Are consumption and government expenditures sustitutes or complements? Morishima elasticity estimates from the fourier flexible form.

Fleissig, Adrian R. ; Rossana, Robert J.

1. INTRODUCTION

Macroeconomists paid relatively little attention to the economic effects of government expenditures and to alternative methods of financing those expenditures until Barro (1974, 1979, 1981) challenged conventional views on the economic impact of government activity. Much of the literature stimulated by these studies addressed the effects of alternative financing methods for a given path of government expenditures. But Barro (1981) also stressed the fact that government expenditures can provide direct welfare to economic agents and that variations in the level of government expenditures may have an impact on the consumption decisions of households. As a result, many studies estimate the extent to which economic agents treat government expenditures as substitutes to or complements with private consumption expenditures (e.g., Kormendi, 1983; Aschauer, 1985; Graham and Himarios, 1991; Ni, 1995). The degree of substitution or complementarity between private consumption and government expenditures is of crucial importance in assessing the impact of fiscal policies on economic welfare. Thus it is important to establish the extent to which federal expenditures and private consumption are substitutes or complements as a guide to the design of fiscal policies.

There is much disagreement about the extent to which consumption and government expenditures are substitutes or complements in use for economic agents. (1) Studies differ in their details, so it is difficult to know what might account for such diverse results. But it seems fair to say that the relationship between government expenditures and consumption is largely an open question. The diversity of the results, however, may be a consequence of a number of restrictive assumptions that have been used in various studies. Many of these assumptions appear hard to justify a priori.

Studies typically estimate the parameters of an equation containing private consumption and government expenditures, often assuming that the relevant choice variable for the representative household is an aggregate consumption measure given by the sum of nondurables and services consumption (e.g., Aschauer, 1985; Reid, 1985; Bean, 1986; Campbell and Mankiw, 1990) or the sum of nondurables, services, and the service flow from durable goods (e.g., Kormendi, 1983; Graham and Himarios, 1991). This implies that each component of private consumption expenditures is a perfect substitute for the household, an assumption for which there is no compelling theoretical basis. Similarly, government expenditures (defense, nondefense, and state and local) are typically aggregated (e.g., Feldstein, 1982; Aschauer, 1985; Reid, 1985; Graham, 1993), with various studies excluding either federal or state and local expenditures. Just as for private consumption expenditures, these aggregation assumptions should be tested, rather t han simply imposed in applied work. In addition, although substitution elasticities are generally not constant in an optimizing framework, they are often assumed to be constant in applications by using a single aggregate of consumption or a utility function that imposes a constant elasticity of substitution (e.g., Aschauer, 1985; and Ni, 1995). Finally, none of the above-mentioned studies computes elasticities of substitution in a manner consistent with microeconomic utility maximization principles. (2)

In this article we provide estimates of elasticities of substitution between consumption and government expenditures based on the demand-systems approach that has been widely used in testing the neoclassical theory of household behavior (see, eg., Deaton and Muellbauer, 1980, and the survey of Barnett et al. 1992). This approach has a number of attractive features, making it a substantial improvement over the applied studies available in the macroeconomics literature.

The estimates in this article are based on an optimizing framework where both government and consumption expenditures are treated as choice variables by the representative household. In contrast, the typical treatment of government expenditures in the macroeconomics literature is that government expenditures are exogenous to optimizing households. But the public elects agents to represent its interests when fiscal policies are set so that it seems inappropriate to regard government expenditures to be completely beyond the control of the public. It is plausible to suppose that at least as a first approximation consumption decisions are jointly determined with decisions on the appropriate level of government activity. Our framework permits us to incorporate the joint nature of these decisions. In addition, we use a more complete array of decision variables compared to many earlier studies in the literature because we incorporate three types of government expenditures (federal defense and nondefense and state a nd local spending) along with consumption of nondurables, services, and the stock of durable goods. Consumption and government expenditures series will not be aggregated in this study, allowing us to compute elasticities of substitution that may differ between each of these expenditure components. Also, we do not require service flows to be linear functions of consumption and government expenditures, unlike many previous studies, nor will we impose separability in the utility function. (3)

Furthermore, we estimate Morishima elasticities of substitution because Blackorby and Russell (1989) show that it is the appropriate measure of curvature or "ease" of substitution between pairs of commodity bundles when there are more than two commodities chosen by the household. This Morishima elasticity measure can be asymmetric and variable over time.

Substitution or complementarity relationships between goods consumed by households must be determined empirically, not as matter of theory. It is thus especially important to use a functional form that does not bias estimates of such relationships. (4) To avoid imposing the restrictive assumptions of earlier studies, we use the flexible Fourier functional form of Gallant (1981) that can globally approximate the true expenditure-share equations and allows elasticities of substitution to vary across our sample data.

We provide short-run and long-run versions of our elasticity estimates by using lagged data in our estimation procedure. Habit formation and non-time-separable preferences (Eichenbaum, et al. 1988; Eichenbaum and Hansen, 1990; Pollak and Wales, 1992) are just two possibilities that motivate the inclusion of lagged data in utility functions and other decision rules used by optimizing households. Consumption is likely to adjust gradually over time in response to shifts in relative prices, so it is reasonable to anticipate that short-run and long-run results will differ to some degree. We can observe those differences in our elasticity estimates.

Our short-run results consistently find that government and private consumption expenditures are net Morishima substitutes. The results display considerable heterogeneity: Estimated elasticities are significantly different from unity, they are asymmetric, and they vary considerably throughout our sample data. There are interesting temporal patterns in our results. All elasticities involving federal defense expenditures display a consistent pattern that is clearly related to the declining level of defense expenditures. For example, defense goods and private consumption goods have become weaker substitutes over time in response to shifts in the relative prices of consumer goods. Our short-run findings suggest that using Cobb-Douglas or Constant Elasticity of Substitution (CBS) functional forms would be a misspecification that can bias statistical inference.

The long-run results display less heterogeneity than our short-run elasticities. Although the estimated long-run elasticities are much closer to unity compared to their short-run counterparts, short-run elasticities never equal unity. Many of our estimated long-run elasticities are insignificantly different from unity. A unitary value for the Morishima elasticity would arise from the solution of a static optimization problem using a Cobb-Douglas utility function.

