## Aggregation

# Aggregation

3. Aggregation and the determination of economic laws

Economic systems are composed of large numbers of individual economic units interacting to determine market prices and quantities of innumerable goods and services bought and sold by each unit. The inherent complexity of such systems has two important consequences for the methodology of economics, as compared with such experimental sciences as physics, chemistry, and even biology. First, one cannot hope to isolate individual economic units from their context, study them experimentally, and establish what could be called elementary laws of economic behavior. Second, even if one could rely on elementary laws of behavior deduced from logical arguments or adopted as plausible working assumptions, it may be extremely difficult, because of the innumerable complex interactions among the individual units, to determine what these laws imply concerning the behavior of the entire system.

Nevertheless, it is absolutely necessary to over-come these obstacles that stand in the way of determining economic relationships to be used within models intended to explain past economic developments or to forecast future developments. The subject matter of aggregation in economics is the study of the extent to which the difficulties presented by such large numbers of economic units and commodities can be surmounted by considering aggregates of subsets of decision-making units (firms, households, etc.) or of commodities or prices.

**Scope of the aggregation problem.** The aggregation problem arises in nearly all economic decisions and analyses. Every economic agent—the smallest *homo economicus,* the manager of a large firm, or the economic theorist—is confronted by numerous economic forces, although he may not be aware of them all. In making his economic decisions or in deducing the consequences of these forces, he must take account of them in the best possible way. Considered from this point of view, the aggregation problem would be confounded with that of economic science and would be devoid of specificity. Actually, a closer examination of the aggregation problem, initiated by Hurwicz (1952) and pursued by Malinvaud (1956a), narrows the role of aggregation in economic analysis.

The first object of economic theories is to reach general conclusions establishing either existence theorems for some classes of economies or the adequacy of particular economic means and structures for achieving particular economic goals. For example, economic theorists may initially be concerned with the existence and stability of a general equilibrium in an economy characterized by decentralized decision making; if a general equilibrium can be shown to exist for such an economy, they may then determine whether the equilibrium position is Pareto optimal in that no change can benefit anyone without injuring someone else. But too often these results are so general that they provide only loose guides for economic policy; even if they are formally correct, they do not provide answers to many fundamental questions encountered in formulating responsible economic policy. For instance, equivalence conditions between a Paretian optimum and a value equilibrium do not afford any knowledge concerning the sociological aspects of the attainable states. On the other hand, an approximate idea of the values and determinants of global economic variables such as aggregate production or aggregate consumption can be much more useful in determining the need for, and appropriateness of, some economic policy.

The degree of complexity and precision needed in analyzing any economic question depends upon the nature of the actions to be undertaken on the basis of the analysis. Thus, it is advisable to make the best possible allocation of resources available for the analysis among (1) the gathering of data necessary to understand the situation, (2) the degree of precision needed in analyzing the situation, (3) the type and detail of decisions to be made, and (4) the choice and extent of the means by which these decisions are to be carried out. Keeping in mind these four general principles that are fundamental to the theory of decision making, it can be said that the specific concern of aggregation in economics is the study of the best way of making this allocation and of the choice of criteria to be considered.

**Directions of research.** Research on aggregation would be greatly facilitated if information on the development of an economy could be gained from a small number of variables that are functions of the actions taken by the numerous decision-making units. These variables should not be mere mental constructs but should lend themselves to actual numerical determination; hence, the functions describing them should be of a simple analytical form and readily determinable statistically. Beyond these general requirements, different writers have made various demands on the variables. May (1946) suggests that it should be possible to deduce *all* relevant economic variables from knowledge of this small number of variables, whereas Malinvaud (1956a) suggests that it need only be possible to determine a few simple, economically significant relationships among the small number of variables. In the following discussion of the approaches of May and Malinvaud, the terminology of aggregation theory will be defined and a historical sketch of developments of the subject matter and methodology of the theory will be given.

