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Compensating and Equivalent Variation and Consumer Surplus

"A little inaccuracy sometimes saves tons of explanation." -- Saki

We want a measure of how much a consumer is hurt by a price increase. Rather than compare (inherently unobservable) utility levels, we would like to describe the harm to the consumer in terms of dollars. There are two ways to measure this harm.

First, we can ask: How much extra money would we have to give a consumer to completely offset the harm from a price increase? This measure of the welfare harm of a price increase is called the compensating variation because we give money to the consumer: compensate the consumer.

Second, we can ask: How much money would we have to take from a consumer to harm the consumer by as much as the price increase? This measure is called the equivalent variation because it is the same—equivalent—harm as that of the price increase.

Compensating variation and equivalent variation are two different answers to the question: How much of a change in income is necessary to offset a change in price so that a consumer's utility remains at a given level? Effectively, they are measuring the pure income effect of a change in the price of one good relative to other goods.

In Chapter 9, we measured the effect of a price increase by a change in consumer surplus, which does not hold a consumer's utility constant. The change in consumer surplus reflects both the substitution and income effects. Consequently, economists usually think of the change in consumer surplus as an approximation to the other two, pure income effect measures.

In this appendix, we compare the three measures of the consumer welfare harm of price increases: compensating variation, equivalent variation, and a change in consumer surplus. We argue that a change in consumer surplus is a good approximation of the other two, theoretically desirable measures.

Indifference Curve Analysis

We can use indifference curves to analyze the welfare effect of an increase in price. Jerome chooses between clothing and medical care in the following figure. The original price of clothing is $1 per unit and the original price of medical care is p1. His budget constraint at the original prices is L1 and has a slope of -p1. His optimal bundle is a.

The price of medical care rises to p2 > p1, so that his new budget line, L2, has a slope of -p2. After the price increase, his optimal bundle is c. Jerome is worse off because of the price increase: He is on a lower indifference curve I2 (utility level U2) instead of I1 (utility level U1).

Compensating Variation. The amount of money that would fully compensate Jerome for a price increase is the compensating variation, CV. After the price increases to p2, Jerome is given enough extra income, CV, so that his utility remains at U1.

At his new income, Y + CV, Jerome's budget line is L*, which has the same slope, -p2, as L2. After the price change and this income compensation, he buys Bundle b.

How large is CV? Because the price of clothing is $1 per unit, the before-compensation budget line, L2, hits the clothing axis at Y, and the after-compensation budget line, L2*, hits at Y + CV. Thus, the gap between the two intercepts is CV.

This analysis is the same as we engaged in to determine the substitution and income effects of a price change. The compensating variation measure is the income (CV) involved in the income effect (the movement from b to c).

Equivalent Variation. The amount of income that, if taken from Jerome, would lower that consumer's utility by the same amount as the price increase is the equivalent variation, EV. The increase in price p2 harms Jerome by as much as a loss of EV would if the price remained at p1. That is, Jerome's income would have to fall by enough to shift the original budget constraint, L1, down to L1*, where it is tangent to I2 at Bundle d. Because the price of clothing is $1, EV is the distance between the intercept of L1 and that of L1* on the clothing axis. The key distinction between these two measures is that the equivalent variation is calculated using the new, lower utility level, whereas the compensating variation is based on the original utility level.

Comparing the Three Consumer Surplus Measures

Which consumer welfare measure is larger depends on the income elasticity. If the good is normal (as medical care is for Jerome), CV > CS > EV. With an inferior good, CV < CS < EV.

Although the three measures of welfare could, in principle, differ substantially, for most goods, they give similar answers. According to the Slutsky equation (Chapter 5), = * - , the uncompensated elasticity of demand, , equals the compensated elasticity of demand (pure substitution elasticity), *, minus the budget share of the good, , times the income elasticity, . The smaller the income elasticity or the smaller the budget share, the closer the substitution elasticity is to the total elasticity, and the closer is the compensated to the uncompensated demand curve. Thus, the smaller the income elasticity or the budget share, the closer the three welfare measures are to each other.

Because the budget share of most goods is small, these three measures are virtually identical. Even for aggregate goods on which consumers spend a relatively large share of their budget, these differences tend to be small. The following table shows estimates of these three measures for various goods based on the elasticity estimates by Laura Ann Blanciforti ["The Almost Ideal Demand System Incorporating Habits: An Analysis of Expenditures on Food and Aggregate Commodity Groups," Ph.D. thesis, U.C. Davis, 1982]. For each good, the table shows the income elasticity, the budget share, the ratio of compensating variation to the change in consumer surplus (CV/CS), and the ratio of the (absolute value of) equivalent variation to the change in consumer surplus (EV/CS) for a 50% increase in price.

The three welfare measures for alcohol and tobacco (which have the smallest income elasticity and budget share) are virtually identical. Because housing has the largest income elasticity and budget share, it has a relatively large gap between the measures. Even for housing, however, the difference between the change in uncompensated consumer surplus and either of the compensated consumer surplus measures is only 7%.

Willig (1976) shows that the differences between the three measures are small for small price changes regardless of the elasticities. Indeed, for the seven goods in the table, if the price change were only 10% (instead of 50% as in the table), the differences between CV or EV and CS are a small fraction of a percentage point for all goods except housing, where the differences are still only about 1%.

Thus, the three measures of welfare give very similar answers even for aggregate goods. As a result, economists frequently use the change in consumer surplus, which is relatively easy to calculate.

Table: Welfare Measures

Alcohol & Tobacco.394 99%100.4%

Source: These calculations are based on estimated linear expenditure system demand curves from Blanciforti (1992).

Figure: Compensating Variation and Equivalent Variation: At the original prices, Jerome's budget constraint is L1 and has a slope of -p1, where p1 is the price of medical care and the price of clothing is $1. He consumes Bundle a. When the price of medical care rises to p2, his new budget line L2 has a slope of -p2, and he consumes c. His utility falls from U1 on indifference curve I1 to U2 on indifference curve I2. If he receives a compensating amount of income, CV, he maintains his original utility level U1 after the price increase and consumes b. At the original prices, if his income falls by EV, he is harmed by as much as by the price increase, and consumes d.

© 2003 Jeffrey M. Perloff. Reprinted by permission.

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