This article is organized in the following way. The next section describes the optimization framework that forms the basis of our empirical work. Section III discusses the data used and covers estimation issues. Section IV reports the short-run and long-run Morishima elasticity estimates, and the last section summarizes our results. Because our elasticity estimates vary at every point in our sample, it is not possible to report each elasticity estimate and its associated standard error in a concise way within the body of the article. As a result, we only report summary measures of our elasticity estimates. An Appendix concludes. The Appendix reports the individual parameter estimates obtained from estimation of the Fourier expenditure-share equations and their associated standard errors.

II. THE FOURIER DEMAND-SYSTEMS FRAMEWORK

Household Behavior

The model underlying our analysis is an optimization framework where it is assumed that a representative household wishes to maximize the utility function

(1) u([c.sup.*.sub.t], [d.sup.*.sub.t], [g.sup.*.sub.t]),

where [c.sup.*] is a vector of service flows of nondurable goods and services, [d.sup.*] refers to a vector of service flows from stocks of durable goods, and [g.sup.*] denotes service flows from government expenditures. The specification in (1) assumes that consumers get utility from the services provided by government as well as from nondurable goods, services, and durable goods.

Data on service flows are generally unavailable and must be estimated in some manner. For example, service flows from nondurable goods and services have often been assumed to be linear functions of current and lagged expenditures on nondurable goods and services. (5) However, these service flows need not be linear functions of consumption expenditures so that, as an alternative specification, service flows are treated in this study as the possibly nonlinear function

(2) [c.sub.t.sup.*] = [c.sub.t.sup.*]([c.sub.t]; [c.sub.t-1], [c.sub.t-2],...),

where lagged consumption expenditures ([c.sub.t-j]) are present in (2) to indicate that contemporaneous service flows are conditional on past consumption expenditures.

Service flows from durable goods are often assumed to be proportional to the stock of durable goods. (6) For example, Ni (1995, 596) assumes that [d.sub.t.sup.*] = [delta]([k.sub.t-1] - [d.sub.t]) and [DELTA][k.sub.t] = [delta][d.sub.t], which together imply that [d.sub.t.sup.*] = [delta][(1 - [delta]).sup.-1] [k.sub.t], where [d.sub.t] denotes purchases of new durable goods, [k.sub.t] is the stock of durable goods, and [delta] is the depreciation rate. More generally, service flows from durable goods may be written as the possibly nonlinear function

(3) [d.sub.t.sup.*] = [d.sub.t.sup.*]([k.sub.t]; [k.sub.t-1], [k.sub.t-2],...),

indicating that service flows are conditional on lagged stocks of durable goods. The lagged stocks in (3) have been interpreted as an indication of habit persistence in house-hold behavior (Eichenbaum and Hansen, 1988).

Similarly, we write the service flows from government expenditures as

(4) [g.sub.t.sup.*] = [g.sub.t.sup.*]([g.sub.t]; [g.sub.t-1], [g.sub.t-2],...)

This expression reflects the fact that many government activities involve the provision of durable goods, providing services to house-holds that persist over time. In our empirical work reported below, the vector [g.sub.t] will contain federal defense expenditures, federal nondefense expenditures, and state and local government expenditures.

Substituting (2)-(4) into (1) gives the utility function

(5) u([c.sub.t], [k.sub.t], [g.sub.t]; [c.sub.t-1], [c.sub.t-2], ... , [k.sub.t-1], [k.sub.t-2], ... , [g.sub.t-1],[g.sub.t-2], ...),

which is the objective function used by the representative household in our analysis. See Fleissig (1997) for further details about the properties of the utility function.

Let [x.sub.t] be a vector consisting of nondurable goods, services, the stock of consumer durable goods, and government expenditures. Then our optimizing framework is

(6) max u([x.sub.t]; [x.sub.t-1], [x.sub.t-2],...) subject to [p.sub.t][x'.sub.t] = [m.sub.t],

where the budget constraint simply states that total expenditures, [m.sub.t], equal the sum of expenditures on each good entering contemporaneously into the utility function.

Elasticities of Substitution

The solution to (6) yields utility-constant commodity demand functions that give optimal demands for goods as a function of relative prices. Alternatively, if we use total expenditures as the normalizing magnitude, then commodity demands are functions of expenditure-normalized prices. In our empirical work reported later, expenditures on each good as a share of total expenditures are a function of these expenditure-normalized prices. Empirical estimates of the relation between consumer goods and services and government expenditure are analyzed using substitution elasticities. In models containing only two commodities, the elasticity of substitution is quite unambiguous: the two commodities must be net substitutes. But when there are more than two commodities, Blackorby and Russell (1989) show that substitution relationships depend on the direction in which the price ratio changes and they show that the Morishima elasticity, not the traditional Allen-Uzawa elasticity, is the correct measure of substitution among commodities. The Morishima elasticity of substitution, denoted by [M.sub.ij], is given by

(7) [M.sub.ij] = [p.sub.i][E.sub.ij](u, p)/[E.sub.j](u, p) - [p.sub.i][E.sub.ii](u, p)/[E.sub.i](u, p) = [[epsilon].sub.ji] - [[epsilon].sub.ii],

where [E.sub.i] and [E.sub.ij] are first and second derivatives of the expenditure function E(u, p). As shown in (7), the Morishima elasticity is the difference between two terms: The first term is the response of the utility-constant demand for good j with the respect to a shift in price [P.sub.i], and the second term is the response of the utility-constant demand for good i with respect to a change in price [p.sub.i]. If this expression is positive, goods i and j are defined to be net substitutes and, if this magnitude is negative, they are net complements. This elasticity measure need not be symmetric ([M.sub.ij] [not equal to] [M.sub.ji]) because the log derivative of the commodity ratio [x.sub.i]/[x.sub.j] with respect to [p.sub.i]/[p.sub.j] depends on whether the change in the price ratio is attributed to a change in [p.sub.i] or [p.sub.j]. The Morishima elasticity will be symmetric if and only if the utility function is an implicit CES function.

