c.6.i. The Compensating Variation and the Equivalent Variation
The consumer in figure 6.i.3 starts at an optimum at point A. If the price of x were to rise,
the budget line would steepen as shown, and the consumer would move to a new
optimum at a point like B. The rise in price has left the consumer on a lower indifference
curve.
Suppose the consumer is a friend of yours, and you would like to compensate him for his
loss. You could do this by giving him DE units of good y. This gift would shift his budget
line upward as shown, enabling him to reach a new optimum at point C. As shown, C is
on the same indifference curve as A, so your gift has made the consumer just as happy as
he was before the price change. Because the gift of DE units compensated him for the
effect of the price change, the gift is called a compensating variation, or CV for short.
Note that the distance DE represents a certain quantity of good y—let us suppose it is 100
units of y. If the price of y is $1 per unit, then we could say that a gift of $100 is enough
to compensate this consumer for the loss he suffers from the price increase.
Figure 6.i.3
y
D
2. Giving him back DE units of y moves
him to C, leaving him back on his old
indifference curve and compensating
him for the rise in price.
Compensating
Variation
E
C
A
B
x
1. A rise in the price of x moves the
consumer from A to B, reducing his utility.
The compensating variation can be described in several different ways:
(1) the number of dollars it would take to compensate the consumer for the price increase.
(2) the dollar loss the consumer suffers from the price increase.
(3) the loss of consumer surplus resulting from the price increase.
Description (3) shows us the importance of the compensating variation: It gives us
another way to express consumer surplus.
Price
Figure 6.i.4
Loss of
Consumer surplus
=compensating variation
$2
$1
D
80
120
X
The demand curve shown in figure 6.i.4 shows that a rise in price, from $1 to $2, reduces
consumer surplus by the shaded area, which in this case is $100. If the rise in price causes
a $100 loss to the consumer, then a $100 gift would be enough to compensate the
consumer for the price increase. In other words, the loss of consumer surplus is, in this
case, the same thing as the compensating variation. We already know the importance of
consumer surplus for cost/benefit analysis. Now we have another tool, the compensating
variation, which can give us a better understanding of consumer surplus. We will see later
that the compensating variation is only approximately equal to the change in consumer
surplus. But for now, we can treat the two things as being equal.
y
Figure 6.i.5
2. Starting from point A, taking away EF
units of y moves him to C, which is on the
same indifference curve as B. The taking
of EF has an effect on his utility that is
equivalent to the price increase.
E
Equivalent
Variation
A
F
B
C
x
1. A rise in the price of x moves the
consumer from A to B, reducing his utility.
Figure 6.i.5 illustrates the equivalent variation, which gives us yet another way of
understanding consumer surplus. Starting from an optimum at point A, the price of x rises
and the consumer moves to a new optimum at point B, which is on a lower indifference
curve. Suppose that you are in a position to prevent the price of x from rising, and that the
consumer would be willing to pay you a bribe to do so. What is the maximum bribe he
would be willing to pay? The answer is that the consumer would be willing to pay up to
EF units of y to prevent the price from rising, because paying EF units leaves the
consumer at point C, which is on the same indifference curve as point B. In other words,
having to pay the bribe of EF hurts the consumer just as much as the price increase. A
bribe of EF units has an effect on his utility that is equivalent to the price increase, so we
call EF the equivalent variation. Since the equivalent variation gives us a measure of how
much the consumer loses because of the price increase, we can see that the equivalent
variation, like the compensating variation, gives us another way of expressing consumer
surplus.
Both the compensating variation and the equivalent variation are only approximately
equal to the change in consumer surplus. We will see later on that the compensating
variation normally gives a lower bound estimate of the change in consumer surplus, while
the equivalent variation normally gives us an upper-bound estimate. Thus we can be
confident that the true value of the change in consumer surplus lies somewhere between
the compensating variation and the equivalent variation.