Suppose that consumer has utility function U(X1, X2)=X½,X½. He originally face price (1,1) and has income 100. Then the price of good1 increase to 2. What are the compensating and equivalent variation? Show your working clearly
To find the compensating variation and equivalent variation, we need to calculate the utility before and after the price change for good 1.
Before the price change:
Utility function: U(X1, X2) = X1^(1/2) * X2^(1/2)
Income: M = 100
Price of good 1: P1 = 1
Price of good 2: P2 = 1
Using the budget constraint, we have:
M = P1 * X1 + P2 * X2
100 = 1 * X1 + 1 * X2
We need to find the optimal bundle (X1, X2) that maximizes utility subject to the budget constraint. This can be done by using the Lagrangian method:
L = X1^(1/2) * X2^(1/2) + λ * (100 - X1 - X2)
Taking partial derivatives and setting them to zero:
∂L/∂X1 = 1/2 * X1^(-1/2) * X2^(1/2) - λ = 0
∂L/∂X2 = 1/2 * X1^(1/2) * X2^(-1/2) - λ = 0
Solving these equations, we get:
X1^(1/2) / X2^(1/2) = 1
X1 = X2
Substituting X1 = X2 into the budget constraint, we have:
100 = 1 * X1 + 1 * X1
100 = 2 * X1
X1 = 50
X2 = 50
So the optimal bundle before the price change is (50, 50).
The utility before the price change is:
U(X1, X2) = X1^(1/2) * X2^(1/2)
U(50, 50) = 50^(1/2) * 50^(1/2)
U(50, 50) = 50 * 50
U(50, 50) = 2500
After the price change:
Price of good 1: P1 = 2
Using the budget constraint, we have:
100 = 2 * X1 + 1 * X2
Again, using the Lagrangian method to find the optimal bundle:
L = X1^(1/2) * X2^(1/2) + λ * (100 - 2*X1 - X2)
Taking partial derivatives and setting them to zero:
∂L/∂X1 = 1/2 * X1^(-1/2) * X2^(1/2) - 2 * λ = 0
∂L/∂X2 = 1/2 * X1^(1/2) * X2^(-1/2) - λ = 0
Solving these equations, we get:
X1^(1/2) / X2^(1/2) = 2
X1 = 2 * X2
Substituting X1 = 2 * X2 into the budget constraint, we have:
100 = 2 * (2 * X2) + X2
100 = 4 * X2 + X2
100 = 5 * X2
X2 = 20
X1 = 2 * 20 = 40
So the optimal bundle after the price change is (40, 20).
The utility after the price change is:
U(X1, X2) = X1^(1/2) * X2^(1/2)
U(40, 20) = 40^(1/2) * 20^(1/2)
U(40, 20) = 2 * 2 = 4
The compensating variation (CV) is the amount of money needed to restore the consumer to their original utility level after the price change. It can be calculated as the difference in income between the two situations:
CV = M - M'
CV = 100 - 100
CV = 0
The equivalent variation (EV) is the amount of money needed to keep the consumer at the new utility level in the initial situation. It can be calculated as the difference in income between the two situations:
EV = M' - M
EV = 100 - 100
EV = 0
Therefore, the compensating variation and equivalent variation in this case are both zero.