Expenditure function

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 says what amount of money is needed to achieve a utility if the n prices are given by the price vector . This function is defined by where is the set of all bundles that give utility at least as good as . Expressed equivalently, the individual minimizes expenditure subject to the minimal utility constraint that giving optimal quantities to consume of the various goods as as function of and the prices; then the expenditure function is (Properties of the Expenditure Function) Suppose u is a continuous utility function representing a locally non-satiated preference relation o on Rn +. Then e(p, u) is

Homogeneous of degree one in p: for all and >0,
Continuous in and
Nondecreasing in and strictly increasing in provided
Concave in
If the utility function is strictly quasi-concave, there is the Shephard's lemma Proof (1) As in the above proposition, note that (2) Continue on the domain : (3) Let and suppose . Then , and . It follows immediately that . For the second statement , suppose to the contrary that for some , Than, for some , , which contradicts the "no excess utility" conclusion of the previous proposition (4)Let and suppose . Then, and , so . (5) The expenditure function is the inverse of the indirect utility function when the prices are kept constant. I.e, for every price vector and income level : There is a duality relationship between expenditure function and utility function. If given a specific regular quasi-concave utility function, the corresponding price is homogeneous, and the utility is monotonically increasing expenditure function, conversely, the given price is homogeneous, and the utility is monotonically increasing expenditure function will generate the regular quasi-concave utility function.

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related concepts (3)

Hicksian demand function

In microeconomics, a consumer's Hicksian demand function or compensated demand function for a good is his quantity demanded as part of the solution to minimizing his expenditure on all goods while delivering a fixed level of utility. Essentially, a Hicksian demand function shows how an economic agent would react to the change in the price of a good, if the agent's income was compensated to guarantee the agent the same utility previous to the change in the price of the good—the agent will remain on the same indifference curve before and after the change in the price of the good.

Indirect utility function

NOTOC In economics, a consumer's indirect utility function gives the consumer's maximal attainable utility when faced with a vector of goods prices and an amount of income . It reflects both the consumer's preferences and market conditions. This function is called indirect because consumers usually think about their preferences in terms of what they consume rather than prices.

Utility

As a topic of economics, utility is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or happiness as part of the theory of utilitarianism by moral philosophers such as Jeremy Bentham and John Stuart Mill. The term has been adapted and reapplied within neoclassical economics, which dominates modern economic theory, as a utility function that represents a consumer's ordinal preferences over a choice set, but is not necessarily comparable across consumers or possessing a cardinal interpretation.