980; Jorgenson and Fraumeni, 1981; Jorgenson and Sullivan, 1981), ii) the use of static expectations = </k.t (Jorgenson' and Siebert, 1972; Diewert, 1980; Cas, Diewert and Ostensoe, 1986) and iii) the use of a forecasting model to predict next period asset prices (Epstein, 1977; Gillingham, 1980). The perfect foresight assumption will be unbiased, but it is appropriate only if the capital gains are realized next period, that is, if the firm sells next period. The second method will generally be incorrect. The third one cannot be ascertained while it presupposes extensive modeling expertise from the part of the producers. The latter approach, however, has the important advantage of smoothing out yearly erratic behavior. Since no option could be assumed correct a priori, the present paper explored a number of ad hoc specifications. Let the expected purchase price of the z-th capital good be expressed as a proportion of the price paid at time t: = (1 + ^ W<* (8-8) where <^.[tt+1] is the expected percentage change in the purchase price of capital. The rental price (8.5) becomes: If the actual percentage of change cj[.[tt+1] = <?kt+i = i^kt+i ~~ ^ kt)/4ttls u s e d , the expecta-tions of next period's purchase price of capital are taken as perfect. If ^.[ t t+1] = 0, static expectations are assumed. A third option used a five-years moving average of the percent-ages change in asset prices, ^.[ t t + 1j = \ E ;=o ^ kt-i^° m o c l e l the producers' expectations. Their expectations of capital prices increases then were assumed to be best implemented using capital appreciations averaged over longer time periods. Land was treated differently from the other assets, since the data seemed to confirm Denison's hypothesis that, Land prices may and often do reflect not only current earnings related to cur-rent marginal products but also the expectation that marginal products will be 118 1961 5.50 1966 6.43 1971 1962 5.44 1967 7.03 1972 1968 5.37 1968 7.87 1973 1964 5.49 1969 8.66 1974 1965 5.63 1970 9.22 1975 higher in the future because of increasing land scarcity (relative to other factors). (Denison, 1969, p.8) Accordingly, no additional capital gains were added in the rental price formula for land: The choice of static expectations was, of course, the option that generated rental prices exhibiting the least volatility. Perfect expectations gave highly erratic rental prices if the sector's own percentage change is asset prices was utilized. Instead, if the producers were assumed to perceive economy-wide rather than sector-specific movements in asset prices, 37 yh K&t+l] = ZX * H-l4 , * H w h e r e = ^ 3 7 'Zh > (81°) h=l 2^/i=l vk{t+l the resulting rental prices became slightly more reasonable even though many of them were still negative. Similarly, when sector-specific percentages change in asset prices were used in the five-years moving average model, very large variations still occured, as in Gillingham's (1980) study. By contrast, when smoothed capital prices expectations were formed using a weighted average (across sectors) of asset prices changes, 4 37 i=0 h=l where the weights are defined as in (8.10), their volatility became quite acceptable. Table 8.3 gives volatility and negativity statistics for three of the models of expectations analyzed: the static expectations, the perfect expectations - computed with percentages-change averaged across sectors-, the smoothed expectations -computed with percentages-change averaged across sectors- models. The volatility statistic is an average (over years) of the absolute value of the percentages of change in capital services prices. To provide a basis of comparison, it is useful to indicate that the capital services prices for manufacturing from the Berndt and Wood (1984) extended data set exhibit a volatility of 11.1295% over the period 1947-81, which comprises a relatively shorter period of high inflation. The volatility of these rental prices is 150% higher that the corresponding average inflation, whereas the volatility of the energy prices in the same data set is only 8% higher than the corresponding average inflation. The negativity statistic indicates the percentage of negative prices over the twenty years period; e.g. 0.10% indicates that there were 2 years where the rental prices were negative. 119 T A B L E 8.3 - Volatility and Negativity of Alternative Capital Prices for Structures Asset Prices Rental Prices Expectations Static Perfect* Smoothed* Sectors Vol. Vol. % < 0 Vol. % < 0 Vol. % < 1- Agr k Fish 7.3296 10.9453 0.0 65.9532 0.0 9.9099 0.0 2- Forest 6.8145 9.8060 0.0 26.7425 0.0 9.0930 0.0 3- Mines, etc. 6.7942 10.4737 0.0 62.4450 0.05 10.5282 0.0 4- Food k Bev 6.2855 9.6329 0.0 106.9167 0.0 9.2502 0.0 5- Tobacco 6.3140 9.9253 0.0 63.9348 0.0 9.1081 0.0 6- Rub k Plas 6.2556 9.7813 0.0 78.2473 0.0 9.6556 0.0 7- Leather 6.3272 10.8480 0.0 34.7458 0.0 10.1350 0.0 8- Textile 6.2728 9.8543 0.0 42.9910 0.0 10.0181 0.0 9- Knitting 6.4593 9.5270 0.0 15.8923 0.0 8.6338 0.0 10- Clothing 6.3215 10.2511 0.0 15.2662 0.0 9.6018 0.0 11- Woods 6.3531 9.6516 0.0 25.5717 0.0 9.0378 0.0 12- Furn k Fix 6.2433 9.6809 0.0 19.8134 0.0 9.2938 0.0 13- Paper, etc. 6.3425 9.2065 0.0 69.4737 0.05 9.8280 0.0 14- Printing 6.2676 9.4405 0.0 56.5427 0.0 9.2679 0.0 15- Pri Metals 6.3151 10.1353 0.0 490.4975 0.05 9.7635 0.0 16- Metal Fab 6.2937 10.5316 0.0 39.8963 0.0 9.5883 0.0 17- Machinery 6.2881 9.6717 0.0 39.2979 0.0 10.0599 0.0 18- Trans Eqp. 6.3281 9.1596 0,0 48.2831 0.0 9.2362 0.0 19- Elec Prod 6.2827 10.1314 0.0 32.9070 0.0 9.1993 0.0 20- Non Metal 6.3038 9.7744 0.0 42.4658 0.0 9.3257 0.0 21- Pet k Coal 6.8848 10.4031 0.0 67.1349 0.05 15.4963 0.0 22- Chemical 6.4773 9.1328 0.0 53.4060 0.05 11.4085 0.0 23- Mise Manuf 6.3210 10.0371 0.0 21.6020 0.0 8.9817 0.0 24- Construct 6.3084 9.3630 0.0 22.9542 0.0 8.2705 0.0 25- Air Tran 6.9493 11.1311 0.0 121.1548 0.10 28.6788 0.0 26- Rail Tran 6.8048 10.1062 0.0 64.2723 0.05 14.8382 0.0 27- Water Tran 6.8429 12.0543 0.0 77.2323 0.05 15.7207 0.0 28- Motor Tran 6.3220 10.6233 0.0 171.7637 0.10 25.8010 0.0 29- Urban Tran 6.8553 9.5877 0.0 156.7101 0.05 26.4066 0.0 30- Storage 6.2720 10.4650 0.0 57.4079 0.05 14.4695 0.0 31- Broadcast 6.6640 8.7861 0.0 68.4239 0.05 8.3290 0.0 32- Telephone 6.8300 11.1463 0.0 63.7707 0.05 13.6298 0.0 33- Elec Power 7.3002 11.3707 0.0 135.2051 0.10 32.5850 0.0 34- Gas Distr 7.2940 11.9053 0.0 81.2348 0.05 16.2158 0.0 35- Trade 6.2801 11.3468 0.0 48.3866 0.0 11.0094 0.0 36- F . I .R.E. 7.1421 11.3604 0.0 65.5304 0.05 11.0109 0.0 37- Comm Serv 6.2027 8.8747 0.0 39.6127 0.0 9.2655 0.0 * (percentages change averaged across sectors as in 8 .10) 120 Of the four alternative capital prices analyzed in table 8.3, only the rental prices assuming perfect expectations will not be retained for further sensitivity analyses, since their volatility is unreasonable. Another difficulty in implementing rental price formulae is the choice and construc-tion of depreciation rates S^r Fortunately, the data items 51 and 52 provided effective depreciation values in the form of capital consumption allowance on stocks of material & equipment and structures. The corresponding depreciation rates for material & equip-ment were between 5 and 10% for most industries (3-5, 8-22, 27, 31, 32, 35-36). For some utilities (33-34), for the storage (30) and railways sectors (26), they were lower at around 2-3%. For a mix of sectors (1, 6-7, 23, 25, 29, 37), including the leather industries and commercial services, the depreciation rates reached the midteens (12-16%), while they skyrocketed in the upper teens (up to 22%) in the sectors of forestry (2), construction (24) and the motor transportation industries (28). The corresponding depreciation rates for structures approached 2-3% for the transportation and services industries as well as for some capital-intensive manufacturing industries (6, 13-14, 25, 27-30, 32-37). In most manufacturing industries, in the agriculture, railways and broadcasting sectors (1^  4-5, 7-8, 15-19, 21-22, 26, 31), they were slightly higher around 3-4%. Forestry (2), mines (3), lighter manufacturing industries (9-12, 20, 23) and the construction industry (24) reported depreciation rates between 4 and 10%. It is interesting to note that a sector (24) that seems more highly taxed than others also exhibits depreciation rates that are much higher. No specific data on the depreciation rates for inventories was available. A weighted aver-age of the depreciation rates for material & equipment and for structures was used as the corresponding proxy. With respect to the quantities of capital flows, the tradition has been to assumed that capital services are proportional to capital stocks, implying a constant utilization rate. Since sector-specific data on utilization rates was not available, the conventional assumption was maintained. 