# What Is the Indirect Utility Function?

## Indirect Utility Function Defined as a Function of Price and Income

A consumer's indirect utility function is a function of prices of goods and the consumer's income or budget. The function is typically denoted as v(p, m) where p is a vector of prices for goods, and m is a budget presented in the same units as the prices. The indirect utility function takes the value of the maximum utility that can be achieved by spending the budget m on the consumption goods with prices p. This function is termed "indirect" because consumers generally consider their preferences in terms of what they consume rather than price (as is used in the function). Some versions of the indirect utility function substitute for m where w is considered income rather than budget such that v(p,w).

## Indirect Utility Function and Microeconomics

The indirect utility function is of particular importance in microeconomic theory as it adds value to the continual development of consumer choice theory and applied microeconomic theory. Related to the indirect utility function is the expenditure function, which provides the minimum amount of money or income an individual must spend to achieve some pre-defined level of utility. In microeconomics, a consumer's indirect utility function illustrates both the consumer's preferences and prevailing market conditions and the economic environment.

## Indirect Utility Function and UMP

The indirect utility function is closely related to the utility maximization problem (UMP). In microeconomics, the UMP is an optimal decision problem that refers to the problem consumers face with regards to how to spend money in order to maximize utility. The indirect utility function is the value function, or the best possible value of the objective, of the utility maximization problem:

v(p, m) = max u(x) s.t. p · ≤ m

## Properties of the Indirect Utility Function

It is important to note that in the utility maximization problem consumers are assumed to be rational and locally non-satiated with convex preferences that maximize utility. As a result of the function's relationship with the UMP, this assumption applies to the indirect utility function as well. Another important property of the indirect utility function is that it is degree-zero homogeneous function, meaning that if prices (p) and income (m) are both multiplied by the same constant the optimal does not change (it has no impact). It is also assumed that all income is spent and the function adheres to the law of demand, which is reflected in increasing income m and decreasing price p. Last, but not least, the indirect utility function is also quasi-convex in price.

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