Question

competitive firm produces output using three fixed factors and one variable factor. The firm’s short-run production function is q = 305x − 2x2, where x is the amount of the variable factor used. The price of the output is £2 per unit and the price of the variable factor is £10 per unit. In the short run, how many units of x should the firm use ?

Answer 1

Solution:

With price per unit of output = £2 per unit

Total revenue = price*quantity

TR = 2*q = 2*(305x - 2x²)

TR = 610x - 4x²

Cost (the variable part) = price of variable input*quantity of variable input

C = 10*x

Assuming fixed cost to be $F.

Total cost = variable cost + fixed cost

TC = 10x + F

(Note that amount of fixed cost has no impact on optimal level of variable factor to be used).

So, profit, Z = total revenue - total cost

Z = 610x - 4x² - 10x - F

Z = 600x - 4x² - F

Optimal amount of x can be found using first order condition: dZ/dx = 0

dZ/dx = 600 - 2*4x - 0

dZ/dx = 600 - 8x

Then, using FOC: 600 - 8x = 0

So, x = 600/8 = 75 units