# A Demographically Modified Demand Model: Behavioral and Welfare Properties

## Abstract

This chapter describes the specification of the Almost Ideal exactly aggregable demand system derived from a cost function demographically modified using Lewbel’s (1985) technique. The original demand system is transformed by both a scaling and translating term. The most common demand systems used in applied work are the Almost Ideal Demand System (AIDS) of Deaton and Muellbauer (1980a,b), and the exactly aggregable Translog of Christensen, Jorgenson, and Lau (1975) along with their quadratic extensions (Banks, Blundell, and Lewbel 1997) and Ryan and Wales (1999). These different specifications have similar fundamental properties. Both systems are flexible in prices and income and have similar aggregation properties. The indirect utility function representing AIDS preferences is fractional and yields, via Roy’s identity, linear demand systems. The opposite is the case for Translog preferences.

## Keywords

Intermediate Good Demand System Budget Share Indirect Utility Function Engel Curve## Preview

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## Notes

- 1.The function
*f(y)*can otherwise be equal to zero, y^{k}, or tan(kln(y)) for some non-zero constant*k*. Google Scholar - 2.Interesting empirical comparisons of flexible functional forms applied to demand analysis are the studies by Castagnini, Cox, and Perali (1999), Cooper and McLaren (1996), Lewbel (1989), Lancaster and Ray (1998), Ryan and Wales (1999).Google Scholar
- 3.Lewbel’s Theorem 8 (1985) characterizes the class of modifying functions yielding affine transformations of budget shares. In our context, we confine our attention to special cases of interest belonging to the same class.Google Scholar
- 4.The choice of the functional form of the demographic functions is not restricted to any particular form. The researcher may specify a more flexible form, such as a Translog, if, for example, the interest is in modeling household economies of scale.Google Scholar
- 6.The work by Bollino, Perali, and Rossi (2000) is an example of Gorman preferences modified by an additive translation term and expressed in quantity space where no parametric restrictions are required for identification given the data set used by the authors.Google Scholar
- 7.Cooper and McLaren (1996) have proposed a globally regular demand system using Modified PIGLOG preferences. Lewbel (1995b) provides sufficient conditions for global regularity of fractional demand systems of both rank two and rank three.Google Scholar
- 8.The Annex presents an application of Lewbel’s (1991c) rank test and an illustration of the use of non parametric and semiparametric tools to construct evidence for a sound specification analysis.Google Scholar
- 9.The virtual price is the price that exactly supports a zero realization of expenditure.Google Scholar
- 10.The demand systems belonging to this class can enhance the degree of flexibility in prices by generalizing the concavity properties of the Translog price aggregator ln
*A(p,d)*as shown in Castagnini, Cox, and Perali (1999).Google Scholar - 11.Note that an index function may depend on individual expenditures. Admissible index functions are additive in functions of individual expenditures and demographics. For example, the Gini coefficient is not properbecause not additive in individual expenditures.Google Scholar
- 12.Assumptions about the functional form of
*S(d)*are*implicit*if any cost function is first modified, then the*IB*restrictions are imposed to separate the preferences of a reference household. On the other hand, if the functional form for a reference household is first chosen, and, in a second step, a functional form is specified for the equivalence scale, then it is exact by construction if assumptions about the*S(d)*function are also*explicitly*made.Google Scholar - 13.The structure of the
*IB*parametric restrictions is conditional on the choice of the demographic modifying technique chosen. For example, Blundell and Lewbel (1991), Pashardes (1995) or Lyssiotou (1997) estimate demand systems using a demographic parameterization of the intercept and income parameters. The form of the restrictions used by these authors differ from the ones provided here that are derived from a Barten-Gorman demographic transformation. Comparisons across outcomes of the*IB-ESE*test should acknowledge these different experimental conditions.Google Scholar - 14.On this point, the interested reader may see Blackorby and Donaldson (1993b) and Lewbel (1993c).Google Scholar