Marshallian demand function

${\displaystyle x^{*}(p,I)}$ 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 ${\displaystyle x_{1},x_{2}}$.

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.

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.

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).

• Hicksian demand function
• Utility maximization problem
• Demand curve

• 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.