Advanced Microeconomics/Demand Correspondence

The demand correspondence vector ${\displaystyle x(p,w)={\begin{bmatrix}x_{1}(p,w)\\x_{2}(p,w)\\\vdots \\x_{L}(p,w)\end{bmatrix}}}$ assigns a set of consumption bundles to each pair ${\displaystyle (p,w)}$. A single valued demand correspondence is a demand function.

1. Homogeneity of degree zero:
   ${\displaystyle x(\alpha p,\alpha w)=x(p,w)\;\forall p,w,\alpha >0}$
2. Walras's law:
   ${\displaystyle p\cdot x=w\;\forall x\in x(p,w),\;p>>0,\;w>0}$

Notice, the homogeneity assumption allows one argument of ${\displaystyle x(p,w)}$ to be normalized.

Holding the price vector constant, the demand correspondence ${\displaystyle x(p={\bar {p}},w)}$ is the Engel function. In ${\displaystyle \mathbb {R} ^{L}}$ the Engel function is known as the wealth expansion path, illustrating changes in the demand correspondence at various levels of wealth. The first derivative of the Engel function with respect to wealth for good ${\displaystyle l}$ ${\displaystyle {\frac {\partial x_{l}({\bar {p}})}{\partial w}}}$ is the wealth effect.

• for normal goods the wealth effect is nonnegative,
  ${\displaystyle {\frac {\partial x_{l}({\bar {p}},w)}{\partial w}}\geq 0}$
• for inferior goods the wealth effect is negative,
  ${\displaystyle {\frac {\partial x_{l}({\bar {p}},w)}{\partial w}}<0}$

For any two goods ${\displaystyle l,k}$ the representation of ${\displaystyle x_{l}(p_{k},{\bar {w}})}$ across all prices ${\displaystyle p_{k}}$ is the offer curve. Define the price effect of good ${\displaystyle k}$ on good ${\displaystyle l}$, ${\displaystyle {\frac {\partial x_{l}(p_{k},{\bar {w}})}{\partial p_{k}}}}$