Oligopolies and Game Theory
Lecture 5

Oligopolies

1. Market Stucture and Game Theory

1. Oligopolies – two or more firms in a market

   • A firm considers the action of others in the market

   • Game theory – strategic decision making

   • Oligopoly must have some market power to influence the price

2. Competitive market – a firm has no influence over the market price

   • No game theory, because there are too many firms to keep an eye on

3. Monopoly - one firm the market

   • No game theory because a firm does not have a competitor to watch

2. Payoff interdependency – optimal choice by firm depends on actions of others

• Example: Netscape invented the web browser

  • Microsoft concentrated on Windows 95

  • Microsoft had caught up to Netscape and offered Internet Explorer for free

  • Netscape dropped its price to zero

Cournot Game

1. Cournot Game

• Assumptions

  • Simultaneous game - imperfect information

    • We do not know how the competitor will respond

  • Products are homogeneous

  • Firms choose output levels

  • No entry by other firms

• Equilibrium – Nash-Cournot Equilibrium

  • Profit functions

• The “c” is for Cournot Equilibrium

• Firm 1 maximizes profits given Firm 2 will play the best strategy

• Similarly,

• Solve for demand function by maximizing profit

• Remember, both firms produce q ₁ + q ₂

• If Firm 2 sets his production level at q ₂, then Firm 1 sets his production level to q ₁

  • The more Firm 1 produces, the lower the market price

• Maximize profits

• If Firm 1 produces one more unit, then the total quantity increases by one more unit

• Similarly,

• Usually the and are plotted

  • Called Reaction Functions

• Solve for q ₁ = R ₁(q ₂)

• Solve for q ₂ = R ₂(q ₂)

• Remember

  • has both q ₁ and q ₂

  • has both q ₁ and q ₂

• Use algebra to solve two equations for two unknowns

2. Problem 1

• Inverse demand function, P(Q) = 50 – 2Q, where Q = q ₁ + q ₂

• Cost functions for each firm, C(q _i) = 2 q _i

• Cournot Competitors

• Find best-response functions

• Similarly,

• Graph them

• Solve for Nash-Cournot Equilibrium

• Similarly, q ₁+½ (8) = 12, thus, q ₁ = 8

• Market price

  • P(Q) = 50 – 2Q, where Q = q ₁ + q ₂

  • Since Q = 8 + 8 = 16

  • Then P(16) = 50 – 2(16) = 18

• Firm’s profit

  • p ₁ = P(Q) q ₁ – 2q ₁

  • p ₁= 18(8) – 2(8) = 128

  • Similarly, p ₂ = 128

• Is the outcome allocatively efficient?

  • No, because P > MC, or 18 > 2

  • P = 18

  • MC = 2

• Only competitive markets are allocatively efficient

3. We can generalize the case with N firms in the market

• Inverse demand function, P(Q) = 50 – 2Q, where Q = N q _i

• The market has N firms and each firm is identical and produces q _i units

• Cost functions for each firm, C(q _i) = 2 q _i

• Trick

• Substitute the Quantity, Q into the inverse demand function

  • P(Q) = 50 – 2Nq _i

  • Re-write equation as, P(Q) = 50 – 2(N – 1)q _i – 2q _i

  • Remember, all firms are identical, so q _i = q _j

  • Substitute the q _j firms into the equation

  • P(Q) = 50 – 2(N – 1)q _j – 2q _i

  • If we did not do this trick, the answer would be the N firms act like a monopoly

• Firm i's profit is:

• Firm i's responds to Firm j is the Firm i's reaction function

• Remember, Firm i and Firm j are identical. Firm j would have an identical reaction function

• Substitute q _j = q _i into Firm i's reaction function, and solve for the quantity that Firm i will produce

Stackelberg Model

1. Stackelberg Model - a price leader moves first

• Stackelberg was an economist

• U.S. law makes collusion illegal

  • Firms cannot collude by forming a cartel

  • A cartel acts like a monopoly, i.e. one seller

• Stackelberg Model - a price leader sets his prices first and sets high prices

  • Other firms follow suit and sets high prices

  • If firms compete, they drive their profits to zero

  • Successful examples: General Motors, Intel

  • Unsuccessful examples: American Airlines

2. Example

• Price leader earns greater profits than other firms

• Price leader has no profit function

• Steps to solve Stackelberg Model

• Problem #3

  • Boeing is Stackelberg leader

    • Its marginal cost is, MC _B = 10

  • Airbus is the follower

    • Its marginal cost is, MC _A = 20

  • The market demand function is, P = 60 - 2(q _B - q _A)

