# Consider the utility function for a utility maximizing individual consuming two goods X and Y. U (X,Y) = X2Y + 15. This person pays 3 dollars for good X and 5 dollars for good Y with an income of 150 dollars. ( 3 X + 5 Y ≤ 150) Budget constraint. • Find the marginal rate of substitution between x and y. • Find the amounts of good X and Y that maximizes this utility. • Compute the utility.

Consider the utility function for a utility maximizing individual consuming two goods X and Y.

U (X,Y) = X2Y + 15.

This person pays 3 dollars for good X and 5 dollars for good Y with an income of 150 dollars.

( 3 X + 5 Y ≤ 150) Budget constraint.

• Find the marginal rate of substitution between x and y.
• Find the amounts of good X and Y that maximizes this utility.
• Compute the utility.

u = X2Y +15

(1) MRS = MUx/MUy

MUx = dU/dx = 2X

MUy = dU/dY = X2

MRS = 2X/X2 = 2/X

(2) Utility is maximized when slope of budget line = MRS

Slope of budget line = Px/Py= 3/5

Equalting 2/X = 3/5

3X = 15 i.e. X =5

if X =5 then Y can be determined using the Budget constraint

3X + 5Y = 150

3(5) + 5 Y = 150

5Y = 135

Y = 27

(3) If x =5 and Y = 27 then utility

U = X2Y + 15 = (5)2 * 27 + 15 = 690

Asked on February 14, 2018 in