Chapter 7Market Structures and Game Theory

Introduction

Chapter 6 derived the supply curve of a competitive firm: produce where $P = MC$. But this result assumes the firm is a price taker — so small relative to the market that it cannot influence the price. Many real markets violate this assumption. A single seller (monopolist) sets its own price. A handful of large firms (oligopolists) must account for rivals' reactions. This chapter maps the spectrum of market structures and introduces game theory as the language for strategic interaction.

By the end of this chapter, you will be able to:
  1. Characterize long-run competitive equilibrium and explain the zero-profit condition
  2. Solve a monopolist's pricing problem and compute deadweight loss
  3. Analyze price discrimination (first, second, and third degree)
  4. Solve Cournot, Bertrand, and Stackelberg oligopoly models
  5. Find Nash equilibria in normal-form games
  6. Apply the prisoner's dilemma to economic settings

Prerequisites: Chapter 6 (cost curves, profit maximization, Lagrangians).

7.1 Perfect Competition: Long-Run Equilibrium

In Chapter 6, we showed that a competitive firm maximizes profit at $P = MC$. In the long run, free entry and exit drives a further result.

Long-run competitive equilibrium. In the long run, entry occurs when existing firms earn positive economic profit (attracting new firms) and exit occurs when firms earn negative profit. Entry shifts the market supply curve right, pushing the price down; exit shifts it left, pushing the price up. The process continues until:
$$P = MC = AC_{min} \quad \text{and} \quad \Pi = 0$$ (Eq. 7.1)
Economic profit vs accounting profit. Economic profit subtracts all costs, including the opportunity cost of the owner's capital and time. Accounting profit subtracts only explicit (monetary) costs. In long-run competitive equilibrium, economic profit is zero but accounting profit is positive.

Zero economic profit does not mean firms suffer. It means they earn a normal return — exactly covering all costs, including the opportunity cost of capital. Accounting profit is still positive.

7.2 Monopoly

Monopoly. A market with a single seller. The monopolist faces the entire market demand curve and chooses quantity (or equivalently, price) to maximize profit.
$$\max_Q \; \Pi = P(Q) \cdot Q - TC(Q)$$ (Eq. 7.2)

where $P(Q)$ is the inverse demand function — it gives the price the monopolist must set to sell $Q$ units. Unlike the competitive firm (which takes price as given), the monopolist recognizes that selling more requires cutting the price.

Marginal Revenue

Marginal revenue. The additional revenue from selling one more unit. For a price-taking firm, $MR = P$. For a firm with market power, $MR < P$ because increasing output requires lowering the price on all units sold.
$$MR = \frac{dTR}{dQ} = P + Q\frac{dP}{dQ}$$ (Eq. 7.3)

This has two terms:

Output effect and price effect. The output effect is the gain from selling one additional unit at the current price. The price effect is the loss from reducing the price on all inframarginal units. Marginal revenue is the net of these two forces: $MR = \underbrace{P}_{\text{output effect}} + \underbrace{Q \cdot dP/dQ}_{\text{price effect}}$.

For a downward-sloping demand curve, $dP/dQ < 0$, so $MR < P$. For linear demand $P = a - bQ$: $TR = aQ - bQ^2$, so $MR = a - 2bQ$. The MR curve has the same intercept as the demand curve but twice the slope.

The Relationship Between MR and Elasticity

$$MR = P\left(1 - \frac{1}{|\varepsilon_d|}\right)$$

A monopolist never produces where $MR < 0$ (she could raise revenue by producing less), so she always operates on the elastic portion of the demand curve.

The profit-maximizing condition:

$$MR = MC$$ (Eq. 7.4)

The Lerner Index

Lerner Index. A measure of market power:
$$\frac{P - MC}{P} = \frac{1}{|\varepsilon_d|}$$ (Eq. 7.5)

The markup over marginal cost equals the inverse of the (absolute) price elasticity of demand. More elastic demand means less market power.

Example 7.1 — Monopoly Pricing

Demand: $P = 100 - 2Q$. Cost: $TC = 20Q$ (constant $MC = 20$).

