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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 *p*_{1}. His budget constraint at the original prices is *L*^{1} and has a
slope of -*p*_{1}. His optimal bundle is *a*.

The price of medical care rises to *p*_{2} > *p*_{1}, so that his new budget line, *L*^{2},
has a slope of -*p*_{2}. 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 *I*^{2}
(utility level *U*_{2}) instead of *I*^{1} (utility level *U*_{1}).

**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 *p*_{2}, Jerome is given enough extra income, *CV*, so
that his utility remains at *U*_{1}.

At his new income, *Y + CV*, Jerome's budget line is *L**, which has the same slope,
-*p*_{2}, as *L*^{2}. 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, *L*^{2}, hits the clothing axis at *Y*, and the after-compensation
budget line, *L*^{2*}, 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 *p*_{2} harms Jerome by as much
as a loss of *EV* would if the price remained at *p*_{1}. That is, Jerome's income would
have to fall by enough to shift the original budget constraint, *L*^{1}, down to *L*^{1*},
where it is tangent to *I*^{2} at Bundle *d*. Because the price of clothing is $1, *EV* is the
distance between the intercept of *L*^{1} and that of *L*^{1*} 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 | ||||
---|---|---|---|---|

IncomeElasticity, | BudgetShare | EV---- CS | CV---- CS | |

Alcohol & Tobacco | .39 | 4 | 99% | 100.4% |

Food | .46 | 17 | 97 | 103 |

Clothing | .88 | 8 | 97 | 102 |

Utilities | 1.00 | 4 | 98 | 101 |

Transportation | 1.04 | 8 | 97 | 103 |

Medical | 1.37 | 9 | 95 | 104 |

Housing | 1.38 | 15 | 93 | 107 |

**Figure: Compensating Variation and Equivalent Variation:** At the original prices,
Jerome's budget constraint is *L*^{1} and has a slope of -*p*_{1}, where *p*_{1} 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 *p*_{2}, his new budget line *L*^{2} has a slope of -*p*_{2}, and
he consumes *c*. His utility falls from *U*_{1} on indifference curve *I*^{1} to *U*_{2} on
indifference curve *I*^{2}. If he receives a compensating amount of income, *CV*, he
maintains his original utility level *U*_{1} 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*.