It is important to note that our definition of substitution and complementarity is not identical to that used in earlier studies. The Morishima elasticity measures changes in government purchases and private consumption in response to changes in relative prices. In contrast, many previous studies define substitution between government purchases and private consumption expenditures to mean that increases in government purchases reduce the marginal utility of private consumption. For example, the utility function U(c + [theta]g) used by Graham and Himarios (1991) implies that the marginal utility of private consumption (c) rises (falls) with an increase in government expenditures (g) if [theta] is negative (positive). Consumption and government expenditures thus are substitutes if [theta] > 0 and complements if [theta] < 0. In our analysis, consumption and government expenditures can be substitutes even if the marginal utility of consumption does not fall with an increase in government expenditures.

The Fourier Flexible Form

An indirect utility function can be approximated by parametric functions such as the Cobb-Douglas, quadratic, CES, or constant relative risk aversion. The difficulty with parametric forms is that they can often restrict the substitution relationship between commodities and often fail to give a good approximation to the utility function, as suggested by the frequent rejection of the corresponding overidentifying restrictions implied by economic models. A more precise approximation to the utility function can be obtained using a flexible functional form. Moreover, flexible functional forms have the convenient property of leaving elasticities of substitution unrestricted (Diewert, 1974).

Diewert (1974) defines a flexible functional form as a second-order local approximation to an arbitrary, twice-continuously differentiable function at a given point. Diewert-flexible functions have some limitations--they can only provide a local approximation in a delta neighborhood of unknown and often small size, and they may fail to measure how accurately partial derivatives of the true utility function are estimated. Estimating partial derivatives is important because the Morishima elasticities are functions of the partial derivatives of the data generating function. Semi-nonparametric functions, such as the Fourier flexible form and the asymptotically ideal model, give global approximations of the unknown data generating function and its partial derivatives and thus are free of the shortcomings of Diewert-flexible functions. (7)

Gallant (1981) augments Diewert's definition of a flexible functional form by requiring the function to globally approximate both the true utility function and its partial derivatives. The semi-nonparametric Fourier flexible form developed by Gallant (1981) and extended by Fisher and Fleissig (1997) will be used in this study to approximate the indirect utility function from the solution of (6), used in deriving the expenditure-share equations that we estimate (see later discussion). This indirect utility function can be written as

(8) f(.) = [u.sub.0] + b' v + (1/2)v' Cv

+ [summation over (A/[alpha]=1)] ([u.sub.0[alpha]] + 2 [summation over (J/j=1)][[u.sub.j[alpha]] COS(j[k'.sub.[alpha]]v)

--[w.sub.j[alpha]] sin(j[k'.sub.[alpha]]v)]),

where

C = -- [summation over (A/[alpha]=1)] [u.sub.0[alpha]][k.sub.[alpha]][k'.sub.[alpha]].

In the expressions, parameters to be estimated are contained in the vectors {b}, {[u.sub.j[alpha]]}, and {[w.sub.j[alpha]]}. The vector of current and lagged expenditure-normalized prices ([p.sub.i]/m) for commodities is denoted by v. The term [k'.sub.[alpha]] is referred to as a multi-index and is an n-vector of integers denoting partial differentiation of the utility function. The parameters A and J determine the degree of the Fourier polynomials and are determined by empirical testing (see the Appendix).

The utility function in (8) has two parts: a quadratic part, given by [u.sub.0] + b'v + (1/2)v' Cv, and a Fourier series component, given by the remainder of the expression. Special cases of (8) generate functional forms that have been popular in applied work. For example, if the vector v consists solely of expenditure-normalized prices for the sum of contemporaneous nondurables and services and if the parameters {[u.sub.j[alpha]]} and {[w.sub.j[alpha]]} are set to zero, the quadratic form in Hall (1978) emerges. If v consists of durable and nondurable goods and if the parameters {[u.sub.j[alpha]]} and {[w.sub.j[alpha]]} are zero, the utility function is that of Bernanke (1985) without adjustment costs. Substitutability between goods in the utility function is determined by the parameters {[u.sub.0[alpha]]}, {[u.sub.j[alpha]]}, and {[w.sub.j[alpha]]}. The elasticity of substitution is free to change over time and does not depend on the precision of a single parameter estimate, as in Bernanke (1985).

Our estimates are based on Fourier expenditure share equations arising from the solution of the optimization problem given in (6). Using Roy's identity, these expenditure-share relationships, y(.), are given by

(9) [y.sub.i](.) = [[v.sub.i][b.sub.i] - [summation over (A/[alpha]=1)] ([u.sub.0[alpha]] v'[k.sub.[alpha]] + 2 [summation over (J/j=1)] [[u.sub.j[alpha]] sin (j[k'.sub.[alpha]] v) + [w.sub.j[alpha]] cos (j[k'.sub.[alpha]]v)]) [k.sub.i[alpha]] [v.sub.i]] / [b'v - [summation over (A/[alpha]=1)] ([u.sub.0[alpha]] v' [k.sub.[alpha]] + 2 [summation over (J/j=1)] j [[u.sub.j[alpha]] sin (j[k'.sub.[alpha]] v) + [w.sub.j[alpha]] cos(j[k'.sub.[alpha]]v)]) [k'.sub.[alpha]]v]

for i = 1,..., n goods. The time notation has been suppressed in (9). This expression, along with the restrictions implied by ordinary consumer theory (budget shares for all six goods must sum to one, and cross-price derivatives of the Hicksian demand functions must be symmetric) is the basis for our empirical work.

Because budget shares must add to unity, estimation of a system of n budget-share equations will result in a residual covariance matrix that is singular. We follow the traditional approach and estimate a system of n-1 share equations and recover parameter estimates for the omitted equation from the estimated n-1 equations. Barten (1969) shows that the parameter estimates are invariant to the choice of share equation omitted from estimation.

III. DATA AND ESTIMATION ISSUES

The government and consumption expenditures data that we use are from the U.S. National Income and Product Accounts (NIPA) and are available in Citibase. We use data on government expenditures for three types of activities: defense, nondefense, and state and local expenditures, all measured in billions of 1992 U.S. dollars. Prices for these magnitudes are measured by their implicit deflators. For private consumption expenditures, we use data on constant-dollar nondurable goods, services, and the stock durable goods. Expenditures and price indexes for nondurable goods and services are from NIPA, also available in Citibase. We constructed the stock of consumer durable goods using a benchmark stock for 1946:4 from Musgrave (1979), combined with data on purchases of durable goods and an assumed depreciation rate of 6% per quarter, as in Ni (1995). The user cost of durable goods is calculated using a depreciation rate of 6% per quarter, the NIPA price index for durable goods, and an interest rate. The one-month T-bill rate, converted into a quarterly series, is the interest rate used in calculating Diewert's (1974) formula for the user cost for durable goods. (8) Resident population was used to convert magnitudes to per capita levels. The data span 1947 to 1996 at quarterly frequency and are adjusted for seasonality.

