The use of conditional cost functions to Generate estimable mixed demand systems.

Consumer demand models have always been estimated under the assumption that either (1) prices are predetermined or (2) quantities are predetermined. The first assumption leads to direct demand (or quantity-dependent) systems such as the Translog form (Christensen, Jorgenson, and Lau 1975) and the Almost Ideal Demand System (AIDS) (Deaton and Muelbauer 1980a). These systems satisfy the conditions that supplies are perfectly elastic and that demands adjust to clear the market. The second assumption leads to inverse demand (or price-dependent) systems such as the Inverse AIDS (Eales and Unnevehr 1994) and the Normalized Quadratic Inverse Demand System (Beach and Holt 2001). It is advantageous to treat quantities as being predetermined in cases where quantities do not adjust in the short run or for nonmarket goods where prices are arbitrarily distorted.

In between the direct and inverse demand systems, Samuelson (1965) argued that there exists a whole family of mixed demands: the prices of some goods are predetermined such that the respective quantities demanded adjust to clear the market, whereas for the remaining set of goods, quantities supplied are predetermined and prices must adjust to clear the market. An attractive property of this system is that it provides a theoretical basis for studying aggregate consumption behavior (Chavas 1984). Furthermore, the systems express demand relationships as a function of a mixed set of prices and quantities, which is ideally suited for the measurement of welfare effects of any policy option related to both quantity and price changes.

The literature on direct and inverse demand systems is voluminous while the literature on mixed demands is much smaller. Only in recent years have the econometric issues arising in mixed demand systems been explored (see, e.g., Moschini and Vissa 1993; Matsuda 2004; Moschini and Rizzi, 2006). Not surprisingly, this model is unlikely to be popular, apparently because knowledge of both direct and indirect utility functions is required to derive the closed forms for mixed demand functions. Consequently, commonly used flexible functional forms such as the Translog and AIDS cannot be employed for empirical application since they typically do not have a closed form dual representation. (1) This problem was acknowledged by Barten (1992), Moschini and Vissa (1993), and Matsuda (2004), leading them to develop some flexible functional forms by approximating the mixed demand functions directly through a differential approach. Though these models are of considerable interest, they do not have exact parametric representations of preferences; that are required for some policy applications (such as welfare analysis).

In this article, we propose a new approach to specifying empirical mixed demand functions, which is based on parametric representations of the Hicksian conditional cost function used in the area of rationed demand (see Neary and Roberts 1980). Provided that preferences are specified in terms of a conditional cost function, then Hicksian mixed demand functions can be derived via Samuelson's Envelope Theorem. Whilst these functions are conditioned on an unobservable variable (utility), in most cases they do not have an explicit closed-form representation as the Marshallian mixed demand functions; that is, in terms of the observable variables such as prices, quantities, and expenditure. With modern hardware and software, however, the aforesaid problem need not hinder specification and estimation of mixed demands. A simple one-dimensional numerical inversion allows us to estimate the parameters of a particular conditional cost function via the parameters of the implied Marshallian mixed demand systems. The advantage of this approach is that it refines the theoretical properties of mixed demand systems and makes them operational for empirical analysis of consumer behavior. Additionally, it opens up a further avenue for ultimately obtaining systems of mixed demand functions, which are simultaneously more flexible and more regular than those currently employed in applied consumer demand analyses. The proposed method will be illustrated with the applications to Japanese meat and fish consumption.

Background Developments

Let x = {[x.sub.1], ..., [x.sub.N]} denote a vector of commodities indexed by the members of the index set I = {1, ..., N}, p = {[p.sub.1], ..., [p.sub.N]) the corresponding price vector, and c a level of expenditure. Partition I so that it is represented by [??] = {[I.sub.A], [I.sub.B]}, where [I.sub.A] = {1, ..., M} and [I.sub.B] = {M + 1, ..., N}. The vector x can then be partitioned analogously as x = {[x.sub.A], [x.sub.B]} with [x.sub.A] = {[x.sub.1], ..., [x.sub.M]} containing commodities chosen optimally, and [x.sub.B] = {[x.sub.M+1], ..., [x.sub.N]} containing commodities in fixed quantities whose prices are optimally determined. Likewise, the price vector p can be partitioned as p = {[p.sub.A], [p.sub.B]} with [p.sub.A] and [p.sub.B] containing the prices of groups A and B commodities respectively.

Suppose that individual preferences are represented by the direct utility function u = U([x.sub.A], [x.sub.B]), satisfying the following regularity conditions RU:

RU1: U is real;

RU2: U is continuous;

RU3: U is increasing in([x.sub.A], [x.sub.B]); and

RU4: U is quasi--concave in([x.sub.A], [x.sub.B]).

