Question

Find



Marginal Utility for x and determine if it is diminishing



Marginal utility for y and determine if its diminishing



Marginal rate of Substitution of x for y (MRS xy) and determine if its diminishing

U(x,y)=2x^(2/3)y^(1/3)

U(x,y)=x^3+4y^(1/4)

Answer 1

(1) U = 2x^2/3y^1/3

MUx = U / x = 2 x (2/3) x (y / x)^1/3 = (4/3) x (y / x)^1/3

As x rises, MUx falls, so MUx is diminishing in x.

MUy = U / y = 2 x (1/3) x (x / y)^2/3 = (2/3) x (x / y)^2/3

As y rises, MUy falls, so MUy is diminishing in y.

MRS_xy = MUx / MUy = 2 x (y / x)

As x rises, MRS_xy falls, so MRS_xy is diminishing in x.

(2) U = x³ + 4y^1/4

MUx = U / x = 3x²

As x rises, MUx rises, so MUx is not diminishing in x.

MUy = U / y = 4 x (1/4) / y^3/4 = 1 / y^3/4

As y rises, MUy falls, so MUy is diminishing in y.

MRS_xy = MUx / MUy = 3x² / (1 / y^3/4) = 3x².y^3/4

As x rises, MRS_xy rises, so MRS_xy is not diminishing in x.