Logarithmic quasi-homothetic preferences
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: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):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 p¯i(pj, y, α) is given by:(note that p¯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.
References (8)
- et al.
Linear-homothetic preferences
Economics Letters
(2000) Increasing returns, monopolistic competition, and international trade
Journal of International Economics
(1979)Economic integration effects on market structure
- et al.
Lectures on Macroeconomics
(1989)
Cited by (8)
Price-cost margins and firm size under monopolistic competition: The case of IES preferences
2017, Research in EconomicsCitation Excerpt :Indeed, Krugman (1980) himself returned to a standard DS structure with its strong implication that neither markups nor firm sizes are affected by trade policy. In fact, an explicit functional form for the Krugman's (1979) type of preferences has only recently been proposed by Bertoletti (1998 and 2006) and Behrens and Murata (2007): see Appendix A. In this paper, we introduce an alternative class of (non-homothetic) preferences characterised by non-constant demand elasticities.
Indirect taxes for redistribution: Should necessity goods be favored?
2016, Research in EconomicsTrade, non-homothetic preferences, and the impact of country size on wages
2015, Economics LettersMonopolistic competition: CES redux?
2014, Journal of International EconomicsSelection effects with heterogeneous firms
2019, Journal of the European Economic AssociationMonopolistic Competition when Income Matters
2017, Economic Journal