$TR = 100Q - 2Q^2$, $MR = 100 - 4Q$.

$MR = MC$: \$100 - 4Q = 20 \implies Q_M = 20$, $P_M = 60$.

$\Pi = (60 - 20)(20) = 800$.

Competitive outcome: $P = MC = 20$, $Q_C = 40$.

$DWL = \frac{1}{2}(60 - 20)(40 - 20) = 400$.

Lerner index: $(60 - 20)/60 = 2/3$. Check: $\varepsilon_d = (dQ/dP)(P/Q) = (-1/2)(60/20) = -1.5$, so \$1/|\varepsilon_d| = 2/3$. ✓

Interactive: Monopoly Pricing

Adjust the marginal cost to see how the monopolist's optimal price, quantity, profit, and deadweight loss change. Toggle the competitive outcome overlay to compare.

\$1 \$15 \$10
Monopoly: QM = 20  |  PM = \$10.00  |  Profit = \$100.00  |  DWL = \$100.00  |  Lerner = 0.667

Figure 7.2. The monopolist restricts output to where MR = MC, charging above marginal cost. The blue rectangle is monopoly profit; the yellow triangle is deadweight loss. Toggle the competitive overlay to see the efficient outcome.

7.3 Price Discrimination

Price discrimination. Charging different prices to different consumers (or for different units) based on willingness to pay, not cost differences.

First-Degree (Perfect) Price Discrimination

The firm charges each consumer their maximum willingness to pay. This extracts all consumer surplus. Output is efficient ($Q = Q_C$) — no DWL — but all surplus goes to the firm.

Second-Degree Price Discrimination

The firm offers different pricing schemes (quantity discounts, bundling, versioning) and lets consumers self-select. Examples: airline tickets (business vs. economy), software (basic vs. pro edition), bulk pricing.

Third-Degree Price Discrimination

The firm identifies groups with different elasticities and charges each group a different price:

$$MR_1 = MR_2 = MC$$ (Eq. 7.6)

The group with more inelastic demand pays the higher price.

Example 7.2 — Third-Degree Price Discrimination

A theater faces two markets. Adult demand: $P_A = 20 - Q_A$. Student demand: $P_S = 12 - Q_S$. $MC = 2$.

Adults: $MR_A = 20 - 2Q_A = 2 \implies Q_A = 9$, $P_A = 11$.

Students: $MR_S = 12 - 2Q_S = 2 \implies Q_S = 5$, $P_S = 7$.

Total profit: $(11-2)(9) + (7-2)(5) = 81 + 25 = 106$.

Interactive: Third-Degree Price Discrimination

Two markets with different demand elasticities. Adjust MC to see how optimal prices and quantities change in each market.

\$1 \$1 \$10
Market A (Adults): Q = 9.0, P = \$11.00, Profit = \$11.00  |  Market B (Students): Q = 5.0, P = \$1.00, Profit = \$15.00  |  Total Profit = \$106.00

Market A (Adults): $P_A = 20 - Q_A$

Market B (Students): $P_S = 12 - Q_S$

7.4 Monopolistic Competition

Monopolistic competition. A market with many firms selling differentiated products. Each firm has some market power (downward-sloping demand due to product differentiation) but faces free entry.

Short run: Firms may earn positive or negative profit. Long run: Entry and exit drive economic profit to zero. Each firm produces where its demand curve is tangent to its AC curve — not at the minimum of AC.

This means monopolistic competition has two "inefficiencies" relative to perfect competition:

  1. Markup: $P > MC$ (market power from differentiation)
  2. Excess capacity: Firms produce below the scale that minimizes AC

Whether these are truly inefficient is debatable. The Dixit-Stiglitz framework shows that consumers value variety — having 50 different restaurants is worth more than 50 identical ones, even if the identical ones are cheaper. The markup above MC is the "price of variety."

7.5 Oligopoly: Cournot Competition

Oligopoly. A market with a few large firms, each aware that its actions affect others. Strategic interaction is the defining feature.

Cournot Model

Cournot competition. An oligopoly model in which firms simultaneously choose quantities. Each firm selects the quantity that maximizes its profit given its belief about the other firms' quantities.

