Question

Suppose that a consumer has a utility function U(x1,x2) = x1 ^0.5 x2^0.5 . Initial prices are p1 =1and p2 =1,andincomeism=100. Now, the price of good1 increases to 2. (a) On the graph, please show initial choice (in black), new choice (in blue), compensating variation (in green) and equivalent variation (in red). (b) What is amount of the compensating variation? How to interpret it? (c) What is amount of the equivalent variation? How to interpret it?

Answer 1

A). The first diagram is of CV and 2nd one is of EV

B). U = X1^1/2 X2^1/2

· In order to find demand fucntions

· MU_X1 = ½ X1 ^-1/2 X2^1/2

· MU_X2 = ½ X1 ^1/2 X2 ^-1/2

· MU_X1 /MU_X2

· ½ X1 ^-1/2 X2^1/2 / ½ X1 ^1/2 X2 ^-1/2

· X1 / X2…..and as we know MU_X1 /MU_X2 = P1/P2( slope of IC = slope of budget line.

· X1 / X2 = P1/P2

· X1P1 = X2P2………(1)

Our income function is

· M = P1X1 + P2X2……….(2)

Put the value of equation 1 in equation 2, we get

· M = 2P1X1

· X1 = M/2P1………(3)

· X2 = M/2P2……….(4)

Eq 3 and 4 is our demand function

And as we know P1 = 1 and P2 = 1

· X1 = 50 and X2 = 50

Compensating variation is a measure of utility change introduced by John Hicks. 'Compensating variation' is the amount of additional money a consumer would need to reach their original utility after a change in prices, a change in product quality, or the introduction of new products.

(2,1) as price change our income which we need to compensate the consumer become “m”

Now demand function

· ( m / 2*2 ) ( m / 2*1 )

· (m / 4) ( m / 2)

We need to compensate the consumer such that the new utility level becomes equal to original utility level.

· (m / 4)^1/2 ( m / 2) ^1/2 = (50)^1/2 (50) ^1/2

· m² / 8 = 2500

· m = 141

CV = 141 – 100 = 41

We need to give the consumer extra 41 income in order to compensate the consumer.

C). Equivalent variation is a measure of welfare changes which rekates with changes in prices. John Hicks is attributed with introducing the concept of compensating and equivalent variationThe equivalent variation is the change in income, at current prices, that would have the same effect on consumer welfare as would the change in prices, with income unchanged.

Price level (1,1)

· Level of utility ( 100 / 4 , 100 / 2)

· ( 25, 50)

· (m / 2)^1/2 ( m / 2) ^1/2 = (25)^1/2 (50) ^1/2

· m = 70

· EV = 100 – 70 = 30

He need only 70 income in (1,1) price level

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