Given this utility function, the Hicksian or compensated conditional cost function ([C.sup.h.sub.A]) is defined by

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [C.sup.h.sub.A] is cost of the least expensive collection of [x.sub.A] capable of achieving utility level u when ([p.sub.A], [x.sub.B]) are given, and the superscript h is to indicate that (1) represents the Hicksian function; that is, the function is conditioned on [p.sub.A], [x.sub.B], and u. The Hicksian conditional cost function will inherit the set of regularity conditions [RC.sup.h.sub.A]:

[RC.sup.h.sub.A] 1: [C.sup.h.sub.A] is positive;

[RC.sup.h.sub.A] 2: [C.sup.h.sub.A] is continuous;

[RC.sup.h.sub.A] 3: [C.sup.h.sub.A] is increasing in [p.sub.A];

[RC.sup.h.sub.A] 4: [C.sup.h.sub.A] is decreasing in [x.sub.B];

[RC.sup.h.sub.A] 5: [C.sup.h.sub.A] is increasing in u;

[RC.sup.h.sub.A] 6: [C.sup.h.sub.A] is homogeneous of degree one (HD1) in [p.sub.A];

[RC.sup.h.sub.A] 7: [C.sup.h.sub.A] is concave in [p.sub.A]; and

[RC.sup.h.sub.A] 8: [C.sup.h.sub.A] is convex in [x.sub.B].

Consider now the possibility of using a conditional cost function to generate systems of estimable mixed demand functions. Take as given a conditional cost function satisfying Conditions [RC.sup.h.sub.A]. As shown in Chavas (1984), Hicksian mixed demand functions are related to the conditional cost function via Samuelson's Envelope Theorem

(2) [X.sup.h.sub.Ai] ([p.sub.A], [x.sub.B], u) = [partial derivative][C.sup.h.sub.Ai] / [partial derivative] [p.sub.Ai], i [member of] [I.sub.A]

[P.sup.h.sub.Bj] ([p.sub.A], [x.sub.B], u) = -[partial derivative][C.sup.h.sub.A] / [partial derivative] [x.sub.Bj], j [member of] [I.sub.B]

where [X.sup.h.sub.Ai] (or [P.sup.h.sub.Bj]) are the Hicksian quantity-dependent (or price-dependent) mixed demand functions. Application of Samuelson's Envelope Theorem after some manipulation also yields the Hicksian total cost function

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which measures the total cost ([p'.sub.A] [x.sub.A] + [p'.sub.B] [x.sub.B) of achieving the utility level u given [p.sub.A] and [x.sub.B].

It is clear that the Hicksian mixed demand functions are not directly estimable since they are defined in terms of the level of unobservable utility u. This makes estimation a bit more complicated, but does not create as many difficulties as one might expect. To motivate what follows, note that if the explicit functional form of the corresponding mixed utility function [U.sup.m] ([p.sub.A], [x.sub.B], c) were available, the Hicksian mixed demand functions could be "Marshallianized" by replacing u by

(4) [U.sup.m] ([p.sub.A],[x.sub.B], c)

to give

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [U.sup.m]([p.sub.A], [x.sub.B], c) in (4) and (5) is the analytical inversion of the identity function c = [C.sup.h]([p.sub.A], [x.sub.B], u), [X.sup.m.sub.Ai](or [P.sup.m.sub.Bj]) are the quantity-dependent (or price-dependent) Marshallian mixed demands corresponding to (2), and the superscript m reminds us that we are considering Marshallian functions. Similarly, the Marshallian mixed demand functions can be converted into the Hicksian mixed demand functions by the following identical relationships

(6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In practice, however, such an explicit inversion between [C.sup.h] and u is not always workable; it depends heavily on the particular parametric form of [C.sup.h], which is itself determined by the particular parametric form of [C.sup.h.sub.A]. This study focuses on the class of [C.sup.h] for which such explicit inversion is not available; that is, solving [C.sup.h]([p.sub.A], [x.sub.B], u) = c for [U.sup.m] ([p.sub.A], [x.sub.B], c) may not be accomplished analytically. Then, the implied Marshallian mixed demand functions has to be expressed implicitly by the following system

(7) [X.sup.h.sub.Ai] ([p.sub.A], [x.sub.B], u) = [partial derivative] [C.sup.h.sub.A] / [partial derivative] [p.sub.Ai], i = 1, ..., M

(8) [P.sup.h.sub.Bj] ([p.sub.A], [x.sub.B], u) = -[partial derivative] [C.sup.h.sub.A] / [partial derivative] [x.sub.Bj], j = M + 1, ..., N

(9) c = [summation over (i')] [p.sub.Ai'] ([partial derivative][C.sup.h.sub.A] / [partial derivative] [p.sub.Ai']) -[summation over (j')] ([partial derivative][C.sup.h.sub.A] / [partial derivative] [x.sub.Bj']) [x.sub.Bj']).

