Elsevier

Economics Letters

Volume 90, Issue 3, March 2006, Pages 433-439
Economics Letters

Logarithmic quasi-homothetic preferences

https://doi.org/10.1016/j.econlet.2005.10.005Get rights and content

Abstract

We study a class of quasi-homothetic preferences, which result in demands that are logarithmic in own prices when these have a negligible impact on aggregate prices (as in monopolistic competition models). Thus marginal revenues are computationally friendly and well behaved.

Introduction

Dixit and Stiglitz (1977, Sections 1 and 3) popularized the use of constant-elasticity-of-substitution (CES) homothethic preferences to model the Chamberlinian “large group” monopolistic competition. However, they also showed that more general additive forms (allowing more general commodity substitutability) highlight different results: see e.g. Dixit and Stiglitz (1977, Section 2) and Krugman (1979). In this paper we consider a class of non-homothetic preferences that explicitly exhibit variable demand elasticities. Preferences are symmetric, and no commodity plays a special role independently from prices and income. In fact, the expenditure function (the indirect utility function) only depends on a price aggregate and a price dispersion index. Preferences are indeed additive, which is both useful, because this implies a parsimonious parameterisation, and restrictive, because it rules out inferior goods and net complements. However, it turns out that commodities can be either gross substitutes or complements, according to the size of consumption (this can be “controlled” using a single parameter). In addition, while the expenditure shares depend on income, as is economically reasonable, the Engel curves are linear, which is formally convenient (i.e., preferences are quasi-homothetic: see e.g. Deaton and Muellbauer, 1980, Section 5.4). Finally, once a large group of commodities is considered (a parameter accounts for any number of them), the uncompensated demands just depend on the logarithm of the own price, which is computationally friendly (marginal revenues are decreasing whenever demands are elastic).

Section snippets

The negative exponential utility function

Suppose that consumer preferences over a number N of commodities can be represented by the following “negative exponential”1 utility function:U(x)=1αh=1Neαxh,where xh  0 indicates the consumption of commodity h = 1, N and α is a positive parameter.

The case of two commodities

To fully grasp the implications of (1), consider the case of only two goods. In any interior solution the Marshallian demand of commodity i reduces to (i  j, i,j = 1,2):xi(p1,p2,y)=αypjlnpipjα(p1+p2).Note that the right-hand side of (5) is always decreasing and convex with respect to pi whenever the left-hand side is positive. Also notice that the choke-off price i(pj, y, α) is given by:pi¯(pj,y,α)=pje(αy)/pj(note that i increases with respect to pj if and only if αy/pj < 1).

There might also

Conclusion

In this paper we have studied the case of symmetric, logarithmic quasi-homothetic preferences. They can account for any number of goods and generate demand curves that are more general than the commonly used ones, which come from CES preferences. In particular, they have demand elasticities (both with respect to income and prices) which are not constant, and exhibit finite choke-off prices. Nevertheless, by exploiting the properties of additive preferences, we can argue that when the number of

Acknowledgements

I am grateful to Guido Ascari and Antonio Lijoi for useful discussions, and especially to Giulia Felice for detailed comments. Any errors are mine.

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