The characterization of demand and excess demand functions, revisited

Abstract

In this note, the characterization problem of individual demand and excess demand functions is revisited. It is assumed that the individual’s income is price dependent. When the income function is homogeneous of degree one, we show that similar conditions characterize both demand and excess demand functions.

A duality theorem

We state the following important theorem from Chiappori and Ekeland (2004). This theorem ensures the existence of a direct utility function $U(z(p))=V(p)$ where V is an indirect utility function and z(p) is excess demand. It thus establishes duality in the excess demand case. As it was pointed out in Chiappori and Ekeland (2004), the one-to-one correspondence between the direct and indirect utility functions is lost in the excess demand case because of homogeneity. The reader is referred to Chiappori and Ekeland (2004) for details.

Theorem 9

(Chiappori and Ekeland 2004) Let V(p) be a $C^k$ function, $k\ge 3$, defined on some neighborhood of ${\bar{p}}$ with $V_p({\bar{p}})\ne 0$. Assume that, on that neighborhood, V(p) is quasi-convex, homogeneous of degree zero, that $V_{pp}(p)$ has rank $n-1$ and the restriction of $V_{pp}(p)$ to $[\text {Span}\{p,V_p\}]^\bot$ is positive definite. Then, there is an open convex cone $\Gamma \subset {\mathbb {R}}^n_{++}$ containing ${\bar{p}}$, such that V is an indirect utility function on $\Gamma$. In fact, take any $C^{k-1}$ function $\lambda (p)>0$ homogeneous of degree $-1$ on $\Gamma$, and define

$$\begin{aligned} z(p)=-\frac{1}{\lambda (p)}V_p(p) \end{aligned}$$

Then, there is a quasi-concave function U(z), defined and $C^{k-1}$ on a convex neighborhood ${\mathcal N}$ of $z({\bar{p}})$, with the following properties

• (a) For every $z\in {\mathcal N}$, the restriction of $U_{zz}(z)$ to $\{U_z\}^\bot$ is negative definite.

• (b) For every $p\in \Gamma$, we have

  $$\begin{aligned} U_z(z(p)=\lambda (p)p,\quad p^\prime z(p)=0 \end{aligned}$$

  and

  $$\begin{aligned} V(p)=U(z(p))=\max \limits _z\{U(z)|p^\prime z\le 0,z\in {\mathcal N}\} \end{aligned}$$