Imperfect Competition – Study Notes

Introduction to Imperfect Competition

• Perfect competition and monopoly are polar cases; real‐world markets usually fall in-between.

  • Perfect competition: many firms, price takers, no need to consider rivals.

  • Monopoly: single seller, faces entire market demand, no rivals to consider.

• Intermediate structures:

  • Monopolistic competition: many firms, free entry/exit, products are close but imperfect substitutes.

  • Oligopoly: small number of firms; strategic interaction is key.

Primer in Game Theory

• Game theory = mathematical study of strategic interaction.

• Relevant to collusion, advertising races, entry decisions, household bargaining, etc.

• Individually optimal (rational) strategies can yield socially inefficient outcomes due to coordination failures.

• Three basic elements of any game:

  • Players

  • Strategies: complete plans of action.

  • Payoffs: numerical values (profits, years in jail, utility) associated with outcomes.

• Key definitions:

  • Dominant strategy: maximises a player’s payoff regardless of rivals’ actions.

  • Equilibrium in dominant strategies: each player selects their dominant strategy; outcome is self-enforcing.

  • Nash equilibrium: strategy profile where no player can gain by unilateral deviation; does not require dominant strategies.

  • Maximin strategy: maximises the minimum payoff achievable (extreme risk-averse / security-level choice).

Prisoner’s Dilemma Illustration

• Payoff matrix (years in prison):

  • Both confess → $5$ years each.

  • One confesses, other silent → confessor $0$, silent $20$.

  • Both silent → $1$ year each.

• Confessing is a dominant strategy for both → equilibrium = both confess (socially sub-optimal).

Cartel Instability Example (Mineral Water)

• Market demand $P = 20 - Q$; marginal cost $MC = 0$.

• Monopoly outcome: $Q<em>M = 10,\; P</em>M = 10.$

• Collusive agreement: each firm produces $5$ units → profit $\pi = 50$ each.

• Deviating firm cuts price to $9$:

  • Deviator sells entire demand (~$Q = 11$) → $\pi = 99$.

  • Other firm gets $0$.

• If both deviate to $9$ they split market: $\pi = 49.5$ each.

• "Defect" (undercut) is dominant; equilibrium = both deviate → lower joint profits.

Repeated Games & Tit-for-Tat

• Strategy: start by cooperate, then mimic opponent’s previous move.

• Can sustain cooperation if horizon is infinite/unknown → threat of future punishment disciplines defection.

Advertising as Prisoner’s Dilemma

• Under perfect competition, no incentive to advertise.

• With product differentiation, advertising can:

  1. Bring new consumers to market (industry demand shifts out).

  2. Steal share from rivals (redistribution).

• Two-firm example: total revenue $TR = 1000$ if no ads. Advertising cost $250$.

  • Both refrain: split $500$ each.

  • One advertises: gets larger share net of cost; rival’s share falls.

  • Both advertise: higher costs cancel gains → both worse off than no-ad outcome.

• Variants can change payoffs; Nash equilibrium depends on relative gains/costs.

Sequential Games & Strategic Entry Deterrence

• Sequential game: players move in turn; backward induction finds equilibrium.

• Cold War analogy, skyscraper examples (Shard vs. Company X): first mover can influence rival expectations.

• Strategic entry deterrence: Incumbent invests in excess capacity / low marginal cost to signal aggressive post-entry pricing, deterring entrants.

  • Even if fixed costs rise, lower $MC$ makes fight credible.

Oligopoly Models Overview

• Focus on duopoly for clarity; symmetry assumed unless stated.

Bertrand (Price) Competition

• Homogeneous product, constant $MC=0$.

• Undercutting rival’s price by tiny amount captures whole market.

• Iterated undercutting -> equilibrium price $P^* = MC$; quantity equals perfect competition outcome.

• Numerical example: $P = 56 - 2Q$, $MC=20$ → $P^* = 20$, $Q^* = 18$ split: $Q<em>1=Q</em>2=9$.

Cournot (Quantity) Competition

• Firms choose quantities simultaneously; each treats rival quantity as fixed.

• Market demand $P = a - b(Q<em>1+Q</em>2)$.

• Firm 1’s marginal revenue: $MR<em>1 = (a - bQ</em>2) - 2bQ_1$.

• Reaction functions:
  $R<em>1(Q</em>2) = \frac{a - bQ<em>2}{2b}, \; R</em>2(Q<em>1) = \frac{a - bQ</em>1}{2b}.$

• Nash equilibrium (symmetric):
  $Q<em>1^* = Q</em>2^* = \frac{a}{3b}, \; Q^* = \frac{2a}{3b}, \; P^* = \frac{a}{3}.$

• Profits positive if MC< P^*.

