Homework Assignment 8

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:

A homogeneous products duopoly faces a market demand function given by $Q=10-2P$, where $Q=Q_{1}+Q_{2}$. Both firms have a constant marginal cost $MC=2$.

  1. Suppose the two firms set their quantities simultaneously by guessing the other firm's quantity choice. Derive the equation of each firm's reaction curve and then graph these curves.

  2. What is the Cournot equilibrium quantity and price in this market for each firm?

  3. What would the equilibrium price in this market be if it were perfectly competitive?

  4. What is the Bertrand equilibrium price in this market?

Question 2:

Consider a duopoly with each firm having different marginal costs. Each firm has a marginal cost curve $MC_{i}=10+2Q_{i}$ for $i=1,2$. The market demand curve is $P=20-Q$, where $Q=Q_{1}+Q_{2}$.

  1. What are the Cournot equilibrium quantities and price in this market?

  2. What would be the equilibrium price in this market if the two firms acted as a profit-maximizing cartel $($i.e., attempt to set prices and outputs together to maximize total industry profits $)$?

  3. What would be the equilibrium price in this market if firms acted as price-taking firms $($there are still only two firms$)$?

  4. What is the Bertrand equilibrium price in this market?

Question 3:

Consider an oligopoly in which firms choose quantities. The inverse market demand curve is given by $P=a-b\left(q_{1}+q_{2}\right)$, where $q_{1}$ is the quantity produced by Firm 1, and $q_{2}$ is the quantity produced by Firm 2. Each firm has a marginal cost equal to $c$.

  1. What is the equilibrium market quantity if the two firms acted as a cartel $($i.e., attempt to set prices and outputs together to maximize total industry profits$)$. How about the equilibrium market price?

  2. Instead of cartel, suppose now firm 1 acts as the leader and firm 2 acts as the follower. What is the Stackelberg equilibrium quantities determined by each firm? What is the equilibrium market price?

  3. Find $\frac{\pi_1}{\pi_2}$ using the Stackelberg quantities and price. Whose profit is higher?

Question 4:

An industry has two firms, a leader and a follower. The demand curve for the industry’s output is given by $p = 50 - 5q$, where q is total industry output. Each firm has zero marginal cost. The leader chooses his quantity first, knowing that the follower will observe the leader’s choice and choose his quantity to maximize profits, given the quantity produced by the leader. What output will the leader choose?