Marshallian demand function

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 refused wealth effects.

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 budget set of affordable packages

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

where ${\displaystyle \langle p,x\rangle }$ is the inner product of the price and quantity vectors. The consumer has a utility function

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

The consumer's Marshallian demand correspondence is defined to be

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

Uniqueness

${\displaystyle x^{*}(p,I)}$ is called a correspondence because in general it may be set-valued - there may be several different bundles that attain the same maximum utility. In some cases, there is a unique utility-maximizing bundle for each price and income situation; then, ${\displaystyle x^{*}(p,I)}$ is a function and it is called the Marshallian demand function.

If the consumer has strictly convex preferences and the prices of all goods are strictly positive, then there is a unique utility-maximizing bundle.[1]^:156 To prove this, suppose, by contradiction, that there are two different bundles, ${\displaystyle x_{1}}$ and ${\displaystyle x_{2}}$, that maximize the utility. Then ${\displaystyle x_{1}}$ and ${\displaystyle x_{2}}$ are equally preferred. By definition of strict convexity, the mixed bundle ${\displaystyle 0.5x_{1}+0.5x_{2}}$ is strictly better than ${\displaystyle x_{1},x_{2}}$. But this contradicts the optimality of ${\displaystyle x_{1},x_{2}}$.

Continuity

The maximum theorem implies that if:

• The utility function ${\displaystyle u(x)}$ is continuous with respect to ${\displaystyle x}$,
• The correspondence ${\displaystyle B(p,I)}$ is non-empty, compact-valued, and continuous with respect to ${\displaystyle p,I}$,

then ${\displaystyle x^{*}(p,I)}$ is an upper-semicontinuous correspondence. Moreover, if ${\displaystyle x^{*}(p,I)}$ is unique, then it is a continuous function of ${\displaystyle p}$ and ${\displaystyle I}$.[1]^:156,506

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 ${\displaystyle a>0,}$

${\displaystyle x^{*}(a\cdot p,a\cdot I)=x^{*}(p,I).}$

This is intuitively clear. Suppose ${\displaystyle p}$ and ${\displaystyle I}$ are measured in dollars. When ${\displaystyle a=100}$, ${\displaystyle ap}$ and ${\displaystyle aI}$ are exactly the same quantities measured in cents. Obviously, changing the unit of measurement should not affect the demand.

Examples

In the following examples, there are two commodities, 1 and 2.

1. The utility function has the Cobb–Douglas form:

${\displaystyle u(x_{1},x_{2})=x_{1}^{\alpha }x_{2}^{\beta }.}$

The constrained optimization leads to the Marshallian demand function:

${\displaystyle x^{*}(p_{1},p_{2},I)=\left({\frac {\alpha I}{(\alpha +\beta )p_{1}}},{\frac {\beta I}{(\alpha +\beta )p_{2}}}\right).}$

2. The utility function is a CES utility function:

${\displaystyle u(x_{1},x_{2})=\left[{\frac {x_{1}^{\delta }}{\delta }}+{\frac {x_{2}^{\delta }}{\delta }}\right]^{\frac {1}{\delta }}.}$

Then ${\displaystyle x^{*}(p_{1},p_{2},I)=\left({\frac {Ip_{1}^{\epsilon -1}}{p_{1}^{\epsilon }+p_{2}^{\epsilon }}},{\frac {Ip_{2}^{\epsilon -1}}{p_{1}^{\epsilon }+p_{2}^{\epsilon }}}\right),\quad {\text{with}}\quad \epsilon ={\frac {\delta }{\delta -1}}.}$

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:

${\displaystyle u(x_{1},x_{2})=x_{1}+x_{2}.}$

The utility function is only weakly convex, and indeed the demand is not unique: when ${\displaystyle p_{1}=p_{2}}$, the consumer may divide his income in arbitrary ratios between product types 1 and 2 and get the same utility.

4. The utility function exhibits a non-diminishing marginal rate of substitution:

${\displaystyle u(x_{1},x_{2})=(x_{1}^{\alpha }+x_{2}^{\alpha }),\quad {\text{with}}\quad \alpha >1.}$

The utility function is not concave, and indeed the demand is not continuous: when ${\displaystyle p_{1}<p_{2}}$, the consumer demands only product 1, and when ${\displaystyle p_{2}<p_{1}}$, the consumer demands only product 2 (when ${\displaystyle p_{1}=p_{2}}$ the demand correspondence contains two distinct bundles: either buy only product 1 or buy only product 2).

See also

• Hicksian demand function
• Utility maximization problem
• Demand curve

References

• Mas-Colell, Andreu; Whinston, Michael & Green, Jerry (1995). Microeconomic Theory. Oxford: Oxford University Press. ISBN 0-19-507340-1.
• Nicholson, Walter (1978). Microeconomic Theory (Second ed.). Hinsdale: Dryden Press. pp. 90–93. ISBN 0-03-020831-9.

1. Varian, Hal (1992). Microeconomic Analysis (Third ed.). New York: Norton. ISBN 0-393-95735-7.