8.4- Intrasectoral Factor Aggregation For subsequent empirical analyses, some factor inputs, in particular capital inputs, may need to be aggregated into a single category. Since the problem of the aggregation 121 over factors is not studied in the present stochastic framework, standard methodologies are used. In the time series context, an attractive procedure uses the Tornquist index formula with the chain principle (Diewert, 1986b). Quantities of aggregate (across factors) capital inputs, k{, for example, are then be obtained from the formula: ^ = £ 2 v« J j + r* J tf l n / " - { 8 " 1 2 ) ,•= 1 1 2-J ,=1 Pk{tKit i= 1 P y - i K i t - i J 1 K i t - i 1 The Tornquist index has been shown to be superior (even "surperlative") both theoret-ically (Diewert, 1976) and empirically (Barnett, 1983) to the more traditional Laspeyres-Paasche indexes. The Tornquist approximation to the Divisia index is an exact index cor-responding to a linearly homogeneous translog production function, which can be viewed as a second-order approximation in the logarithms to an arbitrary production function. The only potential problem with the Tornquist index is that it is only approximately consistent in two-stage aggregation (Diewert, 1978). Empirical studies (Parkan, 1975), however, have shown that the divergence is not important. 122 Chapter 9. Bias Estimation for the Production Function In chapter 5, the aggregation of output outside equilibrium (or equivalently, the aggre-gation of production functions) was shown to be exact, in an average-representative firm sense, only in the case of special linear forms. It was also shown that, in most circum-stances, the exact-aggregation macro parameters would be unstable over time. Considering the first problem of the average representative, the following empirical investigations will attempt i) to find whether the conditions for the existence of an average-representative firm could possibly be met by "real" data; ii) to assess whether the divergences brought by the use of non-linear functional forms are important. Secondly, the question of the in-stability of the exact-aggregation macro parameters will be examined using a simple linear production function. It will be assessed whether "real" shocks (such as the 75-oil shocks) bring about important divergences in the ex ante and ex post coefficients. 9.1. Variables Choice and Description A simple production function of two inputs, capital and labor, is used first to illustrate the bias implications of alternative functional forms. As explained above, intrasectoral capital aggregates of the five capital inputs, were formed using Divisia quantity indices under various specifications of producers' expectations of future capital prices. However, since the volatility of the rental prices under the perfect expectations hypothesis was unreasonably large, the corresponding capital aggregates were discarded from subsequent analyses. The remaining two options of static and smoothed expectations were not altogether satisfactory. They lead to very different capital inputs aggregates, as shown in Tables 9.1 - 9.4. Consequently, three alternatives intrasectoral capital aggregates were retained in the following empirical analyses. The first one (option- a) is based on asset prices, notwithstanding the criticisms of Jorgenson and Griliches (1967), and could correspond to capital data obtained already aggregated. The second (option b) and third (option c) alternative capital aggregates were formed with rental prices computed using formula (8.5) under the static expectations and the smoothed expectations hypotheses, respectively. 123 The labor inputs are given by data item 46. Labor's share of gross output, computed as the ratio of the nominal value of labor inputs to the nominal value of output, varied considerably across sectors. The petroleum and coal industries (21) had the lowest share of labor inputs: it fluctuated between 4% and 9%. At the other extreme, the expenditure on labor inputs exceeded the value of total output in the urban transportation (29) and broadcasting (31) sectors; it totalled 165% of total output in the broadcasting sector in 1961. For most manufacturing industries (6-12, 16-20, 23-24) and the gas distribution sector (34), labor's share of total output oscillated between 25% and 35%. A mixed group of primary (1,3), manufacturing (4-5, 13, 15, 22), utilities (34) and services (36) industries exhibited lower labor's shares between 12% and 25%. Meanwhile, many other sectors, mostly services, had substantially higher labor shares anywhere from 35% to 65% (2, 14, 26-28, 30, 32, 35, 37). Table 9.1 gives averages over time of the quantities of output, labor and of different capital inputs aggregates for each sector. These figures show that distributions across sectors are substantially different between variables, including the capital aggregates. They confirm that differences in modeling the expectations of producers have dramatic effects on the magnitude of the capital aggregates. Table 9.2 gives the means of the variables over time and illustrates the differences in the time profile of the different capital aggregates. The means also evidence dissimilar rates of growth across variables: between 1961 and 1980, output has risen 2.5 fold, labor 1.6 fold and capital 4.5, 6.4, 4.2 fold, depending on the choice of capital aggregate. These figures support the view of increasing substitution of capital for labor. Table 9.3 gives the correlations between the variables over time. The high correlations between the variables indicate that they vary proportionally over time. For a majority of sectors, labor is more highly correlated with output than capital. However, for a few sectors (4, 8, 9, 10, 30), labor is very poorly correlated with output. Similarly, though generally, capital and labor are highly correlated, there are a few sectors (4, 10, 19) for which this correlation is negligible. The negative correlations between labor and the other variables indicate that while output and capital were increasing over time, labor was declining. It can be assumed in those cases that capital and labor are substitutes. Table 9.4 give the intersectoral correlations between output, capital and labor. Here again, labor is the 124 T A B L E 9.1- Means over Time Sectors Output -• Y Capital - K(a) Capital - K(b) Capital - K(c) Labor - L 1- Agr & Fish 4154440. .00 8418813.00 19784100.00 14460260.00 440939.60 2- Forest 1104066. .00 4646649.00 1427641.00 1082285.00 258980.60 3- Mines,etc 4142139 .00 6348970.00 11606970.00 7793943.00 662839.00 4- Food & Bev 7304194. .00 9519515.00 5427312.00 3594190.00 879938.00 5- Tobacco 435099. .10 585856.70 576894.10 385885.10 40630.66 6- Rub & Plas 1206368. .00 1304943.00 617330.30 454380.70 209201.40 7- Leather 332281 .60 1087498.00 302778.00 219048.20 90474.14 8- Textile 1643633 .00 2519927.00 1489627.00 1035588.00 247772.20 9- Knitting 407531. .20 710934.40 215405.00 159692.50 68126.09 10- Clothing 1207986. .00 2888687.00 485915.90 351993.20 291929.00 11- Woods 1703670, .00 3867245.00 1513576.00 1057643.00 380691.00 12- Furn & Fix 645833 .00 1323751.00 284071.10 197179.90 167065.90 13- Paper,etc 3416514. .00 5699322.00 5194289.00 3604232.00 608033.50 14- Printing 1299119. .00 3750630.00 1101057.00 741780.30 421000.00 15- Pri Metals 3938245 .00 5289766.00 5429101.00 3779912.00 613953.70 16- Metal Fab 3008904 .00 5067925.00 2083048.00 1446405.00 690937.70 17- Machinery 1817751 .00 2647112.00 1029263.00 700279.30 406912.10 18- Trans Eqp. 6087591. .00 5717051.00 2497514.00 1638564.00 820069.30 19- Elec Prod 2551179 .00 4328574.00 1555990.00 1082541.00 563167.80 20- Non Metal 1200563 .00 2292502.00 1769470.00 1229382.00 242147.60 21- Pet & Coal 2162347 .00 926167.90 3411184.00 1984857.00 97171.82 22- Chemical 2742031. .00 3476874.00 3212891.00 2138725.00 403046.10 23- Mise Manuf 1010087 .00 2007977.00 516564.50 396391.60 242421.70 24- Construct 10933760 .00 28445830.00 2181945.00 1778180.00 2878616.00 25- Air Tran 1367491 .00 1692132.00 8393821.00 4595004.00 212626.50 26- ' Rail Tran 1869621 .00 7864505.00 13987650.00 8915836.00 517595.60 27- Water Tran 684819 .10 2143298.00 3753706.00 2347266.00 155554.60 28- Motor Tran 2134194 .00 4549567.00 510628.60 368375.10 564393.40 29- Urban Tran 157137 .90 1032930.00 730977.80 424415.60 108299.90 30- Storage 211371 .00 681682.40 1050035.00 641550.20 64194.48 31- Broadcast 116936 .50 1007010.00 235760.80 170357.50 116394.30 32- Telephone 2035060 .00 3142273.00 7519990.00 4786820.00 374628.70 33- Elec power 1767009 .00 1989985.00 22302170.00 11813020.00 265463.00 34- Gas distr 350960 .50 416304.30 1760315.00 1045469.00 43755.53 35- Trade 11390570 .00 34747460.00 18762780.00 13102890.00 4312837.00 36- F I R E . 