A Bayesian approach to aggregation in demand systems: smoothing with perfect substitution priors

Abstract

Demand analysis requires aggregation of commodities. Some are imposed at the data collection level, leaving some for the estimation level. When data are collected, the implicit assumption underlying the aggregation is perfect substitutability: one gallon of gasoline is viewed by consumers as equivalent to another gallon; hence, the two are added together. While such aggregation can be carried out further by the data analyst, it is difficult to incorporate perfect substitutability into the estimation of direct demand systems. Perfect substitution in that context implies discontinuous demand functions, which are not nested within standard empirical demand systems. Perfect substitution is much more easily handled in a system of inverse demands, though an empirical method to impose perfect substitutability in an inverse demand system has not previously appeared in the literature. In this article, we develop such a method, which allows perfect substitutability to be imposed as a prior restriction. We use Leamer’s information contract curve as a tool to flexibly impose the substitution restriction and to investigate consistency between the data and prior. We illustrate the method with an application to inverse demands for fish in Korea.

Keywords:

• substitutability
• marginal values
• inverse demands
• information contract curve

JEL Classification:

• Q22
• D12

View correction statement:

Erratum

Notes

 To our knowledge, four applications of the information contract curve have been published: Burkett and Skegro (1989); Burkett (1992); Rucker et al. (1995); and Thurman et al. (2001). The first two studies apply the information contract curve with a normal prior distribution. The third and fourth studies apply conically uniform priors to analysis of the US tobacco quota system and to environmental regulation of the pulp, paper and paperboard industry. Bayesian methods have previously been applied to the estimation of direct demand systems. See Chalfant et al. (1991) who use the methods of Geweke (1986) to impose inequality restrictions, and Boatwright et al. (1999), who impose smoothness on estimates of sales response to promotion.

 The inverse of the FOC is called the Hotelling-Wold identity. See Anderson (1980) and Weymark (1980).

 One can derive the relation between the Antonelli matrix (A) and quantity elasticities:

where f_ij* denotes compensated quantity elasticities.

 In order to derive this relationship, let b_ij be the Marshallian estimates in our model:

(4.1)

Since the Antonelli equation (analogous to the Slutsky equation) is in terms of elasticity,

(4.2)

where f_i denotes the scale flexibility of good i and the asterisk implies Hicksian, we then have that

(4.3)

and

(4.4)

We combine (4.3) with (4.4) to derive the Marshallian quantity effect or the Marshallian inverse substitution matrix of (4.1).

 The foundation for work on separability was laid by Leontief (1947). Moschini et al. (1994) derived necessary and sufficient conditions for weak separability in ordinary demand functions in terms of the Allen-Uzawa elasticities of substitution. Kim (1997) derived restrictions for homothetic weak separability in inverse demands in terms of the Antonelli elasticities of complementarity, which are a measure of substitutability in terms of quantities. Moro and Sckokai (2002) derived restrictions from non-homothetic weak separability in inverse demands in terms of the Antonelli elasticities of complementarity.

 See Poirier (1995) on concentrating σ² out of the likelihood function.

 As the angle between β in parameter space and the 45-degree line approaches 0, the cosine labelling function approaches 1. As the angle between β and the 45 degree line approaches 90 degree, the cosine approaches 0.

 We categorized data from eight collected fish products and aggregated into five fish categories based on their use in Korean cuisine and culture. Normally the fifth category in Table 1 is served as raw fish for dinner while the other categories are served as cooked fishes for dinner and breakfast. For example, Hairtails are boiled down into soup for dinner, but sometimes are fried.