Answer to Question #273280 in Microeconomics for Katie

Question

Suppose a competitive firm has long run total costs TC = 300 + 5 Q + 3 Q2. MC = 5 + 6 Q Now a tax is imposed on the firm—for each unit of output produced, it must pay $ 15 in taxes. Derive the new MC curve. What is minimum ATC? At what output does this occur?

Answer 1

Solution:

TC = 200 + 5Q + 3Q²

Additional $15 taxes per output

New additional total cost function:

TC = 200 + 5Q + 3Q² + 15Q

MC = $\frac{\partial TC} {\partial Q}$ = 5 + 6Q + 15

The new MC curve = 5 + 6Q + 15

 

Minimum ATC is where MC = ATC

Derive ATC:

ATC = 200 + 5Q + 3Q² + $\frac{150}{Q}$ = $\frac{300}{Q}$ + 5 + 3Q + 15

ATC = $\frac{300}{Q}$ + 5 + 3Q + 15

 

Set MC = ATC

5 + 6Q + 15 = $\frac{300}{Q}$ + 5 + 3Q + 15

Q = 10

 

Minimum ATC = $\frac{300}{10}$ + 5 + 3(10) + 15 = 30 + 5 + 30 + 15 = 80

Minimum ATC = 80

 

Minimum ATC occurs at an output of 10