A framework for aggregation theories

Ricerche

Economiche

(1993) 47,107-135

A framework

for aggregation

theories

EDMONDMALINVAUD CollSge de France,

3 Rue d’Uh,

75005 Paris,

France

Summary The past literature on aggregation theory is surveyed in terms of three nested formal models. The most general one distinguishes exogenous and endogenous variables, both for a micro- and a corresponding macro-model; it shows why the main problem lies in aggregation of exogenous variables; it suffices in particular for Leontief’s results on commodity aggregates. The aggregation of economic relations further assumes that the micro-model is individualized; the case of aggregate production functions shows how varied are the micro-models that may be selected for the discussion. The statistical approach, moreover, introduces the distribution of individual characteristics and individual exogenous variables; it leads to a new view of exact aggregation; more importantly it draws attention to structural stability and to the study of factors explaining shifts in the statistical distribution of micro-variables and -parameters. The success of this approach explains why aggregation theory increasingly requires empirical research. J.E.L. Classification: COO, EOO. Keywords: Aggregation, representative stability.

aggregates,

structural

1. Introduction

Aggregation problems arise in many different forms in economics. First faced in economic statistics, they appear more and more in economic theory, particularly so because the constraints imposed on agents’ activity seldom easily aggregate and, even when they do, the resulting individual behaviour does not obviously translate into aggregate behaviour. In many of our theories, some aggregation problems have now become classical and are systematically studied. They then appear in each case within particular models and with special features. Could these problems not be seen within a common framework, applying to most of them, if not to all? If a positive answer could be given, one could hope to reach general results easily applicable to several domains; or at least to benefit from the understanding of 107 0035~5054/93/020107

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of Venice

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the differences between the formal problems raised by aggregation in various branches of economics. To present such a framework was the motivation in Malinvaud (1956), a long article that was not published in English. Over the years aggregation issues kept worrying me, all the more so as my attention was turning increasingly toward macroeconomics. I could witness the appearance of new topics concerning these issues and of new lines of research for tackling them. I studied some questions concerning the meaning of the relations contained in our macroeconomic models. At times I even reconsidered the general approach proposed in 1956, but again did not publish anything about it in English. This article provides me with the first opportunity to expose my idea to a wide readership. Unfortunately, a general theory of aggregation could only have a limited scope, precisely because aggregation problems may be so different from one another and results with a wide domain of applicability do not seem to go very deep. By far the largest part of aggregation theory will remain domain specific. This is why one had better speak of “a framework” rather than of a general theory. A framework helps the organization of ideas but does not claim to bring directly usable solutions. It mainly contributes to the unification of the scientific language and methodology. A framework seems to imply and to announce a broad model within which all aggregation problems would fit. Such a model will indeed be presented here. But the paper will do in a sense more, in another sense less. It will do more because it will discuss several interrelated models; the main thread of the argument will run through three nested models from the broadest one to the most tightly structured. Particular stress will be placed on the third model which is appropriate for the discussion of the promising “statistical approach” to the aggregation of economic relations. The paper will also do less, because even the most general of these models will not usefully apply to some significant aggregation problems. The framework ought to be particularly relevant for the aggregation of economic relations over individual agents, but it may cover other cases as well. Since it aims at organizing ideas, it has to be examined from various viewpoints, really from the viewpoint of each specific problem. This is why many examples will be considered in the following pages, almost like in a survey on aggregation in economics. Over the period that would have to be covered by such a survey, notably over the past forty years, the main focus of attention often changed, shifting from one type of substantive economic issue to another. The shift concerned at times the domain, at times the question to be answered. The most natural domains belong to macroeconomics: saving behaviour, production functions, price

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and wage equations, mismatch between supply and demand, and so on. But the microeconomic theory of prices and resource allocation also considers how individual demands aggregate. The oldest question arose in economic statistics: namely, how ought aggregates and synthetic indices to be defined? It has not yet been definitely answered and is now often seen as part of a more general type of question: how can aggregate models be justified? Under which conditions do such models exactly derive from microeconomic laws? Under which conditions do they provide a satisfactory approximation? Looking for answers to this second type of question at times appeared to be the only objective of aggregation theory. But economists are also interested in knowing the properties of aggregate laws, once a rationale has been found for the appearance of such laws: under which conditions, for instance, does smoothing by aggregation operate? Or which properties are preserved by aggregation? This third type of question increasingly attracts attention (it was not explicitly brought to the fore in Malinvaud (1956) but was touched upon only in passing in discussions that had different purposes). In applications, as soon as we must aggregate, even if we do not like it because it has no stronger justification than the need for simplicity, the most relevant question is to know the errors that may result. Applications then do not always concern macroeconomic diagnosis, forecasting or policy analysis; they may appear even within theoretical investigations, for instance when the hypothesis of a representative agent is introduced. Answers to this last type of question, about the nature and likely importance of aggregation errors, can hardly ever be neat and elegant, so they may not receive the attention they deserve. The following sections are organized in the natural order, from the most general framework to the most tightly structured. The argument stresses the second type of the above questions, those concerning the justification of the aggregate model. But the argument often stops for the discussion of other questions and of particular cases that either illustrate the approach or show its limitations.

