Question

1.      Given market demand Qd = 50 - P, and market supply P = Qs + 5

A.      Find the market equilibrium price and quantity

2.      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

1a

$Qd= 50 - P, P = Qs + 5 -> Qs = P - 5.$

The market equilibrium price and quantity are:

$Qd = Qs,\\ 50 - P = P - 5,\\ 2P = 55,\\ P = \$27.5. Q = 27.5 - 5 = 22.5 units.$

2)

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)

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