# Marshallian demand function

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In microeconomics, a consumer's Marshallian demand function (named after Alfred Marshall) 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 (named after Léon Walras) or uncompensated demand function instead, because the original Marshallian analysis ignored wealth effects.

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

$B(p, w) = \{x : \langle p, x \rangle \leq w\},$

where $\langle p, x \rangle$ is the inner product of the prices and quantity of goods. 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.

## Example

If there are two commodities, and the consumer has the utility function $U(x_1,x_2) = x_1^{0.5}x_2^{0.5}$ (Cobb-Douglas), he chooses to spend half of its income on each commodity, and its Marshallian demand function is the following:

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

In general, an agent with Cobb-Douglas preferences, given by $U(x_1,x_2) = x_1^{a}x_2^{1-a}$, will use a constant share of his income in order to buy each of the two commodities, as follows:

$x(p_1,p_2,w) = \left(\frac{aw}{p_1}, \frac{(1-a)w}{p_2}\right).$

If we are in a more general case, i.e. $U(x_1,x_2) = x_1^{a}x_2^{b}$, we have:

$x(p_1,p_2,w) = \left(\frac{aw}{(a+b)p_1}, \frac{bw}{(a+b)p_2}\right).$