2. A first conceptualization

To begin with, let us think about the justification of macroeconomic theories. With respect to the real world, such a theory stands at a higher level of abstraction than does or would do a microeconomic theory concerning the same phenomenon. To argue on global entities is to remain away from concrete events. This distance does not change the nature of theory, which in any case is abstract. Nor does it mean that present macroeconomic theories

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are more abstract than existing microeconomic theories, which most often deal with different phenomena. But any methodological study must start from the idea that aggregation increases the degree of abstraction. This being understood, there is no simple way to conceptualize aggregation other than to assume the existence of a formal microeconomic model that is taken as perfectly appropriate for the problems that the macroeconomic theory studies. We know that the assumption is a fiction, but we need it in order to focus attention on the main point, namely that any macroeconomic theory provides an image of a set of underlying microeconomic relations. We may as well speak of a working hypothesis. At this stage we may define the microeconomic model in very general terms. Let x and y respectively stand for the two sets of variables or other abstract objects contained in the model, x for what is taken as exogenous and y for the endogenous elements. Since it pretends to explain y, the model can be written in functional form, Y

This suggests an analogous economic theory,

=f(x>.

formal

(1) representation

of the macro-

where X and Y respectively stand for the exogenous and endogenous elements of the theory. Aggregation is the operation that substitutes model (2) for model (1) as a tool for the representation of the phenomenon. By far the most natural and most frequent form of substitution involves a definition of the aggregate elements of X and Y from the elements of x and y, even more precisely of the elements of X from those of x and of the elements of Y from those of y. These two groups of definitions may be formally represented as

x= g(x),

(3)

Y = h(y).

(4)

It is then interesting to visualize the setting of these definitions by a graph, which, at this stage, gives only the main structure (Figure 1). In other words, aggregation of the microeconomic model (1) is defined by the choice of X and Y, of g and h, and finally of 8’. Broadly speaking, we may say that the problems to be studied concern the accuracy of the image that the macro-model (2) gives of (1).

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f X

-Y

clI X-Y

I

h

F

FIGURE 1.

In such a framework the distinction between exogenous and endogenous elements plays a major role. Indeed, their respective aggregations raise different issues. Let us focus attention first on endogenous variables, the determination of which is the ultimate aim of the models. For those variables, aggregation h must be judged with respect to the uses of the work; it is perfectly appropriate if the user is indifferent to whether y or h(y) is known. In less ideal cases the loss of information when y is replaced by h(j) must be directly judged; aggregation of these endogenous variables is an independent problem from others raised by the substitution of the macro-model to the micro-model. Let us assume then that this particular problem is put aside and let us consider the difficulties that remain when h is taken as given, in the same way as the micromodel. For the determination of the aggregate endogenous variable Y, an “exact theory” is exhibited by Figure 1. This would consist of explaining Y from the microeconomic exogenous elements x using first the micro-model y = f(x) and then aggregating. The alternative determination of Y via the macro-model is often the only available one. It entails an “aggregation error”, which may be symbolically written as+ E(x) = Fg(x)

- h f(x).

(5)

This formula is a reminder that aggregation errors depend in general not only on aggregation formulas but also on the values taken by exogenous variables. Exact aggregation at a particular value of x occurs if E(x) = 0. The ideal case is when exact aggregation holds for all values of X. Having now defined a basic model, let us stop in order to see why it does not usefully cover all cases in which economists are confronted by aggregation issues. t For some components of Y the difference written defined a priori. This secondary issue is not considered

in (5) might here.

not be well

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3. Aggregation

over commodities

Aggregation of quantities and prices has long been discussed from various viewpoints. The relation with our basic model is direct in some cases, more far-fetched in others. Let us first consider the aggregation of inputs in a production function along the lines proposed by Leontief (1947). A vector x of inputs (xl, x,, . . . , xJ gives an output y according to (6)

Y = fcqt x2, * * * 9XJ.

Can we aggregate the last m - M+ 1 inputs into a new quantity in such a way as to still have an exact determination of output a function of x,, . . . , xMPl, X,? The basic model applies, with Y=y, and some appropriate

Xh=x,,

function

X, y as

(7)

for h
gM, where

(There is no aggregation of the endogenous variable y.) As Leontief showed, an exact aggregate model can be so defined for all nonnegative values of the xh if and only if the production function is separable, i.e. if there exist two functions F and g, so that, identically, fbp

x2,. . . , XJ = &l,

x,, * * * , XM- 1, &f&f,.

. ., XJ >.

(9)

Of course similar necessary and sufficient conditions exist for exact aggregation of other inputs or for aggregation at a higher level, for instance aggregation of xMPl and X,. In such cases the formula for the aggregation of inputs cannot be a given a priori, but must be appropriate to the form of the separable production function. Leontief (1936) considered another case permitting exact aggregation of inputs, the case in which the firm minimizes the share of its production cost due to the last m - M+ 1 inputs, for given values the relative of the prices pM, pM+ 1.. .p, and in which, moreover, prices of the last m - M+ i inputs are fixed. Then, any aggregation formula g, permits exact aggregation,? as long as it is such that the following system of m-M+ 1 equations can be solved in xM, xM+l,--.,x,: 7 It is natural to assume that g, is homogeneous of degree 1; this is necessary if one wants to have a sensible interpretation of the quantity X, and to give a price to this aggregate input.

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&f(x,, xM+1,’ . .,x,>=x,-

J

Since the price ratios of the right hand side are fixed, the solution depends only on xi, x2,. . . ,xM-i, XM; it can be written as Xh = &(x1,

* - * >x‘q-

1, X,),

Inserting these values in (6) and taking exact equation of the form

h=M

,..., m.

(7) into account

y=F(xl,...,xM-1, x,>.

(11) leads to an

(9

Our basic model still applies. However, the micro-model is no longer simply the production function (6), but, moreover, the model in which the firm minimizes the share of the production cost due to the last m-M+ 1 inputs. The two cases found by Leontief, have, of course, a more general validity. Exact aggregation can also be obtained outside of the theory of production, if the micro-model is separable or if it has the effect that the range of some exogenous variables of the micromodel is limited to a subspace of lower dimension than the space in which they are defined (the second case permits what is sometimes called “Hicks-Leontief aggregation”). In both cases correspondence between the micro- and the macro-models will conform to our basic framework. This framework will not, on the contrary, be applicable for the statistical theory of price and quantity index numbers. The simplest case, then, is comparison of a current period to a base period. Let pt and qt be the vectors of prices and quantities in the current period, pb and qbthe corresponding vectors of the base period. Let Pt and Q, be the two indices, for prices and quantities respectively. The statistical theory looks for the definition of two functions CP and cy defining the formulas to be applied for the computation of the indices:

pt = dPp qt ;Pb,qb) Qt= t&t, 41i Pb,qb)