10705730 .00 12361140.00 102504100.00 68110660.00 2285561.00 37- Comm serv 7716469 .00 19239050.00 2598009.00 1951113.00 3352658.00 T A B L E 9.2- Means across Sectors Year Output - Y Capital - K(a) Capital - K(b) Capital - K(c) Labor - L 1961 1642680.0 2533425.0 2789697.0 2789697.0 506685.1 1962 1755883.0 2622610.0 2837162.0 2787474.0 524724.2 1963 1861661.0 2723011.0 2895623.0 2752503.0 536123.8 1964 2010009.0 .2812692.0 3023715.0 2749513.0 558007.0 1965 2164515.0 2960806.0 3216847.0 2723642.0 588838.8 1966 2312821.0 3210153.0 3627075.0 2861232.0 609050.0 1967 2370271.0 3435889.0 3963170.0 3048790.0 613367.6 1968 2494181.0 3692690.0 4293856.0 3423956.0 610934.7 1969 2628863.0 3989560.0 4782943.0 3777756.0 626060.1 1970 2656133.0 4290493.0 5154766.0 4157909.0 621068.5 1971 2819883.0 4655945.0 5224746.0 4113058.0 628846.6 1972 3008489.0 5007325.0 5571374.0 4266830.0 650475.6 1973 3264223.0 5572651.0 6253742.0 4171212.0 685908.4 1974 3389922.0 6357771.0 8053072.0 4734743.0 717936.8 1975 3342998.0 7278069.0 9632661.0 5161620.0 720217.6 1976 3531299.0 8254015.0 • 10829760.0 5601196.0 729627.9 1977 3633168.0 8936041.0 11596460.0 5509505.0 738760.6 1978 3795389.0 9461453.0 12896630.0 6553068.0 763350.4 1979 3996666.0 10054900.0 14803780.0 8643052.0 791127.3 1980 4057541.0 11457640.0 17917160.0 11841410.0 805930.8 125 T A B L E 9.3- Correlations over Time Sectors Cor[Y,Lj Cor[Y,K(a)l Cor[K(a),L] Cor[Y,K(b)] Cor[K(b),Lj Cor[Y,K(c)j Cor[K(c),L] 1- Agr & Fish -0.8244 0.9062 -0.7420 0.8714 -0.6284 0.8283 -0.5928 2- Forest -0.7300 0.8651 -0.7730 0.8658 -0.7210 0.8447 -0.6695 3- Mines,etc 0.8931 0.8411 0.9150 0.8193 0.9190 0.7522 0.8713 4- Food & Bev 0.1143 0.9252 -0.0344 0.9108 0.0079 0.8289 0.0727 5- Tobacco -0.9197 0.8442 -0.9341 0.7606 -0.8732 0.4833 -0.6398 6- Rub & Plas 0.9860 0.9312 0.8928 0.8715 0.8331 0.7833 0.7412 7- Leather -0.6312 0.7920 -0.9022 0.7985 -0.8614 0.7670 -0.7720 8- Textile -0.2352 0.8737 -0.5641 0.8409 -0.5488 .0.7534 -0.4485 9- Knitting -0.3391 0.8784 -0.7158 0.8319 -0.7059 0.6238 -0.4881 10- Clothing 0.1571 0.9203 -0.2089 0.9112 -0.1770 0.6883 -0.1537 11- Woods 0.9338 0.9555 0.8546 0.9417 0.8404 0.8529 0.7282 12- Furn & Fix 0.9468 0.8090 0.6920 0.7661 0.6691 0.6960 0.6144 13- Paper,etc 0.9115 0.8746 0.6946 0.8438 0.6842 0.8105" 0.6832 14- Printing 0.9663 0.9824 0.9234 0.9458 0.9210 0.8002 0!8452 15- Pri Metals 0.9361 0.7844 0.7156 0.7452 0.7112 0.7189 0.7032 16- Metal Fab 0.9416 0.8723 0.7408 0.8231 0.7012 0.7346 0.6205 17- Machinery 0.9678 0.9460 0.8480 0.9414 0.8471 0.8473 0.7656 18- Trans Eqp. 0.9354 0.8707 0.7490 0.8044 0.6984 0.6116 0.5718 19- Elec Prod 0.6605 0.8271 0.1844 0.8185 0.2042 0.7513 0.2421 20- Non Metal . 0.8206 0.8626 0.4908 0.8167 0.4502 0.7446 0.3925 21- Pet & Coal 0.8641 0.9455 0.8795 0.9177 0.9131 0.7382 0.8455 22- Chemical 0.9176 0.9539 0.8014 0.9322 0.7811 0.8272 0.6874 23- Mise Manuf 0.9389 0.8880 0.7584 0.8438 0.7205 0.7812 0.6578 24- Construct 0.9006 0.9598 0.8388 0.8851 0.7979 0.8497 0.7500 25- Air Tran 0.9844 0.9398 0.9661 0.9259 0.9645 0.7150 0.7827 26- Rail Tran -0.9210 0.8901 -0.8739 0.8674 -0.8181 0.7906 -0.6867 27- Water Tran -0.8377 0.7576 -0.8299 0.7481 -0.7967 0.6950 -0.7091 28- Motor Tran 0.9844 0.9493 . 0.9451 0.9299 0.9227 0.8389 0.8073 29- Urban Tran 0.8874 0.9051 0.9802 0.8948 0.9601 0.7166 0.6536 30- Storage -0.2178 0.8686 -0.5256 0.8474 -0.4990 0.7919 -0.3559 31- Broadcast 0.9949 0.9947 0.9896 0.9843 0.9821 0.9036 0.8977 32- Telephone 0.9747 0.9973 0.9600 0.9942 0.9528 0.9031 0.8189 33- Elec power 0.9702 0.9657 0.9129 0.9599 0.9306 0.6923 0.6209 34- Gas distr 0.3081 0.9458 0.5034 0.8983 0.5865 0.7500 0.5522 35- Trade 0.9951 0.9485 0.9620 0.9083 0.9321 0.8212 0.8495 36- F.I.R.E. 0.9958 0.9858 0.9724 0.9760 0.9603 0.9207 0.8920 37- Comm serv 0.9973 0.9930 0.9873 0.9644 0.9610 0.8580 0.8503 T A B L E 9.4-- Correlations across Sectors Year Cor[Y,L] Cor[Y,K(a)] Cor[K(a),Lj Cor[Y,K(b)] Cor[K(b),L] Cor[Y,K(c)] Cor[K(c),Lj 1961 0 .8939 0.0572 0.1829 0.5577 0.3610 0.5577 0.3610 1962 0 .8937 0.0471 0.1685 0.5581 0.3647 0.5562 0.3624 1963 0 .8931 0.0432 0.1708 0.5530 0.3668 0.5540 .0.3676 1964 0 .8961 0.0464 0.1746 0.5418 0.3576 0.5458 0.3618 1965 0 .8943 0.0438 0.1681 0.5297 0.3395 0.5394 0.3493 1966 0 .8975 0.0441 0.1590 0.5156 0.3272 0.5301 0.3407 1967 0, .8987 0.0395 0.1430 0.5092 0.3415 0.5289 0.3603 1968 0 .8969 0.0553 0.1625 0.5092 0.3608 0.5246 0.3750 1969 0. .8902 0.0358 0.1423 0.5004 0.3626 0.5154 0.3761 1970 0 .8934 0.0398 0.1368 0.5144 0.3678 0.5263 0.3784 1971 0 .8905 0.0411 0.1349 0.5049 0.3748 0.5207 0.3892 1972 0 .8935 0.0438 0.1365 0.5025 0.3830 0.5166 0.3960 1973 0 .8904 0.0229 0.1100 0.4913 0.3732 0.5093 0.3895 1974 0 .8935 0.0177 0.0970 0.4943 0.3792 0.5113 0.3950 1975 0 .8986 0.0196 0.0852 0.5080 0.3821 0.5254 0.3979 1976 0. .8993 0.0419 0.1123 0.5062 0.3831 0.5221 0.3978 1977 0. .8940 0.0258 0.0960 0.5236 0.3941 0.5410 0.4114 1978 0. .8905 0.0196 0.0924 0.5347 0.3964 0.5471 0.4105 1979 0. .8935 0.0046 0.0712 0.5417 0.3909 0.5497 0.4013 1980 0. .8999 0.0108 0.0817 0.5531 0.3941 0.5571 0.3999 126 variable most highly correlated with output. The differing levels of capitalization across sectors imply that capital is not a good indicator of sectoral output. 9.2. Average-Representative Firm's Bias A first series of empirical evaluations serves to find: how far is the output prediction of the average-representative firm from the "true" output aggregate? As pointed out earlier, the magnitude of the average output aggregation bias is related both to non-linearities in functional form and to the distribution of the production characteristics. 9.2.1. Linear Production Function First, a linear production function of capital and labor was estimated for each firm using the following model: yft = a{ + c/klt + a/ll{ + eft, t=l,...,T, (9.1) where e{ is a normally distributed random variable (over time) with E(e )^ = 0 and E(tfV) = a2 I. These assumptions about the disturbances will be maintained for all sectors, data sets and functional forms. It will facilitate the interpretation of the biases and allow the incidence of alternative rental prices on the error structure to be examined. The rental prices will be found to be responsable for mild divergences from these assumptions. However, the objective here is to assess the importance of the biasedness of the average-representative firm. Ideally, this evaluation would rest on knowledge of the true technologies: B^ (t) = E( Yt) — Ytr, where Yf is the output prediction of the average-representative firm. In the absence of such knowledge, the bias related to the estimated technologies will be computed, B^,{t) = E(Yj) - Yrt, where Yrt is the best prediction available based on the technological parameters. Whereas predictions that include of a more sophisticated specification of the error structure were marginally more accurate, the component attributable to the specification of the production technologies accounted for less than 50% of the prediction. The corresponding technological coefficients could then not be seen as providing a good description of the technologies. Let Yt, Kt, Lt, Ai, Ak, A\ be the random (across firms) variables whose realizations are {y},---,yf), (#, -..,kf), ...,lf), {&\, ••.,&{), [a\, ...,aF), (&}, ...,&f), 127 respectively. Figure 9.1 and 9.2 illustrate the marginal distributions of Ak and Kt, and of A\ and Lt (t=1970), for the three data sets analyzed, as well as their joint distributions. The joint distributions are pictured with cube-histograms. The height of the cubes indicates the number of sectors whose parameter and factor input fall in the intervals indicated on the corresponding walls. The marginal distributions are projections in the parameter and input planes of the joint distributions; they take the form of blank bar-histograms contrasting the grilled background. The expectation (across sectors) of estimates of (9.1) may be written equivalently as: E( Yt) = E(#0 + E{AkKt) + V{A,Lt) + E(e() (9-2) = E ( Y t ) + E(et), and the estimated production function of an average-representative firm is given by: Yrt = E(Ai)+E{Ak)E{Kt)+-E(A,)E{Lt). (9.3) The resulting output aggregation bias then can be estimated equivalently using: B^t) = Cov (4 Kt) + Ca*{Ah Lt) = E( f t) - % (9.4) Table 9.5 gives these bias estimates for the three data sets considered, which differ in their valuation of the capital inputs. The first terms, E(Y t) - E ( Y t ) , measure the average (across sectors) fitting error. It is acceptably small (usually less than 5%) when the capital aggregate is based on asset prices or on the rental prices with smoothed expectations. The corresponding residuals exhibit patterns of serial correlation, each with different cycle lengths. The pattern of residuals is quite different when the capital aggregate is formed using rental prices with static expectations and the average fitting error reaches 12% in 1977. It is instructive to note that alternative capital prices modify the error structures in non-trivial ways. The second terms, E( Yi) — Y\— B^t), is the output aggregation bias of an average-representative firm. Unexpectedly, the aggregation bias is found to be negligible (in the order of the fitting error) with the first data set. As illustrated in figure 9.1 and 9.2, the covariance between the factor inputs and their respective parameters is not important. In case a), where the bias is small, the joint distributions are more randomly spread out than 128 T A B L E 9.5- Average Representative Firm's Bias: Linear Production Function Using Capital Inputs Based on Years E{Yt) - E{?t) 100% E(Yt)-Yrt 100% a) Asset Prices 1961 -87784.1250 -0.053 42958.3203 0.025 1962 -75981.6250 -0.043 52520.9609 0.029 1963 -57102.3984 -0.031 74427.7500 0.039 1964 -44166.4883 -0.022 101356.8750 0.049 1965 -82752.7500 -0.038 138167.0000 0.061 1966 -96704.1875 -0.042 173927.8125 0.072 1967 -77259.0000 -0.033 157667.5000 0.064 1968 2831.0940 0.001 171411.6250 0.069 1969 37950.8398 0.014 158890.8750 0.061 1970 52615.5781 0.020 145477.6250 0.056 1971 147940.1875 0.052 122654.5000 0.046 1972 179045.6250 0.060 131431.1875 0.046 1973 188378.5000 0.058 135819.8125 0.044 1974 93395.8125 0.028 94445.3125 0.029 1975 -12019.8984 -0.004 -2256.4939 -0.001 1976 10077.9805 0.003 -30764.2617 -0.009 1977 9358.2539 0.003 -75265.7500 -0.021 1978 -9273.8555 -0.002 -83386.3125 -0.022 1979 -11793.0117 -0.003 -93046.7500 -0.023 1980 -166759.0000 -0.041 -162575.8750 -0.038 b) Rental Prices with Static Expectations 1961 -74740.1875 -0.045 -361263.5000 -0.210 1962 -66194.1250 -0.038 -343017.3750 -0.188 1963 -43483.8203 -0.023 -305326.6875 -0.160 1964 -36339.2891 -0.018 -268865.3750 -0.131 1965 -88411.2500 -0.041 -203907.8125 -0.091 1966 -98169.8750 -0.042 -181173.5000 -0.075 1967 -67909.6250 -0.029 -226382.1875 -0.093 1968 23008.1797 0.009 -284807.3125 -0.115 1969 62490.1406 0.024 -359851.8125 -0.140 1970 84743.3750 0.032 -435280.1250 -0.169 1971 208631.1250 0.074 -420608.0000 -0.161 1972 265301.8750 0.088 -435292.1875 -0.159 1973 325124.8125 0.100 -384019.6875 -0.131 1974 266914.8750 0.079 -509562.1875 -0.163 1975 251890.0000 0.075 -669836.3750 -0.217 1976 370959.6250 0.105 -766863.5000 -0.243 1977 448277.8750 0.123 -761158.6250 -0.239 1978 446776.5000 0.118 -1002991.0000 -0.300 1979 374640.8750 0.094 -1438226.0000 -0.397 1980 188176.6250 0.046 -2141702.0000 -0.554 c) Rental Prices with Smoothed Expectations 1961 -67479.5000 -0.041 -962233.1250 -0.563 1962 -73060.0625 -0.042 -949612.8125 -0.519 1963 -59135.1602 -0.032 -911224.6250 -0.474 1964 -63611.4102 -0.032 -887065.3750 -0.428 1965 -127160.3125 -0.059 -841501.6250 -0.367 1966 -158703.8750 -0.069 -837452.3125 -0.339 1967 -150690.1875 -0.064 -889766.6250 -0.353 1968 -70432.4375 -0.028 -983944.8750 -0.384 1969 -53080.5117 -0.020 -1099984.0000 -0.410 1970 -37215.0117 -0.014 -1213248.0000 -0.450 1971 78239.3750 0.028 -1192913.0000 -0.435 1972 126689.8125 0.042 -1243541.0000 -0.432 1973 164720.8750 0.050 -1197328.0000 -0.386 1974 72224.9375 0.021 -1397948.0000 -0.421 1975 55408.2305 0.017 -1614851.0000 -0.491 1976 154720.0000 0.044 -1760125.0000 -0.521 1977 217160.8125 0.060 -1737686.0000 -0.509 1978 178363.1250 0.047 -2106166.0000 -0.582 1979 48198.6211 0.012 -2787855.0000 -0.706 1980 -235156.8750 -0.058 -3829614.0000 -0.892 129 a) Asset Prices b) Rental Prices with Static Expectations c) Rental Prices with Smoothed Expectations Figure 9.1 - Joint Distributions of Ak and Kt (t=1970) ; Linear Production Function, Using Capital Inputs based on 130 a) Asset Prices b) Rental Prices with Static Expectations c) Rental Prices with Smoothed Expectations Figure 9.2 - Joint Distributions of At and Lt (t=1970) : Linear Production Function, Using Capital Inputs based on in cases b) and c). For the two other data sets, the biasedness of the average-representa-tive firm seems no longer acceptable, since it ranges between 7% and 55% in case b) and between 34% and 89% in case c). The illustrations of the joint distributions clearly show that the covariance is much more important in the two latter cases. If 5% was considered to be an acceptable level of biasedness, it would be reached if C o v ( 4 Kt) + Cov(Ah Lt) = j ^ f ^ [E(2i) + V(AK)E(Kt) + E ( £ , ) E(L t)] • (9.5) This is, unfortunately, not a very intuitive rule for bias acceptability. 9.2.2. Quadratic Production Function In another set of estimations, a quadratic production function of capital and labor was fitted to the data using the model: ^ = a{ + a ^ + a{/{ + a^ (^ ) 2 + 4 ^ + a ^ ) 2 + e { ; t = 1, . .T, (9.6) where e{ is a disturbance term. Estimates of these parameters show that in most cases, the production function was concave at the observed values. Exceptions were in case a), the Tobacco (5), Broadcasting (31) and Telephone (32) sectors; in case b), the Agricultural (1), Tobacco (5) and Petroleum sectors; in case c), the Agricultural (1), Water (27) and Motor Transportation (28) sectors. Let Yt, Kt, Lt, Ai, Ak, Ai, Akk, Au and Au be the random (across firms) variables whose realizations are (yj,..., yf), (k], . . . , & f ) , •••Jt), (pt\, •••,aF), (6t\, . . . , a f ) , {a}, •••>&?)> •••>«»). (&1u> •••>6iu) a n d ( " J ; • • • • ; « f l ) i respectively. Then, the average (across sectors) output aggregate is given by: E(Yt) = E ( £ x ) + E(AkKt) + V(A,Lt) + E ( ^ X t 2 ) + +E(AaKtLt) + V(AaL2t) + E(et) = E ( Y t ) + E(et), (9.7) and the estimated production function of an average-representative is: Yrt = E ( ^ ) + EfA)E(f f t ) + E(^)E(I t ) + E ( ^ ) E ( X 2 ) + E ( £ H ) E ( K i ) E ( I t ) + E ( £ „ ) E ( £ 2 ) . (9.8) 132 T A B L E 9 . 6 - Average Representative Firm's Bias: Quadratic Production Function Using Capital Inputs Based on ears ' E(Y0 - E ( y t ) 100% E ( Y t ) - Y\ 100% a) Asset Prices 1961 -46853.8906 -0.029 -1268885. -0.751 1962 -32206.3086 -0.018 -1345250. -0.752 1963 -12635.0195 -0.007 -1390618. -0.742 1964 19863.6211 0.010 -1488729. -0.748 1965 -7.9237 -0.000 -1636300. -0.756 1966 -22320.0391 -0.010 -1749285. -0.749 1967 -34302.0898 -0.014 -1813313. -0.754 1968 9750.1758 0.004 -1815476. -0.731 1969 14225.0703 0.005 -1942041. -0.743 1970 -5067.9570 -0.002 -1961539. -0.737 1971 71287.3125 0.025 -2079615. -0.757 1972 96315.6250 0.032 -2251253. -0.773 1973 102046.1250 0.031 -2556471. -0.808 1974 -8615.2227 -0.003 -2912775. -0.857 1975 -117987.1875 -0.035 -3115221. -0.900 1976 -61220.4414 -0.017 -3298410. -0.918 1977 -39305.1797 -0.011 -3442879. -0.937 1978 -24043.5781 -0.006 -3685183. -0.965 1979 47458.4219 0.012 -3996711. -1.012 1980 37903.9219 0.009 -4221356. -1.050 b) Rental Prices with Static Expectations 1961 -57265.2188 -0.035 -7448398. -4.382 1962 -39532.2813 -0.023 -7783198. -4.335 1963 -10531.1211 -0.006 -8047419. -4.298 1964 8861.0547 0.004 -8625179. -4.310 1965 -21240.3789 -0.010 -9504822. -4.349 1966 -53932.6406 -0.023 -10679650. -4.512 1967 -55447.5898 -0.023 -11514730. -4.747 1968 8207.2617 0.003 -12245960. -4.926 1969 -3654.4629 -0.001 -13747420. -5.222 1970 -26192.1484 -0.010 -14696180. -5.479 1971 107279.0000 0.038 -15027890. -5.540 1972 153499.3750 0.051 -16389040. -5.740 1973 170770.0000 0.052 -19117152. -6.180 1974 -17184.9414 , -0.005 -26350944. -7.734 1975 -130693.6250 -0.039 -33709600. -9.704 1976 -32919.1406 -0.009 -40166208. -11.269 1977 -16031.9102 -0.004 -44660688. -12.238 1978 -37386.8008 -0.010 -52925968. -13.809 1979 26597.2500 0.007 -66473264. -16.744 1980 25058.4297 0.006 -92309216. -22.891 c) Rental Prices with Smoothed Expectations 1961 -81108.4375 -0.049 5601232. 3.249 1962 -43197.8984 -0.025 4748141. 2.639 1963 -7127.6211 -0.004 3700516. 1.980 1964 37402.9492 0.019 2528846. 1.282 1965 18912.9883 0.009 411646. 0.192 1966 -14908.4297 -0.006 1429910. 0.614 1967 -37367.3594 -0.016 4356337. 1.809 1968 -20213.0781 -0.008 11400300. 4.534 1969 -73346.5000 -0.028 17750288. 6.569 1970 -125342.3750 -0.047 26395152. 9.490 1971 -8804.5820 -0.003 25010672. 8.842 1972 56430.0117 0.019 27481712. 9.309 1973 137323.1875 0.042 23243104. 7.433 1974 18528.0898 0.005 35619152. 10.565 1975 -37030.4688 -0.011 47366928. 14.014 1976 57492.9492 0.016 60121264. 17.307 1977 139710.5000 0.038 56784368. 16.254 1978 10156.7383 0.003 90751056. 23.975 1979 -57603.4414 -0.014 176731904. 43.592 1980 18772.1406 0.005 351570688. 87.049 133 The output aggregation bias of the average-representative firm can be estimated using: Btly{t) = V(Yt)-Y't. (9.9) An equivalent expression in terms of variances-covariances has been obtained by applying decomposition (5.14) to (9.7) and by substracting (9.8) from it. However, it is overly cum-bersome and has been left out. Table 9.6 lists the bias estimates for the three alternatives capital aggregates. In all cases, the fitting error is reduced, sometimes substantially. But the average output aggregation bias increases dramatically as expected. In the worst case, it is almost 100% greater than with the linear technology. Interestingly, the variability of the aggregation bias increases dramatically also, as if the average-representative firm was at the outskirts of a confidence band for Y t-A separable quadratic production function was also fitted to the data. The resulting bias estimates were similar to the non-separable case. If the covariance terms in the population of characteristics are important, the imposition of separable functional form will not eliminate them; it will simply displace them. In a nutshell, these results show that if the production technology is even slightly non-linear, chances are that the average-representative firm's approach will not hold. If the production technology is linear, however, it is empirically tenable that an average-rep-resentative firm describes the aggregate level of output with reasonable accuracy (within the fitting error). 9.3. Exact-Aggregation Macro Parameters with Stable Microfoundations The second issue to be investigated empirically concerns the time-inconsistency of exact-aggregation macro parameters. In the previous average-representative firm problem, the macro parameters were derived from the micro parameters. Here, the macro parameters are estimated first using a macro relation and then their relation to the micro parameters and variables is established. Let the macro production function be a linear function of input and output aggregates, denned in terms of averages: E( Yt) = a f + a f E(Xt) + af E(Lt) + $. (9.10) 134 T A B L E 9.7- Macroparameters with Stable Microfoundations : (75-76-77) Time Drift Coefficients Linear Production Function, Using Capital Inputs Based on Years ay aj. otj a) Asset Prices 1961 -7845641. 0.024368 15.5215 1962 -7520342. 0.022751 14.5652 1963 -7249876. 0.021354 13.9159 1964 -6787346. 0.019386 12.7904 1965 -6146339. 0.016863 11.3678 1966 -5681666. 0.015054 10.4967 1967 -5638895. 0.014544 10.4124 1968 -5624248. 0.014333 10.4967 1969 -5400527. 0.013491 10.0434 1970 -5512689. 0.013572 10.3179 1971 -5444699. 0.013202 10.1858 1972 -5043435. 0.012269 9.4724 1973 -4442130. 0.011160 8.4646 1974 -4024298. 0.010469 7.8063 1975 -4260461. 0.010762 8.1743 1976 -4181683. 0.010762 8.1743 1977 -4161088. 0.010762 8.1743 1978 -3789602. 