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

(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.]

References

Barren, A.P. 1992. "Estimation of Mixed Demand Systems." In R. Bewley and V.H. Tran, eds. Contributions to Consumer Demand and Econometrics: Essays in Honour of Henri Theil. London: MacMillian Academic and Professional Ltd. pp. 31-57.

Beach, R.H., and M.T. Holt. 2001. "Incorporating Quadratic Scale Curves in Inverse Demand Systems." American Journal of Agricultural Economics 83:230-45.

Blackorby, C., R. Davidson, and W. Schworm. 1991. "Implicit Separability: Characterisation and Implications for Consumer Demands." Journal Economic Theory 55:364-99.

Chavas, J.-P. 1984. "The Theory of Mixed Demand Functions." European Economic Review 24:321-44.

Christensen, L.R., D.W. Jorgenson, and L.J. Lau. 1975. "Transcendental Logrithmic Utility Functions." American Economic Review 70:422-32.

Deaton, A.S., and J. Muellbauer. 1980a. "An Almost Ideal Demand System." American Economic Review 70:312-26.

--. 1980b. Economics and Consumer Behaviour. Cambridge: Cambridge University Press.

Eales, J., and L.J. Unnevehr. 1994. "The Inverse Almost Ideal Demand System." European Economic Review 38:101-15.

Eirheim, O, and T. Terasvirta. 1996. "Testing the Adequacy of Smooth Transition Autoregressive Models." Journal of Econometrics 74:59-75.

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

Holt, M.T. 1998. "Autocorrelation Specification in Singular Equation Systems: A Further Look." Economics Letters 58:135-41.

--. 2002. "Inverse Demand Systems and Choice of Functional Form." European Economic Review 46:117-42.

Kim, H.Y. 1997. "Inverse Demand Systems and Welfare Measurement in Quantity Space." Southern Economic Journal 63:663-79.

Lewbel, A. 1991. "The Rank of Demand Systems: Theory and Non-Parametric Estimation." Econometrica 59:711-30.

McLaren, K.R., P. Rossiter, and A.A. Powell. 2000. "Using the Cost Function to Generate Flexible Marshallian Demand Systems." Empirical Economics 25:209-27.

Matsuda, T. 2004. "Incorporating Generalized Marginal Budget Shares in a Mixed Demand System." American Journal of Agricultural Economics 86:1117-26.

Michelini, C. 1999. "The Estimation of a Rank 3 Demand System with Demographic Demand Shifters from Quasi-Unit Record Data of Household Consumption." Economics Letters 65:17-24.

Moschini, G. 1998. "The Semi-Flexible Almost Ideal Demand System." European Economic Review 42:349-64.

Moschini, G., and D. Moro. 1994. "Autocorrelation Specification in Singular Equation Systems." Economics Letters 46:303-09.

Moschini, G., and A. Vissa. 1993. "Flexible Specification of Mixed Demand Systems." American Journal of Agricultural Economics 75:1-9.

Moschini, G., and P.L. Rizzi. 2006. "Coherent Specification of a Mixed Demand System: The Stone-Geary Model." In M.T. Holt and J.-P. Chavas, eds. Exploring Frontier in Applied Economics: Essays in Honours of Stanley R. Johnson. Berkeley CA: Berkeley Electronic Press.

Neary, J.P., and K.W.S. Roberts. 1980. "The Theory of Household Behavior under Rationing." European Economic Review 13:25-42.

Perroni, C., and T.F. Rutherford. 1995. "Regular Flexibility of the Nested CES Function." European Economic Review 39:335-43.

Pollak, R.A., and T.J. Wales. 1991. "The Likelihood Dominance Criterion: A New Approach to Model Selection." Journal of Econometrics 47:227-42.

Samuelson, P.A. 1965. "Using Full Duality to Show that Simultaneously Additive Direct and Indirect Utilities Implies Unitary Price Elasticity of Demand." Econometrica 33:781-96.

(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]