Question

Charlie's utility function is xAxB. The price of apples used to be $1 per unit and the price of bananas was $2 per unit. His income was $40 per day. If the price of
apples increased to $2.25 and the price of bananas fell to $1.25,

a. compute the optimal consumption bundle for both goods before the price change;

b. Compute the daily income after the price change in order to be able to just afford his old bundle.

Answer 1

Answer)

Utility = X_AX_B

MU_A = X_B

MU_B = X_A

A person consumes at a point where,

Ratio of marginal utility = Ratio of price of goods

X_B/X_A = P_A/P_B

X_B/X_A = 1/2

X_A  = 2X_B

Budget is given as:

P_A(X_A) + P_B(X_B) = M

(2X_B) + 2(X_B) = 40

4X_B = 40

X_B = 40/4 = 10

X_A = 2(10) = 20

Optimal quantity of apple = 20

Optimal quantity of bananas = 10

Answer (b)

New price:

Bananas = $1.25

Apples = $2.25

Cost of old bundle at new prices is : 20(2.25) + 10(1.25) = 45+12.5 = $57.5

Increase in income required = 57.5 - 40 = $17.5

So, the individual must receive $17.5 extra in order to continue consuming the old bundle.