Parameter estimates for the Fourier flexible form were obtained using maximum likelihood. We searched over a wide range of starting values to estimate the expenditure-share equations in the system (see the Appendix). (9) A series of Wald tests were carried out to determine the number of lags for each series. Four lags were used for each variable in our system based on the results from these tests. The economic content of the Fourier parameter estimates is obtained by estimating the Morishima elasticities of substitution. Because there are no costs of adjustment to equilibrium, lagged variables capture adjustment to the ultimate long-run steady state and are used to estimate short-run elasticities of substitution. In the long run, agents have time to fully adjust their consumption decisions, and thus we also analyze the long-run Morishima elasticity. To estimate long-run Morishima elasticities, set [x.sub.t] = [x.sub.t-j] for j = 1,..., 4, then take the parameter estimates and recalculate (7).

The standard errors in nonlinear models are usually calculated using the delta method with analytical derivatives. We calculate the covariance matrix for the estimated Fourier flexible form and the Morishima elasticities. The covariance matrix for k nonlinear constraints for a function of the parameter vector [theta], h([theta]), is calculated as V(h([theta])) = ([partial]h/[partial]([theta]))'V([theta])([partial]h/[partial]([thet a]))', evaluated at the estimated [theta] vector. Using h([theta]) and its variance, the Wald statistic W = h([theta])V(h([theta])).sup.-1] h([theta])' is distributed asymptotically as a chi-square statistic with k degrees of freedom under the null hypothesis.

There was evidence of first-order serial correlation in the estimated demand-share system of equations when the equation system was first estimated. Autocorrelation is commonly found in demand systems and was first analyzed in detail by Berndt and Savin (1975) for the type of system that we estimate. If the disturbance vector, [e.sub.t], has the AR(1) form [e.sub.t] = R[e.sub.t-1] + [u.sub.t] where R is an autoregressive parameter matrix, adding up requires the matrix R to be diagonal with identical diagonal elements. Our estimates correct for this evident serial correlation.

It is useful to be aware of the time-series behavior of our data as an aid to interpreting the elasticity estimates reported later. Although these patterns are known, a brief discussion is given here to facilitate the discussion of our elasticity estimates.

In per capita terms, the salient feature of our data is the systematic decline in defense expenditures that began in the early 1950s. Federal defense expenditures have fluctuated about a negative trend over most of our sample and, measured as shares of government expenditures, there has been a shift in the mix of government spending: defense spending has fallen and nondefense spending has increased modestly, and state and local spending has substantially increased. As for consumption, all series follow positively sloped time paths, but, using shares of total consumption, services have increased while nondurable and durable consumption decline over our sample data.

IV. ESTIMATED ELASTICITIES OF SUBSTITUTION

Estimation results are given in Tables 1 and 2. Our notation is as follows: the Morishima elasticity of substitution between variables i and j is denoted by [M.sub.ij]([M.sub.ji]) when relative prices change due to a change in the price of variable i(j). Elasticities are indexed for i = 1,...,6 where:

Government Expenditures

1 = Defense 2 = Nondefense 3 = State and Local

Consumer Expenditures

4 = Nondurables 5 = Services 6 = Stock of Durable Goods.

The tables contain summary statistics for the Morishima elasticities that we estimate at each point in our sample. The tables contain the mean, median, maximum, and minimum estimate for each elasticity estimate. The same summary data, except for the mean, is provided for the standard errors attached to these elasticity estimates.

Short-Run Estimates

Table 1 contains summary information about the estimated short-run Morishima elasticities. Substitution elasticities are significantly different from zero at the 5% level at every point in our sample (see the Appendix for individual parameter estimates and standard errors). In fact, the Morishima elasticities are estimated very precisely at all sample points. The estimated elasticities are always positive for all pairs of expenditure series. Thus all pairs of government and consumption expenditure series are observed to be net Morishima substitutes.

An important empirical issue is whether our estimated functional form satisfies the regularity restrictions imposed by neoclassical microeconomic theory. These regularity conditions are adding-up, homogeneity, symmetry, nonnegativity, monotonicity, and the curvature conditions (quasi-concavity) on the indirect utility function. Following Christensen et al. (1975) and Fisher et al. (2001), adding-up, homogeneity, and symmetry are imposed on the Fourier prior to estimation. The results (available on request) show that there are no violations of nonnegativity, monotonicity, and quasi-concavity. Thus the Fourier flexible form, for this data set, satisfies neoclassical regularity restrictions. (10)

Additional aspects of our results are illuminated by the estimated Morishima elasticities for defense expenditures and the consumption of services, denoted by [M.sub.15] and [M.sub.51]. The behavior of these estimated elasticities is representative of the characteristics found in many other estimated elasticities, Figures 1 and 2 contain time-series plots of the estimated Morishima elasticities for these series. The dotted lines in the diagrams are the standard error bands associated with each elasticity estimate.

Inspection of the individual elasticity estimates for defense spending and services shows that there is considerable variability in our estimated elasticities. For example, Table 1 reveals that [M.sub.15] ranges in value from a minimum of 1.019 to a maximum of 1.199, whereas [M.sub.51] has a minimum value of 0.804 and a maximum of 0.984. Similar differences may be found in other pairwise comparisons from Table 1. The estimated elasticities over all pairs of goods range in value from a minimum of 0.388 ([M.sub.42]) to a maximum of 1.376 ([M.sub.25]). Figures 1 and 2 confirm this; the figures show substantial changes over time in the elasticities that we estimate.

The summary data clearly suggest that the estimated elasticities are asymmetric. The figures clearly show this asymmetry for [M.sub.15] and [M.sub.51]. One elasticity, [M.sub.15], always exceeds unity, and [M.sub.51] always lies below unity. Inspection of all the estimated elasticities and their standard errors confirms that they are asymmetric for all pairs of goods and statistically different from one. Theoretical assumptions restricting substitution elasticities to be symmetric and constant are not supported by the data.

