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.