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 directly from the definition. I am not too strong on this topic and I am confused what to do. Ignore class formulas thank you!

Answer 1

This is a very simple problem which requires calculation of slope of these curves i.e. MRS

MRS = MUx/MUy = (dU/dx) / (dU/dy) = (Partial derivative of U wrt X) / (Partial derivative of U wrt Y)

a. $\frac{\frac{1}{2}x^{-\frac{1}{2}}y^{\frac{1}{2}}}{\frac{1}{2}y^{-\frac{1}{2}}x^{\frac{1}{2}}} = \frac{y}{x}$

Similarly for others:

b. $\frac{\frac{1}{2}x^{-\frac{1}{2}}}{\frac{1}{2}y^{-\frac{1}{2}}} = (\frac{y}{x})^{\frac{1}{2}}$

c. $\frac{\frac{1}{4}x^{-\frac{3}{4}}}{\frac{1}{4}y^{-\frac{3}{4}}} = (\frac{y}{x})^{\frac{3}{4}}$

d. $(\frac{y}{x})^{2}$