Abstract
This paper derives closed-form solutions for the total consumption-expenditure function (i.e., aggregate consumption function), the savings function and the demand functions from a nonstationary intertemporal utility-maximization problem under uncertainty for a class of demand systems, including the linear expenditure system (LES) from the Klein-Rubin-Samuelson (KRS) utility function, the generalized linear expenditure systems (GLES) from the CES and S-branch-tree utility functions, the Almost Ideal Demand System (AIDS) from the PIGLOG class of preferences, and the indirect addilog demand system (IADS). We do so by following Hicks’ and Tinmer’s method of maximizing a discounted utility function subject to expected constraints rather than the more fashionable method of maximizing an expected discounted utility function subject to stochastic constraints. Furthermore, the preferences are allowed to vary with the time period. Theoretical analyses for these systems are also given in this paper.
Work supported by NSF grant SES-8607652. The comments and suggestions of an anonymous referee of this Volume greatly improved the paper.
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Notes
The inequality in the budget constraint can be removed by assuming preferences to be locally nonsatiated. Note that if the stochastic budget constraint holds with equality, so must the expected budget constraint.
Extension to the general case where b varies with time is possible but the exposition is simpler if attention is restricted to the case where b is constant over time.
We use this terminology in place of the widespread’ stone-Geary utility function’ used by Powell (1973) and which apparently derives from Brown and Heien (1972), who referred to the LES as the’ stone-Geary linear expenditure system’. This attribution is simply not accurate and we do not wish to perpetuate it The form (15) was first obtained by Samuelson (1948) upon integrating the LES demand functions first proposed by Klein and Rubin (1948) — who had shown that this system of demand functions was integrable. Samuelson’s result was subsequently rediscovered — without reference to Samuelson — by Geary (1949), who expressed (15) in the shifted Cobb-Douglas form exp (u). Stone (1954) presented his work as an empirical implementation of the Klein-Rubin (1948) LES, and referred to Samuelson’s (1948) result but not to Geary’s (1949).
As was pointed out in footnote 2, we can still obtain closed-form solutions even if this assumption is dropped.
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Tian, G., Chipman, J.S. (1989). A Class of Dynamic Demand Systems. In: Raj, B. (eds) Advances in Econometrics and Modelling. Advanced Studies in Theoretical and Applied Econometrics, vol 15. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-7819-6_7
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