Definitions

# Hicksian demand function

In microeconomics, a consumer's Hicksian demand function, also called the compensated demand function, is the demand of a consumer over a bundle of goods that minimizes their expenditure while delivering a fixed level of utility. The function is named after John Hicks.

Mathematically,

$h\left(p, bar\left\{u\right\}\right) = arg min_x sum_i p_i x_i$
$\left\{rm such that\right\} u\left(x\right) > bar\left\{u\right\}$

where h(p,u) is the Hicksian demand function, or commodity bundle demanded, at price level p and utility level $bar\left\{u\right\}$. Here p is a vector of prices, and X is a vector of quantities demanded so that the sum of all pixi, is the total expense on goods X.

## Relationship to other functions

Hicksian demand functions are often convenient for mathematical manipulation because they do not require income or wealth to be represented. However, Marshallian demand functions of the form $x\left(p, w\right)$ that describe demand given prices $p$ and income $w$ are easier to observe directly. The two are trivially related by

$h\left(p, u\right) = x\left(p, e\left(p, u\right)\right),$

where $e\left(p, u\right)$ is the expenditure function (the function that gives the minimum wealth required to get to a given utility level), and by

$h\left(p, v\left(p, w\right)\right) = x\left(p, w\right),$

where $v\left(p, w\right)$ is the indirect utility function (which gives the utility level of having a given wealth under a fixed price regime). Their derivatives are more fundamentally related by the Slutsky equation.

The Hicksian demand function is intimately related to the expenditure function. If the consumer's utility function $u\left(x\right)$ is locally nonsatiated and strictly convex, then $h\left(p, u\right) = nabla_p e\left(p, u\right).$