In economics, a consumer's indirect utility function v ( p , w ) gives the consumer's maximal attainable utility when faced with a vector p of goods prices and an amount of income w . 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. A consumer's indirect utility v ( p , w ) can be computed from his or her utility function u ( x ) , defined over vectors x of quantities of consumable goods, by first computing the most preferred affordable bundle, represented by the vector x ( p , w ) by solving the utility maximization problem, and second, computing the utility u ( x ( p , w ) ) the consumer derives from that bundle. The resulting indirect utility function is
v ( p , w ) = u ( x ( p , w ) ) . The indirect utility function is:
Continuous on Rn+++ R+;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.quasi convex in (p,w);Moreover, Roy's identity states that if v(p,w) is differentiable at ( p 0 , w 0 ) and ∂ v ( p , w ) ∂ w ≠ 0 , then
− ∂ v ( p 0 , w 0 ) / ( ∂ p i ) ∂ v ( p 0 , w 0 ) / ∂ w = x i ( p 0 , w 0 ) , i = 1 , … , n . 