Generalized marginal rate of substitution in multiconstraint consumer’s problems and their reciprocal expenditure problems
The aim of this paper is to explore several features concerning the generalized marginal rate of substitution (GMRS) when the consumers utility maximization problem with several constraints is formulated as a quasi-concave programming problem. We show that a point satisfying the first order sufficient conditions for the consumer’s problem minimizes the associated quasi-convex reciprocal cost minimization problems. We define the GMRS between endowments and show how it can be computed using the reciprocal expenditure multipliers. Additionally, GMRS is proved to be a rate of change between different proportion bundles of initial endowments. Finally, conditions are provided to guarantee a decreasing GMRS along a curve of initial endowments while keeping the consumer’s utility level constant.
KeywordsQuasi-concave programming Indirect utility function Marginal rates of substitution Multiple constraint optimization problems
JEL ClassificationC61 D11
- Boadway RW, Bruce N (1989) Welfare economics. Basil Blackwell, OxfordGoogle Scholar
- Cornes R, Milne F (1989) A simple analysis of mutually disadvantage trading opportunities. Econ Stud Q 40: 122–134Google Scholar
- Diamond PA, Yaari M (1972) Implications of the theory of rationing for consumer choice under uncertainty. Am Econ Rev 62(3): 333–343Google Scholar
- Giorgi G (1995) On first order sufficient conditions for constraint optima. In: Maruyama T, Takahashi W Nonlinear and convex analysis in economic theory. Springer, Berlin, pp 53–66Google Scholar
- Jehle GA (1991) Advanced microeconomic theory. Prentice Hall, New JerseyGoogle Scholar
- Lancaster K (1968) Mathematical economics. MacMillan, New YorkGoogle Scholar
- Samuelson P (1974) Foundations of economic analysis. Harvard University Press, CambridgeGoogle Scholar
- Silberberg E (1990) The structure of economics: a mathematical analysis. McGraw-Hill, New YorkGoogle Scholar
- Sydsaeter K (1981) Topics in mathematical analysis for economists. Academic Press, Inc, LondonGoogle Scholar
- Takayama A (1994) Analytical methods in economics. Harvester Wheatsheaf, New YorkGoogle Scholar
- Tobin J, Houthakker H (1951) The effects of rationing on demand elasticities. Rev Econ Stud 18: 140–453Google Scholar
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