May considered a theoretical economic system S such that all its variables can be expressed as functions of *p* parameters only. The *p* parameters may vary freely or may be subject to constraints not expressed in S. This system is called the “microeconomic” system, and the same adjective will qualify the variables and functions in S. Although these microeconomic elements are conceptually defined, they may often be neither measurable nor explicit. The analytical form of S may present many difficulties in solving the system or in studying its qualitative properties. On the other hand, there might exist functions of several microeconomic variables in S, functions to be called “macroeconomic,” that define macroeconomic variables or aggregates. For instance, suppose that after appropriate indexing, S is composed of I times *K* microeconomic relationships, f_{ik}, involving all conceivable microeconomic variables. Then it may be possible to find *I* functions, F_{i} (f_{il}, f_{i2}) – the F_{i} being the aggregates. The aggregates cannot be of an arbitrary nature but must have a simple and important economic meaning. A number *j* › *p* of these aggregates being considered, there are, generally speaking, *j* – *p* independent relationships that constitute a system, s, called the macroeconomic system. By combining microeconomic variables in S to obtain the *j* aggregates, one obtains s. Once the form and main features of *s* have been theoretically recognized, it is extremely desirable in practice that it be possible to determine *s* by statistical methods without relying directly on information contained in S. The great interest in the existence of a macroeconomic system, s, is now easily understood, for the existence of such a system enormously simplifies the study of economic systems and consequently the process of decision making at every level.

Before May, Klein (1946a) had presented the aggregation problem in a similar way except that he restricted himself to some part, say D, of S. Although his exposition did not portray the full generality of the aggregation problem, he nevertheless treated the problem as it is most often met in practice, where difficulties are encountered only for some economic variables or relationships. Aggregations in S restricted to D have a great advantage in that they can be used in some other system, say S’, differing from S but involving the same part D. For instance, S and S’ might present the same description of production, the part D, although they differ in their treatments of consumption, income distribution, etc. An aggregation procedure pertaining to D only could then be used in both S and S’. A general feature of research in aggregation is the twofold preoccupation of solving the aggregation problem in a specified economic framework and of finding partial solutions that may be utilized in many different frameworks.

Malinvaud noted that, as regards their consequences on actual economic actions and realizations, varied decisions might have the same practical implications or implications that can at least be considered equivalent. For example, aggregate fruit consumption may be largely unaltered by some variations in the composition of the fruit bundle; hence, a particular explanation of aggregate fruit consumption may be approximately valid for various decisions that determine the composition of fruit purchases. Consequently, every aggregation of a system S resulting not in exactly the same implications but in equivalent ones must be accepted. This approach usefully enlarges that of May, for it does not vitiate the conclusions derived from an exact system. Malinvaud calls this type of aggregation “intrinsic aggregation”; it includes the types of aggregations considered by Klein and May.

As will be seen below, intrinsic aggregations do not exist in general. Therefore, a solution is sought (1) by imposing restrictions on the range of variation of the microeconomic variables, the restrictions being based on certain economic considerations; or (2) by allowing the aggregates to be tolerably approximate, the permissible range of errors being determined by the uses to be made of the aggregates. The last approach is also the result of Malinvaud’s study, and he calls it “representative aggregation.”

The foregoing discussion suggests that the principal results obtained in the study of aggregation might be grouped under two headings, intrinsic aggregation and representative aggregation; and this dichotomy will be followed below. Systematic studies, consciously noting the difficulties due to aggregation in over-all economic problems, are rare when methodological studies are excepted. They are to be found especially in the literature dealing with the determination and empirical testing of economic laws. Because of their basic importance, they are grouped here in a special section (see section 3 below), although they could technically be placed under the heading of “representative aggregation.”

## 1. Intrinsic aggregation

Intrinsic aggregation has been studied in the context of production relationships and in the context of consumption relationships.

### Aggregation of production functions

Klein (1946a) examined the circumstances necessary for the existence of an aggregate production function that could be employed in the type of macroeconomic analysis developed by John Maynard Keynes in *The General Theory*. In doing so, he presented, probably for the first time, a precise statement of an aggregation problem. The following problem was posed: Let there be *n* microeconomic production functions that may be written in their implicit forms as

R_{i}(**a** _{i}, **b** _{i}, **C** _{i}) = 0 i = 1, 2, …, n,

where **a** _{i} is the vector of outputs, **b** _{i} the vector of labor inputs, and **c** _{i} the vector of capital inputs for the *i*th firm. Can three functions,

*A = A*(**a** _{1}, …, **a** _{n})*B = B*(**b** _{1}, …, **b** _{n})*C = C*(**C** _{l}, …, **C** _{n}),

be found such that a nonvanishing relation *R(A,B,C)* = 0 holds for all values of **a** _{1}, …, **a** _{n}, **b** _{1}, …, **b** _{n}, **c** _{1}, …, **c** _{n} satisfying the microeconomic production functions? Aggregate output, labor input, and capital input are denoted, by A, B, and C, respectively, and R *(A,B,C)* = 0 is the aggregate production function.