(13)

General properties that the couple of functions 9 and I,V should have were and are the main reference (for instance in Fisher, 1922). No micro-model is involved. The economic theory of cost of living indices introduces more structure (see e.g. Diewert, 1987). A representative consumer is supposed to allocate a total budget B, to the purchase of the vector of quantities qt, taking the vector of prices pt as given. The question

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then is to know what ought to be the increase of the budget, from B, to B,, that would lead the representative consumer to obtain the same utility level in the current period t that he was obtaining in the base period b. The appropriate cost of living index P,* is then the ratio between Bt so determined and B,. The basic model may be said to apply in this case if one accepts the following argument. The exogenous x would be the vector pt of prices; the endogenous y would be the vector qt of quantities and the amount B, of the budget; the micro-model would find the correct value of B, for a given specification of the utility u(q), this specification being taken as a parameter. Aggregation h(~) of the endogenous variables would be the simple projection from (qt, B,) to B,. A macro-model could be defined with an exogenous Pt, meant to represent the change in the cost of living, and with B, obtained by the simple relation B,= B, Pt. In order to apply our basic model, one still has to assume that the aggregation of exogenous variables is given by a formula (13) in which qt does not appear (this condition would be met by a Laspeyres price index). The aggregation error would then be the difference between the true required budget B, PF and the estimated budget B, Pt. One could study how this error varies when variations occur in the price vector pt, in the utility function u or in the formula defining the price index P,. Of course, such an application of the basic model would have no significant interest for the economic theory of cost of living indices. But being aware of its feasibility, we get some feeling of how general the framework proposed here really is.

4. Other,

not so friendly,

cases

With this same purpose we may briefly consider still other cases. For instance “aggregation of preferences” is a well-known challenge for the theory of social choice (see e.g. Moulin, 1983). Its microeconomic elements are not organized so as to fit within a causal pattern such as (1). They are n individual preference orders on a given set A of alternatives. If L(A) denotes the set of orders on A, the microeconomic model is simply L(A)“. The social choice problem consists in finding a rule for selecting an alternative in A, depending on what element of L(A)” represents individual preferences. The problem was first approached by the requirement of finding a rule for deriving a social preference order: a “social welfare function” Z would map L(A)” into L(A). One wanted this function to be endowed with desirable properties. It would clearly be artificial to force this problem into our framework, unless one is satisfied to say that the finding of a function Z raises issues that are also sometimes found in the aggregation of exogenous elements of microeconomic models with a more causal structure.

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An important part of the aggregation literature concerns optimal aggregation of variables, usually within linear models and for predictive purpose. This problem will not be stressed here, but may easily be introduced on the particular case of the Leontief input-output model. Let x be a vector of outputs of the n goods and y be a vector of the quantities available for final demand. If 1 and A are respectively the identity matrix of order n and the matrix of input-output coefficients, the microeconomic model has the form (l), with y = (I-

A)x.

04)

A corresponding macro-model concerns N aggregated industries, obtained by a grouping matrix G applying inputs and final demands: X= Gx, The aggregated

input-output

Y= Gy.

model, Y = (I-

then,

A*)X.

goods and to outputs,

05) is (16)

The problem is to know how to cluster goods, i.e. how to best select G and to correspondingly define A* so as to minimize aggregation errors. This problem of optimal aggregation has of course to be defined more precisely and this may be done in various ways, even for the particular case of input-output models (on this, see Malinvaud, 1954; W. Fisher, 1969; Chipman, 1975). Although the scheme of Figure 1 applies, the questions to be answered are somewhat different from those discussed in this article. It is interesting to refer to yet another problem, but one that was not commonly thought to be an aggregation issue, namely whether in a constant returns to scale economy the real wage rate is independent of the composition of final demand and depends only on the rate of profit. One knows the importance assigned to this problem not only by a whole school of economists following Sraffa (1960) but also by all those who find it convenient to refer to a welldetermined factor price frontier. The microeconomic model, which need not be spelt out here, is supposed to determine many variables from the exogenous rate of profit and final demand vector: the techniques to be used, the output of the various sectors, the prices of the various commodities, the wage rate, and other variables as well in some specifications. For this problem, aggregation of variables is a kind of projection: the function g for the exogenous variables retains only the profit rate r and forgets about the final demand vector. The function h for the endogenous variables maps the wage rate and

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the price vector into just one variable, the real wage rate w. The question is to find out whether the microeconomic model implies a well-defined “macroeconomic” relation: w = F(r). One knows that a positive answer depends on whether a non-substitution theorem applies to the microeconomic model (see e.g. Stiglitz, 1970). After the preceding references to a large variety of aggregation problems, each one with its particular features, let us turn our attention to a somewhat more specific framework than the one presented in Section 2, a framework which covers the main class of aggregation issues.

5. The individualized

microeconomic

model

The most frequently discussed cases concern aggregation of individual relations. There is then a number n of individuals i. The activity of individual i determines an endogenous variable yi from some exogenous variables (yi may of course be a vector). Among the latter variables some are specific to individual i; others apply to all individuals, or at least to several. Let us denote by xi the exogenous variable, most often a vector, that is specific to i and does not appear in the determination of the activity of other individuals; let us then denote by z the vector of other exogenous variables appearing in the full model, i.e. other than those making the n variables xi. With respect to the general formalization of the variables in the microeconomic model of Section 2, we may now write

X=(X1,X2,.-, qpk

Y=tY~,Y2,...,Y,)-

(17)

In these cases the microeconomic model is made up of the n relations, or groups of relations, following from the activity of the n individuals. For each individual the corresponding relation or group of relations will be written here as

Model (1) will then take the form of the n equations (18). We may say that the microeconomic model is “individualized”. Aggregation of individual relations most often operates with a simple form of aggregation of variables, which are just added or averaged. Sometimes the functions g(x) and h(y) are a little more complex, while usually still remaining linear. It will suffice here to assume that the endogenous variable Y of the macroeconomic model is the arithmetic average 3 of the endogenous variables of the individual models

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it being understood that this equation may apply to vectors yi and 7. Similarly, we shall consider the average X of the individualized exogenous variables xi and we shall assume that the exogenous variables z are not aggregated, x= g(x) = (2; 2). For such a setting the possibility vidual relations is well characterized

(20)

of exact aggregation of indiby the following theorem.7

THEOREM 1. If xi may be any vector of a set of maximal dimension, exact aggregation is possible at all values of the exogenous variables if and only if the microeconomic model has the form yi=A(z)xi+b,(z),

i=1,2

,..., n.

