Dale Jorgenson

Samuel W. Morris University Professor

WELFARE: Volume One -- Preface

This volume is the first of two volumes containing my empirical studies of consumer behavior. The centerpiece of the volume is an econometric model of demand obtained by aggregating over a population of consumers with heterogeneous preferences. Heterogeneity is captured by allowing preferences to depend on the demographic characteristics of individual households. The two principal streams of empirical research on consumer behavior are unified by implementing the model through pooling of aggregate time series data with cross section data for households.

The second volume, Measuring Social Welfare, presents a new conceptual framework for normative economics that exploits the model of aggregate consumer behavior presented in this volume. The model is a system of aggregate demand functions constructed from underlying individual demand functions. Measures of individual welfare are recovered from the individual demand functions and combined into an indicator of social welfare, reflecting concepts of horizontal and vertical equity. This approach has been successfully applied to the evaluation of economic policy, the measurement of poverty and inequality, and the assessment of the standard of living and its cost.

The exact aggregation approach to modeling aggregate consumer behavior presented in this volume, like the representative consumer approach that preceded it, rests on the foundations of the theory of individual consumer behavior. (1) The exact aggregation approach readily encompasses both the theory of the consumer and heterogeneity of preferences. The essential innovation in this approach is to incorporate the attributes of consumers, such as demographic characteristics, that reflect this heterogeneity into a model of aggregate demand through statistics of the joint distribution of attributes and total expenditures over the population.

The representative consumer approach has persisted for decades in the face of critical deficiencies. The first is that the theory of individual consumer behavior, applied to aggregate behavior, is essentially vacuous. (2) The second is that the main implication of the representative consumer model - he integrability of aggregate demand functions - oes not adequately describe aggregate data. This is not surprising, since conditions required for validity of the representative consumer approach are totally at odds with evidence accumulated over more than a century of empirical research on the behavior of individual households. This evidence reveals heterogeneity of individual preferences that is inconsistent with the model of a representative consumer, but is perfectly consistent with the theory of individual consumer behavior.

The econometric model of aggregate consumer behavior presented in my 1982 paper with Lawrence Lau and Stoker, reprinted in Chapter 8, has successfully extricated demand modeling from the procrustrean bed provided for more than half a century by the highly restrictive framework of a representative consumer. This model has provided a wholly new point of departure for subsequent empirical research, including my 1984 paper with Stoker, reprinted in Chapter 9, my 1983 paper with Daniel Slesnick and Stoker, reprinted in Chapter 11, and my 1987 paper with Slesnick, reprinted as Chapter 5 of Measuring Social Welfare. Stoker (1993) provides a detailed survey of this rapidly growing literature.

The model presented in Chapter 8 unifies two distinct lines of empirical research on consumer behavior. The first issues from the seminal contributions of Henry Schultz (1938), Richard Stone (1954b), and Herman Wold (1953) and focuses on prices and total expenditure as determinants of aggregate demand. The theory of consumer behavior is used to derive a model of a representative consumer. This model is implemented through time series data on prices and per capita quantities consumed and total expenditure.

A second line of research, represented by the classic studies of household budgets by Conrad Leser (1963), Sigmund Prais and Hendrik Houthakker (1955), and Holbrook Working (1943), has focused on demographic characteristics and total expenditure as determinants of household demand. The theory of consumer behavior is used to derive a model of the individual consumer. This model is implemented through cross section data on quantities consumed, total expenditure, and the characteristics of households.

Aggregate time series and individual cross section data have been combined by Stone (1954b) and Wold (1953) in models of aggregate demand based on the model of a representative consumer. Cross section data are used to estimate the effect of total expenditure and time series data to characterize the impact of prices. However, this pioneering research omits a crucial link between individual and aggregate demands arising from the fact that aggregate demand functions are sums of individual demand functions.

Aggregate demand functions depend on prices and total expenditure; however, aggregate demands depend on individual expenditures rather than aggregate expenditure. If individual expenditures are allowed to vary independently, the model of a representative consumer places restrictions on preferences that severely restrict the dependence of individual demands on total expenditure. Alternatively, if individual expenditures are functions of aggregate expenditure, for example, fixed proportions of expenditure, the implications of the theory of consumer behavior for aggregate demand are extremely limited.

One implication of the theory of consumer behavior is that aggregate expenditure is a weighted sum of aggregate demand functions with each function weighted by the corresponding price. A second implication is that aggregate demands are homogeneous of degree zero in prices and expenditures. Erwin Diewert (1977) and Sonnenschein (1973b) have shown that any system of demand functions that satisfies these two conditions, but is otherwise arbitrary, can be rationalized as the sum of individual demand functions with a fixed expenditure distribution.