Provided that Conditions [RC.sup.h.sub.A]2 and [RC.sup.h.sub.A]5 are satisfied, (3) it is feasible to numerically invert (9) to express u as a function of [p.sub.A], [x.sub.B] and c. Note that the dimension of the numerical inversion is not related to the dimension of the commodity vector x = ([x.sub.1], ..., [x.sub.N]) so that the order of numerical complexity does not increase with the number of commodities.

Given a specific functional form for [C.sup.h.sub.A] and a vector of parameters [xi], the corresponding mixed demand system can be written as

(10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [U.sup.m] ([p.sub.A], [x.sub.B], c; [xi]) is the numerical solution of the identity function

(11) c = [C.sup.h]([p.sub.A],[x.sub.B], u; [xi])

for u, solved at the given values of [p.sub.A], [x.sub.B], c, and [xi]. In a maximum likelihood search for the parameters of the mixed demand functions, explicit solution is not necessary: all that is required is that software capable of solving the identity function (11) be imbedded in the maximum likelihood computer routine.

At each iterative step of the maximization of the likelihood function, there is a given set of parameter values. For these parameter values, (11) can be numerically inverted to recover the value of utility consistent with the given values of [p.sub.A], [x.sub.B], and c. Then, this value of utility can be used to eliminate the unknown value of u from the Hicksian mixed demand system.

Define [E.sub.Y(h,r)], z(s) and [E.sub.Y(m,r), z(s)] as two sets of price, quantity, and expenditure elasticities, written as

(12) [E.sub.Y(h,r), z(s)] = [partial derivative] log ([Y.sup.h.sub.r])/[partial derivative] log ([z.sub.s]) and [E.sub.Y(m,r), z(s)] = [partial derivative] log ([Y.sup.m.sub.r]) / [partial derivative] log([z.sub.s]),

where [Y.sup.h] = {[X.sup.h.sub.A1], ..., [X.sup.h.sub.AM], [P.sup.h.sub.BM+1], ..., [P.sup.h.sub.BN]} and z = {[p.sub.A1], ..., [p.sub.AM], [x.sub.BM+1], ..., [x.sub.BN], c}. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for instance, is the Hicksian cross-price elasticity of the ith (group A) good with respect to the price of the kth (group A) good; [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the Marshallian cross-price elasticity of the price of the jth (group B) good with respect to the price of the ith (group A) good; and [E.sub.X(m,Ai),c] is the expenditure elasticity of the ith (group A) good. To facilitate thinking about preferences in terms of a conditional cost function, the price, quantity and expenditure elasticity equations may be written in terms of [p.sub.A], [x.sub.B], and u.

The Hicksian Price/Quantity Elasticities

(13) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The Marshallian Price/Quantity Elasticities

(14) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The Marshallian Expenditure Elasticities

(15) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Empirical Specification

In this section, we examine the three specifications on which our empirical analysis is based. The first model to be considered is Moschini and Rizzi's (2006) Normalized Quadratic Model (NQM), in which the analytical inversion of the total cost function is available. We then move to the second model, namely the Quadratic Almost Ideal Mixed Demand System (QAIMDS), in which the mixed utility function lacks an explicit closed form, demonstrating that the numerical inversion method is feasible for the estimation of this QAIMDS system. As can be seen, the QAIMDS is parametrically similar to the expenditure function underlying Michelini's (1999) Quadratic Almost Ideal Demand (QAIDS). Therefore, most of the desirable theoretical properties attributed to the QAIDS carry over to QAIMDS. Finally, we use the intuition stemming from Holt's (2002) Invese Nonseparable Linear Expenditure System, and Perroni and Rutherford's (1995) N-stage CES form to build up a model, namely the Nested Constant Elasticity of Substitution (CES) form. Note that this specification is particularly attractive for the purpose of modeling complete and multistage mixed demand systems since it can be easily constrained to be regular over an unbounded region, since the numbers of additional parameters to be estimated are small, and since it is general enough to nest a number of well-known functional structures.

The Normalized Quadratic Model (NQM)

Suppose that preferences are represented by the Linear-In-Utility (LIU) mixed cost function proposed by Moschini and Rizzi (2006)

(16) [C.sup.h.sub.A] = F([p.sub.A], [x.sub.B]) + G([p.sub.A], [x.sub.B])H(u),

where F([p.sub.A], [x.sub.B]) and G([p.sub.A], [x.sub.B]) are functions of ([p.sub.A], [x.sub.B]) which are increasing, HD1 and concave in [p.sub.A], and decreasing and convex in [x.sub.B], and H(u) is an increasing function of u. The simplified version of NQM results if F([p.sub.A], [x.sub.B]), G([p.sub.A], [x.sub.B]), and H(u) are specified as (5)

(17) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Applying Samuelson's Envelope Theorem to (16), and after some manipulation, we obtain the NQM budget share equations

(18) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and the Hicksian total cost function [C.sup.h] corresponding to (16) is given by

(19) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Elimination of u by the analytical inversion of (19) at the optimum (setting (19) equal to c) leads immediately to the Marshallian mixed demand system. It is also transparent that, given the values of parameters, the numerical inversion of (19) at the optimum to give u in terms of [p.sub.A], [x.sub.B], and [xi], and its substitution in (18) would give the same results as analytical inversion.