(21)

In other words, the individual relations must be linear and imply the same matrix A(z) of propensities. When this holds, the form of the perfect macro-model is obvious. It is also obvious that essentially the same theorem applies for other forms of linear aggregation of the xi and yi. It is not necessary to dwell here on the fact that Theorem 1 shows how severe the conditions for exact aggregation are. At this stage, we should rather examine how the general framework defined in Section 2, and now made more particular, applies to a number of problems discussed in economic theory.

6. Examples

in production

theory

The important literature on the aggregation of production functions concerns various problems, which are easily placed in the framework considered here. Most of them assume the existence of an individual production function linking output Qi of firm i to its inputs of labour L, and capital Ki. We need to make a clear distinction between this production function and the function fi characterizing the full model of the behaviour of firm i. Let us then write the individual production function as

Let us also note that

all firms produce

the same good from the same

t This is a direct consequence of a somewhat more proof transposes the argument given by Nataf (1948).

general

theorem

whose

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types of labour and capital, these three commodities being perfectly homogeneous (aggregation of commodities is not considered). An aggregate production function would similarly be a relation linking average output & to the average inputs L and K:

However the meaning of this function and its role in the macromodel depends on the full micro-model within which the n individual production functions (22) appear. The following examples concern various familiar cases. EXAMPLE 1 In the first example, no distinction need be made between the production function 6 and the function fi of the micro-model: the vector xi of exogenous individual variables is precisely (&, KJ; no other exogenous variable z appears. Theorem 1 then shows that, if the choice of inputs is arbitrary, exact aggregation would require the individual production functions to be linear and to have the same two coefficients, respectively for the two inputs. We may_ note in passing the case in which all production functions fi would be Cobb-Douglas with the same exponents, i.e. with the same elasticity applying to labour, as well as the same elasticity applying to capital. With a vector xi being now given by (log Li, log KJ, the micro-model for yi = log Qi would have form (21). Exact aggregation would then be possible and lead to a CobbDouglas production function with the given exponents; it would link not the arithmetic averages L, iiT and Q, but the geometric means defined according to (19). We may say that it would solve another aggregation problem than the one usually posed in macroeconomic model building. EXAMPLE 2 A completely different situation occurs if inputs are assumed to be chosen by the firm in a way that can be predicted. Inputs are then not exogenous but endogenous in the model formalizing the choice. The simplest case concerns perfect competition and full factor mobility. Let p, w and I” respectively be the price of output, the wage rate and the rental price of capital. Those three variables are exogenous and common to all firms. The exogenous variable xI disappears. There is no longer any aggregation of exogenous variables. Then

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exact aggregation holds, as shown x = X, such is the case here with

by the graph

119

of Figure

1 when

(24) Under the common convexity assumption, the full micro-model of a firm i is given by the solution of (22) together with the following two equations:

We shall assume that this solution is well determined for arbitrary positive values of p, w and r.7 The macro-model then represents how the aggregate endogenous variables Q, L and K are determined from the values of the exogenous variables p, w and r. It is a rather simple matter to show that the macro-model may be given a similar form to the one of the micro-model:$ this macromodel may be said to determine &, L and K by the solution of an aggregate production function (23) together with the two equations

The aggregate production function P is then determined from the n individual production functions-[. An important question is to know how the characteristics of F relate to those of fi. This question naturally comes after the previous one, concerning the justification of an aggregate production function, has been answered. (One then shifts from the second to the third group of questions listed in the introduction.) Clearly, if all the fi are identical, Qi, Li and Ki are for all i the same three functions ofp, w and r. These functions then also apply to &, L and R, which are equal to Q;, Li and K. for any i. The aggregate production function P is identical in this trivial case to the microeconomic function. It may also be proved that, if all the f^i are CobbDouglas functions with the same exponents, the aggregate production function is Cobb-Douglas, with the same elasticities and with a t The assumption is significant since it rules out the case of constant returns to scale. Dealing with this case does not lead to wholly different conclusions but requires that the aggregate output to be produced by all firms is exogenously given. The micro-model, which takes this exogenous quantity into account, is somewhat more complex than the juxtaposition of n individual systems such as (22) and (25). $ See Malinvaud (1981), Chapter 4, pp. 211-213.

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productivity factor equal to a kind of mean of the factors appearing in the individual functions.? I do not know of any interesting result for other cases. Actually, the relation between the characteristics of the individual production functions and those of the aggregate function are not simple: for instance, the elasticity of substitution of the aggregate function differs from one in general when the individual functions are all Cobb-Douglas, but not with the same exponents. EXAMPLE 3 An important literature exists for the case in which free competition still rules the market for the good and the market for labour, this factor being perfectly mobile; but the amount of capital available in firm i is exogenous.$ The general framework then applies, with xi=&,

z=(p,

w),

yi=(L,,

QJ.

(27)

The micro-model determines L, and Qi as the solution of (22) and the first of the two equations (25). In general there is no macro-model that would exactly determine Q from i?,p and w, no matter what the K, would be. Any model built from an aggregate production function (23) with the first of the two equations (26) would imply aggregation errors for the determination of Q from k, p and w. There is, however, a favourable case, namely the one in which the same individual production function would apply to all firms and would exhibit constant returns to scale. This function could be written as ALi,

KJ=K,q?

