In microeconomics, a consumer's Marshallian demand function (named after Alfred Marshall) specifies what the consumer would buy in each price and income or 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.
Contents
According to the utility maximization problem, there are L commodities with price vector p and choosable quantity vector x. The consumer has income I, and hence a set of affordable packages
where
The consumer's Marshallian demand correspondence is defined to be
Uniqueness
If the consumer has strictly convex preferences and the prices of all goods are strictly positive, then there is a unique utility-maximizing bundle. PROOF: suppose, by contradiction, that there are two different bundles,
Continuity
The maximum theorem implies that if:
then
Combining with the previous subsection, if the consumer has strictly convex preferences, then the Marshallian demand is unique and continuous. In contrast, if the preferences are not convex, then the Marshallian demand may be non-unique and non-continuous.
Homogeneity
The Marshallian demand correspondence is a homogeneous function with degree 0. This means that for every constant
This is intuitively clear. Suppose
Examples
In the following examples, there are two commodities, 1 and 2.
1. The utility function has the Cobb–Douglas form:
the constrained optimization leads to the Marshallian demand function:
2. The utility function is a CES utility function:
then:
In both cases, the preferences are strictly convex, the demand is unique and the demand function is continuous.
3. The utility function has the linear form:
the utility function is only weakly convex, and indeed the demand is not unique: when
4. The utility function exhibits a non-diminishing marginal rate of substitution:
The utility function is concave, and indeed the demand is not continuous: when