The summary data in Table 1 may possibly mask interesting temporal patterns in our results. It is apparent from Figures 1 and 2 that the elasticities for defense and services vary systematically over our sample, with [M.sub.15] displaying a positive trend most of the time and [M.sub.51] displaying a negative trend. It is interesting to see how these temporal patterns match up with the time-series behavior of defense spending and consumption of services. Inspection of the data reveals that the temporal pattern of the elasticities corresponds closely with the time-series behavior of federal defense expenditures. Whenever federal defense expenditures are falling, as they do over most of our sample data, we observe [M.sub.15] ([M.sub.51]) to be rising (falling). Similarly, any other elasticity that involves government defense expenditures displays this same pattern. That is, [M.sub.1j] ([M.sub.j1]) rises (falls) for j = 4, 5, 6. (11) To interpret these results, it is helpful to recall the definition of the Morish ima elasticity, given above in (7). [M.sub.ij] is the net shift (net of the own-price elasticity effect denoted by [[epsilon].sub.ii] in [7]) in the compensated or utility-constant demand for good j when the price of good i changes. If [M.sub.1j] is rising over time, then shifts in the relative price of defense expenditures are generating larger net shifts in the demands for consumer goods. If [M.sub.j1] is falling, then shifts in the prices of consumer goods are inducing smaller net shifts in the demand for defense goods. These results are consistent with the view that with declining resource use in the defense goods sector associated with declining defense expenditures, a given desired increase in the consumption of consumer goods induces a smaller net decline in the demand for defense goods than would have been true in the past. Households may wish to shift fewer resources out of the defense sector because resource use in this sector has fallen for many years. Similarly, if defense goods fall in relative p rice, a rising elasticity of substitution reflects a desire to shift relatively more resources into the defense sector as compared to the past.

There are apparent trends in the other estimated elasticities involving government and consumption expenditures, but these are not as prominent or widespread as the results involving defense expenditures. We find [M.sub.3j] rising over time for j = 4, 5, 6. These are elasticities that involve state and local spending and the private consumption components. These movements reflect the increasing levels of state and local spending that have occurred in our sample. However we do not see falling levels of [M.sub.j3] in our results for reasons that are unclear.

As for the consumption elasticities, these series are all net Morishima substitutes with elasticities that are significantly different from unity. There are no striking patterns in the estimated elasticities with the exceptions of [M.sub.54] and [M.sub.64]. These are elasticities between either services or durable consumption and nondurable consumption, and these elasticities decline over our sample. This implies that if either the relative price of services or durable services were to rise, services and durables are becoming stronger substitutes with nondurable goods so that households would respond with increasingly large shifts in the demand for nondurable goods. These patterns are consistent with the fact that, measured as a share of total consumption, nondurables expenditures decline in our sample data.

To summarize, we find that all pairs of consumption and government expenditure series are net Morishima substitutes in use, statistically different from one, and clearly changing over time. The elasticities associated with federal defense spending closely correspond to movements in defense spending over time. We now turn to our long-run results that enable us to see if these short-run findings continue to hold once households have fully adjusted to variations in relative prices. (12)

Long-Run Estimates

Table 2 provides summary information for the estimated long-run elasticity estimates. The data show that all pairs of goods in our analysis are net Morishima substitutes. As before, inspection of the data in the table suggests that our elasticity estimates are asymmetric. Using [M.sub.15] and [M.sub.51] for illustration, [M.sub.15] ranges from a maximum of 1.325 to a minimum of 0.951, and the median value is estimated to be 1.084. Regarding [M.sub.51], the estimated values range from a maximum of 1.045 to a minimum of 0.686 with a median value of 0.918. Individual estimated elasticities and their associated standard errors show that the elasticities are asymmetric at most sample points.

However, these numerical results actually mask a general tendency for our results to become more homogeneous as compared to our short-run results. This change toward homogeneity may be illustrated graphically. Figures 3 and 4 provide time series plots of [M.sub.15] and [M.sub.51] that may be compared to their short-run counterparts. The figures show that the long-run results are much closer to unity than their short-run values. At a substantial number of sample points, the estimated elasticities are within one standard error of unity and thus are statistically indistinguishable from unity. This tendency is quite evident in the elasticities for other pairs of goods that we estimate. Furthermore, although the elasticities are still asymmetric at most sample points, each elasticity moves closer to unity when we compare short-run and long-run versions of a given elasticity. (13) The degree of asymmetry in our results declines as we move to the long run. The figures also suggest that our long-run estimates often a re not as tightly estimated as our short-run elasticities, which is also true much of the time. When our elasticity results are statistically different from unity, the magnitude of this departure from unity is generally small.

It is also true that the long-run results display far fewer examples of elasticities with distinct trends over time. Thus when we move from the short-run to the long-run version of our results, we eliminate a substantial degree of the heterogeneity evident in our short-run results. (14)

Previous evidence of complementarity or substitutability between consumption and government expenditures appears sensitive to the details of the empirical specification used by the investigator. (15) Our short-run and long-run results are quite consistent (consumption and government expenditures are always found to be net Morishima substitutes), results that are generally in agreement with the general equilibrium analysis of Baxter and King (1993). (16) Furthermore, the results are considerably more uniform than the results in the literature, suggesting that our use of the flexible Fourier form provides at least part of the explanation for the diversity of results in previous research.

V. CONCLUDING REMARKS

Accurate estimates of the economic welfare effects of government expenditures must take account of the changes in household consumption levels that occur in response to variations in government expenditures. Toward this end it is crucially important to have reliable estimates of substitution elasticities between consumption and government expenditures to assess these welfare effects. We estimate Morishima elasticities of substitution between consumption and government expenditures. The Morishima elasticities are the theoretically appropriate measure of substitution when there are more than two choice variables in an optimizing model, as in our study. In addition, the elasticities are estimated using the Fourier functional form to limit the potential biases associated with the parametric functional forms used in previous studies. Government expenditures are treated as endogenous, unlike much of the macro-economics literature studying the economic effects of government spending. By using lagged conditioning var iables, we provide short-run and long-run elasticity estimates, allowing us to observe any differences that might arise when the public has completely adjusted to variations in relative prices.

