Question

Suppose the price of good A is $2 and price of good B is $3. You have $90 to spend and your preferences over A and B are defined as: a^2/3*b^1/3 = U(a,b).

If income changes from $100 to $84, Pa = $2, Pb = $3 calculate and show work on how the optimal choice of A and B change and what the total utility achieved is given the Utility Function.

Answer 1

U(a,b) = a^2/3 b^1/3

MUa / MUb = [2/3* a^-1/3 b^1/3 ] / [1/3*a^2/3 b^-2/3] = 2b / a

optimal choice when

price of good A is $2 and price of good B is $3 and Income = 90

MUa /Mub = Pa / Pb

2b/a = 2/3

a = 3b

put this in budget constraint

90 = 2*3b + 3*(b)

90 = 6b + 3b

90 =9b

b = 10

a =10*3 = 30

optimal choice (a,b) = 30,10

now if income change from 90 to 84, then

a = 3b -- put this is new budget constraint

84 = 9b

b = 9.33

a = 28

optimal choice (a, b) = (28, 9.33)

If income fall from 90 to 84, quantity of both the goods a and b falls.

Total utility when income= 90

U =30^2/3 10^1/3 = 20.8

Total utility when income= 84

U =28^2/3 9.33^1/3 = 19.4