0.010449 7.6578 1979 -3369891. 0.010131 7.1122 1980 -3313239. 0.010203 7.1604 b) Rental Prices with Static Expectations 1961 -6731753. 0.041961 12.9451 1962 -6531920. 0.035684 12.3611 1963 -6375397. 0.030621 11.9963 1964 -6107399. 0.022260 11.3541 1965 -5702581. 0.011085 10.4998 1966 -5426546. 0.003372 10.0204 1967 -5419978. 0.002475 9.9871 1968 -5423149. 0.001759 10.0834 1969 -5319302. -0.000866 9.8530 1970 -5390619. -0.000425 10.0530 1971 -5310688. -0.001615 9.8806 1972 -5070620. -0.005098 9.4371 1973 -4700071. -0.009639 8.8018 1974 -4530927. -0.010823 8.5727 1975 -4758899. -0.008564 8.9079 1976 -4721931. -0.008564 8.9079 1977 -4717786. -0.008564 8.9079 1978 -4611146. -0.009264 8.7739 1979 -4484292. -0.009883 8.6892 1980 -4634427. -0.009022 9.0464 b) Rental Prices with Smoothed Expectation 1961 -4216089. 0.003750 9.0781 1962 -4244187. 0.002797 9.0510 1963 -4263961. 0.002047 9.0708 1964 -4297307. 0.000749 9.0551 1965 -4356772. -0.000887 9.0599 1966 -4382079. -0.002571 9.1042 1967 -4350887. -0.003518 9.0753 1968 -4295981. -0.004422 9.1003 1969 -4253501. -0.005684 9.0101 1970 -4185565. -0.006121 8.9979 1971 -4202537. -0.006380 8.9916 1972 -4232387. -0.006928 8.9591 1973 -4318368. -0.007821 8.9435 1974 -4297503. -0.008731 8.8315 1975 -4241891. -0.008373 8.6871 1976 -4230969. -0.008373 8.6871 1977 -4271646. -0.008373 8.6871 1978 -4184645. -0.008809 8.5719 1979 -3903623. -0.009730 8.3680 1980 -3448230. -0.010143 8.1210 135 The estimated macro production function then is given by: E ( y t ) = d f + d f E(ff t) + d f E(L t ) or E ( ^ ) + E ( £ t K i ) + E ( £ , I t ) = d f + d f V{Kt) + d f E(L f ) . (9.11) Using equation (9.11), the exact-aggregation macro coefficients can be described by the following family of microfoundations: where A,(<) + 7,(() + <5,(£) = 1 , 2 = 1 ; 2, 3, for all t = 1,2,... , T. Using three time periods at a time, that is assuming that the microdefinitions of the macro parameters are identical over three year periods, equations (9.12) can be solved exactly for A, 7 and 6, provided that the corresponding solution matrix is non-singular. The resulting time-drift coefficients have the interesting property of showing corre-sponding components of their respective coefficients that are relatively stable and more important than the non-corresponding components. In that respect, they resemble Theil's solution (5.31). The effects of the 75-76 oil-shock on the macro parameters are illustrated in Table 9.7. Macro parameters with stable microdefinitions have been computed using the (75-76-77) set of time-drifts coefficients. The original macro parameters ( d f and d f ) are found under the corresponding headings in the years 75-76-77. In the cases a) and b), the macro parameters show important drifts outside the confidence intervals of the original parameters. Probably because of expectations smoothing, this anomaly does not appear in the third data set. When time-drift coefficients based on periods of relative economic stability are used, the exact-aggregation macro parameters with consistent microdefinitions do not fluctu-ate outside of their confidence intervals. Note that because the micro parameters were assumed stable over time, the time drifts in the macro parameters measure solely distri-butional effects. Models using aggregate data are known to give statistically acceptable "explanations" of aggregate data patterns. Would the orders of magnitude for the aggre-gation biases in the parameters found in the present analyses be confirmed by a wider Ax(i)E(^) + \2{t)V{AkKt) + \3{t)E{A,Lt), 7 i ( Q E ( ^ ) + -nW{AkKt) + l3(t)E{AiLt) E(A») gi(QE(^x) + 82{t)V(AkKt) + 83(t)E(A,Lt) (9.12) 136 array of tests, this result could be partially attributed to the the relative smallest of biases due to distributional effects. The potential biases due to structural model-changes effects cannot be analyzed in the present empirical setting. 137 Chapter 10. Conclusion The proposed stochastic interpretation of aggregation problems has proven to facilitate the understanding of their origin. But the present analysis merely opened a door on the possibilities to utilize this comprehension. This chapter first presents an example to suggest how the framework could be used to analyze the implications of potential aggregation biases for macroeconometric modeling and policy analysis. A summary of the main results of the dissertation, along with a discussion of the future steps of continuing research, follows. 10.1. Modeling Implications The potential aggregation biases identified above, have different implications whether aggregation is envisaged at the economy level or at the industry level, since the corre-sponding "macro" models serve different purposes. At the economy level, macroecono-metric models are used predominantly for policy evaluation, despite the Lucas critique (1976). The critique argued that since macroeconometric models generally do not capture the primitive (or, one could add the individual) tastes and technologies, their parameters cannot be expected to be stable under a policy change. At the industry level, the more im-portant questions focus on productivity measurement and industrial technology. At both levels, the reliability of the parameters is critical and the presence of significant aggre-gation biases could distort the policy recommendations in undesirable ways. As Geweke (1985) points out "ignoring the sensitivity of aggregators to policy changes seems no more compelling than ignoring the dependence of expectations on the policy regime" (p. 206). This remark could easily be extended to include the sensitivity of the macro parameters. In macroeconometric modeling, the potential aggregative problems are usually ignored. This stand is justified by conveniently assuming that representative agent models with exact aggregation can be constructed. However as shown in chapter 5, exact macro models, that are stable over time, generally may not be derived from micro models using a relation that is time-invariant. The exact-aggregation macro parameters do not depend uniquely on the micro parameters but also on the micro variables. Consequently, the macro parameters will generally be sensitive to policy changes as shown in the following example. 138 10.1.1. Consumption Function, Policy Changes and Aggregation Biases Departing from the producer's context to pick the simplest meaningful macro model, consider a simple textbook consumption function, where consumption is proportional to disposable income. 1 Assume that the macro consumption behavior is representative of the behavior of individual agents in the sense that it leads no total consumption bias. More formally, let each individual consumption behavior follow the model: c ' W y , i=l,2,...,n, (10.1) at a particular point in time (the time subscripts are omitted for clarity), where c1 is the actual consumption of the i-tb. individual, y' is disposable income, k' is the individual's marginal propensity to consume and n is the number of individuals. Let C, K and Y be the random (across individuals) variables whose realizations are (c1,...,cn), (k1,..., kn) and (y1,..., yn), respectively. The full-information aggregate consumption function is then, following an analog to definition (5.5): C = n E ( C ) = nE{KY), (10.2) where aggregate consumption C is evaluated as the total, ]C£=i c '- The macro consumption function, that is, the function of aggregate income, can be written as n E ( C ) = * [ n E ( Y ) ] (10-3) or C = ky, where aggregate income y is evaluated as the total, X^JLi v\ a n < l where k is the macro marginal propensity to consume. Since the total consumption aggregation bias is assumed to vanish, the macro function (10.3) equals the true aggregate (10.2) and the macro parameter k satisfies: S _ « E ( * T ) C o v ( X , y ) * - n E ( Y ) - E ^ + E ( Y ) ' ( 1 ° - 4 ) If the term Cov(K, F ) / E ( Y ) varies over time, say because of a general increase in the standard of living with no correction in consumption habits, the macro marginal propensity to consume will vary over time. Interestingly, one of the tasks of the literature on the 1 This simplification is made to separate the biases related to policy changes from the time-drift problems. 