The implications of aggregation over a population of consumers for the model of a representative consumer are summarized in Section 8.1.A of Chapter 8. First, Terence Gorman (1953) has shown that individual demand functions must be linear in the total expenditure and attributes of households, such as demographic characteristics. Furthermore, if demands are equal to zero when expenditure is zero, all consumers must have identical, homothetic preferences.

Homothetic preferences are inconsistent with well-established empirical regularities in the behavior of consumers, such as Engel's Law, which states that the proportion of expenditure devoted to food is a decreasing function of total expenditure. (3) Identical preferences are inconsistent with empirical findings that expenditure patterns depend on the demographic characteristics of households. (4) Even the weaker form of Gorman's results, that demands are linear functions of expenditure with identical slopes for all individuals, is inconsistent with empirical evidence from budget studies. (5)

Despite the conflict between Gorman's characterization of individual behavior and empirical evidence on households, this characterization has been an important stimulus to empirical research. The linear expenditure system, proposed by Lawrence Klein and Herman Rubin (1947-1948) and implemented by Stone (1954b), has the property that individual demand functions are linear in total expenditure. This system has been widely used in modeling time series. Generalizations that retain the critical property of linearity in total expenditure have also been employed. (6)

John Muellbauer (1975, 1976a) has substantially generalized Gorman's concept of a representative consumer. In Muellbauer's approach individual preferences are identical, but not necessarily homothetic, and demands may be nonlinear functions of total expenditure. An implication of nonlinearity is that aggregate expenditure shares, interpreted as the shares of a representative consumer, depend on prices and a function of individual expenditures that is not restricted to aggregate or per capita expenditure.

Despite the conflict between Muellbauer's assumption of identical preferences and empirical evidence on households, his characterization of individual behavior, like Gorman's, has been an important stimulus to empirical research. Ernst Berndt, Masako Darrough, and Diewert (1977) and Angus Deaton and Muellbauer (1980a) have implemented models of aggregate demand conforming to Muellbauer's characterization. Both studies retain the assumption that preferences are identical for all individuals.

Lau's (1977b, 1982) theory of exact aggregation, summarized in Sections 8.2 and 8.3 of Chapter 8, is the key to surmounting the limitations of the representative consumer model. One of the remarkable implications of Lau's theory is that individual demand functions can be recovered uniquely from a system of aggregate demand functions. This makes it possible to exploit the theory of the individual consumer in specifying a model of aggregate demand - he feature of the representative consumer model that accounts for its widespread application.

Lau permits aggregate demand functions to depend on one or more symmetric functions of individual expenditures and attributes of households such as demographic characteristics. These functions can be interpreted as statistics describing the population of consumers. Lau's (1977b, 1982) Fundamental Theorem of Exact Aggregation, Theorem 8.1 of Chapter 8, provides a characterization of individual and aggregate demand functions that depends only on the existence of well-defined individual demand functions:

1. Individual demand functions for the same commodity must be identical up to the addition of a function that is independent of individual expenditure and attributes.

2. Individual demand functions must be the products of separate functions of prices and of individual expenditure and attributes.

3. Aggregate demand functions depend on statistics of the joint distribution of individual expenditures and attributes; the only admissible statistics are additive in functions of individual expenditures and attributes.

4. Aggregate demand functions are linear functions of these statistics.

Lau's theory of exact aggregation contains the models of a representative consumer of Gorman (1953) and Muellbauer (1975, 1976a) as special cases. For example, Gorman's model includes only one statistic describing the population of consumers, namely, aggregate expenditure. Muellbauer's model involves two such statistics, one of which must be aggregate expenditure. The assumption of identical preferences can be tested by eliminating the attributes of individual consumers from statistics that characterize the population of consumers in Lau's theory.

Gorman (1981) has shown that a system of individual demand functions that is linear in functions of total expenditure depends on at most three such functions. Although he motivates the condition of linearity by its connection with aggregation, he does not derive this condition from a theory of aggregation. Arthur Lewbel (1991) presents a test for determining the number of functions of total expenditure empirically and has applied this test to cross section data for individual households with a common set of demographic characteristics. Neither Gorman nor Lewbel deal with the issues raised by heterogeneity of preferences.

A system of individual demand functions that conforms to the theory of exact aggregation must be linear in one or more functions of total expenditure and the attributes of the individual consumer. In Section 8.3 we incorporate differences in preferences into aggregate demand functions. A simple, but relatively tractable, representation of individual demands involves three functions of the total expenditure and attributes of households. One of these is total expenditure, a second is nonlinear in expenditure and does not depend on attributes, and a third depends on both expenditure and attributes.

