Question

Phil’s utility function is U=X1/3Y2/3, where MU? = 3X2/3 and MU? = 3Y1/3 . In Los Angeles, Phil’s net income would equal $4500, and prices of good X and good Y are $5 and $8, respectively. In Washington, D.C., Phil’s net income would equal $6000, and the prices of good X and good Y are $8 and $10, respectively.

a)Where should Phil live if his goal is to maximize his utility? Show your calculations

b) What is the minimum income Phil would need in London where prices for X and Y are $10 and $12 respectively to achieve the same utility as he would in D.C.?

Answer 1

(a)

Utility maximizing condition:

In order to maximize utility a consumer should consume at a point where Budget line is tangent to IC curve.

Hence mathematically we can write utility maximizing condition as:

MUX/MUY = Px/Py and PxX + PyY = I , where I = income, Px and Py are prices of X and Y respectively

In Washington DC

Px/Py = 8/10

Here MUX = (1/3)(Y/X)^2/3 and MUY = (2/3)(X/Y)^1/3

MUX/MUY = (1/2)(Y/X)

Hence, MUX/MUY = Px/Py => (1/2)(Y/X) = 8/10 => Y = 1.6X

and PxX + PyY = I => 8X + 10Y = 6000--------------------------------------------Budget Constraint

Solving above equations we get:

8X + 10*1.6X = 6000

= X = 6000/24 = 250

Hence Y = 1.6X = 1.6*250 = 400

Hence Utility (U) = X^1/3Y^2/3 = 250^1/3400^2/3  = 342

In Los Angeles

Px/Py = 5/8

Here MUX = (1/3)(Y/X)^2/3 and MUY = (2/3)(X/Y)^1/3

MUX/MUY = (1/2)(Y/X)

Hence, MUX/MUY = Px/Py => (1/2)(Y/X) = 5/8 => Y = 0.8X

and PxX + PyY = I => 5X + 8Y = 4500--------------------------------------------Budget Constraint

Solving above equations we get:

5X + 8*0.8X = 4500

= X = 4500/11.4 = 394

Hence Y = 1.6X = 1.6*250 = 4500

=> X = 394 and Y = 0.8*X = 0.8*394 = 315

Hence Utility (U) = X^1/3Y^2/3 = 394^1/3315^2/3  = 340

Hence, Phil should live in Washington DC in order to maximize his utility.

(b)

Similarly as above

In Los Angeles

Px/Py = 5/8

Here MUX = (1/3)(Y/X)^2/3 and MUY = (2/3)(X/Y)^1/3

MUX/MUY = (1/2)(Y/X)

Hence, MUX/MUY = Px/Py => (1/2)(Y/X) = 10/12 => Y = 1.67X

and PxX + PyY = I => 10X + 12Y = I--------------------------------------------Budget Constraint

Solving above equations we get:

10X + 12*1.67X = I

= X = I/30.04

Hence Y = 1.67X = 1.67*I/30.04

In DC Utility = 342

Hence Utility (U) = X^1/3Y^2/3 = (I/30.04)^1/3(1.67I/30.04)^2/3 = 342

=> I = 342*30.04/(1.67)^2/3 = 7300(approx)

Hence, the minimum income Phil would need in London is $7300