Nataf (1948) established that such aggregates could exist only under the condition that each *R _{i}* can be written as a sum of function A

_{i}of the

**a**

_{i}’s,

*B*

_{i}of the

**b**

_{i}’S, and

**C**

_{i}of the

**c**

_{i}’s, that is: only if

*R*0 is equivalent to A

_{i}=_{i}(

**a**

_{i})+B

_{i}(b

_{i}) + C

_{i}(C

_{i}) = 0. Then the aggregates A, B, and C are, up to a transformation, equal to ΣA

_{i}, ΣB

_{i}, and ΣC

_{i}respectively, and

*R*= 0 is equivalent to A +

*B*+ C = 0. These conditions are obviously unrealistic.

*Aggregation in Leontief systems*. Problems similar to that of Klein have been solved for aggregation of goods in a linear system of industrial relationships of the Leontief type. Ara (1959), Hatanaka (1952), McManus (1956), and Malinvaud (1956b) studied the conditions that must be met in order that the linear production relationships assumed by Leontief continue to hold for some aggregates of microeconomic goods. Aggregation within such systems is necessary in order to apply Leontief’s theory to actual forecasting or planning. The conditions are always extremely strict for aggregation to be possible in the purely algebraic range of validity of these tableaux [*see*Input–Output Analysis].

*Disaggregation problems.* In considering the utilization of interindustrial tableaux, Fei (1956) raised an interesting problem perfectly within the realm of intrinsic aggregation, although the problem presents itself in exactly the reverse form. In practice, examining whether any appreciable information can be gained by disaggregating aggregated variables or relationships may be of more immediate interest than examining whether information is lost in constructing aggregates. Of course, any gain in information realized by disaggregation must be weighed against the costs involved in doing so. Fei assumed there is a known aggregated Leontief matrix **A** , while the ideal unknown matrix is * B .* For computing production levels corresponding to a given final demand, we need the inverse of (

**I – B**). In fact, we can only use that of (

**I – A**). In order to judge the usefulness of determining

*of which only some structural properties are known in excess of its aggregation in*

**B**,**A**, Fei builds matrix operators that disaggregate

**A**to

**A***. The inverse of the transformation that disaggregates

**A**to

**A***will not only aggregate

**A***to

**A**but will also yield (

**I — A**)

^{-1}when applied to (

**I – A***)

^{-1}. With knowledge of a particular

**A***and structural information on

**B**, and using the numerous mathematical studies on approximations and errors in matrix inversion, one can often express bounds on errors resulting from the substitution of

**A***for

**B**. Hence, since

**A***and (

**I — A***)

^{-1}give the same aggregated results as

**A**and (

**I — A**)

^{-1}, one can determine whether the magnitude of the established errors warrants actual determination of

*Similar problems have been studied by Fisher (1958; 1962).*

**B.**### Aggregation in consumption problems

Gorman (1953) considered the problem of aggregating the indifference functions of all individuals in a population into a single function expressing the population’s mean consumption in terms of the population’s mean disposable income. He showed that this is possible only when Engel curves for each individual are straight lines and have the same direction for all individuals for a given set of prices, although the direction of the curves may be a variable function of prices (see also Nataf 1953, p. 20).

Nataf (1958), considering cases where all individuals have the same income or, more generally, where individuals have different incomes that depend on the same unique parameter, found the least number of representative consumers whose behavior is in accord with the theory of choice and whose mean consumption is that of the population. Curiously, this number is equal to the number of goods minus one, that is to say, considerably smaller than the number of individuals in the population. This result suggests that there is some hope of finding a few explanatory relations of consumption.

