Question

Given utility function U=X^0.5Y^0.5 Where P_x=4 Birr and p_y=4 Birr and the income of the consumer M=240

a. Find the utility maximizing combination of X and Y.

b. Calculate marginal rate of substitution of X for Y (MRS_x,y) at equilibrium and interpret your result

Answer 1

Solution:

A)

$$MU_x=0.5 \frac {y^{0.5}}{x^{0.5}}$$ $$MU_y=0.5 \frac {x^{0.5}}{y^{0.5}}$$ $${\frac{MU_x}{p_x}}=\frac {MU_y}{p_y}$$ $$x \times p_x+ y \times p_y=M$$ $$y=3x$$ $$4 \times x+4\times y=240$$ $$x=15, y=45.$$

D)

$$\frac {\partial U}{\partial x \partial y} =\frac {0.25} {(xy)^{0.5}}$$ $$MRS x.y = \frac {\partial U} {\partial x \partial y} = \frac {0.25}{(15 \times 45)^{0.5}}=0.00962$$