Question

A perfectly competitive firm has the cost function TC = 1000 + 2Q + 0.1 Q2. What is the lowest price at which this firm can break even

Answer 1

A perfectly competitive firm will be earning an economic profit of zero in the long run and, therefore, the break-even point will be where the Average cost curve intersects the Marginal cost curve. The lowest price will be at that point where MC = ATC.

ATC is the TC equation divided by q, while MC is the derivative of Total cost with respect to quantity.

First derive MC:

TC = 1000 + 2Q + 0.1Q²

MC = $\frac{\partial TC} {\partial Q} = 2 + 0.2Q$

 

Then derive ATC:

ATC = $\frac{1000 + 2Q +0.1Q^{2} }{Q}$

ATC = $\frac{1000 }{Q} +2Q +0.1Q^{2}$

Now set: MC = ATC

2 + 0.2Q = $\frac{1000 }{Q} +2Q +0.1Q^{2}$

0.2Q – 0.1Q = $\frac{1000 }{Q} +2 -2$

0.1Q = $\frac{1000 }{Q}$

0.1Q² = 1000

Q² = $\frac{1000 }{0.1}$

Q² = 10,000

Take the square root of both sides and find:

Q = 100

We know that the firm produces were Price = MR = MC, so we will derive the lowest price from the MC function:

 

MC = 2 + 0.2Q

We know Q = 100: Substitute in the equation

MC = 2 + 0.2 (100) = 2 + 20 = 22

Price = 22

 

Therefore, the lowest price at which the firm can breakeven is 22.