The QAIMDS

Though the NQM provides a reasonable degree of flexibility in estimation, it is by no means the only feasible functional form of a mixed demand system. Other functional forms exist which also provide local approximations to the underlying conditional cost function. A good example is the QAIMDS, based on a modification by Michelini (1999) of the AIDS of Deaton and Muelbauer (1980a), and results in one of the flexible mixed demand systems. The basic specification of the conditional cost function is

(20) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [delta], [[eta].sub.1], and [[eta].sub.2] are parameters, [Pk.sub.A] (k = 1, 2) are positive functions of [p.sub.A] with [P1.sub.A] HD1 in [p.sub.A] and [P2.sub.A] HD0 in [p.sub.A], and [Xk.sub.B] are positive functions of [X.sub.B], which are linear homogeneous. Using the intuition stemming from Michelini's (1999) QAIDS, we choose the [Pk.sub.A] and [Xk.sub.B] as follows

(21) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [[alpha].sub.Ai], [[alpha].sub.Bj], [[beta].sub.Ai], [[beta].sub.Bj], [[gamma]'.sub.Aii'] and [[gamma]'.sub./Bjj], are parameters.

Applying Samuelson's Envelope Theorem to (20), and after some manipulation, we obtain the QAIMDS budget share equations

(22) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Employing (9) compatibly with [C.sup.h] specified as

(23) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

it is impossible to solve (23) explicitly for the value of u in terms of parameters, [p.sub.A], [x.sub.B], and c. In order to convert (22) to a Marshallian system, the unobservable u in (22) has to be replaced by the numerical inversion of (23) at [C.sup.h] = c.

The NCES

An attractive feature of the NQM and QAIMDS is that they are based on flexible functional forms so that they allow for more possibilities regarding the substitution relationships among commodities. In addition, in the spirit of Lewbel's (1991) definition, both systems are consistent with rank 3 preferences. This is potentially important empirically because it implies that the resulting mixed demand systems are more flexible in describing consumer behavior.

Unfortunately, there are two major problems in the empirical investigation of these two systems. First of all, the implied mixed demand functions do not necessarily satisfy the regularity properties (Conditions [RC.sup.h.sub.A]) of a conditional cost function. One may recall, however, that these conditions are required by microeconomic theory, and they must be met for the estimates to be meaningful and valid for use in applied general equilibrium modeling and in policy analysis. Second, when the number of commodities under consideration is large, the usefulness of these specifications diminishes rapidly due to increased volume of computation, to difficulties in imposing and testing regularity conditions, and to problems of degrees of freedom and multicollinearity. In this subsection, we suggest a new specification (the NCES), which in principle is free from the aforesaid problems but is readily applicable for empirical estimation.

As an alternative to the NQM and QAIMDS, consider the following specification

(24) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [[eta].sub.1], [[eta].sub.2], [delta], [rho], and [kappa] are parameters, [Pk.sub.A] (k = 1, 2) are two price functions satisfying Conditions [RP.sub.A]

[RP1.sub.A]: [Pk.sub.A] is positive;

[RP2.sub.A]: [Pk.sub.A] is continuous;

[RP3.sub.A]: [Pk.sub.A] is HD1 in [p.sub.A];

[RP4.sub.A]: [Pk.sub.A] is increasing in [p.sub.A]; and

[RP5.sub.A]: [Pk.sub.A] is concave in [p.sub.A];

and [Xk.sub.B] (k = 1, 2) are two quantity functions satisfying Conditions [RX.sub.B]

[RX1.sub.B]: [Xk.sub.B] is positive;

[RX2.sub.B]: [Xk.sub.B] is continuous;

[RX3.sub.B]: [Xk.sub.B] is HD1 in [x.sub.B];

[RX4.sub.B]: [Xk.sub.B] is increasing in [x.sub.B]; and

[RX5.sub.B]: [Xk.sub.B] is concave in [x.sub.B].

Suppose that [Pk.sub.A] and [Xk.sub.B] have the following forms:

(25) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [[gamma].sub.Li], [[alpha].sub.Li], and [[beta].sub.Li] (L = A, B) are parameters. The sufficient conditions to ensure (24) to be a regular conditional cost function over the region [kappa] > [X1.sub.B] are: 0 [less than or equal to] [delta], [[alpha].sub.Li], [[beta].sub.Li] [less than or equal to] 1, [[gamma].sub.Li] [greater than or equal to] 0, [rho] [less than or equal to] 1, and [[eta].sub.L] [greater than or equal, to] 0.