Firms choose quantities simultaneously. Each firm's optimal quantity depends on the other firms' quantities.

Setup. Two firms, demand $P = a - b(q_1 + q_2)$, constant marginal cost $c$ for both.

Best response (reaction) function. Firm $i$'s optimal quantity as a function of the rival's quantity: $q_i^*(q_j)$. It solves $\max_{q_i} \Pi_i = (P(q_i + q_j) - c) q_i$. In Cournot equilibrium, every firm is on its best response function simultaneously.

Best response function for firm 1:

$$q_1^*(q_2) = \frac{a - c}{2b} - \frac{q_2}{2}$$ (Eq. 7.7)

Cournot-Nash equilibrium (solving simultaneously):

$$q_1^C = q_2^C = \frac{a - c}{3b}$$ (Eq. 7.9)
$$Q^C = \frac{2(a-c)}{3b}, \quad P^C = \frac{a + 2c}{3}$$ (Eq. 7.10)

With $n$ symmetric firms, $q_i = (a-c)/((n+1)b)$ and $P \to c$ as $n \to \infty$.

Example 7.3 — Cournot Duopoly

Demand: $P = 100 - Q$, $c = 10$. Best responses: $q_i^* = 45 - q_j/2$.

Equilibrium: $q_1^C = q_2^C = 30$. $Q^C = 60$, $P^C = 40$. $\Pi_i = 900$.

StructureOutputPriceIndustry ProfitDWL
Competition901000
Cournot duopoly60401,800450
Monopoly45552,0251,012.5

Interactive: Cournot with N Firms

Slide the number of firms from 1 (monopoly) to 20. Watch total output rise, price fall, and deadweight loss shrink toward zero as the market approaches perfect competition.

Monopoly (1) 10 Competition (20)
N = 2: qi = 30.0  |  Q = 60.0  |  P = \$10.00  |  Per-firm profit = \$100.00  |  DWL = \$150.00

Figure 7.3a. As N increases, the Cournot outcome converges to perfect competition. At N=1, this is monopoly. The bar chart shows how key outcomes change with market structure.

Interactive: Cournot Reaction Functions

Adjust each firm's marginal cost to see how their reaction functions shift and the equilibrium moves. Asymmetric costs lead to asymmetric output.

\$1\$10\$10
\$1\$10\$10
Equilibrium: q1 = 30.0, q2 = 30.0  |  Q = 60.0  |  P = \$10.00

Figure 7.3b. Each firm's reaction function is downward-sloping: more output by the rival reduces the optimal response. The intersection is the Cournot-Nash equilibrium. Drag the cost sliders to see how asymmetric costs shift the reaction functions and move the equilibrium.

7.6 Bertrand Competition

Bertrand competition. An oligopoly model in which firms simultaneously choose prices. Consumers buy from the lowest-price firm; if prices are equal, demand is split evenly.

In the Bertrand model, firms choose prices simultaneously (rather than quantities). With identical products and equal marginal costs:

$$P^B = c \quad \text{(Bertrand paradox)}$$ (Eq. 7.11)
Bertrand paradox. With two firms selling identical products at equal marginal cost, the unique Nash equilibrium is $P = MC$ — the perfectly competitive outcome. The paradox is that just two firms suffice to eliminate all market power, contradicting the Cournot prediction that market power persists with few firms.

With just two firms, price competition reproduces the perfectly competitive outcome. This is the Bertrand paradox: the Cournot model says you need many firms for competition; the Bertrand model says two suffice.

When the paradox dissolves:

Example 7.6 — Bertrand with Differentiated Products

Two firms sell differentiated goods. Demand for firm $i$: $q_i = 100 - 2p_i + p_j$ (products are substitutes but not identical). Marginal cost: $c = 10$.

Firm 1 maximizes: $\Pi_1 = (p_1 - 10)(100 - 2p_1 + p_2)$.

FOC: \$100 - 4p_1 + p_2 + 20 = 0 \implies p_1^*(p_2) = \frac{120 + p_2}{4} = 30 + p_2/4$.

