Question

Given utility function U= where PX = 12 Birr, Birr, PY = 4 Birr and the income of the consumer is, M= 240 Birr. A. Find the utility maximizing combinations of X and Y. B. Calculate marginal rate of substitution of X for Y (MRSX,Y) at equilibrium and interpret your result

Answer 1

$U= X^{0.5}Y^{0.5}$

$MU_x​=0.5\frac{Y^{0.5}}{X^{0.5​}}$

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

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

$\frac{Y}{X}= \frac{12}{4}$

$P_xX+ p_yY=M$

y= 3x

$X= \frac{1}{3}Y$

Plug into the below equation;

12X+4Y=240

x=10, y=30

b)

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

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