In economics

Economics

Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...

, a consumer's indirect utility function
gives the consumer's maximal utility

Utility

In economics, utility is a measure of customer satisfaction, referring to the total satisfaction received by a consumer from consuming a good or service....

 when faced with a price level

Price level

A price level is a hypothetical measure of overall prices for some set of goods and services, in a given region during a given interval, normalized relative to some base set...

  and an amount of income

Income

Income is the consumption and savings opportunity gained by an entity within a specified time frame, which is generally expressed in monetary terms. However, for households and individuals, "income is the sum of all the wages, salaries, profits, interests payments, rents and other forms of earnings...

 . It represents the consumer's preferences over market conditions.

This function is called indirect because consumers usually think about their preferences in terms of what they consume rather than prices. A consumer's indirect utility can be computed from its utility function by first computing the most preferred bundle by solving the utility maximization problem

Utility maximization problem

In microeconomics, the utility maximization problem is the problem consumers face: "how should I spend my money in order to maximize my utility?" It is a type of optimal decision problem.-Basic setup:...

; and second, computing the utility the consumer derives from that bundle. The indirect utility function for consumers is analogous to the profit function for firms.

Formally, the indirect utility function is:

• Continuous on Rⁿ₊₊+ R₊;
• Decreasing in prices;
• Strictly increasing in income;
• Homogenous
  Homogeneous function
  In mathematics, a homogeneous function is a function with multiplicative scaling behaviour: if the argument is multiplied by a factor, then the result is multiplied by some power of this factor. More precisely, if is a function between two vector spaces over a field F, and k is an integer, then...
  
   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.
• quasi convex
  Quasiconvex function
  In mathematics, a quasiconvex function is a real-valued function defined on an interval or on a convex subset of a real vector space such that the inverse image of any set of the form is a convex set...
  
  in (p,w);

Moreover,

• Roy's identity: If v(p,w) is differentiable at (p⁰, w⁰) and (∂v(p,w))/∂w≠0, then
  

-(∂v(p⁰,w⁰)/(∂p_i))/(∂v(p⁰,w⁰)/∂w)=x_i (p⁰,w⁰), i=1,…,n.