The estimates show that consumption and government expenditures are net Morishima substitutes. Further evidence indicates that elasticities of substitution are not equal to unity in the short run, nor are these elasticities constant over our sample data. There is a general tendency for elasticities of substitution between consumption and government expenditures to rise and fall with the level of federal government defense expenditures, suggesting that defense goods and private consumption goods have become weaker substitutes in response to variations in the price of consumer goods. In the long-run version of our results, we find evidence that elasticities of substitution become somewhat less heterogeneous, approaching unitary elasticities for some goods. Thus our long-run results move towards but are still different from the results one would find if the optimization problem were solved using a Cobb-Douglas utility function with no lagged data.

The virtue of our results is that they are quite uniform compared to the literature, consistently implying that consumption and government expenditures are net Morishima substitutes. This fact would seem to be relevant information for government policy makers making government expenditure decisions.

APPENDIX

A series of Wald tests were used to determine the number of lags of the goods in the utility function and the parameters A and J. The share equations were estimated using maximum likelihood in version 43 of TSP with a convergence criterion of 0.00001. The number of lags was set to four with A = 4 and J = 1. Because the Fourier flexible form is nonlinear, we searched over a wide range of starting values. The estimated parameters are given in Table A1.

Table A1

Parameter Estimates of Fourier Share Equations

Parameter Estimate Standard Error

[b.sub.1] .013378 .020531
[b.sub.2] .013199 .015228
[b.sub.3] -.006461 .010212
[b.sub.4] .057732 .024753
[b.sub.5] -.108541 .019809
[u.sub.01] .000119 .000027
[u.sub.11] .000005 .000008
[v.sub.11] .000003 .000008
[u.sub.02] .000097 .000026
[u.sub.12] -.000006 .000008
[v.sub.12] .000004 .000008
[u.sub.03] -.000198 .000034
[u.sub.13] .000014 .000006
[v.sub.13] -.000003 .000006
[u.sub.04] .000126 .000030
[u.sub.14] -.000005 .000009
[v.sub.14] -.000021 .000009

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

TABLE 1

Short-Run Morishima Elasticities of Substitution

 [M.sub.14] [M.sub.15]

Consumption and government expenditures

Mean 1.080 1.094
Median 1.085 .0064 1.095
Max 1.159 .0108 1.199
Min 1.023 .0025 1.019

 [M.sub.34] [M.sub.35]

Mean 1.014 1.031
Median 1.009 .0037 1.029
Max 1.057 .0080 1.057
Min 0.996 .0031 1.017

 [M.sub.42] [M.sub.52]

Mean 0.651 0.791
Median 0.663 .0221 0.795
Max 0.776 .0398 0.887
Min 0.388 .0165 0.639

 [M.sub.45] [M.sub.46]

Consumption expenditures

Mean 1.021 0.996
Median 1.021 .0021 0.999
Max 1.053 .0033 1.081
Min 0.990 .0015 0.960

 [M.sub.16]

Consumption and government expenditures

Mean 1.073
Median .0068 1.076 .0064
Max .0121 1.165 .0113
Min .0016 0.992 .0022

 [M.sub.36]

Mean 1.013
Median .0048 1.011 .0050
Max .0083 1.046 .0086
Min .0042 1.000 .0043

 [M.sub.62]

Mean 0.691
Median .0163 0.698 .0192
Max .0265 0.800 .0345
Min .0110 0.471 .0130

 [M.sub.54]

Consumption expenditures

Mean 0.990
Median .0013 0.990 .0009
Max .0025 1.015 .0017
Min .0011 0.965 .0007

 [M.sub.24] [M.sub.25]

Consumption and government expenditures

Mean 1.191 1.208
Median 1.185 .0233 1.206
Max 1.368 .0400 1.376
Min 1.104 .0171 1.117

 [M.sub.41] [M.sub.51]

Mean 0.901 0.909
Median 0.896 .0061 0.908
Max 0.971 .0106 0.984
Min 0.804 .0022 0.804

 [M.sub.43] [M.sub.53]

Mean 0.969 0.981
Median 0.980 .0027 0.981
Max 0.999 .0077 0.994
Min 0.896 .0023 0.959

 [M.sub.56] [M.sub.64]

Consumption expenditures

Mean 0.982 1.007
Median 0.982 .0012 1.004
Max 0.984 .0019 1.041
Min 0.968 .0009 0.987

 [M.sub.26]

Consumption and government expenditures

Mean 1.188
Median .0244 1.185 .0236
Max .0405 1.361 .0398
Min .0172 1.101 .0164

 [M.sub.61]

Mean 0.909
Median .0065 0.907 .0055
Max .0116 1.006 .0108
Min .0013 0.799 .0013

 [M.sub.63]

Mean 0.983
Median .0026 0.987 .0021
Max .0053 1.003 .0045
Min .0022 0.945 .0019

 [M.sub.65]

Consumption expenditures

Mean 1.023
Median .0006 1.022 .0017
Max .0025 1.039 .0026
Min .0004 1.008 .0013

Notes: Table entries are short-run Morishima elasticities of
substitution, denoted by [M.sup.ij], between consumption and government
expenditures. Summary measures of estimated standard errors are given
adjacent to summary data on elasticity estimates. Subscript notation: 1
= defense, 2 = nondefense, 3 = state and local, 4 = nonduable goods, 5 =
services, and 6 = the stock of durable goods.

