Hicksian demand function

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In microeconomics, a consumer's Hicksian demand function or compensated demand function for a good is their quantity demanded as part of the solution to minimizing their expenditure on all goods while delivering a fixed level of utility. Essentially, a Hicksian demand function shows how an economic agent would react to the change in the price of a good, if the agent's income was compensated to guarantee the agent the same utility previous to the change in the price of the good—the agent will remain on the same indifference curve before and after the change in the price of the good. The function is named after John Hicks.

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 h(p,u) is the Hicksian demand function, or commodity bundle demanded, at price vector p and utility level ${\displaystyle {\bar {u}}}$. Here p is a vector of prices, and x is a vector of quantities demanded, so the sum of all p_ix_i is total expenditure on all goods. (Note that if there is more than one vector of quantities that minimizes expenditure for the given utility, we have a Hicksian demand correspondence rather than a function.)

Hicksian demand functions are useful for isolating the effect of relative prices on quantities demanded of goods, in contrast to Marshallian demand functions, which combine that with the effect of the real income of the consumer being reduced by a price increase, as explained below.