The indirect utility function is defined as the maximum utility that can be attained given money income and goods prices.

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 w for m where w is considered income rather than budget such that v(p,w).

Let qi denotes commodity i and pi is the price of that commodity. Let y denotes money income of the consumer. Suppose vi = pi/y. The budget constraint now may be written as

n

1 = ∑ viqi -------------------- (a)

i =1

Since optimal solutions in the demand functions are homogeneous of degree zero in income and prices, nothing essential is lost by this transformation to “normalised” prices. The utility function U = f (q1, qn) together with equation (a) gives the following first order conditions of utility maximisation:

fi - λvi = 0 for all i = 1,…, n

n

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