CV and EV- Examples1

Before we do any examples, let’s make sure we have CV and EV straight. They’re similar,

but not the exact same thing, and which is which can get confusing. Both deal with changes

in price and how they affect utility - the difference lies in which prices (old or new) and

which utility level you use to calculate them.

In both situations, we have a price change. Price can go up or down. In the descriptions

below, the parts in parenthesis apply to decreases in prices, while the non-paranthetical parts

apply to increases in price.

Compensating Variation (CV) - CV is how much money we would have to give to (or

take away from) the consumer to get them back to the same level of utility that they

had before prices changed. So to calculated CV, you try to get the consumer to the

initial utility level at the new prices by changing income.

The process for calculating CV is generally as follows:

1. Find demand functions

2. Plug demand functions into utility function

3. Solve for utility level achieved at the old income and old prices

4. Set the value found in step 3 equal to the utility function using new prices and

unknown new income, and solve for new income

5. Subtract old income from income found in step 4 - that’s CV

Equivalent Variation (EV) - EV is how much money the consumer would be willing to

give up (or be paid) to prevent prices from changing - it is the change in income that

would get them to the same new utility level as the change in price would if it happened.

Thus to get EV, we get the consumer to the different utility level under old prices

by changing income.

The process for calculating EV is generally as follows:

1. Find demand functions

2. Plug demand functions into utility function

1Disclaimer : This handout has not been reviewed by the professor. In the case of any discrepancybetween this handout and lecture material, the lecture material should be considered the correct source.Despite all efforts, typos may find their way in - please read with a wary eye.

Prepared by Nick Sanders, UC Davis Graduate Department of Economics 2007

3. Solve for utility level achieved at the old income and new prices

4. Set the value found in step 3 equal to the utility function using old prices and

unknown new income, and solve for new income

5. Subtract income found in step 4 from old income - that’s EV

What follows are four examples of calculating CV and EV, using the preference relations

and utility functions we’ve seen most in class. The prices and incomes will be the same in all

examples: p1 = 1, p′1 = 4 (where p′

1 is the price of good 1 after the price change), p2 = 2, and

m = 200. Income after transfers is m′. Utility calculated using old prices will be referred to

as uold. Utility calculating new prices will be referred to as unew.

1 Calculating CV and EV

1.1 Perfect Substitutes Example - u(x1, x2) = 2x1 + 2x2

1.1.1 CV

Remember for CV, we use initial utility and new prices. That means we’ll have to calculate

the utility level under the original price vector p = (1, 2). Remember that with perfect

substitutes, we have three different possible demand functions, which vary with prices.

x1(p1.p2, m) =

0 if p1 > p2

mp1

if p1 < p2

Any (x1,x2) that satisfies p1x1 + p2x2 = m if p1 = p2

and similarly for good 2 (with the inequalities reversed). Since at the original prices p1 < p2,

the consumer will demand only good 1. Utility before the price change will be uold =

2(200) + 2(0) = 400.

After the price change, we’ll have p′1 > p2, so the consumer will shift all their demand

over to good 2. We now have to solve for a m′ that gives them a utility equal to uold under

the new prices.

uold = 400 = 2(0) + 2m′

2

400 = m′

To get their CV, subtract m from m′;

CV = m′ −m = 400− 200 = 200

2

What does it all mean? CV tells us, after prices have changed (p1 has gone up to p′1) how

much money would be required to get the consumer back to the utility level they had before

the price change. Put another way, the consumer is indifferent between the world where

they face prices (1, 2) and have income of 200 and the world where they face prices (4, 2)

and have an income of 400.

1.1.2 EV

To calculate EV we use new utility and old prices. That means we should find what their

new utility level would be if they faced the price vector (4, 2) and still had their old income.

Since they face the new prices, their demand will be positive only for good 2.

unew = 2(0) + 2200

2= 200

Now, what level of new income m′ would get them to that same level of utility at the old

prices? At the old prices, p1 < p2, so they’ll only have positive demand for good 1.

unew = 200 = 2(m′

1+ 2(0)

m′ = 100

To get EV, subtract m′ from m.

m−m′ = 200− 100 = 100

EV tells us, at the old prices (before the change) how much money we would have to take

away from the consumer to get them to the same new utility level they would reach if they

had their old income but faced the new prices (4, 2). Put another way, the consumer is

indifferent between facing the price change and keeping their old income and giving up the

amount m−m′ but keeping prices at their old level.

Still confusing? Let’s see if more examples clear it up.

