2.4 Consumer Welfare and Surplus

In the previous section, we saw that when income or the price of a good changes, the consumer’s optimal choice shifts from one indifference curve to another, causing a change in welfare whose direction is obvious: the consumer is better off if income increases or a price decreases, worse off if income decreases or a price increases. However, establishing whether the consumer is better or worse off without also measuring this change in welfare is not enough, nor does it make sense to do so using utility as a unit of measurement, since utility is a purely ordinal concept. For consumer theory to be useful not only in explaining behavior but also in comparing different market structures or assessing the social impact of a new technology or government intervention, what we need is to measure in monetary terms the changes in welfare.

To understand why, suppose the government expects an inevitable future increase in food prices, for example due to a pest damaging crops, and wants to prepare to give families a subsidy We will discuss government interventions such as taxes and subsidies in Chapter 7. or bonus (effectively, an increase in income) that ensures them a level of welfare equal to the current one. What should the amount of the bonus be? And if the government could instead avoid the price increase, The government would not know what to do if, besides the cost of the campaign (money to spend on necessary treatments), it did not also evaluate the benefit in monetary terms, i.e., if it did not answer the question on the left. Note: the evaluation should also include the benefits that businesses would gain from the campaign, but for simplicity here we are assuming that the government cares only about the welfare of families. for example through a pest control campaign, how much would families be willing to pay (i.e., how much income they would be willing to give up, paying it to the government as taxes) to finance the campaign?

It should be clear that to answer questions like these, and many others, it is necessary to have a monetary measure of welfare changes caused by changes in income or the price of a good. In the case of an income change, we already have what we need, since the variation is already expressed in euros. And in the case of a price change, how do we proceed? The idea is that every change in welfare caused by a price change can be offset by an appropriate change in income. Being able to calculate this latter, we can then use it as a monetary measure of the welfare change caused by the price variation.

Consumer Surplus

In Chapter 1 we introduced the concept of consumer surplus and accepted the idea of using the change in surplus caused by a price change as a monetary measure of the corresponding change in consumer welfare.

In the second part of this section, we will provide a foundation for that idea. In particular, we will show that the change in surplus lies halfway between (and thus provides a reasonable approximation to) two exact monetary measures of welfare change called compensating variation and equivalent variation.

Before doing this, let us review the concept of consumer surplus, assuming preferences represented by the utility function $U(X,Y)=XY$. In the previous section we saw that in this case the optimal choice is to spend half of the income, which we assume to be $M=360$, on each of the two goods. The demand function for good $X$ is therefore $X=180/P_X$.

Compensating Variation and Equivalent Variation

Continuing our example, suppose income is $M=360$ and initially prices are $P_X=9$ and $P_Y=18$. The optimal choice is then the bundle $A=(20,10)$, which yields utility $U(20,10)=200$. Now suppose the price of good $X$ increases to $P_X=36$. The optimal choice becomes bundle $B=(5,10)$, which yields utility $U(5,10)=50$.

We can think of two ways to measure exactly and in monetary terms the consumer’s loss of welfare:

• Compensating Variation ($CV$): the additional income (subsidy) that would allow the consumer to reach, at the new prices $P_X=36$ and $P_Y=18$, the utility level of the old bundle $A$, i.e., $U=200$.

• Equivalent Variation ($EV$): the reduction Equivalent variation thus measures how much the consumer would be willing to pay to avoid the price change. in income (tax) that, at the old prices $P_X=9$ and $P_Y=18$, would bring the consumer to the utility level of the new bundle $B$, i.e., $U=50$.

The following figure illustrates the calculation of compensating and equivalent variations and shows their relation with the change in consumer surplus.