Question

A firm has the following function Q = 2(xy)^0.5 where x is maize and Y is rice. The cost of maize is k10 and the cost of rice is k40 the firm has a budget of k80 to spend on two goods.
i). Formulate the firms optimization problem
ii). Compute the optimal input combination of x and y
iii). What level of output is associated with the optimal input combination
iv). What is the impact of a k1 increase in the budget

Answer 1

Solution:

i)

$$Q=2\times(xy)^{0.5}$$

$$P_x=10, P_y=40, I=80$$

$$P_xQ_x+P_yQ_y=I$$

$$Q_x=x, Q_y=y$$

ii)

$$10x+40y=80$$

$$\frac{\partial Q}{\partial x}=2y^{0,5}x^{-0.5}$$

$$\frac {\partial Q}{\partial y}=2x^{0.5}y^{-0.5}$$

$$\frac {\frac {\partial Q}{\partial x}}{P_x}=\frac {\frac{\partial Q}{\partial y}}{P_y}$$

$$\frac {2 y^{0.5}}{10 x^{0.5}}= \frac {2 x^{0.5}}{40 y^{0.5}}$$

$$x=4, y=1$$

iii)

$$Q=2(4\times 1)^{0.5}=4$$

iv)

An increase in budget for k1 will lead to a decrease in budget for k2