Editing Marshallian demand function (section)

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

1. The utility function has the [[Cobb–Douglas|Cobb–Douglas form]]:
:<math>u(x_1,x_2) = x_1^{\alpha}x_2^{\beta}.</math>

The constrained optimization leads to the Marshallian demand function:
:<math>x^*(p_1,p_2,I) = \left(\frac{\alpha I}{(\alpha+\beta)p_1}, \frac{\beta I}{(\alpha+\beta)p_2}\right).</math>

2. The utility function is a [[Constant elasticity of substitution#CES utility function|CES utility function]]:
:<math>u(x_1,x_2) = \left[ \frac{x_1^{\delta}}{\delta} + \frac{x_2^{\delta}}{\delta} \right]^{\frac{1}{\delta}}.</math>

Then
<math>x^*(p_1,p_2,I) = \left(\frac{I p_1^{\epsilon-1}}{p_1^{\epsilon} + p_2^{\epsilon}}, \frac{I p_2^{\epsilon-1}}{p_1^{\epsilon} + p_2^{\epsilon}}\right), \quad \text{with} \quad \epsilon = \frac{\delta}{\delta-1}.</math>

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 utility|linear form]]:
:<math>u(x_1,x_2) = x_1 + x_2.</math>
The utility function is only weakly convex, and indeed the demand is not unique: when <math>p_1=p_2</math>, 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:
:<math>u(x_1,x_2) = (x_1^{\alpha} + x_2^{\alpha}), \quad \text{with} \quad \alpha > 1.</math> 
The utility function is not convex, and indeed the demand is not continuous: when <math>p_1<p_2</math>, the consumer demands only product 1, and when <math>p_2<p_1</math>, the consumer demands only product 2 (when <math>p_1=p_2</math> the demand correspondence contains two distinct bundles: either buy only product 1 or buy only product 2).