Advanced Microeconomics/Demand Correspondence

Demand Correspondence
The demand correspondence vector $$x(p,w) = \begin{bmatrix}x_1(p,w)\\x_2(p,w)\\\vdots\\x_L(p,w)\end{bmatrix}$$ assigns a set of consumption bundles to each pair $$(p,w)$$. A single valued demand correspondence is a demand function.

Assumptions on demand correspondences

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

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

The Engel Function
Holding the price vector constant, the demand correspondence $$ x(p=\bar{p},w) $$ is the Engel function. In $$\mathbb{R}^L $$ 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 $$l$$ $$\frac{\partial x_l(\bar{p})}{\partial w}$$ is the wealth effect.
 * for normal goods the wealth effect is nonnegative, $$\frac{\partial x_l(\bar{p},w)}{\partial w}\geq 0$$
 * for inferior goods the wealth effect is negative, $$\frac{\partial x_l(\bar{p},w)}{\partial w} < 0$$

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