Marshallian vs Hicksian demand function. Derivation , Properties

Demand Functions
Economists use mathematical equations (functions) to model consumer demand. The causal relationship is between quantity demanded by the consumer, which is the dependent variable, and the price of a good and consumer income, which are the independent variables.

Hicksian or Compensated Demand
The Hicksian demand function (after British economist Sir John R. Hicks) shows the relationship between the price of a good, P1, and the quantity purchased on the assumption that other prices, P2, and utility, U0, are held constant. This consumer demand function is obtained by minimizing the consumer's expenditures subject to the constraint that his/her utility (the satisfaction a consumer derives from a particular market basket) is fixed at level U0. Hicksian and Compensated Demand functions are the same and are represented by the following equation: h1(P1, P2, U0).

Marshallian, Ordinary, or Uncompensated Demand
The Marshallian demand function (after British economist Alfred Marshall) shows the relationship between the price of a good, P1, and the quantity purchased, Q1, on the assumption that other prices, P2, and the consumer's budget (or income), Y0, is held constant. The demand function is obtained by maximizing the consumer's utility subject to the constraint that the customer's budget is fixed at the level Y0 and so are other prices. Marshallian, Ordinary, and Uncompensated Demand functions are the same and are represented by the following equation: Q1= f (P1, P2, Y0).
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