• Step 1 - calculate Airbus's reaction function

  • Note - Boeing does not have a reaction function; it is the leader

• Step 2 - substitute Airbus's reaction function into Boeing's profit function

  • Boeing takes Airbus's reaction functin as a given

• Step 3: - calculate Airbus's response to Boeing's production level, using Airbus's reaction function

• Step 4 - Calculate the market price

• Boeing earns $225 profits while Airbus earns 12.5

Lerner Index

1. Refer to Lecture 3, since derivation is very similar

2. Market share is defined as s _i = q _i / Q

3. Duopoly has less market share than a monopoly

• Monopoly is s _i = 1

  • 

• Duopoly is 0 < s _i < 1

  • 

4. If more firms enter the market, then s _i becomes smaller

• Pure competition,

• 

• Note – if firms are exactly the same size, then s _i = 1/ N

  • N is the number of firms in the market

  • If N = 2, then s _i = ½ or 50%

  • If N = 4, then s _i = ¼ or 25%

5. Monopoly profits are higher than a Cournot

• Two firms could collude to create a “monopoly”

• If the firms are identical, then firms split the profits 50/50

• Both firms have to limit their production using quotas

• Incentive for cheating

  • One firm could cheat, and sell and produce a little more

  • Both firms end up cheating on the quotas

  • Collusive agreements are not a Nash equilibrium

Bertrand Game

1. Bertrand – Cournot game is wrong, because firms compete with prices and not quantities

• Same assumptions as Cournot

• Each firm has the demand function,

• D is market demand, while D ₁ is demand for Firm 1’s product

• Four outcomes

  1. P ₁ > P ₂ > MC

     • Marginal cost (MC)

     • Not an equilibrium

     • Firm 2 captures the whole market

     • Firm 1 has zero demand and must lower the price

  2. P ₁ > P ₂ = MC

     • Not an equilibrium

     • Firm 2 captures the whole market

     • Firm 1 has zero demand and must lower the price

  3. P ₁ = P ₂ > MC

     • Not an equilibrium

     • Each firm has half the market

     • Each firm tries to lower its price to capture the whole market

  4. P ₁ = P ₂ = MC

     • This is a Nash equilibrium

     • Each firm has half the market

     • Each firm earns zero economic profit

     • Each firm cannot lower its price to capture the market

     • Called Bertrand’s Paradox, because this market is allocative efficient; same result as a competitive market

2. Change assumptions

1. Firms have fixed costs, f

   • The rule P ₁ = P ₂ = MC is MR = MC, and thus does not include fixed costs

   • Eventually, one firm must leave the market because profits are negative, because of fixed costs

   • A monopoly market forms

2. Firms have different marginal costs, such as c ₂ > c ₁

   • Firm 1 can lower its price below Firm 2’s marginal cost

   • Eventually Firm 2 has to leave the market

   • Firm 1 becomes a monopoly

Kinked Demand Function

1. Kinked demand curve - some economists debate the existence

• An oligipolist faces two demand curves

• The rival's reaction determines which demand function the oligopolist faces

  1. If the oligoplist raises his price

     • The rivals do not increase their prices

     • Oligoplist faces an elastic demand function, d ₁

     • Consumers are sensitive

     • Oligoplist loses consumers to rivals

  2. If the oligopolist lowers his price

     • The rivals match this price decrease, so they do not lose consumers

     • Oligopolist faces an inelastic demand function, d ₂

     • Consumers are not sensitive

• We erase the portion of demand and marginal revenue that the oligopolist could never operate

  • Oligopolist produces at q* and market price is P*

  • If oligopolist increases production and causes the market price to fall, the oligopolist could trigger a price war

  • If oligopolist decreases production and causes the market price to rise, the oligopolist loses consumers because rivals do not match price increase

  • Oligopolist uses non-price competiton