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In microeconomics, the expenditure function gives the minimum amount of money an individual needs to spend to achieve some level of utility, given a utility function and the prices of the available goods. Formally, if there is a utility function that describes preferences over n commodities, the expenditure function
While there are several ways to derive the Slutsky equation, the following method is likely the simplest. Begin by noting the identity (,) = (, (,)) where (,) is the expenditure function, and u is the utility obtained by maximizing utility given p and w.
Formally, the expenditure function is defined as follows. Suppose the consumer has a utility function defined on commodities. Then the consumer's expenditure function gives the amount of money required to buy a package of commodities at given prices that give utility of at least ,
The indirect utility function is: Continuous on R n + × R + where n is the number of goods; Decreasing in prices; Strictly increasing in income; Homogenous with degree zero in prices and income; if prices and income are all multiplied by a given constant the same bundle of consumption represents a maximum, so optimal utility does not change;
E.g., the commodity is a heterogeneous resource, such as land. Then, the utility functions are not functions of a finite number of variables, but rather set functions defined on Borel subsets of the land. The natural generalization of a linear utility function to that model is an additive set function.
Standard utility functions represent ordinal preferences. The expected utility hypothesis imposes limitations on the utility function and makes utility cardinal (though still not comparable across individuals). Although the expected utility hypothesis is standard in economic modelling, it has been found to be violated in psychological experiments.
Roy's identity reformulates Shephard's lemma in order to get a Marshallian demand function for an individual and a good from some indirect utility function.. The first step is to consider the trivial identity obtained by substituting the expenditure function for wealth or income in the indirect utility function (,), at a utility of :
Given a utility function u(x,y), to calculate the MRS, one takes the partial derivative of the function u with respect to good x and divide it by the partial derivative of the function u with respect to good y. If the marginal rate of substitution is diminishing along an indifference curve, that is the magnitude of the slope is decreasing or ...