Functions (24) and (25), on application of Samuelson's Envelope Theorem, generate the following system of Hicksian budget share equations

(26) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and the Hicksian total cost function used for numerical inversion is

(27) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It is important to note that the NCES is general enough to leave several forms of functional separability and preference structure as hypotheses to be tested rather than maintained. First of all, we note that imposing implicit separability in Partition [??] is equivalent to [gamma] being zero. In this case, (24) will collapse to (will be referred to as the Implicitly Separable NCES or ISNCES)

(28) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the subfunction [T.sub.B] may be interpreted as the quantity index of Group B commodities which is positive, continuous, increasing, and concave in [x.sub.B], and decreasing in u. (6) Second, the restrictions [delta] = 0, [rho] = 1, and [[eta].sub.1] = 0 give the form

(29) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which is a special case of Moschini and Rizzi's (2006) LIU mixed cost function (will be referred to as the LIU).

Brief Remarks on the Database

Budget share systems (18), (22), and (26) were estimated using time series data for Japanese fish and meat consumption. (7) The data consist of thirty-eight types of fish and meat products, and they were aggregated into six categories comprised of:

Group A

1. [x.sub.A1] = Salted and dry fish;

2. [x.sub.A2] = Bonito fillets and fish flakes;

3. [X.sub.A3] = Processed meat (including ham, sausages, bacon, and other meat products)

Group B

4. [x.sub.B4] = Fresh fish;

5. [xB.sub.5] = Fresh meat; and

6. [x.sub.B6] = Shellfish.

The first three categories (salted fish, fillets, and processed meat) are easily stored so that it is acceptable to treat their prices as given in the consumer problem. On the other hand, due to the highly perishable nature and biological production lags, supply of fresh fish, fresh meat, and shell fish (categories 4-6) is often inelastic in the short run, which implies that for these categories, equilibrium should be characterized by exogenously determined quantities with prices adjusting to clear the market. Thus, this dataset appears to fulfill the basic assumptions underlying the applicability of a mixed demand system.

The raw data, gathered from Annual Report on the Family Income and Expenditure Survey, consist of monthly data averaged over 8,000 households throughout the country. These households keep journals of prices paid (per 100 grams) and expenditures on a large number of fish and meat products and other food commodities. The sample period covers January 1985 through December 2003 for a total of 228 monthly observations. The data were further aggregated to quarterly frequency resulting in seventy-six usable observations, and were deseasonalized and mean centered prior to estimation.

Estimation and Stochastic Specification

The computation of the maximum-likelihood estimates reported below is feasible because the GAUSS language used to program the estimators handles the implicit representation of functional relationships well. All budget share systems are estimated by using the GAUSS 3.6.27 computer package with the modules NLSYS and CML. For purposes of estimation, an error term [e.sub.it] is appended additively in all systems. One equation in (18), (22), and (26), which is the budget share equation for fillets, is deleted to ensure nonsingularity of the error covariance matrix. As usual, the estimation should be independent of which equation is excluded.

Results of initial estimation revealed that the computed Durbin-Watson (DW) statistics were low while the computed approximate Lagrange multiplier (ALM) test statistics were high, suggesting significant positive serial correlation. We therefore introduce the first-order autoregressive scheme based on an order N parameterization of the autocovariance matrix using the full information maximum likelihood algorithm of Moschini and Moro (1994). (8)

Empirical Results and Their Interpretation Analysis of Measures of Fit

All models were estimated with adding up, homogeneity, and symmetry restrictions imposed. Consider first the nested tests of the general NCES against its nested specifications (the ISNCES and LIU). These tests have been done by using the chi-squared ([chi square]) based likelihood ratio test, and the results are summarized in figure 1. It happens that the LIU specification is heavily rejected in favor of NCES and ISNCES; that means the freeing up of [delta], [rho], and [[eta].sub.1] is desirable on statistical grounds. As can be seen, the computed [chi square] is 41.31, which far exceeds the critical values (5.99 and 7.82) of [chi square] for the 5% significance level. On the other hand, the implicit separability hypothesis ([delta] = 0) maintained by the ISNCES cannot be rejected by the data, indicating that the ISNCES is not statistically inferior to the NCES in which it is nested. From figure 1, we note that the computed [chi square] value (0.00) is obviously less than the critical [chi square] value (3.84) at the 5% level of significance, thereby suggesting that assuming implicit separability is appropriate in modeling Japanese consumer preferences. The preferred NCES specification is therefore based on ISNCES.

[FIGURE 1 OMITTED]

Comparative results for the NQM, QAIMDS, and ISNCES are presented in table 1. System measures of fit reported in this table include the system log-likelihood values (L), Schwartz Criterion (SC), Akaike's Information Criterion (AIC), and Hannan-Quinn Criterion (HQC). (9) Generally speaking, all three specifications fit the data reasonably well, given that estimation is in share form: the share equation [R.sup.2] values range from 37% for Fillet (implied by the ISNCES) to 98% for Shellfish (implied by the NQM). The [R.sup.2] values for the share equations of Fillet over all specifications are low relative to the other share equations. This may be caused by the failure to allow for imperfect adjustment to price and quantity changes as the shares of Fillet have reasonable high amounts of variation. The serial correlation properties of the error terms as shown in the DW and ALM test statistics are not severely pathological. (10) Especially the ALM test statistics show no evidence of remaining autocorrelation in the residuals since the test statistics for all the specifications are far less than the critical value of the 5% significance level (9.488).