The second step in specifying a model of aggregate consumer behavior is to incorporate the properties of individual demand functions generated by utility maximization, subject to a budget constraint. Such demand functions are said to be integrable. A complete characterization of integrable demand functions is given, for example, by Hurwicz (1971) and summarized in Section 8.1.C of Chapter 8 as follows:

1. Homogeneity. Individual demand functions are homogeneous of degree zero in prices and expenditure.

2. Summability. A weighted sum of the individual demand functions with each function multiplied by the price of the corresponding commodity is equal to total expenditure.

3. Symmetry. The matrix of compensated own- and cross-price effects must be symmetric.

4. Nonnegativity. The individual quantities demanded are nonnegative.

5. Monotonicity. The matrix of compensated own- and cross-price effects must be nonpositive definite for all prices and total expenditure.

Summability and homogeneity simplify the representation of individual demands considerably. The nonlinear function of expenditure must be the product of expenditure and either the logarithm or a power function of expenditure. Demands also depend on the attributes of individual households such as demographic characteristics. Aggregate demand functions depend on prices as well as the joint distribution of expenditures and attributes among individuals.

The theory of exact aggregation enables us to specify individual demands as functions of expenditure and attributes very precisely. To incorporate the implications of the theory of the individual consumer, we must also specify the dependence of these demands on prices. In Section 8.4 we show that only the transcendental logarithmic or translog indirect utility function introduced in my 1975 paper with Laurits Christensen and Lau (1975), reprinted as Chapter 1, combines flexibility in the representation of preferences with parsimony in the number of parameters that must be estimated. (7)

In the model presented in Chapter 1 we incorporate the implications of the integrability into the model of a representative consumer and test the resulting restrictions. Previous tests had employed additive and homothetic preferences in formulating these tests. As a consequence, rejections could be attributed either to failure of integrability or restrictions on preferences. We introduced the indirect translog utility function as a means of discriminating between these two alternative interpretations.

In Chapter 1 we carry out a nested sequence of tests, beginning with tests of integrability and continuing with tests of additivity and homotheticity of preferences, jointly and in parallel, conditional on integrability. The tests reject integrability and, conditional on integrability, reject both additivity and homotheticity. The results have strongly adverse implications for models of consumer demand based on the model of a representative consumer and, more specifically, for models incorporating additive or homothetic preferences.

My 1975 paper with Lau, reprinted as Chapter 2, introduced time-varying preferences into the model of a representative consumer and developed a detailed methodology for choosing an appropriate set of restrictions on preferences, including separability, homotheticity, and neutrality or independence of time. We consider fifty-four different combinations of these restrictions and, conditional on the validity of the representative consumer model, select a relatively parsimonious specification for the United States . We reject restrictions such as additivity, homotheticity, and neutrality. My 1978 paper with Klaus Conrad, reprinted in Chapter 6, carries out a parallel set of tests for Germany with similar results.

My 1977 paper with Lau, reprinted as Chapter 3, and Lau (1977a) show that integrable systems of demand functions in common use - uch as the linear logarithmic demand system employed by Schultz (1938), Stone (1954b), and Wold (1953) - mply that expenditure shares are constant, so that underlying utility function is neutral and linear logarithmic. This utility function embodies the restrictions implied by additivity, homotheticity, and neutrality. Similar results obtain for the Rotterdam system of Anton Barten (1964a) and Henry Theil (1965). These systems can be used to test integrability only in conjunction with neutral linear logarithmic utility.

My 1979 paper with Lau, reprinted as Chapter 4, considers tests for integrability for the model with time-varying preferences. These tests are based on precisely the same formulation of integrability as in Section 8.4 of Chapter 8; however, the implied restrictions are incorporated into the model of a representative consumer. By contrast with the approach of my 1975 paper with Christensen and Lau, tests of homogeneity and summability of demand functions are separated from the test of symmetry. In addition, my 1979 paper with Lau presents tests of nonnegativity and monotonicity. The monotonicity test is based on Lau's (1978) methodology for testing the semi-definiteness of a real symmetric matrix.

My 1986 paper with Lau, reprinted as Chapter 5, carries out the tests of integrability proposed in our 1979 paper for U.S. aggregate time series for the period 1947-1971. We first test homogeneity and summability in parallel without imposing symmetry. We then test symmetry, nonnegativity, and monotonicity, given homogeneity and summability. We reject each of these implications of integrability. My 1979 paper with Conrad, reprinted in Chapter 7, carries out a parallel system of tests for Germany with similar results.

My paper with Christensen and Lau and my four papers papers with Lau, reprinted as Chapters 1-5, and my parallel studies with Conrad, reprinted as Chapters 6 and 7, provide an exhaustive treatment of econometric modeling within the representative consumer framework. We made it possible to discriminate between the two possible interpretations of rejections of this framework. The first is that integrability must be rejected. The other is that functional form restrictions - such as additivity, homotheticity, or neutrality - ust be rejected. Our results show that integrability must be rejected, setting the stage for development of the exact aggregation approach in Chapter 8.

