Question

A firm has the following production 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 the two goods.
i. Formulate the firms’ optimization problem.
ii. Compute the optimal input combination of good 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:

$$Q=2(xy)^2$$

$$p_x=10, p_y=40, I=80$$

i)

$$\frac{\frac{\partial Q}{\partial x}}{p_x}=\frac{\frac{\partial Q}{\partial y}}{p_y}$$

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

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

ii)

$$4y=x$$

y=1; x=4.

iii)

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

iv)

An increase in budget x will lead to a decrease in production y.