139 consumption function has been to try to explain the empirical instability of the macro marginal propensity to consume. This decomposition could be used fruitfully for that purpose. Now, assume that there is a policy change consisting of an increase in income of xi. If the increase in income results from a reduction in the income tax rate, it is reasonable to assume that the increase x* varies across individuals. Even if it were not to vary, the same bias implications would follow. Let X be the random variable whose realizations are (a:1,... , a;n), then income Y can be replaced by Y + X and the macro function becomes: C = nE{C) = k[nE{Y + X)] = knE{Y) + knE{X), (10.5) where k is the macro marginal propensity to consume after the policy change. It predicts an increase in total consumption of k riE(X). For there to be no aggregation bias in total consumption, the macro function has to equal the full-information consumption aggregate and the macro marginal propensity to consume needs to satisfy the analog to expression (5.25), implying: ? nE[K(Y+X)} Cov(KlY) + Cov(K,X) k - nE(Y+X) ~ E W + E(Y) + E(X) " (1°-6) Under a policy change, the macro parameter k that leads no total consumption bias, without further assumptions, will be different from the exact-aggregation parameter k without the policy. In effect, the bias in the marginal propensity to consume is: = E(X)Cov(K,Y)-E(Y)Cov(K,X)  k E(Y)[E(Y) + E(X)] ' / " 1 If the policy change corresponds to a transfer, say from the rich to the poor, such that E(X) = 0, and is not related to the individuals' marginal propensity to consume, such that Cov(if, X) = 0, then the bias in the parameter will vanish. If the more general assumption of the stochastic independence of the marginal propen-sity to consume and income is held, then the macro parameters become: ~ nE(K)E(Y)_ ? n[E(K)E(Y)+E(K)E(X)] k~ nE(Y) ^ k~ »[E(Y) + E(X)] " ^ ^ The parameters are now invariant under a policy change. However, such an assumption is empirically contradicted by the well-known heteroscedasticity problem. Note that requir-ing that the marginal propensity to consume be the same for all individuals, is a special 140 case of stochastic independence between income and the marginal propensity to consume. It is theoretically not "the only case" in which the parameters of an exact-aggregation macroeconomic equation are solely determined by individual behavioral parameters, con-trary to Stoker's affirmation (1986b, p. 768). 2 Zellner's (1969) random coefficient model studied in section 5.6.1 provides a counter-example. For example, let the marginal propensities to consume of individuals, kl, vary around a mean k in relation with their attitude towards risk 5' . In terms of the corresponding random variables: K — k-\- D. Assume further that their attitude towards risk does not depend on their level of income, such that Cov(P, Y) = 0, and that the risk-prone responses are as important as the risk-averse, such that E(P) = 0. Then, the exact-aggregation macro marginal propensity to consume can be derived from the individual behavioral parameters even if these are not equal for all individuals: If an exact-aggregation representative agent approach is adopted, policy changes may be assumed to lead to shifts in the macro parameters, unless the policy changes are stochas-tically independent from the functional parameters and lead no total distributional changes. The more stringent condition requiring that the functional parameters be stochastically in-dependent from the microvariables also will entail macro parameters that are stable under a policy change. If exact aggregation is not required and the macro parameter is chosen to be say k", obviously there will be an aggregation bias in total consumption equal to: Geweke (1985) has suggested that the exact aggregation approach does not always need to be preferred to an approach that is unaffected by the policy regime. However, ignoring the total consumption aggregation bias would seem justifiable only if its magnitude was known to be very small, i.e. in the order of the fitting error. At this point, it is thus proposed that macromodelers perform simulated tests to evaluate the magnitude of this bias. 2 Stoker's remark may have been a voluntary simplification, since he had examined the case of random-coefficient micro models previously (Stoker, 1984, p.890.). C = nE(KY) = nE[(A+ D)Y) = k[nE(Y)\. (10.9) (10.10) 141 Because it includes only one macro parameter that is determined completely at any point in time, this simple consumption example oversimplifies the empirical issues of bias evaluation. When more than one parameter is involved, these will be completely determined over a number of time periods. Thus, the time-drifts problems singled out in section 5.5 will intermix with the biases due to policy changes. A more thorough examination of these perverse implications for macroeconometric modeling will be the subject of future work. However, this simple example clearly shows that such implications should not be ignored. 10.2. Main Results The present research has provided answers to many aggregation puzzles that until now were mere observations. Aggregation problems have been reformulated in a stochastic framework, which admits the generalization of traditional exact-aggregation solutions. The provision of a model which enables empirical analysts and macro modelers to analyze and, ultimately, evaluate the magnitude of potential aggregation errors is an equally important product. In effect, the present empirical analyses have help put aggregation biases in perspective. The application of the framework to a simple macro model has illustrated the effects of aggregation distortions on macroeconometric policy evaluation. The use of a stochastic setting has yielded a variety of results. First, a moments decomposition of "true" aggregate relations provided an alternative explanation to why functional-form restrictions, that impose the equality of parameters across agents, may lead to exact aggregation. These functional-form restrictions of exact-aggregation models (Gorman, 1953, 1968 a; Blackorby and Schworm, 1984, 1988) can be interpreted as spe-cial cases of more general stochastic restrictions. For example, Gorman's (1953) condition for a representative consumer to exist requires the equality across agents of the marginal propensity to consume. This condition now can be seen as a special case of a stochastic restriction requiring that the covariance between utility and the marginal propensity to consume be null. In the case of an average-representative firm, empirical results have con-firmed that covariance terms between variables and coefficients can be negligible, thereby relaxing in a concrete way the famous stringency of exact-aggregation conditions. How-ever, the relative smallness of these sources of bias is an empirical matter and cannot be 142 assumed a priori. Nevertheless, empiricists with good hand knowledge of their data may be able to hold informed opinions about this magnitude. The moments decomposition also can be used to show how additive separability con-ditions (Nataf, 1948; Gorman, 1968a; Blackorby and Schworm, 1984, 1988) simplify the aggregation biases by eliminating covariance terms between variables. However, since the separability restrictions do not require that variance (or higher moments) terms vanish, in a stochastic framework, they no longer entail the annihilation of aggregation biases, if non-linear functional forms are considered. These exact-aggregation conditions are not suf-ficient, if traditional aggregates are used. Furthermore, they are no longer necessary since the annihilation of covariance components is a weaker sufficient condition. While addi-tive separability is a testable hypothesis that implies simpler aggregation biases, empirical tests show that the imposition of a separable functional form will not reduce the output aggregation bias; it will simply displace it. However, if separability is not be rejected, it implies that the underlying covariances are null and aggregation is possible. But, it is too strong a condition to imposed on disaggregated units. In effect, an important implication of a distributional understanding of aggregation structures is that consistent aggregation does not depend uniquely on the functional form of individual relations. Properties of the distribution of individual characteristics can validate the aggregation of functional forms that otherwise could not be aggregated exactly. The Gorman (1968 a) conditions which embody the above two forms of restrictions — commonality across firms of specific marginal components and separability in the capital goods — are another example of exact-aggregation restrictions that need to be reevalu-ated. First, a zero-covariance condition between appropriate components can replace both commonality and separability conditions. This weaker condition generalizes the linear in-dependence condition found in Lau's (1982) fundamental theorem of exact-aggregation. Secondly, the Gorman conditions use totals of firm-valued capital inputs as aggregates which require perfect knowledge of the micro relations. Thus, if the available aggregates are the usual totals and averages, this special exact-aggregation form cannot be utilized. Modeling aggregation problems in an environment where the distribution over indi-viduals matters, enlarges the set of exact (and approximate) aggregation possibilities. A particular stunning set of circumstances that minimized aggregation biases is given by 143 economic diversification. Even if understood outside of a strictly aggregative context, it may have been foreseen by Cournot (1838) as a condition that guaranteed the feasibility of market demand functions: But the wider the market extends, and the more the combinations of needs, of fortunes, or even of caprices, are varied among consumers, the closer the function F(p) [the demand function] will come to varying with p in continuous manner. (Cournot, 1838, p.50). [emphasis added] Interpreted stochastically, assume that the individual demand functions, di(p,xl) = hi(p)g\xi) i=l,...,n, (10.11) can be factored into a component that is price-dependent and a component that is attrib-ute-dependent, where z* is a set of individual attributes and caprices. Let H{p) and G(X) be the random variables whose realizations are (/i 1(p) ;... , hn(p)) and (/(a; 1),... , gn(xn)). Then, under the assumptions that E[G(a;)] = x and that the attributes and caprices are varied enough to ensure that Cov[H(p), G(X)] = 0, the "true" aggregate demand function will be a function of price only, as long as the average attribute remains the same: V = E\H(p) • G(X)} = xE[H(p)} = F^p). (10.12) The diversity assumption allows the construction of demand functions that are robust to mean-preserving changes in tastes and incomes, which relax somewhat the traditional ceteris paribus assumptions. Thereafter, aggregation theorists singled out similarity among agents as the only condition that facilitates the description of market behavior. 