A Flexible Multistage Demand System Based on Indirect Separability.

• Author: Moschini, GianCarlo

GianCarlo Moschini [*]

The notion of indirect separability is exploited to derive a new multistage demand system. The model allows a consistent parameterization of demand relations at various budgeting stages and it fulfills the requirement of flexibility while satisfying separability globally. Two propositions are derived to characterize flexible and separable functional forms, which lead to the specification of a flexible and separable translog (FAST) demand system. The model is particularly attractive for modeling large complete demand systems and is illustrated with an application to Canadian food demand.

1. Introduction

   To be useful for most policy analysis applications, demand systems need to be specified in terms of disaggregated commodities based on relevant conditioning variables, that is, usually one needs to specify a large complete demand system. If at the same time one wants to use parametric specifications that are not too constraining, such as standard flexible functional forms (FFF), then the data requirement may be prohibitive. A workable solution of this problem entails imposing restrictions on the problem solved by consumers, typically by assuming a separable structure for consumer preferences (Blackorby, Primont, and Russell 1978) [1] In particular, certain separability conditions allow the consumer's expenditure allocation problem to satisfy two-stage (multistage) budgeting rules. As shown by Gorman (1959), a simplified two-stage budgeting is possible under two alternative conditions: homothetic weak separability of the direct utility function, or strong separability (block additivity) of the direct utility function with group sub-utility functions the dual of which have the so-called generalized Gorman polar form (GGPF). Such perfect price aggregation conditions underlie a number of multistage complete demand systems. Models relying on strong separability cum GGPF include Brown and Heien (1972), Blackorby, Boyce, and Russell (1978), Anderson (1979), and Yen and Roe (1989). Homothetic weak separability was used by Jorgenson, Slesnick, and Stoker (1997). [2]

   Because the conditions for perfect price aggregation are often deemed too restrictive in empirical applications, attempts have been made to model demand based on the hypothesis of direct weak separability only. Although direct weak separability (DWS) per se is neither necessary nor sufficient for standard two-stage budgeting, it does provide the necessary and sufficient conditions for the existence of conditional (second-stage) demand functions defined only on group prices and group expenditure allocations (Pollak 1971). Because such conditional demand functions typically depend on a small set of variables (data on which is easily found for most applications), some empirical studies have pursued the estimation of second-stage demand functions in isolation (say, demand for food items as function of food prices and expenditure allocated to food). Examples include Barr and Cuthbertson (1994), Gao, Wailes, and Cramer (1997), Kinnucan et al. (1997), and Spencer (1997). But such a widespread approach is highly que stionable because the conditional demand parameters thus estimated are rarely of interest for policy analysis (Hanemann and Morey 1992). In certain instances, a conditional analysis can provide useful information (Browning and Meghir 1991). But in general, the economic question at hand requires one to recover unconditional demands and, under DWS, that necessitates estimation of both first-stage and second-stage expenditure allocation functions.

   If one insists on not weakening the assumption of DWS, then estimation of a complete demand system requires a consistent parametric specification for the two budgeting stages. But whereas one can easily derive second-stage demand functions by Roy's identity from a specification of the (separable) group indirect subutility function, derivation of first-stage expenditure allocation functions requires an explicit solution of the conditional utility maximization problem (Blackorby, Primont, and Russell 1978, chapter 5). It follows that an internally consistent parametric specification of the two budgeting stages is difficult when relying only on DWS (especially if one wants to satisfy a notion of flexibility). If one is willing to accept an approximate solution to the first-stage income allocation under direct weak separability, a useful approach is outlined by Gorman (1995). But implementations of such an approximation are somewhat unsatisfactory. [3]

   This article explores an alternative route to specifying a two-stage flexible demand system by assuming that it is the indirect utility function (rather than the direct utility function) that is weakly separable. This study makes two contributions. First, it shows how indirect weak separability (IWS) can be used to specify a complete demand system that is amenable to a recursive structure typically associated with multistage budgeting. In particular, IWS still permits a meaningful definition of conditional (second-stage) demand functions. Of course, first-stage income allocations functions here cannot be specified in terms of only one price index per group (the same consideration holds for DWS) but will instead depend on all prices and total expenditure. But unlike DWS, in this setting, it is straightforward to specify simultaneously first-stage and second-stage demand functions because, under IWS, both can be obtained by Roy's identity from a well-specified (and separable) indirect utility function. Furthermore, this specification strategy ensures that the parametric structur e of the resulting system of unconditional demand functions is internally consistent. It should also be emphasized that the recursive demand system permitted by IWS does not require the restrictive assumptions of perfect price aggregation. Specifically, it does not require strong separability (additivity with respect to the subutility functions) nor does it require homotheticity of the separable subutility functions. In this sense, it is a genuine generalization of existing multistage demand systems.

