Cobb-Douglas Utility - Eventually!
Consider the following two opinions, both of which can be found in the literature of consumer demand systems: (a) As the real income of a consumer becomes indefinitely large, re-mixing the consumption bundle becomes irrelevant: having chosen the ultimately satisfying budget shares at any given set of relative prices, the superlatively wealthy continue to allocate additional income in the same proportions. With very large and increasing per capita income, ultimately the utility function becomes indistinguishable from Cobb-Douglas. (b) Consumer demand systems in which the income elasticities monotonically approach one (from above, in the case of luxuries; from below, in the case of necessities) are unsatisfactory both theoretically and empirically. For instance, a necessity with a low (less than 1) income elasticity may very well become less elastic with further increases in income. The issue is important for CGE modelers because explicit direct additivity (as in the linear expenditure system [LES]) is often the modeler's default choice: this leaves us firmly in the world of (a). Hanoch's implicit direct additivity exhibits very flexible Engel properties. Rimmer and Powell's AIDADS system belongs to this class. Within such a system it is possible to satisfy the motivations underlying both (a) and (b), as illustrated by the Engel (income) elasticities for a 3-commodity AIDADS system shown in Figure 1 below. Whilst the system eventually converges to Cobb-Douglas, some income elasticities can be effectively zero at any imaginable actual income level. Inferiority can also be accommodated over a range of incomes. This paper discusses the above in more detail, strengthening the case for implicit direct additivity. An experimental calibration of a database to the AIDADS system is illustrated with modifications made to the ORANI-G teaching model. A technical appendix establishes the effectively global regularity of AIDADS. Related computer files may be found here.
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295, Center for Global Trade Analysis, Department of Agricultural Economics, Purdue University.
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