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Compensating and Eqivalent VariationsCompensating Variation: how much money should we give the consumer in orderto compensate her for the reduction in her well-being due to an increases in theprice of a good? This amount of money is called the compensating variation inincome. Note that we may not actually pay the consumer any money, we are just

trying to find a monetary measure of a loss in well being.

Look at Figure 1. Suppose the consumer is initially at point A. Then the price ofgood 1 increases and the BL rotates and becomes blue. The consumer is worse offsince she moves to a lower IC. How much money should we pay her to bring herback to her initial IC. In other words, how much money should we pay her to shifther BL to the green position? This amount of money is called compensatingvariation.

Figure 1

C

AB

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Equivalent Variation: This is the reduction in consumer income thatresults in a reduction in well being that is equaivalent to a loss of well beingresulting from a price incresae. Look at Figure 2. If the price of good 1 increases,the consumer will move from bundle A to B and will become worse off. If insteadof this price change, we had taken away some income from the consumer and as aresult the consumer had moved to the same indifference curve, that amount ofincome would have been called the equivaelnt variation.

Figure 2Equivalent Variation

A B

C

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Algebra of Compensating and Equivalent Variations:

Cobb-Douglas Utility Functions

Suppose we have:

U = X1.8X2

.2

M = $100P1 = $2P2 = $4

Therefore,

X1* = .8 100/2 = 40

X2* = .2 100/4 = 5

These are point A in Figure 8. Now P1 to $4. If we dont do anything, theconsumer will move to point B in Figure 8. She will be worse off because she will

be on a lower IC. How much money should we pay the consumer to compensateher for this price increase? Well, what is her utility at the initial bundle (point A)?

U = (40).8(5).2 = 26.40

If we give her some money to raise her income to M, how much will sheconsume at the new prices?

X1* = .8 M/4

X2* = .2 M/4

We want this new bundle to give the consumer the same amount of utility as theinitial bundle. So

U = (.8 M/4).8(.2 M/4).2 = 26.40orM(.15) = 26.40M = 174.11So the compensating variation is:

M - M = 174.11 - 100 = $74.11

Now the equivalent variation:

As a result of the increase in the price of good 1 to $4 the consumers optiomalbundle becomes

X1* = .8 100/4 = 20

X2* = .2 100/4 = 5

So her utility becomes

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U = (20).8(5).2 = 15.16Now, what level of income with the old prices would give the same amount ofutility? Let this level of income be M. Then

U = (.8M/2).8(.2M).2 = 15.16

M=57.43So the equivalent variation is100 - 57.43 = $42.57

Note in graphing that in the above cases when P1 changes, this change only affectsX1 and not X2. See Figure 10 below for the compensating vartaion. The case ofequivalent variation is similar. This is because of the nature of the utility function.If we had a different utility function, we would get a different result.

Compensating and Equivalent Variation:

Quasi-Linear Utility Functions

A consumer has the following utility function:

u = 2x10.5 + x2 (Quasi-Linear Utility Function)

Assume that initially m = $100, p1 = $2, and p2 = $10. Then p1 increases to $5.

Find the CV and EV.

1

x

u

2

=

1

1

2

1 1

1

1

x

u

x

u

x

xMRS ==

=

2

1

1

1

p

p

x= (MRS = Slope of budget line)

From this we find:

252

10

p

px

22

1

2*

1 =

=

= Plug this in the budget line to get:

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52

10

10

100

p

p

p

mx

1

2

2

*

2 === (verify this)

( ) 155252 5.* =+=u

Now let p1 = $5.

Increase consumer income to m such that with the new price the consumer can

reach u* = 15.

45

102

*'

1 =

=x

210

m'

5

10

10

m'x *'2 ==

( )

130m'

15210

m'42u

.5*

=

=+=

VC= 130 100 = 30

Now EV:

128)4(2

85

10

10

100

45

10

5.*

*

2

2

*

1

=+=

==

=

=

u

x

x

How much income should we take away from the consumer so that at the old

prices the consumer reaches u*= 12?

122

10

10

m'2(25)u .5* =+=

so, m = $70 and

EV = 100 - 70 = 30, same as CV.

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2

1

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Figure 3

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