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# Indirect utility function

In economics, a consumer's indirect utility function $v\left(p, w\right)$ gives the consumer's maximal utility when faced with a price level $p$ and an amount of income $w$. 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 $v\left(p, w\right)$ can be computed from its utility function $u\left(x\right)$ by first computing the most preferred bundle $x\left(p, w\right)$ by solving the utility maximization problem; and second, computing the utility $u\left(x\left(p, w\right)\right)$ 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:

• Non-increasing in prices, because an increase in prices cannot open up an available bundle that would provide more utility;
• Non-decreasing in income, because when income rises, at worst you could consume the same bundle;
• 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).

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