Question

Given utility function U= X0.5Y0.5 where Px= 12 Birr, Bir, Py=4 Birr and the income of
the consumer is, M= 240 Birr.
A. Find the utility maximizing combinations of X and Y.
D. Calculate marginal rate of substitution of X for Y (MRSX.Y) at equilibrium and inierpiet
your result.

Answer 1

Solution:

A)

$$MU_x=0.5 \frac {y^{0.5}}{x^{0.5}}$$

$$MU_y=0.5 \frac {x^{0.5}}{y^{0.5}}$$

$${\frac{MU_x}{p_x}}=\frac {MU_y}{p_y}$$

$$x \times p_x+ y \times p_y=M$$

$$y=3x$$

$$12 \times x+4 \times y=240$$

$$x=10, y=30.$$

D)

$$\frac {\partial U}{\partial x \partial y} =\frac {0.25} {(xy)^{0.5}}$$

$$MRS x.y = \frac {\partial U} {\partial x \partial y} = \frac {0.25}{(10 \times 30)^{0.5}}=0.015$$