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

[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