Problem Set 3

Econ 502: Advanced Microeconomics

Market Structure and Welfare

Consider a market with inverse demand \(P = 150 - Q\), where \(Q\) is total output. There are \(n\) identical firms, each with constant marginal cost \(c = 30\) and no fixed costs.

Denote each firm’s output as \(q_i\) for \(i = 1, \ldots, n\), so that total output is \(Q = \sum_{i=1}^n q_i\).

Part I: Benchmarks

  1. Perfect competition. Suppose firms are price-takers (\(P = MC\)). Find the equilibrium price, total output, consumer surplus, and total surplus.

  2. Monopoly. Suppose there is a single firm (\(n = 1\)). Derive the marginal revenue function. Find the profit-maximizing quantity, price, and profit. Calculate consumer surplus and the deadweight loss relative to perfect competition.

Part II: Price Competition

  1. Bertrand. Suppose \(n \geq 2\) firms compete by simultaneously setting prices. Consumers buy from the cheapest firm; if prices are tied, they split the market equally. What is the Nash equilibrium? Compare the outcome to your answer in part (a) and briefly explain the economic logic.

Part III: Quantity Competition

  1. Cournot duopoly. Now suppose \(n = 2\) firms compete by simultaneously choosing quantities, with total output \(Q = q_1 + q_2\). Write down firm 1’s profit function and derive its best response function \(q_1^*(q_2)\). Find the Cournot-Nash equilibrium: each firm’s output, the market price, and per-firm profit.

  2. Cournot with \(n\) firms. Generalize part (d) to \(n\) identical firms. Show that in the symmetric Cournot-Nash equilibrium, per-firm output is given by:

\[q^* = \frac{120}{n+1}\]

  1. Convergence. Using the result from part (e), fill in the following table and compute the markup \(\mu\), which is defined price relative to marginal cost: \[ \mu = \frac{P^*}{c} \]
\(n\) Per-firm output (\(q^*\)) Price (\(P^*\)) Per-firm profit (\(\pi^*\)) Markup (\(\mu\))
1
2
5
10
50

What happens to the markup as \(n\) gets large?

Part IV: Welfare

  1. Deadweight loss. Show that the deadweight loss from Cournot competition with \(n\) firms is given by:

\[DWL = \frac{7{,}200}{(n+1)^2}\]

How does the DWL change as \(n\) increases? Compare the monopoly DWL (\(n = 1\)) to the Cournot duopoly DWL (\(n = 2\)).