Deriving a flexible mixed demand system: the normalized quadratic model.

A major thrust of the system-wide approach to empirical demand analysis has been to specify models that are integrable into well-behaved preferences. The restrictions of consumer theory help, at the estimation stage, to reduce the number of parameters to be estimated and thus increase efficiency, but adherence to theory also permits meaningful use of the resulting estimates for welfare and policy analysis. The earlier work of Stone (1954), leading to the linear expenditure system (LES), was subsequently extended by models that capture more general and flexible representations of preferences, such as the translog model of Christensen, Jorgenson, and Lau (1975), the differential (Rotterdam) model of Theil (1975), the almost ideal (AI) demand system of Deaton and Muellbauer (1980), the quadratic AI demand system (Banks, Blundell, and Lewbel 1997), and the semiflexible AI demand system (Moschini 1998).

A common feature of these specifications is to represent quantity demanded as a function of market prices and total expenditure (income for short). Whereas this approach corresponds directly to the usual formulation of the individual consumer problem, its use within an econometric model requires some additional identifying assumptions--essentially, what is to be assumed as exogenous or predetermined. The standard specification of demand models with quantities as the dependent variable assumes that prices are predetermined. That is, if one thinks of the data at hand as the outcome of a market equilibrium model, the implicit assumption is that supply functions are perfectly elastic so that demands adjust to clear the market. While this condition may hold for market data in some situations (e.g., tradeable goods for a small open economy), it obviously is not universal. Indeed, Geary (1948-1949), in his derivation of the utility function underlying the LES, noted that "From the regression viewpoint, however, it would be equally logical to regard prices as dependent variables and quantities as independent variables ..." This view has been occasionally implemented in terms of inverse demand systems, as in the (inverse) translog (Christensen, Jorgenson, and Lau 1975), the (inverse) Rotterdam model (Theil 1975, 1976; Barten and Bettendorf 1989), the linear inverse demand system (Moschini and Vissa 1992; Eales and Unnevehr 1994), and the inverse normalized quadratic (NQ) (Holt and Bishop 2002).

The choice of which variables to assume as predetermined in empirical demand models has nontrivial implications. To illustrate, if a direct demand system is specified when in fact an inverse demand specification is called for, then the duality between direct and inverse demand systems implied by consumer theory means that, for nontrivial preferences, the direct demand system will be affected by a nonlinear errors-in-variables problem. For such a case it is notoriously difficult to obtain estimators that are consistent. In fact, instrumental variable estimators are also inconsistent (Amemiya 1985), and their application in standard system estimation is bound to give inconsistent estimates of the underlying preference parameters (Moschini 2001). Unlike other instances of errors-in-variables in demand models (e.g., Lewbel 1996), however, the question in our setting is not about a suitable estimation technique. Rather, the question is one of choosing the appropriate assumption on what to take as predetermined in order to identify the underlying preference parameters. Given that, a mixed demand system approach provides an appealing framework of analysis.

In mixed demand functions, first analyzed by Samuelson (1965), the prices of some goods and the quantity of all the others are predetermined, so that some quantities and some prices adjust to clear the market. This class of models has obvious econometric appeal for the purpose of estimating demand behavior, because it encompasses a spectrum of possibilities between the polar cases of direct and inverse demand functions. Hence, mixed demands allow for a much richer set of options about what is to be assumed as exogenous or predetermined, which permits the identifying assumptions of demand models to be tailored to the nature of the data at hand. Despite this attractive attribute, mixed demand functions have received comparatively little attention in applied studies. Moschini and Rizzi (2006) derive and estimate a mixed demand system for the special case of Stone-Geary preferences. More general representations of mixed demand systems have essentially been confined to the Rotterdam specification. Barten (1992) appealed to mixed demand to illustrate the choice of which variables to assume as exogenous but actually estimated a standard Rotterdam model, while taking into account the endogeneity of some of the prices in formulating the likelihood function. Moschini and Vissa (1993) and Brown and Lee (2006), by contrast, formulated and estimated true differential mixed demand systems. (1)

