Imperfect Competition — Comprehensive Study Notes

Introduction to Imperfect Competition

• Perfect competition and monopoly are extreme benchmarks; real-world markets usually fall in between.
  • Perfect competition: many firms, price takers, no strategic interaction.
  • Monopoly: single firm, no rivals, no strategic interaction.
• Two relevant intermediate forms:
  • Monopolistic competition: many firms, free entry/exit, products are close but imperfect substitutes.
  • Oligopoly: small number of firms; strategic interaction is decisive.
• Course roadmap
  • Primer on game theory (foundational tool for strategic interaction).
  • Specific oligopoly models: Bertrand, Cournot, Stackelberg.
  • Monopolistic competition (Chamberlin model).

Primer in Game Theory

• Game theory: mathematical analysis of strategic interaction.
  • Used in economics, sociology, political science, etc.
  • Illustrates how individually optimal actions can yield socially inefficient outcomes (coordination failure).
  • Remedies: bargaining, threats, trust, social norms, legal/institutional constraints.

Elements of a Game

• Players.
• Strategy set for each player (complete plan of action).
• Payoffs for every strategy combination.
• Optimal strategy: maximises a player’s expected payoff.

Dominant Strategy & Equilibrium in Dominant Strategies

• Dominant strategy: best regardless of rivals’ actions.
• Equilibrium in dominant strategies: every player chooses a dominant strategy; no one conditions on rival behaviour.

Classic Prisoner’s Dilemma (Table 13.2)

• Two prisoners (X, Y) choose Confess or Remain Silent.
  • (Confess, Confess): 5 years each.
  • (Confess, Silent): 0 years for confessor, 20 for the silent.
  • (Silent, Confess): symmetric.
  • (Silent, Silent): 1 year each.
• Dominant strategy = Confess ⇒ equilibrium yields 5 yrs each ⇒ Pareto-inferior.

Cartel Instability Example

• Market demand: P = 20 - Q, MC = 0.
• Monopoly outcome: QM = 10, PM = 10, each firm produces 5, profit \pi = 50.
• If one firm defects by pricing at 9:
  • Defector sells 11 units, \pi = 99; other gets 0.
• If both defect and price 9:
  • Split demand 11 ⇒ 5.5 units each, \pi = 49.5.
• Defect is dominant; equilibrium ≈ Prisoner’s dilemma.

Repeated Games & Tit-for-Tat

• Infinite or uncertain horizon enables cooperation.
• Tit-for-tat strategy: cooperate initially, then mimic opponent’s previous move.

Advertising as Prisoner’s Dilemma

• Under perfect competition no incentive to advertise.
• With differentiation, advertising can:
  1. Inform new consumers (industry demand ↑).
  2. Steal rivals’ customers (share shifting).
• Two-firm payoff matrix (Tables 13.4 & 13.5):
  • Market revenue TR = 1000 without ads.
  • Advertising cost 250.
  • Simultaneous advertising resembles PD – both may end up spending 250 while profits fall.

Maximin Strategy

• Choose action that maximises the minimum payoff obtainable (extremely risk-averse rule).

Nash Equilibrium (NE)

• Strategy profile where no player can gain by unilateral deviation.
• Dominant strategies not required.

Dominant Strategies vs. Nash Equilibrium

• Dominant-strategy equilibrium: strategy optimal regardless of others.
• NE: strategy optimal given the specific strategies chosen by others.

Example: Airline Advertising Matrix

• Payoffs (L = Lufthansa, A = Alitalia):
  • (Increase, Increase): (\piL,\piA)=(30,20)
  • (Increase, Maintain): (80,30)
  • (Maintain, Increase): (40,50)
  • (Maintain, Maintain): (50,20)
• Questions: existence of dominant strategy? identify NE, consider maximin.

Sequential Games & Backward Induction

• Players move sequentially; later movers observe earlier actions.
• Backward induction: solve from final node backwards.
• Illustration: Cold-War arms race; "tallest building" example (Fig. 13.17).

Strategic Entry Deterrence

• Incumbent may build excess capacity (high FC, low MC) to credibly threaten price cuts that make entry unprofitable.
• Figures 13.18 show payoffs with/without credible threat.

Oligopoly Models

Common Features

• Small number of firms, mutual interdependence.
• Focus on duopoly for tractability.

Bertrand Competition (Price Competition)

• Assumptions:
  • Homogeneous product.
  • Identical costs (MC = c, often =0).
  • Each firm believes rival’s price is fixed when choosing its own.
• Logic:
  1. If one firm’s price exceeds the other’s, it sells nothing.
  2. If equal prices, firms split demand.
  3. Undercutting by an infinitesimal amount captures the whole market ⇒ incentive to undercut until P = MC.
• Bertrand NE: identical to perfect competition.

Numerical Example

• Demand P = 56 - 2Q, MC = 20.
• NE price: P^* = 20.
• Total quantity: Q^* = 18 ⇒ each firm supplies 9.

