Question

3. Given utility function U= X0.5Y0.5 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

a)

$$MU_x=\frac{y^{0.5}}{2x^{0.5}}$$

$$MU_y=\frac{x^{0.5}}{2y^{0.5}}$$

$$12x+4y=240$$

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

$$\frac{y^{0.5}}{24x^{0.5}}=\frac{x^{0.5}}{8y^{0.5}}$$

$$y=3x$$

$$x=10, y=30$$

b)

$$MRS_{x,y}=\frac{MU_x}{MU_y}$$

$$MRS_{x,y}=\frac{y}{x}=\frac{30}{10}=3$$