3 Whether diversity in this extreme form is more likely to be observed in the market than the extreme similarity demanded by exact-aggregation models is questionable. But if approximate satisfaction leads to aggregation biases of acceptable magnitude, this result may widen the conditions under which aggregation is reasonable. Another set of results concerns the connections between the parameters of macro rela-tions (optimized or not) and their microfoundations. In the linear case, these connections were studied by Theil (1954, 1971) at the sample level; here, they were given population 3 Cf. Chapter 1.2, 10.1 144 equivalents. A general exact-aggregation manifold relating micro characteristics and macro parameters was derived; it applies to linear as well as non-linear relationships. General conditions under which macro parameters possess stable microfoundations were obtained. Fisher's (1971) simulation findings can be interpreted as an application of those condi-tions: under constant returns to scale, the stability over time of exact-aggregation macro Cobb-Douglas parameters is shown to be insured by the relative constancy of labor's share of total output over time. Unless the population of inputs changes over time in proportion to their productivity, exact aggregation implies that the macro parameters will be unstable if their microdefini-tions remain the same over time. However, if this condition is not satisfied, empirical anal-yses have shown that there are many instances where the diverging microfoundations will have no important effects. In the present empirical analyses, the exact-aggregation macro parameters with stable microdefinitions do not fluctuate outside the confidence intervals of the original parameters when the related time-drift coefficients are based on periods of relative economic stability. This indicates that the biases in the exact-aggregation macro parameters due to distributional changes in the micro variables were not important in those instances. However, this result does not hold in the presence of shocks (e.g. the 75-76 oil shock). The distortions brought about by distributional effects alone, are sufficiently significant to state that macro models estimated prior to important disturbances will not predict accurately the aftermath. In a stochastic framework, the aggregation of optimized relations is not intrinsically different from the aggregation of non-optimized relations. When all agents are assumed to face the same prices, these emerge as a simplifying device to gather individual allocative decisions. Economic relations that run in term of common prices can be aggregated more easily. Thus, perfect competition will reduce aggregation problems considerably. However, it is not a panacea. If the ultimate aim is the construction of a microfounded macro function, perfect competition alone will not guarantee that the macro parameters can be derived from the corresponding micro parameters. 4 In particular, it does not necessarily imply that an average-representative firm will be an exact-representative. Furthermore, the time-consistency of the exact-aggregation macro parameters may be questioned even 4 As demonstrated in section 6.4.1 145 under perfect competition, unless the variations in profit, input and output prices are all proportional. For example, if labor and output prices have different inflation rates, the exact-aggregation parameters of the corresponding macro profit function will be unstable over time. A potential benefit of perfect competition may occur in the long run. If competitive pressures have force firms to become alike, that is, to equate the marginal productivity of their entrepreneurial abilities, then the increased similarity will ease the derivation of a representative technology. The problem of capital aggregation is studied as an intermediate case between the aggregation of non-optimized and optimized relationships. Because of the interdependence of the micro parameters and capital inputs, biases in the restricted profit case are much less likely to vanish in the restricted case than in the unrestricted case. The conditions that established the stability over time of the macro parameters now involve movements in capital aggregates; they are more difficult to devise and satisfy. The choice of capital aggregates and of functional form for the variable profit function is crucial to this stability. Whereas in the completely optimized case, the Generalized Leontief was found to support an exact average representative, under partial optimization, the derivation of such a form may not be attainable. As this study of aggregation biases has indicated, if the restrictions that ensure exact-aggregation (including the less stringent ones evidenced here) are not satisfied, the macro modeler may face aggregation disorders of varying importance. It would then be apppropriate to explore possible cures or at least prescriptions to reduce the symptoms. In that respect, the proposed research is very much preliminary; causes have been identified but remedies are more difficult to devise. The theoretical reasons why the "macro" description of aggregate economic relations can lead to aggregation biases have been identified. First, the stochastic interdependence of the agents' characteristics may imply the existence of covariance terms, even in linear relationships, that are not taken into account by usual aggregate data. Secondly, if the underlying relationships are non-linear, their aggregation will entail the existence of second and higher moments of the characteristics, which again are not described by aggregate data. The usual aggregates, the totals and averages, are informationally inadequate in those circumstances. The same remark applies to more sophisticated aggregates (such as Divisia 146 indexes) that, nevertheless, do not capture the essential features of the microrelations. In particular, in macroeconometric modeling, the omission of covariance terms be-tween functional parameters and explanatory variables may result in aggregation biases with respect to policy changes. If an exact-aggregation representative agent approach is adopted, policy changes may be assumed to lead to shifts in the macro parameters, unless the policy changes are stochastically independent from the functional parameters and are mean-preserving. 10. S. Future Research Strategies After having identified a syndrome, such as the inconsistency of exact representative agent macro models, the next question to ask is whether its consequences are important or can be neglected. The preliminary results presented here may not be widely transferable, but they serve as a warning. In times of relative economic stability, the time-drifts of macro parameters with stable microfoundations were comparable to the fitting error. By contrast, in the presence of shocks (and maybe policy changes), the macro parameters with consistent microfoundations were significantly different from the pre-shock coefficients. Predictions using the latter would be erroneous. The circumstances leading to such biases and the magnitude of the errors need to be qualified. A useful goal of future research would be to provide tests or procedures whereby users of aggregate data could evaluate the seriousness of the potential aggregation problems facing them. Then they would be better placed to formulate alternate modeling strategies. Assuming that the presence of aggregation distortions is troubling by itself, it is useful to explore possible solutions to the informational inadequacy of aggregate data. Partial remedies using existing data may be devised, as will be suggested below, but the ultimate cure is hopelessly trivial: correct the deficiency, supplement the existing aggregate data. Even if this solution seems radical, it is worth mentioning since it was the last resort of previous exact-aggregation models (Nataf, 1948; Gorman, 1968a). Distributional information, in the form of time series of estimated parametric or non-parametric density functions, could be provided by the statistical agencies who compile aggregate data. Data on concentration ratios and number of firms per industry are already available. Informationally richer data items would take the form of an estimate of the probability 147 distribution of production/profit characteristics for each sector or industry, which could be described parametric ally with a few variables. This obvious solution is costly. More importantly, it would require a remodeling of macroeconomic theories, as suggested by works such as Blinder and Gregory (1984). Nevertheless, the present research hopes to further the case for increasing the availability of richer aggregate data. More practical alternatives need to be considered. As illustrated in the case of the av-erage-representative firm, the magnitude of the aggregation biases will be related directly