An issue of importance here is whether the underlying preferences should be approximated by the NQM, QAIMDS, or ISNCES. Overall, the results provide mixed evidence of performance and fit. From table 1, we find that the QAIMDS dominates the other two systems on the basis of comparisons of likelihood values (L), though the ISNCES is slightly preferred to the other two systems in terms of SC and AIC. Of interest is that the NQM, while containing five (or twelve) more free parameters than the QAIMDS (or ISNCES), has the highest SC, AIC and HQC values. On prima facie grounds, it might be concluded that the NQM specification is not supported by the data, whereas the QAIMDS and ISNCES are preferred.

To obtain further insights into the relative performance of the NQM, QAIMDS, and ISNCES, Pollak and Wales' (1991) Likelihood Dominance Criterion (LDC) test is performed. The results of this test are summarized in table 2. When testing the ISNCES (the null model) against the QAIMDS (the alternative model), the test statistic is 10.428 to be compared with a 5% critical value of 4.375 (the upper limit of the critical value). Clearly there is a decisive outcome: the QAIMDS is preferred to the ISNCES. In all other cases, the LDC test statistics are less than the lower limit of the critical values, which means the models with fewer parameters (QAIMDS and ISNCES) are preferred to the model with more parameters (NQM). Consequently, the LDC comparisons suggest that QAIMDS is preferred to ISNCES and NQM, while ISNCES is preferred to NQM.

Analysis of Estimated Welfare Change

Following Kim (1997) in the context of inverse demands, the estimated mixed demand functions for group B commodities ([x.sub.B]) may be used to estimate welfare losses caused by forced reduction in predetermined quantities. Specifically, an exact measure of compensating variation (CV) associated with a change in [x.sub.B] from [x.sup.0.sub.B] to [x.sup.1.B] is given by

(30) CV = [C.sup.h.sub.A] ([p.sub.A], [x.sup.0.sub.B], [u.sup.0]) - [C.sup.h.sub.A] ([p.sub.A], [x.sup.1.sub.B], [u.sup.0]).

In integral form using the Fundamental Theorem of Calculus and Samuelson's Envelope Theorem, we have

(31) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

wherein the base utility [u.sup.0] is defined implicitly from c = [C.sup.h]([p.sub.A], [x.sup.0.sub.B], [u.sup.0]). Intuitively speaking, CV is the amount of additional expenditure required for consumers to reach the utility level [u.sup.0] while facing the quantity [x.sup.1.sub.B]. A positive (negative) value for CV indicates that consumers are worse (better) off while facing quantities [x.sup.1.sub.B].

In a similar manner, the equivalent variation (EV) for a change in quantity from [x.sup.0.sub.B] to [x.sup.1.sub.B] is defined as

(32) EV = [C.sup.h.sub.A]([p.sub.A], [x.sup.0.sub.B], [u.sup.1]) - [C.sup.h.sub.A]([p.sub.A], [x.sup.1.sub.B], [u.sup.1])

which can be defined equivalently as

(33) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [u.sup.1] is defined implicitly from c = [C.sup.h]([p.sub.A], [x.sup.1.sub.B], [u.sup.1]). In here, EV is the amount of additional expenditure that would enable consumers to maintain the new utility level [u.sup.1] while facing the initial quantities [x.sup.0.sub.B]. As for CV, a positive (negative) value for EV suggests that consumers are worse (better) off under [x.sup.1.sub.B] than under [x.sup.0.sub.B].

Equations (30) and (32) are used in conjunction with the estimated ISNCES satisfying the curvature conditions, although the QAIMDS performs marginally better, to obtain estimates of welfare changes (CV and EV). (11) We compute the consumer welfare loss associated with an arbitrary 10% restriction in the supply of fresh fish, fresh meat, and shellfish, which are reported in table 3. The following comments are in order. First, the estimated CV and EV indicate that Japanese consumers are made worse off after the reduction in their supply. For example, the CV for a 10% reduction in the supply of fresh meat is 4,591 yens per capita in 1985, and the resulting welfare loss that Japan families would experience is 6.7 % when measured as CV. (12) Second, on average the largest welfare loss (in %) associated with the supply reduction is for flesh fish rather than fresh meat markets. Third, the numerical differences between the CV and EV estimates are rather small, amounting to no more than 1% in all instances. Finally, the fluctuations over time in CV and EV estimates as a percentage of total expenditure (% CV and % EV) are observed mainly for the recent years. For example, % CV estimate associated with a 10% reduction in the supply of shellfish decreases from 0.95% in 1985 to 0.91% in 2003 whereas that of the others was not much changed over time.

