EV and CV Examples - It is a nice tutorial about welfare analysis using EV and CV with numerical PDF

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It is a nice tutorial about welfare analysis using EV and CV with numerical examples. ...

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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 1

Disclaimer : This handout has not been reviewed by the professor. In the case of any discrepancy between 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, p1′ = 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 1.1 1.1.1

Calculating CV and EV Perfect Substitutes Example - u(x1 , x2 ) = 2x1 + 2x2 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.  0 if p1 > p2   m x1 (p1 .p2 , m) = if p1 < p2 p1   Any (x1 ,x2 ) that satisfies p1 x1 + p2 x2 = 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 p1′ > 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) + 2

m′ 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 p1′ ) 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) + 2

200 = 200 2

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. m′ + 2(0) 1 m′ = 100

unew = 200 = 2(

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 m p1 + 2p2 2m x2 (p1 , p2 , m) = p1 + 2p2 x1 (p1 , p2 , m) =

3

1.2.1

CV

Utility before the price change is uold = min{2

2m 2 ∗ 200 m , }= 5 1 + 2(2) 1 + 2(2)

uold = 80 To find m′ , set the utility function using the demand functions and the new prices equal to 80 2m′ m′ } , 80 = min{2 4 + 2(2) 4 + 2(2) 2m′ 80 = 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 m 2m 2 ∗ 200 unew = min{2 , }= 8 4 + 2(2) 4 + 2(2) 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′ 2m′ } , 1 + 2(2) 1 + 2(2)

2m′ 5 ′ m = 125 50 =

Thus EV is EV = m − m′ = 200 − 125 = 75 4

1.3

Cobb-Douglas - u(x1 , x2 ) = x13x22

Using our Cobb-Douglas demand trick, we know that 3m 5p1 2m x2 (p1 , p2 , m) = 5p2

x1 (p1 , p2 , m) =

We also know that we can apply the monotonic transformation of raising this utility function 2 3 to the power of 15 and getting a new utility function u′ (x1 , x2 ) = x12 x25 1.3.1

CV

Utility before the price change is u′old

=



3 ∗ 200 5∗1

53 

2 ∗ 200 5∗2

 25

= 17.68 ∗ 4.37 ≈ 77 Finding m′ 2 3  3 ∗ m′ 5 2 ∗ m′ 5 77 ≈ 5∗2 5∗4   35   25 2 3 77 ≈ m′ 10 20 

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. ′ unew

=



3 ∗ 200 5∗4

 35 

′ unew ≈ 34

5

2 ∗ 200 5∗2

 25

To get that utility under the old prices, income would have to be 34 ≈



3 ∗ m′ 5∗1

 53 

2 ∗ m′ 5∗2

2

5

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 functions one more time. Assuming we have an interior solution, we can set the MRS equal to the negative of the price ratio. −

p1 1 /1 = − p2 x1 p2 x1 = p1

We now plug that into our budget constraint to solve for x2 . p1

p2 + p2 x2 = m p1

p2 (1 + x2 ) = m m x2 = −1 p2 1.4.1

CV

At the old prices, utility is uold = ln

  2 200 −1 + 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   m′ 2 + 99.7 ≈ ln −1 4 2 m′ 101.4 ≈ 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   200 2 + unew = ln −1 4 2 ≈ −0.7 + 99.5 = 98.3 That same level of utility would be achieved if prices were unchanged but income were changed to m′   m′ 2 + 98.3 ≈ ln −1 1 2 m′ 98.6 ≈ 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 2

Hal 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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