By symmetry: $p^* = 30 + p^*/4 \implies p^* = 40$.

Each firm: $q^* = 100 - 80 + 40 = 60$. $\Pi^* = 30 \times 60 = 1{,}800$.

With differentiated products, equilibrium price (\$10$) exceeds marginal cost (\$10$). The Bertrand paradox dissolves because a small price cut no longer captures the entire market.

7.7 Stackelberg Competition

Stackelberg competition. A sequential oligopoly model in which one firm (the leader) chooses its quantity first, and the other firm (the follower) observes the leader's choice before selecting its own quantity.

In the Stackelberg model, one firm (the leader) moves first, choosing its quantity. The follower observes the leader's choice and then optimizes. The leader internalizes the follower's reaction function.

$$q_1^S = \frac{a - c}{2b}, \quad q_2^S = \frac{a - c}{4b}$$ (Eq. 7.12–7.13)
First-mover advantage. The strategic benefit of committing to an action before rivals can respond. In the Stackelberg model, the leader commits to a large quantity, forcing the follower to accommodate by producing less. The leader earns higher profit than in the simultaneous (Cournot) game.

The leader produces the monopoly quantity, and the follower produces half of that. Total output exceeds Cournot; the price is lower. The first-mover advantage comes from committing to a large quantity before the follower chooses.

Example 7.4 — Stackelberg

$P = 100 - Q$, $c = 10$:

$q_1^S = 45$, $q_2^S = 22.5$. $Q^S = 67.5$, $P^S = 32.5$.

$\Pi_1 = 1{,}012.5$ (leader), $\Pi_2 = 506.25$ (follower).

Leader's profit exceeds Cournot (\$1{,}012.5 > 900$). Follower is worse off (\$106.25 < 900$).

Interactive: Stackelberg vs Cournot

Toggle between simultaneous (Cournot) and sequential (Stackelberg) to compare quantities and profits using $P = 100 - Q$, $c = 10$.

Cournot: q1 = 30.0, q2 = 30.0  |  Q = 60.0, P = \$10.00  |  Π1 = \$100, Π2 = \$100

Figure 7.4. Compare Cournot (symmetric) and Stackelberg (leader advantage). The Stackelberg equilibrium is below-right of Cournot on the reaction function diagram: the leader produces more, the follower less.

7.8 Introduction to Game Theory

Game in normal (strategic) form. Consists of: (1) Players $i = 1, 2, \ldots, n$; (2) Strategies $S_i$ for each player; (3) Payoffs $u_i(s_1, \ldots, s_n)$ for each strategy combination.

Nash Equilibrium

Nash equilibrium. A strategy profile $(s_1^*, s_2^*, \ldots, s_n^*)$ such that no player can increase their payoff by unilaterally changing their strategy. Every player is best-responding to the strategies of all other players.
$$u_i(s_i^*, s_{-i}^*) \geq u_i(s_i, s_{-i}^*) \quad \forall s_i \in S_i, \; \forall i$$ (Eq. 7.14)

Each player is best-responding to the others. No one has a reason to deviate, given what everyone else is doing.

The Prisoner's Dilemma

Dominant strategy. A strategy that yields a weakly higher payoff than any alternative, regardless of what other players do. If $s_i^*$ is dominant, then $u_i(s_i^*, s_{-i}) \geq u_i(s_i, s_{-i})$ for all $s_i$ and all $s_{-i}$.
Prisoner's dilemma. A two-player game in which each player has a dominant strategy to defect, yet mutual cooperation yields a higher payoff for both. The Nash equilibrium (Defect, Defect) is Pareto-dominated by (Cooperate, Cooperate), illustrating the tension between individual rationality and collective welfare.
Player 2: CooperatePlayer 2: Defect
Player 1: Cooperate(3, 3)(0, 5)
Player 1: Defect(5, 0)(1, 1)

Dominant strategy: Defect is best regardless of the other's choice. Nash equilibrium: (Defect, Defect) with payoffs (1, 1). Both are worse off than mutual cooperation (3, 3), but neither can unilaterally improve.