Conclusions

This article demonstrates the feasibility of the conditional cost function approach to the specification and estimation of mixed demand systems. This approach allows both flexible and regular specifications of mixed demand systems, and in this context, overcomes some major problems associated with common flexible functional forms. It is shown that differentiation of a chosen conditional cost function with respect to prices and quantities yields the mixed demand system. While this system is explicit in the utility level, it may lack a closed-form representation in terms of the observable variables. As pointed out by McLaren, Powell, and Rossiter (2000) in the context of the cost function, this problem need not hinder estimation. A simple one-dimensional numerical inversion allows estimation of the parameters of any conditional cost function via the parameters of the implied Marshallian mixed demand functions.

The implementation of the proposed method relies on some simple and flexible functional forms to specify the NQM, QAIMDS, and NCES. These models were illustrated with an application to Japanese meat and fish consumption. Specifically, we allow for three goods (categories of fresh fish, fresh meat, and shellfish) to be represented by predetermined supply, with prices adjusting to clear the market, and for three goods (categories of processed fish and meat) to have standard representation (prices are given and quantities adjust). Results generally indicate all models fit the data well, but the LDC test favors the QAIMDS and ISNCES, with the QAIMDS doing the best. We further find that the hypothesis of implicit separability is not rejected by the data, suggesting that this separable structure is not a bad approximation for Japanese consumer preferences.

[Received August 2004; accepted July 2006.]

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(1) For example, if one were to specify the indirect utility function in terms of AIDS, no closed form dual function can consistently and simultaneously represent the direct utility function.

(2) Indeed, this was exactly the procedure followed by Moschini and Rizzi (2006) in deriving their Normalized Quadratic Model, whereby they first specified the conditional cost function, then derive the Hicksian mixed demand and total cost functions, and finally inverted the total cost function to give implied mixed utility function that was used to eliminate the unobservable u.

(3) In estimation, these conditions have to be maintained in order to rule out the possibility that c = [C.sup.h]([p.sub.A], [x.sub.B], u) has multiple roots.

(4) The Marshallian price, quantity and expenditure elasticities may be derived from the identities

(i) [X.sup.m.sub.Ai] ([p.sub.A], [x.sub.B], c) = [X.sup.h.sub.Ai] [[p.sub.A], [q.sub.B], [U.sup.m]([p.sub.A], [x.sub.B], c)], and

(ii) [P.sup.m.sub.Bj] ([p.sub.A], [x.sub.B], c) = [P.sup.h.sub.Bj] [p.sub.A], [x.sub.B], [U.sup.m] ([p.sub.A], [q.sub.B], c)].

For instance, differentiating (i) with respect to the i'th price yields the following equation:

(iii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and the term [partial derivative] [U.sup.m] /[partial derivative] [p'.sub.Ai] can be rewritten as -([partial derivative] [C.sup.h] / [partial derivative] [p'.sub.Ai]) / [partial derivative] [C.sup.h] / [partial derivative] u), which is obtained by differentiating the identity c = [C.sup.h] [[p.sub.A], [x.sub.B], [U.sup.m] ([p.sub.A], [x.sub.B], c)] with respect to [p'.sub.Ai]. Multiplying through (iii) by [p'.sub.Ai] / [x.sub.Ai] and rearranging gives:

(iv) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which is the Marshallian cross price elasticity equation.

(5) In Moschini and Rizzi (2006), the second terms of F([p.sub.A], [x.sub.B]) and G([p.sub.A], [x.sub.B]) are written as ([[summation]'.sub.i], [[alpha]'.sub.i] [p'.sub.Ai]). [[summation]'.sub.j], [[mu]'.sub.j] [x'.sub.Bj]) and [[summation]'.sub.i], [[alpha]'.sub.i] [p'.sub.Ai]). [[summation]'.sub.j], [[phi]'.sub.j] [x'.sub.Bj]), respectively, where [[phi].sub.j] ([not equal to][[mu].sub.j]) are parameters.

(6) According to Blackorby, Davidson, and Schworm (1991), the direct utility function U([x.sub.A], [x.sub.B]) is implicitly separable with respect to Partition [??] if and only if there exists an implicit representation of U as [T.sub.A]([x.sub.A], u) = [T.sub.B]([x.sub.B], u), which implies that the corresponding conditional cost function can be written in the form

(v) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Clearly an implicitly separable mixed demand functions depend directly on u through their utility argument, but they also depend indirectly on u through the [T.sub.B] functions in their quantity index argument. Furthermore, this structure is identical to Deaton and Muellbauer's (1980b) implicitly separable preference structure when goods are patitioned into S groups with price subvectors ([p.sup.1.sub.7], ..., [p.sup.S]).