(28)

(2). i

Then, the same factor proportion LJK, would be chosen by all firms as the same function w(w) of the real wage rate w (from now on we assume p = 1 for simplicity of the equations and without loss of generality). The function (v(w) is the inverse function of q’, since competition leads to equality between the real wage rate and the marginal productivity of labour, ‘i VI -& =w. ( > t See Malinvaud 1 See F. Fisher

(1981), Chapter (1969).

4, pp. 212.

(2%

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The solution

of the micro-model,

then,

Aggregation

of the endogenous

variables

hence

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121

is

leads to

also to

The second of these two equations defines the exact aggregate production function; it is identical to the individual production function. This special case of exact aggregation may be somewhat generalized if one accepts a change in the definition of the capital aggregate. Indeed, suppose the production function of firm i is not quite (28) but

flLi, Ki)= ui Kiq (A) ; 1 1

(33)

ui being an exogenous parameter characterizing the productivity of firm i in its use of capital. Then the above argument applies all the way through, as soon as Ki is replaced by ui Ki and R by the aggregate J=’

f: UiKi ni,l

(34)

(the capital of firm i is weighted by its productivity). It is noteworthy that, in this favourable case of perfect aggregation, the aggregate production function has the same form as the individual production function. EXAMPLE 4 Among the many cases of imperfect competition, it will suffice for the present purpose to select the one in which firms buy their inputs of labour and capital at competitive prices but can sell only an exogenously given quantity of output Qi. Labour and capital are perfectly mobile and Qi is proportional to aggregate demand n&. Qi = a,&,

(35)

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the sum of the given coefficients ai being equal to n. The micro-model of firm i is then given by (22) and (36) It follows that the mean inputs E and R are functions of & and w/r. Elimination of w/r between the two equations defining these inputs leads to an aggregate production function. It may be shown? that equilibrium corresponds to an aggregate input mix such that the aggregate marginal rate of substitution is equal to w/r. It is noteworthy that the aggregate production function depends on the coefficients ai, hence not only on technological characteristics of the micro-model. 7. Representative

aggregates:

the statistical

approach

Returning now to the general problem of aggregation of individual relations, we must pay attention to an important feature, which has SO far been neglected in this article but gives the real clue to a proper understanding of the significance of many aggregate relations used in economic theory and applications. It is already clear with the framework of Section 2 and with its graphic representation in Figure 1 that aggregation of the exogenous variables, while unavoidable in practice, is also the main source of difficulties. But, when looking for a solution that would hold for all possible values of the exogenous variables, we are often asking for more than is needed. The values of the exogenous variables are not wholly arbitrary, particularly for those variables that are specific to individual agents, those making the xi of Section 5. When these agents are numerous, they ought not to matter individually for phenomena that are seen from a global perspective. Only their statistical distribution seems to matter. If such is the case, the aggregation problem may also be much simplified. According to an old idea which we shall come back to in Section 10, statistical distributions of economic agents change little through time or space, except for some easily identified main movements or differences. In general, a high degree of “structural stability” seems to prevail, even under contemplated changes in policy or economic environment. Once this stability is recognized, it is natural to think that aggregates play the role of indicators that represent the full set of underlying microeconomic variables. We may then speak of “representative aggregates”. We may also conjecture that knowing the values of these aggregates is suffit See Malinvaud

(1981), Chapter

4, pp. 221-224.

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cient to infer the values of the aggregate endogenous variables. A proper macroeconomic model would give this determination precisely. An important subject for aggregation theory is to rigorously study the topic and to see under which conditions the above conjecture is valid. This is precisely what has to be done here. We shall then speak of the “statistical approach” to the aggregation of economic relations.? Looking only at statistical distributions means losing the identity of individual agents. This does not matter if the agents are interchangeable, i.e. if they have identical behaviour under identical circumstances. However such a full interchangeability seldom holds over all individuals. Behaviour also depends on individual characteristics, which vary from some individuals to others. This fact does not, however, prevent application of the approach using representative aggregates; it simply makes it a little more delicate. Indeed, there is structural stability in the distribution of individual characteristics also, even stability in the joint distribution of individual characteristics and exogenous variables. Let us then come back to the model of Section 5 and replace (18) by Yi = &;

2; 4);

(37)

t; being a parameter that characterizes the behaviour of agent i. Now the same function f applies to all individuals (there is no risk of confusion with the notation f used in Section 2 for a formal representation of the microeconomic model). For simplicity we shall deal with the case when xi, yi and ti are numerical variables; but generalization to the case of vectors of such variables would raise no difficulty in principle. We shall also assume that, except for z, the aggregate variables are the means Z and J. Let us then designate by G(<, 0; y) the proportion of individuals i for which

This joint cumulative distribution function depends on y, a vector of parameters. The model then implies that the mean J is determined by a Stieljes integral, which may be written with the usual notation as

7=sm2;0)dW, 0;Y>*

cw

t As will be explained in Section 11, this approach was implicitly accepted in the econometric literature dealing with the aggregation problems raised by the estimation of aggregate relations. However, its usefulness is more general and also concerns many purely theoretical issues.

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It is a function

of z and y, which

we may write

as

y = D(z; y).

(40)

In other words, the endogenous macroeconomic variable is a function of the exogenous variables that are not subject to aggregation (z) and of the parameters of the statistical distribution of the individual exogenous variables and characteristics; these parameters of course matter at the aggregate level. Expression (40) does not look like the macro-model we should want to consider: it does not seem to contain the macroeconomic variable X. We may say that the variable is implicit in the expression. Indeed, changes in 5 mean shifts in the statistical distribution of the xi, hence changes in the vector of parameters y. More precisely, they are changes concerning the location of the bivariate distribution and occurring along the x axis. In order to make the presence of j; explicit we just need to isolate the component of y that most directly relates to this location. We may denote this component as p, using r~ for the vector of other components of y, i.e.

