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