A key contribution of Chapters 1-5 was to introduce and characterize the indirect translog utility function and the corresponding system of individual demand functions. (8) This system provided a flexible and parsimonious approach for modeling individual demand with the representative consumer framework. These advantages are immediately transferable to modeling individual and aggregate demands within the exact aggregation framework. The characterization of integrability in my 1979 paper with Lau and the analysis of the structure of preferences in our 1975 paper also carry over directly.

The final contribution of the papers presented in Chapters 1-5 is the successful application of new statistical methods for systems of nonlinear simultaneous equations to modeling aggregate demand. These methods are essential for avoiding simultaneous equations bias in the estimation of price effects. Methods of estimation for the nonlinear simultaneous equations model are developed in my 1974 paper with Jean-Jacques Laffont and methods for statistical inference in my 1979 paper with Ronald Gallant. These methods are extended to the exact aggregation framework in my 1986 paper with Stoker, reprinted in Chapter 10.

To incorporate differences in individual preferences into a model of aggregate consumer behavior in Chapter 8 we allow the indirect translog utility function for each individual to depend on attributes, such as demographic characteristics, that vary among individuals. The theory of exact aggregation requires that individual demand functions must be linear in functions of the expenditure and attributes of the individual. We impose the resulting restrictions on the indirect utility function for each individual.

Integrability requires that the indirect utility function for each individual must be homogeneous, monotone, and quasiconvex. In Section 8.5 we impose the corresponding restrictions on the indirect translog utility functions for all individuals. To construct a system of individual demand functions we apply Rene Roy's (1943) Identity to these indirect utility functions. (9) The resulting demand functions express the shares of total expenditure allocated to each commodity group as linear functions of the logarithm of expenditure and the attributes of the individual.

Our model of aggregate consumer behavior is a weighted average of individual demand functions for the population as a whole. Weights are given by the share of each individual in aggregate expenditure. This model expresses the share of expenditure allocated to each commodity group as a function of prices. These shares also depend on weighted averages of the logarithms of individual expenditures and attributes.

We find it convenient to represent the attributes of individuals, such as demographic characteristics, by variables equal to unity for an individual with the corresponding attribute and zero otherwise. The weighted average of each attribute for the population as a whole is the share of individuals with that attribute in aggregate expenditure. Our model of aggregate demand includes these shares as explanatory variables to capture the effect of change in the demographic characteristics of the population. Similarly, we incorporate a weighted average of the logarithm of individual expenditures to encompass the impact of changes in the distribution of total expenditure over the population.

We could implement the model of aggregate demand presented in Section 8.5 from individual cross section data on expenditure shares, prices, total expenditure, and demographic characteristics. Alternatively, we could implement this model from aggregate time series on expenditure shares, prices, shares of demographic groups in aggregate expenditure, and a weighted average of the logarithms of expenditure. However, if prices take the same values for all individuals and cross section observations are limited to individual expenditure shares and total expenditure, the best estimation strategy is to pool aggregate time series with individual cross section data along the lines outlined in Section 8.6.

Econometric methodology for pooling aggregate time series with individual cross sections in models based on exact aggregation is presented in detail in my 1986 paper with Stoker, reprinted as Chapter 10. Assuming that prices faced by all households are the same, the first step is to estimate the parameters of a model for a single cross section. This model can be estimated by ordinary least squares. Since the number of cross section observations is typically large, a considerable number of parameters associated with total expenditure and the demographic characteristics of individual households can be identified in this way.

The second step in pooling is to incorporate time series of prices and statistics describing the joint distribution of total expenditures and attributes over the population. Instrumental variables methods are essential for mitigating simultaneous equations bias in the estimation of price effects. Our inequality-constrained, nonlinear three-stage least squares estimator generalizes the nonlinear three-stage least squares estimator for simultaneous equations introduced in my 1974 paper with Laffont. Tests of equality and inequality restrictions generalize the tests of equality restrictions introduced in my 1979 paper with Gallant. (10)

Implementation of nonlinear three-stage least squares requires a sequence of iterations. Under exact aggregation these iterations are greatly simplified by the fact that individual cross section observations are not required for the pooled estimator. This estimator depends only on moments calculated in evaluating the least squares estimator at the initial step of the iterative process. Imposition of inequality constraints requires nonlinear programming techniques.

In Section 8.7 we estimate the parameters of the transcendental logarithmic model of aggregate consumer behavior for the United States. We analyze the allocation of aggregate demand among five categories of goods and services - nergy, food and clothing, consumer services, capital services, and other nondurable goods. We employ a breakdown of households by family size, age of head, region, race, and urban versus rural residence. We implement this model by pooling cross section data for U.S. households from the Consumer Expenditure Survey for 1972 with annual U.S. time series on aggregate demand for the period 1958-1974.

