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

(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];