Why the prisoner's dilemma matters:

Interactive: 2×2 Game Payoff Explorer

Enter any payoffs for a 2×2 game. The tool auto-identifies dominant strategies, Nash equilibria, and Pareto-optimal outcomes. Green cells are Nash equilibria; blue borders mark Pareto-optimal outcomes.

Player 2: L Player 2: R
Player 1: U (, ) (, )
Player 1: D (, ) (, )

Blue = Player 1 payoff  |  Red = Player 2 payoff

Analyzing...

Other Classic Games

Coordination game:

B: LeftB: Right
A: Left(2, 2)(0, 0)
A: Right(0, 0)(1, 1)

Two Nash equilibria: (Left, Left) and (Right, Right). The challenge is coordination, not conflict.

Battle of the sexes:

B: OperaB: Football
A: Opera(3, 1)(0, 0)
A: Football(0, 0)(1, 3)

Two pure-strategy Nash equilibria with different preferred outcomes for each player.

Example 7.5 — Nash Equilibria in an Advertising Game

Two firms choose whether to Advertise (A) or Not Advertise (N):

Firm 2: AFirm 2: N
Firm 1: A(4, 4)(7, 2)
Firm 1: N(2, 7)(5, 5)

Step 1 — Check for dominant strategies.

Firm 1: If Firm 2 plays A, Firm 1 gets 4 (A) vs 2 (N) → A is better. If Firm 2 plays N, Firm 1 gets 7 (A) vs 5 (N) → A is better. So A is a dominant strategy for Firm 1. By symmetry, A is dominant for Firm 2.

Step 2 — Find Nash equilibria.

The unique Nash equilibrium is (A, A) with payoffs (4, 4). Both firms advertise, even though (N, N) = (5, 5) Pareto-dominates. This is a prisoner's dilemma: individual incentives to advertise lead to a collectively worse outcome.

Repeated Games

Repeated game. A game in which the same stage game is played multiple (or infinitely many) times by the same players. Repeated interaction allows strategies to condition on history (e.g., "cooperate until someone defects"), potentially sustaining cooperation that is impossible in a one-shot game.

When the prisoner's dilemma is played repeatedly (and players are patient), cooperation can be sustained. The threat of future punishment (reversion to defection) makes current cooperation self-enforcing. This is the folk theorem.

The intuition: cooperating today sustains the relationship. Cheating gives a short-run gain but triggers punishment forever. If the discount factor $\delta$ is high enough, the long-run cost of punishment outweighs the short-run gain.

Interactive: Repeated Game — Cooperation Threshold

In the standard prisoner's dilemma (payoffs: CC=3, CD=0, DC=5, DD=1), cooperation via grim trigger requires the discount factor $\delta$ to exceed a threshold. Slide $\delta$ to see whether cooperation is sustainable.

Impatient (0) 0.50 Very patient (1)
Calculating...

Figure 7.5. The horizontal line shows the minimum discount factor $\delta^*$ required for cooperation. When $\delta > \delta^*$, the long-run value of cooperation exceeds the one-shot temptation to defect. The chart compares the present value of perpetual cooperation vs defecting once then being punished forever.

Comparing Market Structures

Market structure# of firmsPriceOutputProfitDWLStrategic?
Perfect competitionMany$P = MC$HighestZero (LR)NoneNo
Monopolistic competitionMany$P > MC$Below comp.Zero (LR)SmallNo
Cournot oligopolyFew$MC < P < P_M$BetweenPositiveModerateYes (Q)
StackelbergFewLower than CournotHigherLeader > CournotLessYes (seq.)
Bertrand (identical)Two$P = MC$CompetitiveZeroNoneYes (P)
MonopolyOneHighestLowestHighestLargestNo

Thread Example: Maya's Enterprise

A rival, Nate, opens a lemonade stand across the street. Both have the same cost structure. The neighborhood demand is $P = 5 - (Q_M + Q_N)/20$, with $MC = 1.50$.

Cournot equilibrium: $Q_M^* = Q_N^* = 23.3$ cups. $P = 2.67$. Maya's profit: \$17.2$/day (materials only).