Y=(Pd Then the mean 2 is a function

of the parameters

(41) p and a;

ii =

tdG(& 8; y) = C(y) = C(p, g) s Since p is a location parameter along the x axis, it is natural to assume that the function C can be inverted in order to give the value of this parameter that corresponds to the value of the macrovariable 2, p = I?(?, a).

Inserting this value into (40) leads to the interesting the macro-model; j = F(X,

2; CT).

(43) expression

of

(44)

We see how general the derivation of an exact macro-model can be, as soon as stability of the structures is taken into account. The exact model, however, contains whichever parameters of the statistical distribution may be subject to change and are not represented by the macro-variable 2. This statistical approach to the aggregation of economic relations was used on various occasions and is likely to become more and more common. It indeed fits well the main rationale justifying

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confidence in relations between aggregates. When it is studied in such and such particular context, the question is usually not to know whether a macroeconomic relation holds; this is taken for granted; but to derive the properties of this relation. They depend not only on the form of the microeconomic model f but of course also on the form of the statistical distribution G and on its assumed or proven degree of stability. For instance, in one of the first applications of the approach, Houthakker (1955) showed that the short term aggregate production function would have the Cobb-Douglas form if the statistical distribution of the (small) individual firms had a particular shape. More precisely, each firm was assumed to have a unit production capacity and to require an amount xi of labour per unit of output; markets for the product and for labour were assumed to be competitive, so that firm i would operate, and then at full capacity, if and only if the real wage rate was lower than l/xi; the statistical distribution of the xi was assumed to be well approximated by an exponential distribution; at equilibrium aggregate output was then proved to be a power function of the aggregate labour input. More recently, Hildenbrand (1989) argues that, notwithstanding misgivings generated by microeconomic theory, the “Law of Demand” could safely be accepted at the aggregate level, even though it may not hold at the individual level. Considering the actual statistical distribution of individual expenditures, he proved that conditions were approximately fulfilled for the aggregate demand laws to have a negative definite Jacobian matrix of price derivatives. Other recent results in the same area of research are to be found in these special issues of Ricerche Economiche. Some of these contributions are not only interesting in themselves, but also show the way for a very promising trend. Much is indeed to be learned if theoretical reasoning properly incorporates empirical results on the form of the statistical distributions of individual variables and characteristics. When so doing, theory will become more specific, hence more useful.

8. A new view of exact aggregation

The statistical approach opens new possibilities even when no reference is made to empirical results about structural stability. In order to understand these possibilities, one may imagine that one starts from a very general form of the statistical distribution, i.e. from a function G with a very large vector y of unknown parameters (for instance many statistical moments). It is then possible that aggregation by (39) results in a function D that depends only on a small number of components of y, or on a small number of functions of y. If such a case occurs, it is natural to think

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of a macroeconomic model in which the relevant components or functions would be explicitly taken as exogenous variables. For instance, if only two components of y matter in D, one concerning the location of the distribution, the other its dispersion, then the dispersion component may be taken as an exogenous variable of the macroeconomic model, besides 2 and z. After this extension, one may speak of “exact aggregation”, because the macroeconomic model exactly derives from the microeconomic model. In Sections 5 and 6, where the individualized model was studied with no reference to the statistical distribution of the exogenous variables xi, exact aggregation concerned the case when the macroeconomic exogenous variables would be limited to (2; z) and the macroeconomic model had to hold for all conceivable values of the microeconomic exogenous variables. Recent theoretical research led to the view that such a concept of exact aggregation was too narrow. (It is now apparent that this view inspired W. Gorman at a rather early stage of his continued research on aggregation.) Assume for instance that the function fhas “a finite basis” in the sense that there are J couples of functions uj(z) and qj(x, t) such that /lx; z; t)= fj uj(Z)Vj(x,t)*

(45)

j=l

Then (39) implies Y = i uj(z>cOj~

(46)

j=l

where the ej are the average values of the qj. If these values can be observed, they may be taken as exogenous variables for an exact macroeconomic model. If for instance the qj are polynomials of degrees zero, one and two, then the gj will be constants, mean values and second order moments (perhaps up to a linear transformation). If moreover x: and t are just one-dimensional real variables, the exact macroeconomic model will contain not only 2, and characteristics of the behavioural parameters, t, var (t), but also the variance of x and the covariance of x and t over the population of individuals. The presence of higher order polynomials in the basis of f would similarly imply the presence of higher order moments in the macroeconomic model. This approach has been much used recently in consumer demand analysis and has been dealt with so well in several articles of this issue that I do not need to dwell on it. References to the relevant literature can also be found in these articles. Particular attention must be given here to Heineke (1993, this issue), who provides a precise definition of what one may call “exact aggregation” in this

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new context (it would not be possible to speak of aggregation if for instance the set of the qj had as many elements as the set of microeconomic variables xi and parameters tJ. It is also noteworthy that Lau (1982) followed by Heineke and Shefrin (1988) proved that existence of a finite basis, i.e. the form (45) of the microeconomic model, is not only sufficient but also necessary for exact aggregation if G can be any statistical distribution. This last result shows the limit of the new approach to exact aggregation. If one does not want to rely on structural stability of statistical distributions, if one can only accept a low dimensional set of extra variables qj, then the microeconomic model still has to be narrowly restricted for exact aggregation to hold. However, the new approach also opens the possibility that reliance on structural stability is made less exacting. In order to understand this point well, one has to come back to the modelization of Section 7 and to see why it does not give free access to safe aggregation under all circumstances. 9. Potential

As equation contain a changes in represented Neglecting value of g, aggregation

difficulties

(44) reminds us, the macro-model should in general variable 0, or a vector CJ of variables, tracing those the statistical distribution of micro-units which are not by the macroeconomic exogenous variables such as 2. such changes would amount to using an incorrect for instance to use go instead of cl, hence to make an error, E= F(X, 2; 00) - F(X, 2; al).