Our model of aggregate demand depends on eighty-two unknown parameters. Sixty-four of these describe the impact of attributes of individual households, four represent income effects, ten give price effects, and the remaining four are intercepts in equations for the expenditure shares. With a relatively modest number of time series observations these parameters could not be identified from aggregate data alone. Similarly, the absence of price variation in data for households makes it impossible to identify the price effects from cross section data. All eighty-two parameters can be identified through pooling of time series and cross section data.

As total expenditure increases, we find that the share of the consumer budget allocated to capital services rises, while shares of the other four commodity groups decline. Demographic effects are very important for all categories of consumer expenditures, ruling out both Gorman (1953) and Muellbauer (1975, 1976a) forms of the representative consumer model. Demographic effects are also very important in determining the impacts of total expenditure and prices.

Family size effects are almost a mirror image of total expenditure effects. As family size increases from one to seven or more, the shares of energy, food and clothing, and other nondurables increase, while shares of capital services and consumer services decrease. The difference between consumption patterns for families of size one, unrelated individuals, and larger families is substantial. Two person families, for example, spend a larger portion of their budget on necessities, such as energy and food and clothing. A similarly abrupt change occurs for families of size seven or more, a category that includes families of larger sizes, such as families with twelve or even fifteen members.

The impact of age reflects the high degree of correlation between age of head and the ages of children in the family. For example, the share of food and clothing rises until approximately age 40, levels off, and then declines. The pattern for the share of capital services is the opposite of that for food and clothing, while the share of consumer services declines relatively smoothly with age. Region and urban versus rural residence have substantial impacts on shares of capital services and energy with residents of North Central and Southern regions spending more on capital and energy. White-nonwhite differentials are not substantial, except for a smaller share devoted to capital services and a slightly larger share to consumer services for nonwhites.

Increases in total expenditures reduce expenditure elasticities and enlarge price elasticities, except for capital services. By contrast increases in family size for a given level of expenditure reduce price elasticities and enlarge expenditure elasticities. Differences in price and expenditure elasticities associated with age of head, region, race, and urban versus rural residence are also considerable. These differences are entirely neglected in models of aggregate demand based on the model of a representative consumer.

One of the most important features of our econometric model of aggregate demand is that individual demand functions can be recovered uniquely from the system of aggregate demand functions. This makes it possible to incorporate the implications of the theory of individual consumer behavior into our aggregate model. The same feature enables us to derive measures of welfare from individual demand functions in Section 8.5. These are the compensating and equivalent variations introduced by John Hicks (1942).

My 1987 paper with Slesnick, reprinted as Chapter 5 of Measuring Social Welfare, implements the translog model of aggregate consumer expenditures for a greatly extended time series data set, including U.S. aggregate time series data for the period 1947-1985. This paper provides an alternative interpretation of the role of demographic characteristics by incorporating them into household equivalence scales. All households have identical indirect utility functions when total expenditure is expressed relative to the number of household equivalent members.

Under exact aggregation the translog indirect utility functions employed in Section 8.5 provide cardinal measures of welfare that are fully comparable among individuals. The compensating and equivalent variations of Hicks (1942) give only ordinal measures that are not interpersonally comparable. Cardinality and interpersonal comparability were not exploited in my 1982 paper with Lau and Stoker, but are the key to development of measures of social welfare in my 1983 paper with Slesnick, reprinted as Chapter 2 of Measuring Social Welfare.

The translog indirect utility function used in generating our model of aggregate demand was originally introduced to test the implications of the framework of a representative consumer. The introduction of the attributes of households, such as demographic characteristics, made it possible to overcome the limitations of this framework. Our model of aggregate demand is obtained by exact aggregation over a population of consumers with heterogeneous preferences. An important remaining issue is aggregation over commodity groups.

The primary purpose of demand modeling is to determine price and expenditure elasticities empirically for specific commodities. These elasticities play a critical role in projecting future demands and evaluating the impact of economic policies on consumer welfare. For example, the own-price and cross-price elasticities of demand for energy and nonenergy commodities and specific types of energy are essential for the evaluation of energy policies. Unfortunately, the number of own- and cross-price elasticities increases with the square of the number of commodity groups.

The proliferation of elasticities as the number of commodity groups increases has necessitated the development of modeling strategies based on two-stage budgeting. At each stage the number of commodity groups can be reduced to manageable size. The usefulness of two-stage budgeting is considerably enhanced by Gorman's (1959) detailed characterization of the corresponding restrictions on preferences. (11) This characterization suggests two alternative approaches to modeling consumer behavior. The first is based on a utility function for each consumer that is additive in subutility functions for commodity groups. (12) The second, employed in Chapter 8, is based on homothetic separability. The utility function is not required to be additive, but subutility functions for commodity groups must be homothetic.

