The Expenditure Minimization Problem

$$\begin{array}{rl}\underset{x\ge 0}{\mathrm{Min}}& p\cdot x\\ & \mathrm{s}.\mathrm{t}.& & u(x)\ge u\end{array}$$

The EMP is the “dual” problem to the UMP.

Proposition 3.E.1 (the formal relationship between EMP and the UMP). Suppose that $u(\cdot )$ is a continuous utility function representing a locally nonsatiated preference relation $\succsim$ defined on the consumption set $X={\mathbb{R}}_{+}^L$, and the price vector is $p\gg 0$. We have

1. If $x^{\ast }$ is optimal in the UMP when wealth is $w>0$, then $x^{\ast }$ is optimal in the EMP when the required utility level is $u(x^{\ast })$. Moreover, the minimized expenditure level in this EMP is exactly $w$.
2. If $x^{\ast }$ is optimal in the EMP when the required utility level is $u>u(0)$, then $x^{\ast }$ is optimal in the UMP when wealth is $p\cdot x^{\ast }$. Moreover, the maximized utility level in this UMP is exactly $u$​.

This duality (and the propositions) can be captured by the fundamental identities

$$h(p,u)\equiv x(p,e(p,u))\text{ and }x(p,w)\equiv h(p,v(p,w)).$$

Other identities can be written using the indirect utility function and expenditure function

$$e(p,v(p,w))\equiv w\text{ and }v(p,e(p,u))\equiv u$$

As with the UMP, when $p\gg 0$ a solution to the EMP exists under very general conditions.

The Expenditure Function

Definition (expenditure function). Given prices $p\gg 0$​ and required utility level $u>u(0)$​, the value of the EMP is denoted $e(p,u)$​. The function $e(p,u)$​ is called the expenditure function. Its value for any $(p,u)$​ is simply $p\cdot x^{\ast }$​, where $x^{\ast }$​​ is any solution to the EMP.

Proposition 3.E.2 (basic properties of the expenditure function).

Suppose that $u(\cdot )$ is a continuous utility function representing a locally nonsatiated preference relation $\succsim$ defined on the consumption set $X={\mathbb{R}}_{+}^L$​​, and that for any $p\gg 0$. The expenditure function $e(p,u)$​ is

1. Homogeneous of degree one in $p$.
2. Strictly increasing in $u$ and nondecreasing in $p_{\ell }$ for any $\ell$.
3. Concave in $p$​.
4. Continuous in $p$ and $u$​.

The Hicksian (or Compensated) Demand Function

Definition (Hicksian/compensated demand correspondence/function). The set of optimal commodity vectors in the EMP is denoted $h(p,u)\subset {\mathbb{R}}_{+}^L$​ and is known as the Hicksian, or compensated, demand correspondence, or function if single-valued.

Proposition 3.E.3 (basic properties of Hicksian demand). Suppose that $u(\cdot )$ is a continuous utility function representing a locally nonsatiated preference relation $\succsim$​ defined on the consumption set $X={\mathbb{R}}_{+}^L$​, and that for any $p\gg 0$. The Hicksian demand correspondence $h(p,u)$​​ possesses the following properties:

1. Homogeneity of degree zero in $p$ : $h(ap,u)=h(p,u)$ for any $p$, $u$ and $a>0$​.
2. No excess utility: For any $x\in h(p,u),u(x)=u$.
3. Convexity/uniqueness: If $\succsim$ is convex, then $h(p,u)$ is a convex set; and if $\succsim$ is strictly convex, so that $u(\cdot )$ is strictly quasiconcave, then there is a unique element in $h(p,u)$​.

Using Proposition 3.E.1, we can relate the Hicksian and Walrasian demand correspondences as follows:

$$\begin{array}{}\text{(3.E.4)}& h(p,u)=x(p,e(p,u))\text{and}x(p,w)=h(p,v(p,w)).\end{array}$$

The first of these relations explains the use of the term compensated demand correspondence to describe $h(p,u)$:

• Definition (Hicksian wealth compensation). As prices vary, $h(p,u)$ gives precisely the level of demand that would arise if the consumer’s wealth were simultaneously adjusted to keep her utility level at $u$.

As with the value functions of the EMP and UMP, the relations in (3.E.4) allow us to develop a tight linkage between the properties of the Hicksian demand correspondence $h(p,u)$ and the Walrasian demand correspondence $x(p,w)$.

Hicksian Demand and the Compensated Law of Demand

Proposition 3.E.4 (Hicksian Demand and the Compensated Law of Demand). Suppose that (1) $u(\cdot )$ is a continuous utility function representing a locally nonsatiated preference relation $\succsim$, and that (2) $h(p,u)$ consists of a single element for all $p\gg 0$. Then the Hicksian demand function $h(p,u)$​​​ satisfies the compensated law of demand: For all $p^{\prime }$ and $p^{″}$​,

$$\begin{array}{}\text{(3.E.5)}& (p^{″}-p^{\prime })\cdot [h(p^{″},u)-h(p^{\prime },u)]\le 0.\end{array}$$