(47)

Potential difficulties may concern either the identification of the characteristics that ought to be so considered, or the estimation of their values in any application of the aggregate relation. This relation is often used for comparative statics assessment, which can be exposed to serious error if some of the induced changes in the statistical distribution G are either ignored or incorrectly estimated. The point is sufficiently important for it to be spelt out in a particular case. Let us suppose that the population of individuals is made up of m subgroups (h = 1,2,. . . , m), with nh individuals in subgroup k, so that a proportion x:h of individuals belong to k, “k=-.

Typically,

the

groups

would

nk

(48)

n

differ

because

of both

a different

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distribution of individual characteristics and a different exposure to business conditions. Let us moreover assume that changes in the cumulative distribution G, of the units belonging to k are well represented by just the mean 2, of the exogenous variable xi for units belonging to k. We may then write

(4% k=l

No error will be involved if the vector y of parameters is identified with the vector of the m group averages and of m - 1 proportions nh, y=(~c1,~62,...,X,;711,712,...,71,-1),

(50)

if these averages and proportions are observed and if they appear as exogenous variables in the macro-model. However, it often happens that the relevant groups are not identified or that the group averages E, and proportions zk are not observed. Then the macro-model contains only the exogenous variables 2 and z, where 2 is the overall average. We may say that the model neglects a vector of parameters 0 with the 2(m- 1) components 36,/Z,. . . ,X,-J%; zl,. . . , nmwl. How sensitive J is to variations in X,JX and zk can be understood if we refer to (39) and (40) showing that we may write

(51) k=l

with

Variations in 5,/g or 7rk would not matter if the m functions D, were all the same linear function of Xk. However, for reasons that will be dealt with in Section 11, the derivatives of D, with respect to ZCCk vary from one group to another if the behaviour parameters ti tend to differ and if they significantly affect the propensity df/Gxi with respect to the exogenous variable xi. Such a case turns out to be frequent as soon as one precisely studies particular behaviour laws. Neglecting changes in jl,Jj; and 7zkis therefore likely to lead to errors in forecasting, which could be avoided when some such changes can be anticipated. It may also be dangerous in simulations intended to predict the effects of changes in some of the exogenous variables represented here as components of z, because these changes may react on the ratios Zk/% or the proportions 7th. If, for instance,

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taking 5$/X and nh as constant when z varies may lead to a serious miscalulation of the effect on J of a given change in z. (The article by Blundell, Meghir and Weber that will be published in the second special issue, 47(3), shows how some proportions that matter in consumer demand analysis are cyclical, hence correlated with exogenous variables that one may want to consider.) Since potential difficulties affect the use of macroeconomic models justified by the general notion of structural stability, one ought to be precise on the hypotheses underlying such models. One should never avoid a serious discussion about the joint statistical distribution of exogenous micro-variables and behaviour characteristics.

10. The facts of structural

stability

Structural stability is meant to be an empirical fact. Many features of observed statistical distributions about economic agents appear to be fairly invariant. As is well known, Pareto (189&1897) was the first economist to discover the phenomenon, when looking at a considerable number of distributions of personal incomes; he showed they could all be represented with a fair degree of accuracy by the same mathematical formula, with most often only two parameters, in some cases four. Commenting on this achievement, Schumpeter (1949) wrote: “Few if any economists seem to have realized the possibilities that such invariants hold out for the future of our science . . . nobody seems to have realized that the hunt for, and the interpretation of, invariants of this type might lay the foundations of an entirely novel type of theory” (reprinted 1951: p. 127). The lognormal distribution was also often found to perform very well for the representation of many economic statistical distributions, in particular for distributions of firms by size (Gibrat, 1931; Hart & Prais, 1957). Of course, like many other stylized facts in economics, structural stability is only approximate. If closely examined, the exact shapes of statistical distributions of personal incomes or of firm sizes do change somewhat through time and space; these changes cannot all be accurately traced by variations in the few parameters that were first identified as required for a rough universal fit. Moreover, if foundations for aggregate laws are to be justified, one needs also to look at other statistical distributions of individual characteristics, Thus, a large field of research is open for investigation. This is an area in which economists can now reap profit from the

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statistical investments made in the post-war decades and from modern computing facilities. Individual data sets and panels are available, which can reveal a good deal of the information needed for assessing the facts of structural stability. These special issues of Ricerche Economiche show that a number of economists are now fulfilling the wish expressed by Schumpeter almost fifty years ago. The recent work about exact aggregation, which was reported in Section 8, suggests moreover that empirical regularities ought to be searched for and gauged not only with respect to the distributions of agents but also with respect to agent behaviour, which may be more or less favourable for the validity of aggregate laws. Again, this remark is very much in the spirit of what is being done at present. For instance, Lewbel (1991) shows that average Engel curves, which depend on both the distribution of individual preferences and on the form of individual Engel curves, can be well fitted to a finite basis model of rank 3 (J=3), and even fairly well approximated by such a model of rank 2. Similarly, the article by Hildenbrand and Kneip (1993, this issue) empirically establishes the increasing dispersion of expenditure shares; this property may be seen as a joint result of the form of individual demand laws and of the distribution of individual tastes. Knowing where lapses from relevant structural stability occur also matters, because we need to improve on our present knowledge of the types and importance of aggregation errors that are likely to affect the conclusions drawn from aggregate models. In this respect, progress comes from research such as that reported in the next issue by the articles of Browning and of Blundell, Meghir and Weber.