The model of aggregate demand presented in Chapter 8 includes five commodity groups - nergy, food and clothing, consumer services, capital services, and other nondurable expenditure. The allocation of total expenditure embodies the restriction that the indirect utility function is homothetically separable in the commodities within each group. More specifically, we assume that the price of each commodity group is a homogeneous translog function of its components. The price index for the group has growth rate equal to a weighted average of growth rates of the components with weights given by the average value shares. (13)

The objective of my 1984 paper with Stoker, reprinted as Chapter 9, is to model the allocation of energy expenditure among different types of energy within the exact aggregation framework. This model incorporates time series data on aggregate quantities of energy consumed and energy prices. It also includes time series data on the level and distribution of energy expenditure and the demographic characteristics of the population. Finally, the model incorporates individual cross section data on the allocation of energy expenditure among types of energy for households with different demographic characteristics. This model has been integrated into a model for the allocation of total expenditure between energy and nonenergy commodities in my 1983 paper with Slesnick and Stoker, reprinted as Chapter 11.

The econometric model presented in Chapter 9 is based on two-stage allocation of total expenditure. In the first stage total expenditure is allocated between energy and nonenergy commodities. The first-stage allocation depends on the price of energy, prices of all nonenergy commodities, and the level of total expenditure. In the second stage allocation energy expenditure is allocated among individual types of energy. The second-stage allocation depends on the prices of individual types of energy and the level of energy expenditure determined at the first stage.

The key assumption that underlies two-stage allocation, summarized in Section 9.1.2A of Chapter 9, is homothetic separability in energy prices. Under homothetic separability the price of energy for each consumer is a function of the prices of different types of energy. This assumption was introduced in the model of aggregate demand presented in Chapter 8. Our model of energy demand in Chapter 9 allows the price of energy to depend on household attributes. Although the prices for individual types of energy faced by consumers are the same for all households, preferences are heterogeneous, so that the price for the energy aggregate differs among households.

We define a quantity index of energy is the ratio of energy expenditure to the price index for energy, so that the product of price and quantity indexes is equal to energy expenditure. This quantity index is an indirect subutility function representing preferences that are homothetically separable in energy. Given price and quantity indexes, we define a two-stage allocation process for each household. The allocation of total expenditure between energy and nonenergy commodities depends on the price index for energy and the prices of nonenergy commodities, as well as the level of total expenditure. Energy expenditure determined at the first stage is then allocated among types of energy at the second stage.

Our model of the two-stage allocation process results in two systems of individual demand functions. The first stage of the process generates a system for allocation between energy and nonenergy commodities. The second stage produces a system for allocation of energy expenditure among types of energy. Since preferences are homothetically separable in energy, demands for all types of energy are proportional to energy expenditure with proportions that depend on energy prices. The final step in formulating the two-stage model of individual demand for energy is to incorporate restrictions implied by integrability into both systems of demand functions.

Our model of aggregate demand is obtained by exact aggregation over each of the two systems of individual demand functions. This generates two systems of aggregate demand functions, each a weighted average of the corresponding system of individual demand functions. For the first stage the weights are the shares of each individual in aggregate expenditure. For the second stage these weights are shares in energy expenditure. The aggregate energy shares are linear in the logarithms of energy prices and the shares of demographic groups in aggregate energy expenditure.

The model of demand for energy presented in Chapter 9 is implemented by pooling aggregate time series with five cross section surveys of individual demand for energy. These include the Consumer Expenditure Surveys (CES) of 1960/61, 1972, and l973 and the Lifestyle and Household Energy Use Surveys (LHES) of 1973 and 1975. Altogether there are 34,424 household observations in these five surveys. This massive data set is used to characterize patterns of consumer demand for electricity, natural gas, gasoline, fuel oil and other fuels.

The three CES cross sections are reasonably balanced in terms of demographic composition, so that the cross section estimates can be considered representative of the underlying structure of energy expenditures. Our model of individual consumer behavior implies that estimates of demographic effects should be the same in all three surveys. Similarities should be taken as support for the model, while dissimilarities indicate violations of the underlying assumptions. In fact, the impacts of demographic characteristics on energy use are virtually identical. Similarities among the three surveys are striking, confirming that they are appropriate for pooling with aggregate time series.

Two important differences between the CES and LHES surveys are apparent. The 1973 CES and LHES surveys can be compared directly; substantial differences between average gasoline and fuel oil use are reported in these surveys. The differences can be attributed to alternative procedures for measuring consumption of these two forms of energy. Since the demographic effects estimated from the two surveys are similar, only moments that are unaffected by differences in measurement procedures are used in pooling these results with aggregate time series. Finally, the impacts of race and urban versus rural residence are reversed in the 1975 LHES, relative to the other surveys, so that we model structural change in these coefficients explicitly in pooling these data with time series.

