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

Equation 1

  • The “c” is for Cournot Equilibrium

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

  • Similarly,

Equation 2

  • Solve for demand function by maximizing profit

Equation 3

  • Remember, both firms produce q1 + q2

  • If Firm 2 sets his production level at q2, then Firm 1 sets his production level to q1

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

  • Maximize profits

Equation 4

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

  • Similarly,

Equation 5

  • Usually the Equation 6 and Equation 7 are plotted

    • Called Reaction Functions

A Reaction Function

  • Solve Equation 6 for q1 = R1(q2)

  • Solve Equation 7 for q2 = R2(q2)

  • Remember

    • Equation 6 has both q1 and q2

    • Equation 6 has both q1 and q2

  • Use algebra to solve two equations for two unknowns

2. Problem 1

  • Inverse demand function, P(Q) = 50 – 2Q, where Q = q1 + q2

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

  • Cournot Competitors

  • Find best-response functions

Equation 8

  • Similarly,

Equation 9

  • Graph them

Reaction Functions

  • Solve for Nash-Cournot Equilibrium

Equation 10

  • Similarly, q1+½ (8) = 12, thus, q1 = 8

  • Market price

    • P(Q) = 50 – 2Q, where Q = q1 + q2

    • Since Q = 8 + 8 = 16

    • Then P(16) = 50 – 2(16) = 18

  • Firm’s profit

    • p1 = P(Q) q1 – 2q1

    • p1= 18(8) – 2(8) = 128

    • Similarly, p2 = 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 qi

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

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

  • Trick

  • Substitute the Quantity, Q into the inverse demand function

    • P(Q) = 50 – 2Nqi

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

    • Remember, all firms are identical, so qi = qj

    • Substitute the qj firms into the equation

    • P(Q) = 50 – 2(N – 1)qj – 2qi

    • 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 maximizes profit

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

Firm i's reaction function

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

  • Substitute qj = qi into Firm i's reaction function, and solve for the quantity that Firm i will produce

Firm i production level

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, MCB = 10

    • Airbus is the follower

      • Its marginal cost is, MCA = 20

    • The market demand function is, P = 60 - 2(qB - qA)

  • Step 1 - calculate Airbus's reaction function

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

Deriving reaction function for follower

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

    • Boeing takes Airbus's reaction functin as a given

Calculate the leader's production level

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

Calculate the follower's production level

  • Step 4 - Calculate the market price

Calculate the follower's production level

  • Boeing earns $225 profits while Airbus earns 12.5

Lerner Index

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

Equation 11

2. Market share is defined as si = qi / Q

3. Duopoly has less market share than a monopoly

  • Monopoly is si = 1

    • Equation 12
  • Duopoly is 0 < si < 1

    • Equation 13

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

  • Pure competition, Equation 14

  • Equation 15

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

    • N is the number of firms in the market

    • If N = 2, then si = ½ or 50%

    • If N = 4, then si = 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,

Equation 16

  • D is market demand, while D1 is demand for Firm 1’s product

  • Four outcomes

    1. P1 > P2 > MC

      • Marginal cost (MC)

      • Not an equilibrium

      • Firm 2 captures the whole market

      • Firm 1 has zero demand and must lower the price

    2. P1 > P2 = MC

      • Not an equilibrium

      • Firm 2 captures the whole market

      • Firm 1 has zero demand and must lower the price

    3. P1 = P2 > MC

      • Not an equilibrium

      • Each firm has half the market

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

    4. P1 = P2 = 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 P1 = P2 = 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 c2 > c1

    • 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, d1

      • 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, d2

      • Consumers are not sensitive

Kinked demand function

  • 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

Kinked demand function

 

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