Hicksian demand function

In microeconomics, a consumer's Hicksian demand function or compensated demand function for a good is his quantity demanded as part of the solution to minimizing his 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, where h(p,u) is the Hicksian demand function, or commodity bundle demanded, at price vector p and utility level . Here p is a vector of prices, and x is a vector of quantities demanded, so the sum of all pixi 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. Hicksian demand functions are often convenient for mathematical manipulation because they do not require income or wealth to be represented. Additionally, the function to be minimized is linear in the , which gives a simpler optimization problem. However, Marshallian demand functions of the form that describe demand given prices p and income are easier to observe directly. The two are related by where is the expenditure function (the function that gives the minimum wealth required to get to a given utility level), and by where 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.