A possible explanation for the paucity of applications of mixed demand systems resides in the specific difficulties that arise in this context. That is, in Samuelson's (1965) formulation, knowledge of both direct and indirect utility functions is required to characterize the demand properties. This means that many commonly used flexible functional forms--such as the translog indirect utility function, or the PIGLOG cost function of AI systems--cannot be used to specify a mixed demand system because these flexible functional forms do not have a closed-form dual representation. That is why the only flexible true mixed demand system that has been proposed to date relies on approximating the mixed demand equations directly through a differential approach. Even though such a Rotterdam-type mixed demand system is of considerable interest (e.g., Matsuda 2004), for some applications (such as welfare analysis) it may be desirable to have an exact parametric representation of preferences.

With this article we hope to advance the applicability of mixed demand systems in an empirical setting by making three main contributions. First, we briefly review the theoretical framework of mixed demands that makes it explicit why the procedure used in numerous applications of direct and inverse demand systems is not particularly useful. Second, we suggest a new approach to specifying a mixed demand system, based on the restricted expenditure function used in the related area of rationed demand (Gorman 1976; Neary and Roberts 1980). Mixed demands and rationed demands share important similarities, but a major difference is that for the latter some markets do not clear. In the case of mixed demands, on the other hand, the virtual prices of the quantity-predetermined goods do enter the budget constraint, so that it is in principle possible to solve for the mixed utility function implied by the restricted expenditure function and thus derive integrable mixed demand equations. We identify a class of restricted cost functions for which an explicit solution of the mixed demand equations is possible. Third, to make the approach operational, we develop a new NQ parameterization that is locally flexible, and that can satisfy homogeneity, symmetry, and curvature properties. The model is illustrated with an application to vegetable demand in Italy.

Mixed Demands

In the mixed demand setting, we consider consumers are price takers for all goods, but at the market level the prices of only a subset of goods is predetermined, whereas for the remaining goods it is the aggregate quantities that are predetermined. For an explicit definition of mixed demands, first introduced by Samuelson (1965) and analyzed by Chavas (1984), partition the consumption bundle into (m + n) goods. Let x [equivalent to] [[x.sub.1], [x.sub.2],..., [x.sub.n]] denote the vector of commodities chosen optimally and let z [equivalent to] [z.sub.l], [z.sub.2],..., [z.sub.m]] denote the vector of commodities in fixed quantity whose prices are optimally determined. Correspondingly, [p.sub.i] denotes the nominal price of [x.sub.i], whereas [q.sub.k] denotes the nominal price of [z.sub.k]. Total consumer expenditure (income, for short) is y. Mixed demands are then derived from the constrained optimization problem (Samuelson 1965, p. 791):

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where U(*) and V(*) are the direct and indirect utility functions, respectively, which are assumed quasi-concave and quasi-convex in their respective arguments, as well as satisfying standard monotonicity properties. (2) The optimality conditions for an interior solution of problem (1) are:

(2) [partial derivative]U ([x.sup.*]) / [partial derivative][x.sub.i] - [lambda][p.sub.i] = 0, = i = 1,..., n

(3) - [partial derivative] V ([p, [q.sup.*], y) / [partial derivative][q.sub.k] - [lambda][z.sub.k] = 0, = k = 1,..., m

(4) p * [x.sup.*] + [q.sup.*] * [z = y.

The solutions to (2)-(4) give the Marshallian mixed demand vectors [x.sup.*] = x(p, z, y) and [q.sup.*] = q(p, z, y). Clearly, at the optimum, U([x.sup.*], z) = V(p, [q.sup.*], y) [equivalent] [V.sup.M](p, Z, y), where [V.sup.M] (*) is the mixed utility function. The mixed demand functions x(p, z, y) and q(p, z, y) satisfy Walras's law (the adding-up condition). Moreover, the functions x(p, z, y) and q (p, z, y) are homogeneous of degree zero and degree one in (p, y), respectively [and thus the mixed utility function is homogeneous of degree zero in (p, y)]. The symmetry property applies to compensated mixed demand functions, which are the same as the compensated demands under rationing (Chavas 1984) and may be characterized in terms of the restricted cost function (Gorman 1976)