1.2 Perfect Compliments Example - u(x1, x2) = min{2x1, x2}

With this utility function, our demand functions will be

x1(p1, p2, m) =m

p1 + 2p2

x2(p1, p2, m) =2m

p1 + 2p2

3

1.2.1 CV

Utility before the price change is

uold = min{2 m

1 + 2(2),

2m

1 + 2(2)} =

2 ∗ 200

5

uold = 80

To find m′, set the utility function using the demand functions and the new prices equal to

80

80 = min{2 m′

4 + 2(2),

2m′

4 + 2(2)}

80 =2m′

8

m′ = 320

Again, CV is the new income minus the old income.

CV = m′ −m = 320− 200

= 120

1.2.2 EV

Utility after the price change is

unew = min{2 m

4 + 2(2),

2m

4 + 2(2)} =

2 ∗ 200

8

unew = 50

To find m′, we find what income would be required to get the consumer to a utility level of

50 under the old prices.

50 = min{2 m′

1 + 2(2),

2m′

1 + 2(2)}

50 =2m′

5

m′ = 125

Thus EV is

EV = m−m′ = 200− 125

= 75

4

1.3 Cobb-Douglas - u(x1, x2) = x31x

22

Using our Cobb-Douglas demand trick, we know that

x1(p1, p2, m) =3m

5p1

x2(p1, p2, m) =2m

5p2

We also know that we can apply the monotonic transformation of raising this utility function

to the power of 15

and getting a new utility function u′(x1, x2) = x321 x

252

1.3.1 CV

Utility before the price change is

u′old =

(3 ∗ 200

5 ∗ 1

) 35(

2 ∗ 200

5 ∗ 2

) 25

= 17.68 ∗ 4.37

≈ 77

Finding m′

77 ≈(

3 ∗m′

5 ∗ 4

) 35(

2 ∗m′

5 ∗ 2

) 25

77 ≈ m′(

3

20

) 35(

2

10

) 25

m′ ≈ 458

Thus

CV ≈ 458− 200 = 258

1.3.2 EV

Remember, EV deals with the new utility after the price change but with old income.

u′new =

(3 ∗ 200

5 ∗ 4

) 35(

2 ∗ 200

5 ∗ 2

) 25

u′new ≈ 34

5

To get that utility under the old prices, income would have to be

34 ≈(

3 ∗m′

5 ∗ 1

) 35(

2 ∗m′

5 ∗ 2

) 25

m′ ≈ 88

So EV would be

EV ≈ m−m′ = 200− 88

≈ 112

1.4 Quasilinear - u(x1, x2) = lnx1 + x2

Quasilienar preferences are tricky, so let’s run through the process of finding demand func-

tions one more time. Assuming we have an interior solution, we can set the MRS equal to

the negative of the price ratio.

− 1

x1

/1 = −p1

p2

x1 =p2

p1

We now plug that into our budget constraint to solve for x2.

p1p2

p1

+ p2x2 = m

p2(1 + x2) = m

x2 =m

p2

− 1

1.4.1 CV

At the old prices, utility is

uold = ln

(2

1

)+

200

2− 1

≈ 0.7 + 99

≈ 99.7

6

To reach that level of utility under the new price vector p′, the income m′ would have to

be

99.7 ≈ ln

(2

4

)+

m′

2− 1

101.4 ≈ m′

2

m′ ≈ 202.8

CV ≈ 202.8− 200 = 2.8

1.4.2 EV

Utility at the new price level with the old income would be

unew = ln

(2

4

)+

200

2− 1

≈ −0.7 + 99.5 = 98.3

That same level of utility would be achieved if prices were unchanged but income were

changed to m′

98.3 ≈ ln

(2

1

)+

m′

2− 1

98.6 ≈ m′

2

m′ ≈ 197.2

EV ≈ 200− 197.2 = 2.8

Note that unlike in the last three examples, here we have a situation where CV = EV.

That’s because we’re dealing with quasilinear preferences. Since quasilinear preferences

have indifference curves that are parallel to each other, we’ll always have the CV = EV (see

figure 14.5 in chapter 14 of Varian for a graphical illustration of this result)2.

2 So What?

Sometimes you have to wonder why economists calculate these kinds of things. We already

have utility functions to tell us that if prices change, people’s utilities changes . . . isn’t

2Hal Varian, Intermediate Microeconomics : A Modern Approach (New York: W. W. Norton & Company,2006 7th ed)

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that enough? The problem with utility is there’s no easy translation into ”real world” units.

We can tell that one situation is better than another, but that’s about all (remember, utility

rankings are only ordinal, not cardinal, so you can’t say how much better or worse something

is using utility).

CV and EV are nice because they give us a monetary measure of that abstract change

in utility. So when someone asks “how much worse off would a consumer be because of the

price change?”, you can say “They’re as worse off as they would have been if we had taken

EV away from their income” or “They’re worse off such that we would need to give them

CV after prices changed to make them just as happy as they were before”. Both statements

are a mouthful, but they’re more informative than being stuck with a response like “All we

can say is they have a lower utility now”.

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