TABLE 2

Long-Run Morishima Elasticities of Substitution


 [M.sub.14] [M.sub.15]

Consumption and government
expenditures

Mean 1.084 1.097
Median 1.077 0.0356 1.084 .0376
Max 1.300 0.0663 1.325 .0732
Min 0.936 0.0135 0.951 .0082

 [M.sub.34] [M.sub.35]

Mean 1.018 1.035
Median 1.011 0.0204 1.032 0.0239
Max 1.141 0.0492 1.121 0.0432
Min 0.936 0.0157 0.959 0.0194

 [M.sub.42] [M.sub.52]

Mean 0.629 0.771
Median 0.629 .1157 0.775 .1196
Max 1.111 .2089 1.382 .2060
Min 0.010 .0828 0.202 .0792

 [M.sub.45] [M.sub.46]

Consumption expenditures

Mean 1.019 0.994
Median 1.019 .0044 0.999 .0026
Max 1.074 .0089 1.029 .0096
Min 0.964 .0031 0.928 .0017


 [M.sub.16] [M.sub.24]

Consumption and government
expenditures

Mean 1.076 1.211
Median 1.062 .0340 1.206 .1430
Max 1.304 .0676 1.781 .2358
Min 0.940 .0053 0.640 .1010

 [M.sub.36] [M.sub.41]

Mean 1.017 0.898
Median 1.013 0.0208 0.911 0.0294
Max 1.109 0.0435 1.029 0.0569
Min 0.946 0.0154 0.686 0.0110

 [M.sub.62] [M.sub.43]

Mean 0.671 0.963
Median 0.682 .1100 0.976 .0157
Max 1.171 .1900 1.031 .0423
Min 0.096 .0733 0.809 .0116

 [M.sub.54] [M.sub.56]

Consumption expenditures

Mean 0.992 0.982
Median 0.993 .0038 0.984 .0023
Max 1.045 .0094 1.001 .0062
Min 0.944 .0026 0.949 .0014


 [M.sub.25] [M.sub.26]

Consumption and government
expenditures

Mean 1.228 1.208
Median 1.218 .1448 1.199 .1422
Max 1.782 .2326 1.768 .2320
Min 0.663 .0966 0.649 .0940

 [M.sub.51] [M.sub.61]

Mean 0.907 0.907
Median 0.918 0.0353 0.924 0.0294
Max 1.045 0.0703 1.031 0.0583
Min 0.686 0.0075 0.691 0.0043

 [M.sub.53] [M.sub.63]

Mean 0.977 0.980
Median 0.979 .0179 0.984 .0149
Max 1.058 .0382 1.044 .0341
Min 0.868 .0146 0.870 .0108

 [M.sub.64] [M.sub.65]

Consumption expenditures

Mean 1.009 1.023
Median 1.005 .0027 1.021 .0029
Max 1.076 .0098 1.057 .0064
Min 0.976 .0018 1.003 .0020

Notes: Table entries are long-run Morishima elasticities of
substitution, denoted by [M.sub.ij], between consumption and government
expenditures. Summary measures of estimated standard errors are given
adjacent to summary data on elasticity estimates. Subscript notation: 1
= defense, 2 = nondefense, 3 = state and local, 4 = nondurable goods, 5
= services, and 6 = the stock of durable goods.

(1.) Ni (1995, 595) surveys estimates of substitution elasticities between consumption and government expenditures. Seater (1993) describes the literature on Ricardian equivalence and the economic effects of government expenditures.

(2.) Baxter and King (1993) and Ni (1995) do have explicit utility functions underlying their analyses. However, Ni (1995) uses functional forms that restrict the range of substitution elasticities. The Baxter and King (1993) study is the most complete analysis of the relation between consumption and government expenditures in a general equilibrium framework, but they only provide a simulation study giving quantitative measures of substitution between consumption and government expenditures.

(3.) It wilt be assumed that the labor supply decision is separable from other choices made by the household, which follows the bulk of the literature in this area. Relaxing this separability assumption is thus left as a possible avenue for future research.

(4.) For example, Cobb-Douglas, CES, or quadratic forms for the utility function impose a constant Morishima elasticity of substitution at all sample points.

(5.) See, for example, Dunn and Singleton (1986), Eichenbaum et al. (1988), and Eichenbaum and Hansen (1990).

(6.) See, for example, Deaton and Muellbauer (1980) and Eichenbaum and Hansen (1990).

(7.) Elbadawi et al. (1983) define a semi-nonparametric function as a series expansion dense in a Sobolov norm. We could use the asymptotically ideal model specification (Barnett and Jonas, 1983) but the dimensionality of our economic model precludes the use of this approach. See Fleissig and Swofford (1997) for further discussion.

(8.) This user cost is [p.sub.it] - [(1 + [R.sub.t]).sup.-1] (1 - [delta]) [E.sub.t][p.sub.it] + 1 where R is the nominal interest rate, [delta] is the depreciation rate, [p.sub.it] is the price of good i, and [E.sub.t][p.sub.it] + 1 is the expected price of good i. Following Diewert (1974), we use static expectations.

(9.) The equations were estimated in International TSP version 4.3 with the cross-equation restrictions imposed.

(10.) Evaluating regularity conditions for different functional forms is discussed in detail by Christensen et al. (1975) and Fisher et al. (2001). Briefly, nonnegativity requires that the estimated demand functions are nonnegative for all data points. The gradient vector of the estimated indirect utility function is used to determine if the indirect utility function is monotonically decreasing. Quasi-convexity of the indirect utility function is evaluated from the parameter estimates of the estimated indirect utility function, provided monotonicity holds.

(11.) In fact, this temporal pattern is also found in the Morishima elasticities for federal defense spending and the other two government expenditure series.

(12.) There are no consistent cyclical patterns in the estimated elasticities that we can observe. For example, looking at postwar peak-to-trough values of the estimated elasticities using NBER reference dates, the estimated elasticities display different time patterns across recessions and differ in their behavior in a given recession.

(13.) A unitary value for the Morishima elasticity would arise if economic agents were solving a static utility optimization problem without any lags entering into the optimizing framework and if they had a Cobb-Douglas utility function.

(14.) The only elasticities that unambiguously display trends over time are [M.sub.43], [M.sub.45], [M.sub.46], [M.sub.54], and [M.sub.64].

(15.) Ni (1995) provides a discussion of the sensitivity in estimation results to various elements of specification.

(16.) As pointed out by an anonymous referee, there is an important difference between our analysis and that of Baxter and King (1993) regarding the mechanism that generates substitution or complementarity between consumption and government expenditures. In this earlier study, consumption choices by the public respond to the income effects associated with variations in exogenous government expenditures, whereas in our analysis variations in relative prices are the driving forces behind the consumption and government expenditure choices of the public.

REFERENCES

Aschauer, D. "Fiscal Policy and Aggregate Demand." American Economic Review, 75(1), 1985, 117-27.

Barnett, W, and A. B. Jonas. "The Muntz-Szatz Demand System: An Application of a Globally Well-Behaved Series Expansion." Economics Letters, 11(4), 1983, 337-42.