• Numerical example (demand $56-2Q$, $MC=20$): reaction lines $Q<em>1=9-\frac{Q</em>2}{2}$; equilibrium $Q<em>1=Q</em>2=6$, $P^*=32$.

Stackelberg (Quantity Leadership)

• Sequential: Leader chooses $Q_1$ first; follower best-responds via Cournot reaction.

• Follower’s reaction: $Q<em>2^* = \frac{a - bQ</em>1}{2b}.$

• Leader internalises this when maximising profit; effective residual demand:
  $P = a - b\left(Q<em>1 + \frac{a - bQ</em>1}{2b}\right) = a - \frac{bQ_1}{2}.$

• With $MC=0$: $MR= a/2 - bQ<em>1 = 0 \Rightarrow Q</em>1^* = \frac{a}{2b}, \; Q_2^* = \frac{a}{4b}.$

• Output order: Q{Stackelberg} > Q{Cournot} > Q_{Monopoly} for leader; price between Cournot and Bertrand.

• Numerical example (demand $56-2Q$, $MC=20$):

  • Follower reaction: $Q<em>2 = 9 - Q</em>1/2$.

  • Leader chooses $Q<em>1=9$; follower $Q</em>2=4.5$; total $Q=13.5$; $P^*=29$.

Model Comparison (MC=0 benchmark)

Model Comparison (MC=0 benchmark)

• Monopoly: $<em> QM= a/2b, \; P</em>M = a/2.$

• Cournot: $Q<em>C = 2a/3b, \; P</em>C = a/3.$

• Stackelberg: $Q<em>S = 3a/4b, \; P</em>S = a/4.$

• Bertrand/Perfect competition: $Q<em>B = a/b, \; P</em>B = MC.$

• Price descending order: Monopoly > Cournot > Stackelberg > Bertrand.

• Industry with many symmetric firms producing close (but not perfect) substitutes.

• Each firm faces a downward-sloping individual demand curve; more elastic than market demand due to substitution possibilities.

• Two demand concepts:

  • $dd$: demand if rivals keep prices unchanged.

  • $DD$: demand if all firms change prices symmetrically; less elastic than $dd$.

Short-Run Equilibrium

• Firm maximises profit like a monopolist: set $MR = SMC$.

• Chooses quantity $Q^<em>$ on $dd$, price $P^</em>$ on $DD$.

• Positive economic profit possible → attracts entry.

Long-Run Equilibrium

• Entry shifts each firm’s demand curve inward (toward origin) until profits are zero.

• Firms produce where price equals long-run average cost but not at minimum of $LAC$ → excess capacity.

Efficiency Considerations

• Perfect competition is Pareto efficient; Chamberlinian outcome has higher prices, lower output, and excess capacity (productive inefficiency).

• However, consumers enjoy greater product variety, partly offsetting welfare loss.

• Long-run profits: zero in both perfect and monopolistic competition (free entry eliminates rents).

Ethical / Practical Implications & Real-World Links

• Cartel instability explains why legal enforcement (antitrust) matters; self-interest alone makes collusion fragile.

• Advertising arms races waste resources yet persist without coordination mechanisms or regulation.

• Strategic entry deterrence raises fixed costs, potentially reducing market contestability; policy may limit credible threats (e.g., capacity commitments scrutinised by antitrust agencies).

• Price wars (Bertrand) seldom persist in reality due to product differentiation, capacity limits, collusion, or tacit coordination.

Key Equations & Symbols Recap

• Market demand (general): $P = a - bQ.$

• Monopoly MR: $MR = a - 2bQ.$

• Cournot reaction: $R<em>i(Q</em>j) = \frac{a - bQ_j}{2b}.$

• Stackelberg follower: same as Cournot reaction.

• Profit: $\pi<em>i = (P - MC)Q</em>i$ (if fixed costs zero).

Connections to Earlier Principles

• Marginal analysis (set $MR=MC$) underpins monopoly, Cournot, Stackelberg, and Chamberlin decisions.

• Perfect competition benchmark still critical for welfare comparisons.

• Game-theoretic reasoning extends basic profit maximisation when strategic interdependence exists.

Recommended Reading

• Frank & Cartwright (2013) "Microeconomics and Behavior", Chapter 13.

Quick Memory Aids

• "Bertrand = Price War → $P=MC$."

• "Cournot = Quantity Guess → each produces 1/3 monopoly output."

• "Stackelberg Leader Steals a March → produces double follower’s output."

• "Prisoner’s Dilemma logic underlies cartel & ad wars."