Stackelberg (Maya leads): $Q_M^S = 35$, $Q_N^S = 17.5$. $P = 2.375$. Maya's profit: \$10.6$/day — slightly better due to first-mover advantage.

With Nate in the market, Maya's output drops from 45 to 23.3 cups, and the price drops from \$1.75 to \$1.67.

Summary

Key Equations

LabelEquationDescription
Eq. 7.1$P = MC = AC_{min}$, $\Pi = 0$Long-run competitive equilibrium
Eq. 7.2$\max \Pi = P(Q)Q - TC(Q)$Monopolist's problem
Eq. 7.3$MR = P + Q(dP/dQ)$Marginal revenue
Eq. 7.4$MR = MC$Monopoly profit max condition
Eq. 7.5$(P-MC)/P = 1/|\varepsilon_d|$Lerner index
Eq. 7.6$MR_1 = MR_2 = MC$Third-degree price discrimination
Eq. 7.7–7.8Best response functionsCournot reaction functions
Eq. 7.9$q_i^C = (a-c)/(3b)$Cournot symmetric equilibrium
Eq. 7.10$P^C = (a+2c)/3$Cournot price
Eq. 7.11$P^B = c$Bertrand equilibrium (identical products)
Eq. 7.12–7.13$q_1^S = (a-c)/(2b)$, $q_2^S = (a-c)/(4b)$Stackelberg quantities
Eq. 7.14$u_i(s_i^*, s_{-i}^*) \geq u_i(s_i, s_{-i}^*)$ for all $s_i$Nash equilibrium

Exercises

Practice

  1. A monopolist faces $P = 50 - Q$ and has $MC = 10$. Find the monopoly price, quantity, profit, and DWL. Compute the Lerner index and verify it equals \$1/|\varepsilon_d|$.
  2. A monopolist sells in two markets: $P_1 = 24 - Q_1$ and $P_2 = 16 - 2Q_2$, with $MC = 4$. Find the profit-maximizing price and quantity in each market. Which market has more elastic demand?
  3. Two Cournot duopolists face $P = 80 - Q$, with $c_1 = c_2 = 8$. Find: (a) each firm's output, (b) the market price, (c) each firm's profit. Compare total industry output and profit to the monopoly case.
  4. Repeat Exercise 3 as a Stackelberg game with firm 1 as leader.
  5. Find all pure-strategy Nash equilibria:
    B: XB: Y
    A: X(3, 3)(1, 4)
    A: Y(4, 1)(2, 2)
    Is this a prisoner's dilemma? Why or why not?

Apply

  1. Why doesn't the Bertrand paradox hold for Coca-Cola and Pepsi? Identify three specific features of the real soft drink market that prevent price from falling to marginal cost.
  2. Two gas stations sit on opposite corners of an intersection. They sell identical gasoline and observe each other's prices daily. Explain why the Bertrand model predicts $P = MC$, and then explain why in practice gas stations can maintain prices above MC.
  3. A pharmaceutical company holds a patent (monopoly) on a drug. When the patent expires, generic competitors enter. Using the perfect competition model, predict what happens to: price, quantity, producer surplus, consumer surplus, and deadweight loss. Is the patent system efficient?
  4. Consider a market with one incumbent firm and a potential entrant. The incumbent can set a "limit price" — a low price that makes entry unprofitable — or a high monopoly price. Analyze this as a sequential game. Under what conditions is limit pricing credible?

Challenge

  1. Derive the Cournot equilibrium for $n$ symmetric firms with demand $P = a - bQ$ and constant marginal cost $c$. Show that as $n \to \infty$, $P \to c$ and the outcome converges to perfect competition. At what $n$ does the Cournot price reach within 10% of the competitive price?
  2. In a Cournot duopoly, firms consider forming a cartel. (a) Find the cartel output and profit. (b) Show that each firm has an incentive to cheat. (c) What discount factor $\delta$ makes cooperation sustainable in an infinitely repeated game with Cournot reversion?
  3. Prove that a monopolist never operates on the inelastic portion of the demand curve. (Hint: Show that if $|\varepsilon_d| < 1$, the monopolist can increase profit by reducing output.)
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