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

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