Marshallian demand function

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In microeconomics, a consumer's Marshallian demand function specifies what the consumer would buy in each price and wealth situation, assuming it perfectly solves the utility maximization problem. Marshallian demand is sometimes called Walrasian demand or uncompensated demand function instead, because the original Marshallian analysis ignored wealth effects. Milton Friedman, however, argues^{[citation needed]} that Marshall was misunderstood, that he did account for wealth effects, and that therefore, what is commonly called Marshallian demand is no such thing.

According to the utility maximization problem, there are L commodities with prices p. The consumer has wealth w, and hence a set of affordable packages

$B(p, w) = \{x : p \cdot x \leq w\}$ .

The consumer has a utility function

$u : \textbf R^L_+ \rightarrow \textbf R$ .

The consumer's Marshallian demand correspondence is defined to be

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

If there is a unique utility maximizing package for each price and wealth situation, then it is called the Marshallian demand function. See the utility maximization problem entry for a discussion of this definition.

[edit] Example

If there are two commodities, then a consumer that always chooses to spend half of its income on each commodity would have the Marshallian demand function

$x(p, w) = \left(\frac{w}{2p_1}, \frac{w}{2p_2}\right).$