Question

3. Suppose that a consumer has a utility function u(x₁, x₂) = x₁ + x₂. Initially the consumer faces prices (1, 2) and has income 10. If the prices change to (4, 2), calculate the compensating and equivalent variations. [Hint: find their initial optimal consumption of the two goods, and then after the price increase. Then show this graphically.]

please do step by step and show the graph

Answer 1

Perfect substitute preference

U = X1 + X2

MRS = MU1/MU2 = 1

Initially , P1/P2 = 1/2

Now as, MRS > P1/P2

MU1/P1> MU2/P2

Only X1 is consumed

so X1* = M/P1= 10/1 = 10

(X1, X2) = (10,0)

.

after price change, P1'/P2' = 4/2 = 2

so MRS < P1'/P2'

MU1/P1' < MU2/P2'

only 2 is consumed

X2*' = 10/2 = 5

(X1, X2)*' = (0,5)

now for CV : the income should be increased in a way that original utility level is restored at new price levels

so U1 = 10

U2 = 5

So, let new income level = M"

Then, 0 + M"/P2 = U1

M" = 10*2 = 20

CV = M"-M

= 20-10

= 10

_________________________

now EV ,let new income = M'

EV = M-M'

So, at original prices, new utility should be maintained

So, M'/1 = U2 = 5

M' = 5

EV = 10-5 = 5