﻿ Production Economics - Lecture 5

# 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

• 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 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

• 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 q1 = R1(q2)

• Solve for q2 = R2(q2)

• Remember

• has both q1 and q2

• 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

• Similarly,

• Graph them

• Solve for Nash-Cournot Equilibrium

• 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'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 qj = qi 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

• 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

• 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 si = qi / Q

3. Duopoly has less market share than a monopoly

• Monopoly is si = 1

• Duopoly is 0 < si < 1

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

• Pure competition,

• 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,

• 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

• 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