Hicksian demand function

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In microeconomics, a consumer's Hicksian demand function (or compensated demand function) represents the quantity of a good demanded when the consumer minimizes expenditure while maintaining a fixed level of utility.

The Hicksian demand function illustrates how a consumer would adjust their demand for a good in response to a price change, assuming their income is adjusted (or compensated) to keep them on the same indifference curve—ensuring their utility remains unchanged. Mathematically,^[1]

${\displaystyle h(p,{\bar {u}})=\arg \min _{x}\sum _{i}p_{i}x_{i}}$

${\displaystyle {\rm {subject~to}}\ \ u(x)\geq {\bar {u}}}$.

where ${\displaystyle h(p,u)}$ is the Hicksian demand function or commodity bundle demanded, at price vector ${\displaystyle p}$ and utility level ${\displaystyle {\bar {u}}}$. Here ${\displaystyle p}$ is a vector of prices, and ${\displaystyle x}$ is a vector of quantities demanded, so the sum of all ${\displaystyle p_{i}x_{i}}$ is the total expenditure on all goods.

The Hicksian demand function isolates the effect of relative prices on demand, assuming utility remains constant. It contrasts with the Marshallian demand function, which accounts for both the substitution effect and the reduction in real income caused by price changes. The function is named after John Hicks.