11. The linear

case

Once the statistical approach to aggregation is accepted, attention turns, as we have seen, towards the form of the aggregate relations. In particular, one often wonders whether this form is similar to the one of the corresponding microeconomic relations. The reliance on the representative agent hypothesis in macroeconomic theory gives an obvious importance to the question. One cannot go very far in elucidating it while sticking to the level of generality adopted in this paper. Smoothing by aggregation may be said to occur naturally in the transition from equation (37) to equation (39); but even this argument ought to be looked at more precisely with respect to particular phenomena. However, it may still be instructive for other purposes to consider here the case in which the microeconomic relations are themselves so smooth that they are linear. Whilst it may often appear too simple, this

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case also gives insight of more general value, and was particularly studied in the literature on aggregation (for instance by Theil, 1954). I shall restrict the general model here in two ways.7 Not only will the microeconomic relation (37) be assumed to be linear in xi, i.e. yi = a(z, t,)x; + b(z, ti),

(54)

but also the statistical distribution of the individual exogenous variables xi will be assumed to depend only on a general scale parameter, which may be taken as being the mean 2, thus

G(t, 0; Y>= G(W; 0). Aggregation

then leads to the linear

macroeconomic

~=A(z)~+B(z),

relation

(56)

with

(57)

W4=j-b@, WG(C, e>,

(58)

these last two equations using the new variable [= c/Z. The intercept B(z) is simply the mean of the individual intercepts b,(z) = b(z, tJ, but the slope deserves attention. It may be written as

-44 = A,(4 + A,(z),

(59)

A,(z) is the mean of the individual slopes a,(z), whereas AZ(z) is the covariance of these slopes and the relative values of the individual exogenous variables xi/2 (note that the mean of [ is equal to 1). The slope of the macroeconomic relation deviates from the average value of the slopes of the individual relations. The deviation A,(z) is positive when high values of the exogenous variable xi tend to be associated with high values of the marginal propensity t Moreover, I do not deal here with aggregation in the time dimension, a subject that was much studied within a linear framework. The articles by Granger and by Gonzalo in the next issue concern that subject.

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a,(z). The presence of this second term is all the more relevant as it might be neglected. The fact that it may depend on the other exogenous variables represented by z should also be kept in mind, since it may affect comparative statics assessments. The study of linear aggregation in the 1950s was motivated mainly by the concern for the econometric problems of macroeconomics: how can results obtained from a cross-section of individual observations be entered into an aggregate relation? At the time, econometric methodology was also focusing on the linear case. In most writings no explicit reference was made to structural stability and to the statistical approach here considered. However, it was well shown that coefficients of aggregate relations, such as the slope A(z), differed from the average of the corresponding individual coefficients and that the difference depended on the parameters’ fitted values in regressions of the individual exogenous variables, such as xi, with respect to the aggregate exogenous variables, such as X. Reliance on a given macro-relation then followed from the idea that the regressions could be taken as stable, which was clearly a hypothesis on the statistical distribution of the individual exogenous variables.

12. Research

prospects

At the end of this article it is tempting to wonder whether the general theory of aggregation ought not to investigate some new topics or some developments of old ones. In particular, the statistical approach, as presented in Section 7, has accepted some restrictive hypotheses, which ought to be removed if the theory is to properly cover all applications. Such an objective cannot really be reached, because most aggregation problems encountered in practice appear in mathematical structures that are quite specific to the particular field considered. But extending the domain of validity of the general analysis would help to identify issues and contribute to our understanding, in the spirit accepted throughout this article. The hypothesis that only one endogenous variable and one individual exogenous variable appear in the microeconomic model (37) is not serious. Clearly the argument of Section 7 would apply as well to several such variables. However, one would then stick to the form chosen for (37) i.e. to what is commonly called a “reduced form”: each individual endogenous variable is expressed as a function of only exogenous entities. Most often, models with several endogenous variables are more conveniently given in a “structural form”. Even for the macro-model, such a form often looks more illuminating than a system of reduced equations. But, already in the linear case, aggregating a system of struc-

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tural equations is somewhat subtle. Consider for instance a case with two endogenous variables, yli and yZi, with one individual exogenous variable xi and two other exogenous variables, z1 and z2. Let the behaviour of the individual agent be ruled by the two following equations,

yzi= czl(ti)Y1i+ a,(tJ”i + e,(&

+ bJtJ~

(63)

Aggregation of this structural system may be thought of as leading to a macro-system of the same form, with the first equation written as j1 = C,, jz + A,% + IQ, + B,.

(64)

But, after our treatment of the preceding section, we understand that C,, will not be a simple average of the cJtJ; it will also depend on the covariance of these coefficients and the individual values yzt. But the latter are endogenous. Hence, C,, will be a complex function of all the exogenous entities. The behavioural meaning of C,, will be blurred with respect to the corresponding meaning of the c&tJ. In the previous section something similar occurred about the meaning of A(z), because of A,(z); but the phenomenon is here still more serious. We shall not go more deeply into the study of this case. It points to a useful type of inquiry that was approached by Theil (1959) but seems to have been neglected since then. The statistical approach also has a role to play in cases in which the micro-model may not be quite of the form studied from Section 5 on, namely a system made of the juxtaposition of n individualized micro-systems (18). The footnote concerning constant returns to scale in Example 2 for the justification of aggregate production functions showed that, for some relevant cases, the aggregation of economic relations may involve a somewhat more complex structure. Again, one may wonder whether other subclasses of the general micro-model (1) than the one given by (18) ought not to be studied. As for the aggregation of variables, we considered only arithmetic means. Sums would of course be dealt with in exactly the same manner. But the statistical approach could cope with other aggregation rules. The main point is to know whether, in the general formulas (3) and (4) given at the beginning of the article, the individual variables are interchangeable in the definition of the functions g(x) and My). By this it is meant that, for instance, if only one numerical endogenous micro-variable yi appears, then the macro-variable Y= h(y) keeps the same value when y; changes for only two individuals j and k with the value first given to y; being

134

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attributed to yk and the value given to yh attributed to yj. Whether many interesting aggregation problems require such an extension remains, however, to be seen.

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Stiglitz, J. (1970). Non substitution theorems with durable capital goods. Reuiew of Economic Studies, October. Theil, H. (1954). Linear Aggregation of Economic Relations. Amsterdam: North Holland. Theil, H. (1959). The aggregation implications of identifiable structural macrorelations. Econometrica, January.