Question

6) For each of the following utility functions derive directly from the definition not using the formula(s) from class or the tert - the MRS (marginal rate of substitution) of y for r at (2,4) (a) ua(z, y)= 1/2y1/2 (b) uo(x, y) 2y/2 (b) u(r, y)4y4 (d) udr,y)-1/ 1/y

Please explain in slow steps how MRS can be derived using definition of MRS from the ratio of partial derivatives. No specific information is needed about class formulas.

Answer 1

MRSyx = Marginal utility of x/ marginal utility of y

Marginal utility of x = du/dx

Marginal utility of y = du/dy

(a) MU_x = du/dx = 1/2 (y/x)^1/2

MU_y = du/dy = 1/2 (x/y)^1/2

MRS = y/x

MRS at (2,4) = 4/2 = 2

(b) MUx = 1/2 1/ x^1/2

MUy = 1/2 1/ y^1/2

MRS = (y/x )^1/2 = 2^1/2

c) MUx = 1/ x^3/4

MUy = 1/ y^3/4

MRS = (y/x)^3/4 = (2)^3/4 = 2× 2^1/2

d) MUx = 1/ x²

MUy = 1/ y²

MRS = (y/x)² = 4