Editing Marshallian demand function (section)

{{Short description|Microeconomic 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 <math> L </math> commodities with price vector <math> p </math> and choosable quantity vector <math> x </math>.  The consumer has income <math> I </math>, and hence a [[budget set]] of affordable packages

:<math>B(p, I) = \{x : p \cdot x \leq I\},</math>

where <math> p \cdot x = \sum_i^L p_i x_i </math> is the [[dot product]] of the price and quantity vectors. The consumer has a [[utility function]]

:<math>u : \mathbb R^L_+ \rightarrow \mathbb R.</math>

The consumer's '''Marshallian demand correspondence''' is defined to be

:<math>x^*(p, I) = \operatorname{argmax}_{x \in B(p, I)} u(x) </math>