Cournot Competition (Quantity Competition)

• Firms choose quantities simultaneously; each treats rival quantity as fixed.
• Demand: P = a - b(Q1 + Q2).
• Firm 1’s MR: MR1 = (a - bQ2) - 2bQ_1.
• With MC = 0, profit-maximisation MR = MC ⇒ reaction functions:
  R1(Q2) = \frac{a - bQ2}{2b},\quad R2(Q1) = \frac{a - bQ1}{2b}.
• Intersection ⇒ Cournot NE quantities:
  Q1^* = Q2^* = \frac{a}{3b}.
• Market outcomes:
  • Total Q^* = \frac{2a}{3b}.
  • Price P^* = \frac{a}{3}.
  • Revenue per firm TR = \frac{a^2}{9b}; here also profit if MC=0.

Numerical Example (MC = 20)

• Demand P = 56 - 2Q.
• Reaction function (derived): Q1 = 9 - \frac{Q2}{2} (symmetry for firm 2).
• Solving: Q1^* = Q2^* = 6.
• Total Q = 12; price P^* = 32.

Stackelberg Competition (Sequential Quantity Choice)

• Leader (firm 1) chooses quantity first; follower (firm 2) observes and best-responds via Cournot reaction.
• Follower’s reaction: Q2 = R2(Q1) = \frac{a - bQ1}{2b}.
• Substitute into demand faced by leader:
  P = a - b(Q1 + R2(Q1)) = a - bQ1/2.
• Leader’s MR: MR1 = a/2 - bQ1.
• With MC = 0 ⇒ Q_1^* = \frac{a}{2b},\; P^* = \frac{a}{4}.
• Follower output: Q_2^* = \frac{a}{4b}.
• Compared to Cournot, leader gains, follower loses, total output rises, price falls.

Numerical Example (MC = 20)

• Demand: P = 56 - 2(Q1 + Q2).
• Follower’s reaction: Q2 = 9 - Q1/2.
• Leader’s perceived demand: P = 38 - Q_1.
• MR: MR1 = 38 - 2Q1.
• Set MR = MC = 20 ⇒ Q_1^* = 9.
• Q_2^* = 4.5 ⇒ Q = 13.5, P^* = 29.

Comparative Outcomes (homogeneous good, MC=0)

• Monopoly: QM = \frac{a}{2b},\; PM = \frac{a}{2}.
• Cournot: QC = \frac{2a}{3b},\; PC = \frac{a}{3}.
• Stackelberg: QS = \frac{3a}{4b},\; PS = \frac{a}{4}.
• Bertrand / Perfect Competition: QB = \frac{a}{b},\; PB = MC.
• Ordering: P{Monopoly} > P{Cournot} > P{Stackelberg} > P{Bertrand}.

Monopolistic Competition (Chamberlin Model)

• Large number of symmetric firms; products are close but imperfect substitutes.
• Each firm faces downward-sloping (but highly elastic) individual demand.
• Two relevant demand curves for firm i (Fig. 13.1):
  • dd: assumes rivals keep prices fixed when i changes its price (own-price demand).
  • DD: assumes all firms adjust prices symmetrically (market-share demand); less elastic than dd.

Short-Run Equilibrium

• Profit maximisation: MR = SMC on the dd curve.
• Determines Q^ and P^; positive economic profit possible.

Long-Run Equilibrium

• Positive profits induce entry ⇒ individual demand curves shift inward (firms split market).
• Entry continues until profits are zero; tangent point between demand and ATC where P = ATC but P > MC.
• Firms do not produce at minimum LAC ⇒ excess capacity ⇒ inefficiency.

Efficiency Comparison

• Perfect competition is Pareto-efficient (produce where P = MC = min\,LAC).
• Monopolistic competition: higher price, lower output, excess capacity, but greater product variety.
• Long-run profit = 0 in both structures due to free entry.

Key Definitions & Concepts

• Dominant Strategy: strategy yielding highest payoff irrespective of opponents’ moves.
• Equilibrium in Dominant Strategies: outcome when every player plays dominant strategy.
• Nash Equilibrium: strategy profile where no unilateral deviation profitable.
• Maximin Strategy: maximise the minimum payoff achievable.
• Reaction Function: best-response quantity/price as a function of rival’s choice.
• Backward Induction: solving sequential games from the end backwards.
• Strategic Entry Deterrence: actions by incumbents to alter entrants’ expectations (e.g., excess capacity).
• Excess Capacity: operating to the left of minimum LAC; hallmark of monopolistic competition.

Ethical & Practical Implications

• Cartel agreements benefit firms but harm consumer surplus; unstable without enforcement.
• Advertising races wasteful if purely redistributive; informative advertising can enhance welfare.
• Strategic entry deterrence may raise fixed costs and deter competition, prompting antitrust scrutiny.
• Product differentiation under monopolistic competition enriches consumer choice but at efficiency cost.

Numerical & Algebraic Summary

• Bertrand price equilibrium: P^* = MC.
• Cournot NE: Q_i^* = \frac{a}{3b},\; P^* = \frac{a}{3}.
• Stackelberg leader quantity: Q_1^* = \frac{a}{2b}.
• Monopoly MR = MC condition: a - 2bQ = MC.

Recommended Reading

• Frank, R. H. & Cartwright, E. (2013). Microeconomics and Behavior, Ch. 13. Figures/tables sourced from McGraw-Hill instructor slides.