Question

U(X,Y)=X² Y²

(A) If we put x on the horizontal axis and y on the vertical axis, what is stevens marginal rate of substitution as a function of x and y?

(B) Suppose stevens in come $600 and the price of good X and Y are Px=2 and Py=6 respectively. what is stevens optimal consumption bundle, (X*,Y*) (that maximizes his utility)?

(C) Now suppose the price of good y changes. with new prices, we observe that stevens marginal rate of substitution at the optimal bundle is 1. does this imply that the price of good y has gone up or gone down? explain your answer?

Answer 1

U = X²Y²

(A) MRS = MPX / MPY

MPX = U / X = 2XY²

MPY = U / Y = 2X²Y

MRS = MPX / MPY = Y / X

(B) Budget line: I = X.Px + Y.Py

600 = 2X + 6Y [300 = X + 3Y]

Utility is maximized when MRS = Px / Py = 2/6 = 1/3

Y / X = 1/3

X = 3Y

Substituting in budget line,

300 = X + X = 2X

X* = 150

Y* = X/3 = 150/3 = 50

(C) New budget line: 600 = 2X + Y.Py [Py: New price of Y]

New MRS = 1

Y / X = 1

X = Y

Substituting in budget line,

600 = 2X + Y.Py

600 = 2X + X.Py [Since y = X]

600 = X.(2 + Py)

2 + Py = 600/X = 600/150 = 4

Py = 4 - 2 = 2

Therefore, Price of good Y has gone down (From 6 to 4).