Barnett, W., D. Fisher, and A. Serletis. "Consumer Theory and the Demand for Money." Journal of Economic Literature, 30(4), 1992, 2086-119.

Barten, A. "Maximum Likelihood Estimates of a Complete System of Demand Equations." European Economic Review, 1(1), 1969, 7-73.

Barro, R. "Are Government Bonds Net Wealth?" Journal of Political Economy, 82(6), 1974, 1095-117.

_____. "On the Determination of the Public Debt." Journal of Political Economy, 87(5), 1979, 940-71.

_____. "Output Effects of Government Purchases." Journal of Political Economy, 89(6), 1981, 1086-121.

Baxter, M., and R. G. King. "Fiscal Policy in General Equilibrium." American Economic Review, 83(3), 1993, 315-34.

Bean, C. "The Estimation of Surprise Models and the Surprise Consumption Function." Review of Economic Studies, 53(4), 1986, 497-516.

Bernanke, B. "Adjustment Costs, Durables, and Aggregate Consumption." Journal of Monetary Economics, 15(1), 1985, 41-68.

Berndt, E., and N. E. Savin. "Estimation and Hypothesis Testing in Singular Equation Systems with Autoregressive Disturbances." Econometrica, 43(5- 6), 1975, 937-57.

Blackorby, C., and R. R. Russell. "Will the Real Elasticity of Substitution Please Stand Up? (A Comparison of the Allen/Uzawa and Morishima Elasticities)." American Economic Review, 79(4), 1989, 882-88.

Campbell, J., and N. G. Mankiw. "Permanent Income, Current Income, and Consumption." Journal of Business & Economic Statistics, 8(8), 1990, 265-79.

Christensen, L., D. W Jorgenson, and L. J. Lau, "Transcendental Logarithmic Utility Functions." American Economic Review, 65(3), 1975, 367-83.

Deaton, A., and J. Muellbauer. Economics and Consumer Behavior. Cambridge: Cambridge University Press, 1980.

Diewert, W "Intertemporal Consumer Theory and the Demand for Durables." Econometrica, 42(3), 1974, 497-516.

Dunn, K, and K J. Singleton. "Modelling the Term Structure of Interest Rates under Nonseparable Utility and the Durability of Goods." Journal of Financial Economics, 17(1), 1986, 27-55.

Eichenbaum, M., and L. Hansen. "Estimating Models with Intertemporal Substitution Using Aggregate Time Series Data." Journal of Business & Economic Statistics, 8(1), 1990, 53-69.

Eichenbaum, M., L. Hansen, and K Singleton. "A Time Series Analysis of Representative Agent Models of Consumption and Leisure." Quarterly Journal of Economics, 103(1), 1988, 51-78.

Elbadawi, I., A. R. Gallant, and G. Souza. "An Elasticity Can Be Estimated Consistently without Prior Knowledge of Functional Form." Econometrica, 51(6), 1983, 1731-51.

Fisher, D., and A. R. Fleissig. "Monetary Aggregation and the Demand for Monetary Assets." Journal of Money, Credit, and Banking, 29(4), 1997, 458-75.

Fisher, D., A. R. Fleissig, and A. Serletis. "An Empirical Comparison of Flexible Demand System Functional Forms." Journal of Applied Econometrics, 16(1), 2001, 59-80.

Feldstein, M. "Government Deficits and Aggregate Demand." Journal of Monetary Economics, 9(1), 1982, 1-20.

Fleissig, A. "The Consumer Consumption Conundrum: An Explanation." Journal of Money, Credit, and Banking, 29(2), 1997, 177-91.

Fleissig, A., and J. Swofford. "A Dynamic Asymptotically Ideal Demand for Money." Journal of Monetary Economics, 37(2), 1996, 371-80.

-----. "Dynamic Asymptotically Ideal Models and Finite Approximation." Journal of Business and Economic Statistics, 15(4), 1997, 482-92.

Gallant, A. "On the Bias in Flexible Functional Forms and an Essentially Unbiased Form: The Fourier Flexible Form." Journal of Econometrics, 15(2), 1981, 211-45.

Graham, F. "Fiscal Policy and Aggregate Demand: Comment." American Economic Review, 83(3), 1993, 659-56.

Graham, F., and D. Himarios, "Fiscal Policy and Private Consumption: Instrumental Variables Tests of the Consolidated Approach." Journal of Money, Credit, and Banking, 23(1), 1991, 53-67.

Hall, R. "The Stochastic Implications of the Life-Cycle Permanent Income Hypothesis." Journal of Political Economy, 6(6), 1978, 971-87.

Kormendi, R. "Government Debt, Government Spending, and Private Sector Behavior." American Economic Review, 73(5), 1983, 994-1010.

Musgrave, J. "Durable Goods Owned by Consumers in the United States, 1925-77." Survey of Current Business, 59(3), 1979, 17-25.

Ni, S. "An Empirical Analysis on the Substitutability between Private Consumption and Government Purchases." Journal of Monetary Economics, 36(3), 1995, 593-605.

Pollak, R., and T Wales. Demand System Specification and Estimation. New York: Oxford University Press, 1992.

Reid, B. "Aggregate Consumption and Deficit Financing: An Attempt to Separate Permanent from Transitory Effects." Economic Inquiry, 23(3), 1985, 475-86.

Seater, J. "Ricardian Equivalence." Journal of Economic Literature, 31(1), 1993, 142-90.

RELATED ARTICLE: ABBREVIATIONS

CES: Constant Elasticity of Substitution

NIPA: U.S. National Income and Product Accounts

ADRIAN R. FLEISSIG and ROBERT J. ROSSANA *

* We are indebted to Basma Bekdache, Shawn Ni, and John J. Seater for comments on an earlier draft of this article. Two anonymous referees provided many useful suggestions. The usual disclaimer applies regarding the limitations of our analysis.

Flessig: Associate Professor, Department of Economics, California State University at Fullerton, Fullerton, CA 92834, Phone 1-714-278-3816, Fax 1-714-278-1387, E-mail afleissig@fullerton.edu

Rossana: Professor, Department of Economics, 2074 FAB, Wayne State University, Detroit, MI 48202. Phone 1-313-577-3760, Fax 1-313-577-0149, E-mail r.j.rossana@wayne.edu