In Chapter 9 we present pooled time series and cross section estimates for all five surveys. Our most important finding is that there are strong demographic effects in the determination of aggregate energy expenditures, implying that an energy price index that does not incorporate the attributes of individual households results in errors of aggregation. Second, we test and reject the hypothesis that gasoline is separable from the other three types of energy, so that we are unable to disaggregate energy demand between transportation and household use. Finally, we find evidence for structural change in the final household survey, which occurred after the first oil crisis in 1975.

Our complete model of energy demand consists of two systems of aggregate demand functions. This model has been implemented by Jorgenson, Slesnick, and Stoker (1983), reprinted as Chapter 11. An important link between the two systems of demand functions is provided by the aggregate price index for energy. This takes the simple and intuitively appealing form of a weighted average of price indexes for individual households with weights given by the relative share of each household in aggregate expenditure. This can be expressed in terms of the prices of individual types of energy and the shares of demographic groups in aggregate expenditure.

The second link between the two systems of demand functions is the shares of demographic groups in energy expenditure. Aggregate energy expenditure and expenditure for each demographic group are determined in the first stage of the two-stage allocation process. Shares of demographic groups in the aggregate can be substituted into the model for the second stage of the process. The complete model is estimated by pooling aggregate time series for the United States for the period 1958 to 1978 with cross section data from the 1972 Consumer Expenditure Survey. The pooling process incorporates both links between the two stages of the overall allocation process.

Our two-stage model of aggregate demand employs the same breakdown of energy and nonenergy commodity groups used by Jorgenson, Lau, and Stoker (1982) and the classification of individual types of energy employed by Jorgenson and Stoker (1984). The same classification of households by demographic characteristics is used in all three studies. The results for the first stage are similar to those of Jorgenson, Lau, and Stoker. Expenditure shares for energy and food increase with total expenditure, while the shares for consumer goods, capital services, and other consumer services decrease. The effects of an increase in family size are the reverse of those for total expenditure. The effects of age of head, region, race, and urban versus rural residence are very substantial for most commodity groups. Similarly, own- and cross-price effects are highly significant for all commodity groups, but differ considerably among demographic groups.

Results for the second stage of the two-stage model are similar to those of Jorgenson and Stoker (1984). The effects of family size are substantial only for gasoline. The addition of a second member of a family usually involves a second driver and increases the share of gasoline considerably. Additional members of the family are mainly children, so that the share of gasoline gradually declines. By contrast with family size, the impact of age of head of household is quite dramatic. The share of gasoline declines with age with corresponding increases in the shares of electricity, natural gas, and fuel oil. Effects of region, race, and rural versus urban residence are all quite substantial.

Finally, we compare own- and cross-price elasticities, first holding energy expenditure fixed and then holding total expenditure fixed. As expected, the price elasticities holding total expenditure fixed are uniformly larger, reflecting substitution not only among different types of energy, but also between energy and nonenergy commodities. Focusing on price elasticities with total expenditure constant, we find that gasoline is the least price elastic of the four types of energy, while electricity is the most price elastic form of energy. Price elasticities for natural gas are similar to those for gasoline, while price elasticities for fuel oil are intermediate between those for gasoline and electricity. Price elasticities vary substantially with demographic characteristics of households.

The main contribution of the econometric model of aggregate consumer behavior presented in Chapter 8, was to implement the theory of exact aggregation, making it possible to dispense with the model of a representative consumer. The representative consumer framework was thoroughly tested and rejected in Chapters 1-7. The key feature of the exact aggregation approach, recoverability of individual demand functions, made it possible to adapt the specifications of models of individual consumer behavior in Chapters 1-7 to the exact aggregation framework. These specifications included integrability of demand functions and restrictions on preferences, such as those required by homothetic separability.

The second contribution of the model presented in Chapter 8 was the successful implementation of a statistical methodology for pooling aggregate time series and individual cross sections. This grew out of the methodology for systems of nonlinear simultaneous equations developed by Jorgenson and Laffont (1974) and Gallant and Jorgenson (1979). While the model could be identified from either type of data, the relative paucity of time series data and the absence of price variation in cross sections mandated pooling the two data sources. The statistical methodology for pooling is presented in detail by Jorgenson and Stoker (1986) in Chapter 10.

Finally, aggregation over individuals was successfully combined with aggregation over commodity groups in the models presented in Chapters 9 and 11. Gorman (1959) provided the appropriate modeling framework for individual demand functions by introducing methods for two-stage allocation. This approach is employed in Chapter 8 for constructing price indexes for commodity groups. The approach is extended to price indexes based on heterogeneous preferences in Chapters 9 and 11. This implies links between the two stages of the allocation process that can be exploited in estimation and testing.

