Welfare Evaluation of Economic Changes

The normative side of consumer theory, called welfare analysis. Welfare analysis concerns itself with the evaluation of the effects of changes in the consumer’s environment on her well-being.

Assumptions

1. Consider a consumer with a rational, continuous, and locally nonsatiated preference relation $\succsim$.
2. Assume that the consumer’s expenditure and indirect utility functions are differentiable.
3. We assume that the consumer has a fixed wealth level $w>0$ and that the price vector is initially $p^0$.

We wish to evaluate the impact on the consumer’s welfare of a change from $p^0$ to a new price vector $p^1$​.

If $v(p,w)$​ is any indirect utility function derived from $\succsim$​, the consumer is worse off if and only if $v(p^1,w)-v(p^0,w)<0$​.

Money metric indirect utility functions

One class of indirect utility functions deserves special mention because it leads to measurement of the welfare change expressed in dollar units.

These are called money metric indirect utility functions and are constructed by means of the expenditure function.

• Starting from any indirect utility function $v(\cdot ,\cdot )$​​, choose an arbitrary price vector $\bar{p}\gg 0$​​​, and consider the function $e(\bar{p},v(p,w))$​​. This function gives the wealth required to reach the utility level $v(p,w)$ when prices are $\bar{p}$​.

$e(\bar{p},v(p^1,w))-e(\bar{p},v(p^0,w))$​​ provides a measure of the welfare change expressed in dollars.

A money metric indirect utility function can be constructed in this manner for any price vector $p\gg 0$.

Equivalent and compensating variation

Definition (the equivalent variation (EV) and the compensating variation (CV)). Two particularly natural choices for the price vector $p$ are the initial price vector $p^0$ and the new price vector $p^1$​. These choices lead to two well-known measures of welfare change, the equivalent variation (EV) and the compensating variation (CV).

• Formally, letting $u^0=v(p^0,w)$ and $u^1=v(p^1,w)$, and noting that $e(p^0,u^0)=e(p^l,u^1)=w$​, we define

$$EV(p^0,p^1,w)=e(p^0,u^1)-e(p^0,u^0)=e(p^0,u^1)-w$$

and

$$CV(p^0,p^1,w)=e(p^1,u^1)-e(p^1,u^0)=w-e(p^1,u^0).$$

The equivalent and compensating variations have interesting representations in terms of the Hicksian demand curve.

Suppose, for simplicity, that only the price of good 1 changes, so that $p_1^0\ne p_1^1$ and $p_{\ell }^0=p_{\ell }^1={\bar{p}}_{\ell }$ for all $\ell \ne 1$.

Because $w=e(p^0,u^0)=e(p^1,u^1)$ and $h_1(p,u)=\partial e(p,u)/\partial p_1$, we can write

$$\begin{array}{rl}EV(p^0,p^1,w)=& e(p^0,u^1)-w\\ =& e(p^0,u^1)-e(p^1,u^1)\\ \text{(3.I.3)}& =& {\int }_{p_1^1}^{p_1^0}{h}_1(p_1,{\bar{p}}_{-1},u^1)d{p}_1\end{array}$$

where ${\bar{p}}_{-1}=({\bar{p}}_2,\cdots ,{\bar{p}}_L)$​.

Similarly, the compensating variation can be written as

$$\begin{array}{}\text{(3.I.4)}& CV(p^0,p^1,w)={\int }_{p_1^1}^{p_1^0}{h}_1(p_1,{\bar{p}}_{-1},u^0)d{p}_1\end{array}$$

Marshallian consumer surplus

Definition (Marshallian consumer surplus). In this case of no wealth effects, we call the common value of CV and EV the change in Marshallian consumer surplus.

$$\begin{array}{}\text{(3.I.8)}& CS(p^0,p^1,w)={\int }_{p_1^1}^{p_1^0}{x}_1(p_1,{\bar{p}}_{-1},w)d{p}_1\end{array}$$

Compare

For changes in price of one good (1)

$$min\{EV,CV\}\le CS\le max\{EV,CS\}$$

For a normal good:

$$CV<CS<EV$$

If commodity 1 is normal the Marshallian demand $x_1(·,w)$​ is steeper than the Hicksian demand $h_1(·,u)$ (by Slutsky):

$$\frac{\partial x_1}{\partial p_1}-\frac{\partial h_1}{\partial p_1}=-\frac{\partial x_1}{\partial w}{x}_1<0$$

For a inferior good, $EV<CS<CV$.

When the income effect is zero, $EV=CS=CV$, this is the case for quasi-linear utility functions $u(x_1,x_2)=u(x_1)+x_2$ where $\partial x_1/\partial w=0$.

In summary, if we know the consumer’s expenditure function, we can precisely measure the welfare impact of a price change; moreover, we can do it in a convenient way (in dollars).