Can you please solve this step by step. Thank you so much!

6. Assume a consumer's preferences are given by the following utility function U(X,Y) = X^1/3Y^2/3

A) Find the marginal rate of substitution between good X and good Y. Show all work

MRS(X,Y) = (U/X)/(U/Y) => MRS(.) = *(Y/X)

B) Let P_xbe the price of good X, P_ybe the price of good Y, and M be income. Find the individual consumer demand functions for good X and good Y.Show all work

2 Conditions for utility maximization: 1) P_xX + P_yY = M and 2) MRS(.) = *(Y/X) = P_x/ P_ysolving 1) and 2) simultaneously yields the following demand functions for good X and good Y

X* = 1/3(M/P_x) and Y* = 2/3(M/P_y)

C) If P_x=P_y=$1 and M=$100 how many units of good X and good Y will maximize this consumersutility S.T. the budget constraint? What is the relative price of good X in terms of good Y?

X₁* = 33 1/3 and Y₁* = 66 2/3.U₁*(33 1/3, 66 2/3) = (33 1/3)^1/3* (66 2/3)^2/3= 53.60

The relative price of good X in terms of good Y is 1 unit of good Y.

D) Show your solution to part C) graphically (be careful).** SEE PAGE 4 **

E) Assume now the price of good X rises to $2 and nothing else changes, redo parts C) - D).

(use the same graph).

X₂* = 16 2/3 and Y₂* = 66 2/3.U₁*(16 2/3, 66 2/3) = (16 2/3)^1/3* (66 2/3)^2/3= 42.59

The relative price of good X in terms of good Y is 2 units of good Y.