My papers with Lau and Stoker (1980, 1981, 1982) employed the exact aggregation framework to construct ordinal measures of individual welfare for households with stipulated characteristics and level of total expenditure and applied the results in analyzing the impact of economic policies. However, the exact aggregation framework implies cardinal and interpersonally comparable measures of individual welfare that can be incorporated into an indicator of social welfare. These implications were developed in a series of papers initiated by Jorgenson and Slesnick (1983), reprinted in Chapter 2 of Measuring Social Welfare and summarized in my Presidential Address to the Econometric Society in 1990, reprinted as Chapter 1 of that volume.

The model of aggregate consumer behavior presented in Chapter 8 is incorporated into an intertemporal general equilibrium model for the United States in a series of papers initiated by my 1990 paper with Peter Wilcoxen. (14) This general equilibrium model has been used to project aggregate consumer demand in the United States along with future prices and the future level and distribution of total expenditure. The model has also been used to generate projections for groups of individuals, classified by total expenditure and demographic characteristics. Finally, the model has been utilized in my 1992 paper with Slesnick and Wilcoxen, reprinted as Chapter 9 of Measuring Social Welfare, for assessing the impact of alternative economic policies on social welfare and the welfare of individuals with given characteristics.

Much remains to be done to exploit the exact aggregation approach for modeling aggregate demand. One promising opportunity for extending the econometric models presented in this volume is to encompass labor-leisure choice as well as choice among goods. This will require the recognition of heterogeneity of abilities of different consumers in the labor market as well as heterogeneity of the preferences that underly labor-leisure choice. Another promising extension is to intertemporal choice. Econometric models encompassing heterogeneity of preferences can be used in projecting aggregate demand, labor supply, and saving.

Extensions of the exact aggregation approach offer exciting new possibilities for enriching normative economics as well as the econometric modeling of aggregate behavior. By recovering the underlying models of individual behavior, consumer preferences can be brought to bear on a wider and wider range of issues in economic policy. In addition, measures of economic performance - uch as poverty and inequality and the cost and standard of living - an be extended to encompass leisure as well as goods and lifetime patterns of consumption of both goods and leisure.

I would like to thank June Wynn of the Department of Economics at Harvard University for her excellent work in assembling the manuscripts for this volume in machine-readable form. Renate d'Arcangelo of the Editorial Office of the Division of Applied Sciences at Harvard edited the manuscripts, proofread the machine-readable versions and prepared them for type-setting. Warren Hrung, then a senior at Harvard College, checked the references and proofread successive versions of the typescript. William Richardson and his associates provided the index. Gary Bisbee of Chiron Incorporated typeset the manuscript and provided camera-ready copy for publication. The staff of The MIT Press, especially Terry Vaughn, Victoria Richardson, and Michael Sims, has been very helpful at every stage of the project. Financial support was provided by the Program on Technology and Economic Policy of the Kennedy School of Government at Harvard. As always, the author retains sole responsibility for any remaining deficiencies in the volume.

Endnotes:

1. The canonical formulation of this theory is that of John Chipman, Leonid Hurwicz, Marcel Richter, and Hugo Sonnenschein (1971).

2. This perspective is presented, forcefully, by Alan Kirman (1992) along with references to the literature. See Thomas Stoker (1993) for a detailed discussion of the implications for demand modeling.

3. See, for example, Houthakker (1957) and the references given there.

4. Surveys of empirical evidence on the impact of demographic characteristics on demand are presented by Angus Deaton (1986) and Martin Browning (1992).

5. This evidence is surveyed by Deaton (1986).

6. See Charles Blackorby, Richard Boyce, and Robert Russell (1978) and the references given there.

7. Surveys of the representation of the effects of prices on demand are given by Deaton (1986), Lau (1986), and Richard Blundell (1988).

8. Translog price and production functions for modeling producer behavior were introduced by Christensen, Jorgenson, and Lau (1971, 1973). This functional form provides the basis for the econometric methodology for modeling producer behavior summarized by Jorgenson (1986).

9. The specification of a system of individual demand functions by means of Roy's Identity was introduced in a pathbreaking study by Houthakker (1960). Lau (1977a) provides a detailed survey of econometric models of consumer demand based on this approach.

10. A unified framework for tests of equality and inequality restrictions has been proposed by Frank Wolak (1989).

11. Gorman's theory of two-stage budgeting is discussed in detail by Blackorby, Primont and Russell (1978).

12. This approach is employed by Blackorby, Boyce, and Russell (1978), who provide extensive references to the literature

13. Diewert (1976) has shown that this index number exactly reproduces a homogeneous translog price function.

14. Jorgenson and Wilcoxen (1993) provide a survey and detailed references.