Marshallian demand function

In microeconomics, a consumer's Marshallian demand function (named after Alfred Marshall) is the quantity they demand of a particular good as a function of its price, their income, and the prices of other goods, a more technical exposition of the standard demand function. It is a solution to the utility maximization problem of how the consumer can maximize their utility for given income and prices. A synonymous term is uncompensated demand function, because when the price rises the consumer is not compensated with higher nominal income for the fall in their real income, unlike in the Hicksian demand function. Thus the change in quantity demanded is a combination of a substitution effect and a wealth effect. Although Marshallian demand is in the context of partial equilibrium theory, it is sometimes called Walrasian demand as used in general equilibrium theory (named after Léon Walras).

According to the utility maximization problem, there are ${\displaystyle L}$ commodities with price vector ${\displaystyle p}$ and choosable quantity vector ${\displaystyle x}$. The consumer has income ${\displaystyle I}$, and hence a budget set of affordable packages

${\displaystyle B(p,I)=\{x:p\cdot x\leq I\},}$

where ${\displaystyle p\cdot x=\sum _{i}^{L}p_{i}x_{i}}$ is the dot product of the price and quantity vectors. The consumer has a utility function

${\displaystyle u:\mathbb {R} _{+}^{L}\rightarrow \mathbb {R} .}$

The consumer's Marshallian demand correspondence is defined to be

${\displaystyle x^{*}(p,I)=\operatorname {argmax} _{x\in B(p,I)}u(x)}$