Very similar results appear as by-products in studies by Stone (1954), Fourgeaud and Nataf (1959), and others. Their results are more precise because more restricted families of demand functions are considered. In determining individuals’ demand curves, these authors find that the curves obtained imply that the total amount of every good consumed by the population may be expressed as a function of simple aggregates of individual incomes and of coefficients or parameters contained in the individual functions. These properties might facilitate explicit determination of the aggregate coefficients from global statistical data and permit testing of the goodness of fit of the type of individual functions involved.

*Aggregation of sets of goods*. Hicks (1939) and Leontief (1936) have studied a group of goods whose prices always change by the same proportion. Leontief showed that the total expenditure of an individual consumer on such a group of goods can be determined from an indirect utility function that depends only on his total income, prices of other goods, and the proportionality coefficient of variation of prices of goods in the group. however, it is not possible to determine the individual’s consumption of each of the goods in the group.

Gorman (1959), following an initial study by Strotz (1957), defined aggregates A, B, C, … of goods and examined the conditions that must be satisfied by an individual’s utility function in order that determination of the individual’s consumption bundle be possible, either rigorously or approximately, in two stages. In the first stage, expenditures on each of the aggregates are determined given total money income and price indexes for each of the aggregates. In the second stage, expenditures on each good in aggregate A are determined given total expenditures on A and the prices of every good in A; and similarly for B, C, … . Employing realistic approximations, Gorman concluded that the utility function required can be expressed as a summation of (1) arbitrary functions of each of the aggregates A, B, C, …, *J* and (2) a unique arbitrary function of the aggregates K, L, …, each aggregate K, L, … being a homogeneous function of degree one of its component goods. This result is valid for the case where the distribution of expenditures on the aggregates applies locally for small deviations of prices and income from a known origin. Since this is the case generally met in practice, these approximations, as Gorman noted, are more interesting than the rigorous solution holding for any changes in prices and income.

## 2. Representative aggregation

Representative aggregation encompasses studies of the errors introduced into an economic theory or a practical decision when an aggregated model is substituted for a detailed model.

### Macroeconomic validity of a microtheory

Morishima and Seton (1961) attempted to determine the extent to which aggregation phenomena might partly or wholly invalidate Marx’s microeconomic results on the comparison of price and value systems. They not only found the types of aggregation that left Marx’s results invariant to the level of aggregation (a matter relevant rather to intrinsic aggregation) but also searched for aggregation procedures that resulted in deviations of the same order of magnitude between aggregates as between microeconomic elements, allowing for random deviations within the microeconomic system. Their study could equally well be discussed in section 3, since it is an elaborated investigation of methods of verifying economic laws deduced from a given theory.

### Aggregation of intertemporal relationships

Simon and Ando (1961) studied an intertemporal model with the following properties: (1) the current values of the variables in the model depend on their past values and are obtained through a linear stationary transient transformation of the past values, and (2) the variables for any period can be subdivided into groups in such a way that the values of the variables in a particular group depend only on the values of the variables in that group in the preceding period. Consequently, the transformation that gives the current values of all the variables in the model from their values in the preceding period is a matrix consisting of block submatrices on the principal diagonal, all other elements of the matrix being very small. Each submatrix corresponds to a particular group of variables. Due to the stationarity property of the transformation, the authors were able to define representative aggregates—one for each of the block submatrices—and to deduce the behavior of the aggregates from a simpler system than the original one.

### Errors due to aggregation

Fisher, proceeding in the line of thought of Fei, considered not only the problem of aggregating sets of microeconomic variables but also the problem of going back from aggregates to a microeconomic set in such a way that the deviations from the true values of the microeconomic variables would be as small as possible from a certain point of view. He studied these problems for the classical cases of interindustrial exchanges (1958) and also for cases where stochastic features of the problems might be of relevance in selecting the aggregate model (1962).

Balderston and Whitin (1954) studied numerically the errors resulting from different aggregations associated with different treatments of imports in a Leontief system. They found the errors to be quite important, thus stressing the need to take careful account of aggregation phenomena.