   The second main contribution of this article concerns the parametric specification of the separable demand system. It is known that maintaining or testing separability with commonly used FFF entails difficulties. Blackorby, Primont, and Russell (1977) showed that separability conditions on FFF introduce unwanted restrictions that typically destroy the flexibility of the function. The standard way to deal with this issue has been to require separability to hold only at a point (Jorgenson and Lau 1975; Denny and Fuss 1977). [4] The validity of this procedure has originated considerable debate, especially when the focus is on testing the separability assumption (Aizcorbe 1992). This article proposes an appealing alternative by deriving a general procedure to specify flexible and separable functions. This procedure, applied to an indirectly separable utility function, leads to a demand model that satisfies the standard definitions of flexibility and satisfies the postulated separable structure globally. The flex ible and separable translog (FAST) demand system thus derived is illustrated with an application to a complete demand system for Canada that emphasizes food consumption.

2. Why Indirect Separability

   The attractive features of IWS for the purpose of specifying a complete demand system are best illustrated by comparison with the more common assumption of DWS. To recall briefly these separability notions, let q [equivalent] [[q.sub.1], [q.sub.2], ..., [q.sub.n]] denote the vector of goods demanded by the consumer and p [equivalent] [[p.sub.1], [p.sub.2], ..., [p.sub.n]] the corresponding vector of all nominal prices. Let U(q) represent the direct utility function, which is assumed continuous, nondecreasing and strictly quasiconcave. If I = {1, 2, ..., n} denotes the set of indices of the n goods, order these goods in N [less than] n separable groups defined by the mutually exclusive and exhaustive partition I = {I [I.sup.1], [I.sup.2], ..., [I.sup.N]} of the set I. Then (symmetric) DWS holds if U(q) can be written as

   U(q) = [U.sup.0]([U.sup.1]([q.sup.1]), [U.sup.2]([q.sup.2]),..., [U.sup.N]([q.sup.N])).

   This structure on the utility function is sufficient to guarantee the existence of conditional demand functions [c.sub.i]([p.sup.r]/[y.sub.r]) [forall] i [epsilon] [I.sup.r] (r = 1, ..., N), where [p.sup.r] is the vector of prices in the partition [I.sup.r] (such that p = [[p.sup.1], [p.sup.2],..., [p.sup.N]]) and [Y.sub.r] is the optimal allocation of expenditure to the goods in the rth group. These conditional demand functions can be derived by Roy's identity from the indirect subutility function that is dual to U([q.sup.r]), say [V.sup.r]([p.sup.r]/[y.sub.r]). Because they only require data on the particular separable group, the second-stage demand functions [c.sub.i]([p.sup.r]/[y.sub.r]) are often estimated in applied work.

   It is clear that such conditional demand functions cannot provide the parameters (i.e., elasticities) that are typically of interest for policy questions. This is because the optimal allocation of expenditure to the goods in any one partition depends on all prices and total expenditure. That is, the optimal solution to the first-stage expenditure allocation problem has the functional representation [y.sub.r](P/y), r = 1, 2, ..., N, where y is total expenditures. [5] Hence, the consumer's unconditional demand functions [q.sub.i](p/y) satisfy [q.sub.i](p/y) [equivalent] [c.sub.i]([p.sup.r]/[y.sub.r](p/y)). To say anything meaningful about a consumer's response to change in any one price, say, one needs to know how the optimal allocations [y.sub.r](p/y) are affected by this change, that is, one needs to estimate the first-stage expenditure allocation functions as well. Given a standard FFF representation for the groups' indirect utility...