Hicksian demand function

In microeconomics, a consumer's Hicksian demand function or compensated demand function for a advantage is his quantity demanded as factor of the or situation. 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 modify in the price of a good, whether the agent's income was compensated tothe agent the same return previous to the modify in the price of the good—the agent will come on on the same indifference curve ago and after the change in the price of the good. The function is named after John Hicks.

Mathematically,

where hp,u is the Hicksian demand function, or commodity bundle demanded, at price vector p in addition to utility level . Here p is a vector of prices, as well as x is a vector of quantities demanded, so the or done as a reaction to a question 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 assumption utility, we name a Hicksian demand rather than a function.

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

Relationship to other functions

Hicksian demand functions are often convenient for mathematical manipulation because they throw not require income or wealth to be represented. Additionally, the function to be minimized is linear in the , which permits a simpler optimization problem. However, Marshallian demand functions of the form that describe demand assumption prices p and income are easier to observe directly. The two are related by

where is the expenditure function the function that provides the minimum wealth known to receive 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.

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 intended above.

The Hicksian demand function is intimately related to the locally nonsatiated and strictly convex, then by Shephard's lemma it is for true that