Homework Assignment 7

Intermediate Microeconomics (Econ 100A)
Kristian López Vargas
UCSC

Instructions:

You will scan your homework into a single PDF file to be upload.
Only legible assignments will be graded.
Late assignments will not be accepted.

Only two randomly-chosen questions will be graded.

Question 1:

Suppose that the monopolist faces a linear demand curve, $P(Q) = A - BQ$. Further suppose that the monopolist has the marginal cost function: $MC = Q$.

  1. Find the revenue as a function of Q.
  2. Find the marginal revenue as a function of Q.
  3. Find the quantity that maximizes the monopolist's profit as a function of A and B.
  4. Find the equilibrium price as a function of A and B.
  5. Let's use some numbers. Suppose $A=10$ and $B=2$. Solve for the profit-maximizing quantity and price.
  6. Using $A=10$ and $B=2$, draw a demand curve and a marginal revenue function as well as marginal cost. Shade the deadweight loss. Also label clearly the profit-maximizing quantity and price chosen by the monopolist.
  7. What would have been the competitive equilibrium price and quantity $($hint: equate MC and the demand function$)$? Label the competitive equilbrium point. Also compute the size of the deadweight loss due to inefficiency casued by the monopolist behavior.

Question 2:

Suppose a monopolist faces a market demand of $Q^D=500-P$ and has a total cost function of $TC\left(Q\right)=4Q^{2}$.

  1. What is the equilibrium price and quantity decided by the monopolist?

  2. What is the average cost at the equilibrium quantity?

  3. How much profit does the monopolist make at the equilibrium price and quantity?

Question 3:

Suppose a monopolist faces a demand curve $Q^d = 200 - P $ and that the monopolist has a constant marginal cost of $c$ where $ 0 < c <200$. Find the monopolist’s profit-maximizing quantity and price; and describe how they vary with the marginal cost $c$.

Question 4:

A profit-maximizing monopolist faces a downward-sloping demand curve that has a constant elasticity of -2. The firm finds it optimal to charge a price of $8 for its output. What is its marginal cost at this level of output?