Answer to Question #198929 in Microeconomics for robel legese

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

First derive MC:

$TC = 1000 + 2Q + 0.1Q^2$

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

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

$0.2Q – 0.1Q = \frac{1000 }{Q}$

$0.1Q = \frac{1000 }{Q}$

$0.1Q^2 = 1000$

$Q2 = \frac{1000 }{0.1}$

$Q2 = 10,000$

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.