Question

Consider the market for good Q. The inverse demand function is p(Q) = 24 – 2Q, where p denotes the price of good Q. The production costs of the representative firm are C(Q) = 4Q. In addition, production causes environmental damage of D(Q) = 12Q.
a) Determine the socially optimal output level Q*. Discuss the optimality condition and illustrate your solution in a diagram.
b) Assume that there is no government intervention. Calculate the market equilibrium in the case of (i) a competitive market and (ii) a monopoly. Compare the output levels in the two equilibria with each other and with the socially optimal output level in part a), and explain the reasons for the differences between these output levels. Use a diagram to illustrate the two equilibria.
c) Assume now that the government levies a tax on output. Calculate the optimal tax level in the case of (i) a competitive market and (ii) a monopoly. Use a diagram to illustrate your solutions. Compare the tax levels, and explain why the tax differs in the two cases.

Answer 1

You are the manager of a small manufacturing company and an overpaid consultant has provided estimates of your firm's total revenue and total cost functions:
R(Q) = 2500Q - 3Q2
C(Q) = 100 + 2Q2.

a. What level of Q maximizes profits? How did you determine this?
First, find profit function:
PROF(Q) = R(Q) - C(Q)
PROF(Q) = 2500Q - 3Q2 - (100 + 2Q2)
PROF(Q) = 2500Q - 5Q2 - 100

Now, find the marginal profit function by taking the derivative of the profit function with respect to Q:
MAR_PROF(Q) = 2500 - 10Q
Finally, set marginal profit function equal to 0 & solve for Q:
2500 - 10Q = 0
Q* = 250