to the complexity of the functional form used to describe the economic relations and to the distribution of the characteristics. The complexity in functional form can be reduced by the choice of a simpler functional form, but this may not always be desirable. Since the variance-covariance terms cannot be eliminated completely by functional form restrictions, it seems essential to study their magnitude. A comprehensive study would require extensive statistical analyses of the distributions of the agents' characteristics, especially that of covariances between functional parameters and explanatory variables. In the absence of appropriate data to conduct such tests, roundabout methods have to be devised. In the context of consumers, Stoker (1986b) used cell proportion data, that is observations over time on the percentage of agents in classified group categories, to study distributional effects in macroeconomic relations. In the context of producers, data that cover the total population of firms may be more difficult to obtain. Public surveys often include only the biggest firms or the firms registered on stock exchanges and provide incomplete coverage. Exceptionally, Dunne and Roberts (1986) have utilized data on 300,000 to 350,000 plants to analyze firm entry and exit, covering the total population of U.S. manufacturing industries. Unfortunately, this longitudinal data from the U.S. Census of Manufactures is collected at five years intervals and does not provide capital data. Access to confidential business survey data collected by govermental agencies would be required to complete a comprehensive study of the firms' production/profit characteristics The time allocated for the present research project precluded such a course of action. Alternatively, it may be possible to supplement available data with simulated data. Where concentration ratios are available, these could be used to compute proxies of the variance of output across the industry. If a specific functional form for the density function 148 could be assumed, simulated samples of the population of firms could be reconstitued. Such industry-specific data could be appropriately used to study the aggregation of firms across industries. In particular, the study of the relation between industrial structure and types or magnitudes of aggregation problems may be an interesting path of future research. Obviously, industries where oligopoly prevails or where the theory of optimal firm size applies are likely to be better described by an average-representative firm than an industry made of firms of various sizes and technologies. At one extreme, a monopoly induces no aggregation problem for its industry. At the other, the macro description of a highly diversified industry may not be required to satisfy stringent separability restrictions and thus correspond to empirical findings. In Hazilla and Kopp (1985), a translog cost function was found to be weakly separable in quasi-fixed inputs and variable factors for the Paper and Allied Products sector. This assumption was strongly rejected for the Miscellaneous Manufacturing Industries, as well as for most other sectors. It may be argued that in the Miscellaneous Manufacturing Industries, there is a diversity of technologies. Thus, covariances terms between equipment &c structures and labor, for example, may well be negligible and a mixed multiplicative term could be allowed in the industry's cost function. 5 On the other hand, in the Paper and Allied Products sector, technologies may be similar enough to require separability of the cost function. Obviously, simulations studies are needed to evaluate what levels of similarity and diversity generate acceptable aggregation biases. However, the possible yields of simulation studies should not be overestimated. Their empirical validity is jeopardized by the enormity of the array of simulation choices. The difficult task of picking plausible distributions for the factor inputs and their coefficients is exemplified by some simulation experiments conducted by Fisher and others. In a set of simulation experiments, Fisher, Solow and Kearl (1977) explored the aggregation of three CES production functions: f= 1,2,3, (10.13) where (f measures the elasticity of substitution, 6s is a distribution parameter, kf is the firm's capital stock, Is is the amount of labor assigned to the f-th firm and T/ is the resulting 5 In that particular study, the use of a translog actually complicates this point. 149 output. They used 10 different combinations of elasticities of substitution. For each of these choices, 22 runs were performed where the distribution parameters were chosen in two sets, one being positively correlated with the substitution elasticities, the other one negatively correlated. The underlying input data were generated from a constant-returns economy; the labor series were the same for all experiment, while the capital stocks were chosen in eleven different ways. Even if this aggregation problem was relatively simple (3 production functions with 2 parameters and 2 inputs), the final results were so convoluted that the authors concluded that "the estimated parameters (macro) themselves are sometimes quite far from anything one could sensibly describe as roughly characterizing the real —i.e. the model — world" (p.319). Relying on a distributional understanding of the firms' technological parameters may help to construct meaningful simulation experiments. At this time, it is thus suggested that simulation experiments be used to extend existing data bases. For example, using actual levels of capital and labor inputs, levels of output could be generated under different distributions of Cobb-Douglas parameters. The resulting biases could be linked to the variance of these distributions, and to the level of technological similarity among firms. Another important question that could be tentatively answered using simulated dis-tributional data is the relation between differing levels of aggregation and the resulting biases. It would enable researchers to choose a level of aggregation that satisfy their in-formational needs. For example, in a set of simulation experiments, Orcutt, Watts and Edwards (1968) observed that moving from fully (1 unit) to semi-aggregated data (4 units) increased the information much more than moving from the semi-aggregated level (4 units) to the disaggregated one (16 units). 6 Understood in a stochastic framework, going from one level of aggregation (say, the "L" level with 191 industries) to another level (say, the "M" level with 43 industries) reduces the number of firms per cell if the aggregation is done across industries of similar sizes and would reduce the variance significantly. If the aggregation is done in an heterogeneous way, combining smaller firms with bigger firms, the effect on the variance of the data could be even more dramatic. Ultimately, when the aggregation combines all firms, the variance is completely annihilated. Consequently, 6 The information was measured by the M S E of the coefficients of a macro consumption model. 150 homogeneously aggregated data is more likely to reflect its underlying population than data aggregated following other arbitrary sensible processes, such as the aggregation per output commodity. Note, however, that cell aggregation for one data item, say output, may be heterogeneous for another, say capital, since differing degrees of capitalization across firms are not uncommon. Futhermore, it seems difficult to predict a priori what are the effects of the aggregative process on the covariances between the characteristics. Here, Orcutt, Watts and Edwards (1968) have found that explanatory variables were correlated much more closely at the fully aggregated level than at the dissagregated one. As these exploratives remarks suggest, there is a profusion of problems related to the question of differing levels of aggregation. With reference to the present data base, it is not known whether the aggregative process that generated the data has preserved the distributional properties of the true population. Given the high level of aggregation, it would be reasonable to assume that the distribution of the characteristics at that level does not resemble in any way that of the original population of firms. This statement reiterates the introductive warning that the bias evaluation pertains to the aggregative process modeled, only. It is useful to restate it now to underline that even though the time-drifts of the macroparameters in the illustrative results were related to conditions facing the Canadian Economy, their magnitude accounts for going from 37 sectors to 1 sector (the macro one). These biases should not be viewed as representative of "true" biases that would be obtained by considering the aggregation of individual firms. There is a tremendous amount of empirical work needed to specify the magnitude of aggregation problems with any reasonable level of realism. But the theoretical model also could be amended to reflect better the realities of the economic system. For example, the relationships between individual uncertainties and aggregation problems is a worthy research topic. The present section seems to open numerous avenues of research apparently enlarging rather than reducing the potential aggregation problems. 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