## 3. Aggregation and the determination of economic laws

Economic laws generally emerge from an abstract microeconomic analysis, and their validity can seldom be tested by basic experiments and measurements. Rather, theories are tested by examining the extent to which their global implications are verified. Depending upon the type of problems studied and the aggregations made, various questions of aggregation theory will arise, belonging, technically speaking, to representative aggregation. Theil (1954) initiated the study of these questions. He assumed that microeconomic relationships are stochastic linear functions of microeconomic variables and that linear combinations of microeconomic variables are used as aggregates. He then examined whether there exist stochastic relationships among the aggregates of the same general form as the microeconomic relationships. The aggregates were shown to be tied by relationships involving functions of other unaggregated microeconomic variables of the system. This result points out the difficulties encountered in testing the validity of elementary laws (the microeconomic relationships) through aggregated relationships.

In other works, Theil (1957; 1959; 1960) and Mundlak (1961) have studied other examples where aggregation may generate errors or losses of information impeding the determination or testing of economic theories. For example, Theil (1959), using the two-stage least-squares estimation method, attempted to evaluate the loss of information in an estimate of a relationship directly based upon already aggregated variables, rather than the loss resulting from aggregation of the corresponding microeconomic relationships.

In a linear regression model for monthly data, Mundlak (1961) investigated the error in the regression coefficient when monthly data were grouped in yearly aggregates. Mandelbrot (1960; 1961) was able to demonstrate how some assumptions regarding stochastic variations of individual incomes over time result asymptotically in a general Pareto distribution of income.

## Conclusions

Choice of a convenient aggregation is an absolutely fundamental problem in practical applications and even in the material interpretation of economic theories. It is generally impossible to build aggregated rigorous models of an economy valid through time and space, a fact that must be kept in mind (1) in interpreting real economic laws and (2) in conceiving the methodology of applied economic research. As for the first point, it seems that the results of aggregation have not yet been fully turned to account, and they might shed much light on such important problems as imperfect competition due to lack of complete information on the part of buyers and sellers, resulting in poor aggregate knowledge of their reciprocal reactions. This subject is probably a very fruitful one that has yet to be explored.

On the other hand, constructive ideas concerning the second point have appeared. Hurwicz’ and Malinvaud’s concepts provide a framework for rationally estimating the useful degree of precision in the elaboration and application of economic models. But before these ideas can be translated into practical methods, knowledge of representative aggregation must be enlarged for economic variables such as production, consumption, and investment. Some progress has been made in this matter; but much work, essentially in two directions, remains to be done. First, more studies of representative aggregation problems must be undertaken, a task related to obtaining much better knowledge of actual microeconomic laws regarding these phenomena. Second, these sectoral studies of production, consumption, etc., must be synthesized and incorporated in economic models to be tested as explanations of past economic developments or to be applied in economic planning at the sectoral, national, or even international level. Before progress can be made in these important areas, the results of theoretical analysis must be related to knowledge of the practical problems to be dealt with and solved. Theoreticians and practical economists should be aware of these problems and closely cooperate in resolving them.

AndrÉ Nataf

## BIBLIOGRAPHY

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## aggregation

**aggregation** **1.** The group of animals that forms when individuals are attracted to an environmental resource to which each responds independently. The term does not imply any social organization.**2.** The process in which soil particles coalesce and adhere to form soil aggregates. Aggregation is encouraged by the presence of bonding agents such as organic substances, clay, iron oxides, and ions (e.g. calcium and magnesium).**3.** The progressive attachment of particles (e.g. ice or snow) or droplets around a nucleus, thereby causing its growth.

## aggregation

**aggregation** **1.** Process in which soil particles coalesce and adhere to form soil aggregates. The process is encouraged by the presence of bonding agents such as organic substances, clay, iron oxides, and ions (e.g. calcium and magnesium).

**2.** Progressive attachment of particles (e.g. ice or snow) or droplets around a nucleus, thereby causing its growth.

## aggregation

**aggregation** A process in which soil particles coalesce and adhere to form soil aggregates. The process is encouraged by the presence of bonding agents such as organic substances, clay, iron oxides, and ions (e.g. calcium and magnesium).

## Aggregation

# Aggregation

a collection of parts of a whole; a natural group or body of human beings. See also **assemblage, collection, gathering.**

*Examples:* an aggregation of believers, 1638; of isolated settlements, 1863; of species.

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