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,

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.

Hicksian demand functions are often convenient for mathematical manipulation because they do not require representing income or wealth. Additionally, the function to be minimized is linear in the ${\displaystyle x_{i}}$, which gives a simpler optimization problem. However, Marshallian demand functions of the form ${\displaystyle x(p,w)}$ that describe demand given prices p and income ${\displaystyle w}$ are easier to observe directly. The two are related by

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

where ${\displaystyle v(p,w)}$ 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.

Whereas Marshallian demand comes from the Utility Maximization Problem, Hicksian Demand comes from the Expenditure Minimization Problem. The two problems are mathematical duals, and hence the Duality Theorem provides a method of proving the relationships described above.