Question

Problem 3. A price-taking firm produces an output Q which sells for a price of $20 per unit. The firm’s short-run costs are as follows: Fixed costs = 50 Variable cost = 0.1Q2 + 10Q Marginal cost = 0.2Q + 10 a. What output should the firm produce? Don’t forget to consider the possibility that the firm might choose Q = 0. b. What are its profits? Please be thorough

Answer 1

A price taking firm maximizes profit at the point where P = MC
P = $20

So, P = MC gives,
20 = 0.2Q + 10
So, 0.2Q = 20 - 10 = 10
So, Q = 10/0.2 = 50
So, Q = 50

Profits = Total revenue - Total cost = P*Q - (Fixed cost + Variable cost) = (20*50) - 50 - 0.1Q² - 10Q = 1000 - 50 - 0.1(50)² - 10(50) = 1000 - 50 - 250 - 500 = 200

When Q = 0,
Profits = Total revenue - Total cost = P*Q - (Fixed cost + Variable cost) = (20*0) - 50 - 0.1Q² - 10Q = 1000 - 50 - 0.1(0)² - 10(0) = -50

So, firm will produce 50 units and earn a profit of $200.
It will not choose Q = 0 because it will be incurring loss = fixed cost.