Question

Consider a competitive firm with the short-run cost function

C(q) = 20 + 6q + 5q²

The firm faces a market price of p for its output.

a. Derive the firm's profit maximizing condition. Is the sufficient second order condition satisfied?

b. Suppose a specific tax of t (t < p) is levied on only this firm in the industry. What is the profit maximizing level of output as a function of p and t? (Assume the price is high enough that the firm does not shut down)

c. How does the output change as the tax increases? Use calculus to determine the relevant comparative static.

d. How does the firm's profit chance as the tax increases? Again, use calculus to determine the relevant comparative static. Show that profit decreases as t increases.

Answer 1

a)

Given

C(q)=20+6q+5q²

Total Revenue=R(q)=p*q

Profit is given by

For find the maximum profit, put

--------------------------------(Profit maximizing condition)

Let us check second order constion

we can see that value of second derivative is negative for

So, it satisfies the second order condition for maximization.

b)

Suppose a specific tax is imposed. Total cost is increased by tq

C(q)=20+6q+tq+5q²

Total Revenue=R(q)=p*q

Profit is given by

For find the maximum profit, put

c)

We have derived that

Let us find dq/dt

We can see that profit maximizing output will decrease by (1/10) units for every $1 increase in tax.

d)

It means that profit decreases as tax increases

Profit will decrease by $q as tax increases by $1