(7) These empirical mixed demand systems were constructed under the condition that the fish and meat product groups are directly and weakly separable from other commodities. This separability assumption needs to be held with an aim of keeping the estimation process manageable by merely dealing with certain aspects of the static demand model.

(8) See Holt (1998) for alternative autocorrelation parameterizations.

(9) For reasons of brevity, the detailed parameter estimates of the NQM, QAIMDS, and ISNCES are not reported below but are available on line as Readers' Appendix A at http://au.geocities. com/garywong21/parameter.pdf.

(10) This article adopts Fisher, Fleissig, and Serletis' (2001) approach to calculate the approximate LM test statistics (equivalent to score test statistics in the case of linear models) for autocorrelation. As pointed out by a reviewer, the models being examined here are non-linear in the variables, and thus the adopted approach used to construct the LM test statistics may not be fully appropriate (see Eitrheim and Terasvirta 1996). Future research on deriving and estimating mixed demand systems may focus on this issue, but such an investigation goes well beyond the scope of this study.

(11) The curvature conditions of the mixed demand models were checked by calculating the eigenvalues of the matrices of compensated price and quantity effects. Results indicate that the NQM and QAIMDS fail to satisfy the curvature conditions for some observations in the sample period, while the ISNCES satisfies the curvature properties over the whole sample period. This leads to the conclusion that the ISNCES is regular and therefore may be used to compute the exact welfare measures. Though constrained estimation (as has been done by Moschini [1998]) could be a simple option for NQM and QAIMDS, it is of course possible that the regularity problem may be due to the level of aggregation of the data, and more investigation along these lines would be justified in searching for a regular yet flexible representation of the data.

(12) Similar interpretations apply for fresh fish and shellfish.

K. K. Gary Wong is assistant professor, Department of Economics, The University of Macau, Macau SAR, China. Hoanjae Park is associate professor, Department of Economics, Catholic University of Daegu, Daegu, South Korea. The authors thank two anonymous reviewers for helpful comments. Any remaining errors and omissions are the sole responsibility of the authors. Table 1. Single Equation and System Measures of Fit

NQM QAIMDS ISNCES No. of free parameters 24 17 12 [R.sup.2] Salted fish 0.949 0.954 0.960 Processed meat 0.946 0.947 0.943 Fillet 0.444 0.385 0.367 Fresh meat 0.707 0.741 0.700 Fresh fish 0.858 0.896 0.864 Shellfish 0.982 0.978 0.980 L 1,631.320 1,641.731 1,631.303 SC -40.445 -40.795 -40.874 AIC -41.204 -41.333 -41.349 HQC -41.386 -41.462 -41.462

Residual diagnostics Durbin-Watson statistics Salted fish 2.528 2.359 2.692 Processed meat 2.849 2.865 2.735 Fillet 1.910 2.567 2.287 Fresh meat 2.619 2.417 2.639 Fresh fish 2.633 2.391 2.664 Shellfish 2.191 2.253 2.328 ALM test statistics for autocorrelation ([[chi square].sub.4,0.05] = 9.488) Salted fish 2.368 2.399 1.686 Processed meat 2.954 3.282 2.540 Fillet 3.025 1.999 4.047 Fresh meat 4.682 4.100 4.958 Fresh fish 6.058 3.457 6.470 Shellfish 2.438 1.450 3.157 Table 2. Summary Statistics for Nonnested Comparisons Comparison Test LDC Critical Values

Statistic (5% Significance Level) ISNCES--null model (rejected) V.S. 10.428 (3.254, 4.375) QAIMDS--alternative model QAIMDS--null model V.S. -10.411 (4.392, 5.833) NQM--alternative model (rejected) ISNCES--null model V.S. 0.017 (7.645, 9.260) NQM--alternative model (rejected) Table 3. Compensating and Equivalent Variations for a 10% Reduction in Supply of Fresh Meat, Fresh Fish, and Shellfish (Yens for Annual) Fish Category CV (Yens) %CV EV (Yens) %EV 1985 Fresh meat 4,591.148 6.7% 4,387.749 6.4% Fresh fish 7,918.718 10.1% 7,350.310 9.4% Shellfish 750.633 0.96% 745.162 0.95% 1994 Fresh meat 4,212.659 6.1% 4,029.895 5.8% Fresh fish 7,598.123 10.7% 7,052.650 10.0% Shellfish 678.717 0.96% 673.917 0.96% 2003 Fresh meat 3,330.433 6.1% 3,187.869 5.9% Fresh fish 5,952.340 10.2% 5,532.382 9.5% Shellfish 544.480 0.93% 540.671 0.91% Average Fresh meat 4,147.784 6.2% 3,967.787 5.9% Fresh fish 7,403.172 10.5% 6,874.234 9.7% Shellfish 690.548 1.1% 685.581 1% Note: The column titled % CV denotes compensating variation as a percent of total expenditure on meat and fish. while the column headed %EV is similarly defined for equivalent variation.