Diminishing Marginal Value of Income Without Apology

This paper considers value functions for maximization problems where the objective function is maximized subject to several constraints. I show that such value functions exhibit diminishing marginal rates of substitution (MRS) between the resource endowments which define the constraints. For the household, this implies that an ordinally defined marginal value of income falls as income rises. I also derive conditions under which the value function is homothetic in the endowments, so that the MRS between two resources is a monotone function of the ratio of the endowments of those resources and under which the MRS between two resources declines when one resource is increased with the other held constant, i.e., under which the marginal rate of substitution declines when the households moves from one indirect indifference curve to another. I apply these results to a model with two linear constraints and to a model of asset choice under uncertainty