178.307 Markets, Firms and Consumers

Lecture 11: Competition

Overview

Firms interact with other firms.

This is often in various forms of competition.

Keywords– Cournot Model– Stackleberg Model– Betrand Model– Hotelling Model– Salop Model– Predatory Pricing

Neoclassical Models

Both Cournot and Betrand Models can be solved by Game Theory.

Cournot Model has firms competing on output.

Bertrand Market has firms competing on price.

Collusion is much less stable in a Betrand market.

Cournot Model

See tutorial exercises

Betrand Duopoly

02

0))(1())(1(

))((

cbppa

cpbppap

bqpacpcqqp

bppaq

ji

ijii

i

jiiiiii

jii

Reaction Functions

jji pbca

pR22

)(

Solve (simultaneously) for all firms and find that all optimal prices are: (a+c)/(2-b)

Note: this implies a firm will increase (decrease) its price if its competitors increase (decrease) theirs.

Stackelberg Model

Game is similar to Cournot

Outputs are no longer selected simultaneously.

We make one firm the leader.

The solution is by ‘backward induction’.

We use the terminal point of the game to determine the follower’s output.

This used to derive the leader’s output.

Stackelberg Game

02

2

1

2

)(

2)(

)(

11

1

1112111

11122111

121

122

1121111

bqca

q

cqqqb

cabbqaq

cqqqbRbqaq

qb

caqRq

cqqbqbqacqPq

Solutions

b

caq

b

caq

42 21

Leader obtains larger share of market (first-mover advantage)

In Cournot model the share is (a-c)/3b each.

Simple Spatial Competition

Hotelling Model (1929)– See tutorial

Salop Model- Introduction

Depositors are uniformly located along a circle.

There are n banks, indexed by i= 1,..,n.

Banks invest cash in a riskless technology with a return of r.

Depositors don’t have access to technology.

Transport costs of αx are incurred by each depositor, where x is ‘distance’.

•Each depositor has 1 unit of cash.

•The Total Length ofthe circle = 1

•Total mass of depositors

= D

Optimal Organisation

The most distance any consumer will travel to a bank is 1/2n (halfway round the circle).

The sum of all transport costs are

n

n

DxDdxn 2

1

0 42

Note: you don’t need to prove this.

Optimal number of banks

Let the unit cost of setting up a bank is F. Optimal number is found by minimising setup and

transport costs:

n

DnF

n 4min

Solution

F

Dn

F

Dn

F

Dn

n

DF

2

1

24

04

2

2

Define the marginal depositor

nx

rr

nxrr

xn

rxr

xn

rxr

i

iD

iD

iiD

iD

iiDi

iD

iiDi

iD

2ˆ

2

ˆ2

ˆˆ

ˆ1

ˆ

1

1

1

1

Volume of Deposits

2

21

22

1ˆ

11

1

iD

iD

iD

i

iD

iD

i

rrr

nDD

rr

nx

Total volume of deposits are ‘doubled’ to take account of banks both sides of the bank i.

Profit of Bank i

01)(

2

21)1(

2

21)(

11

11

iD

iD

iD

iD

iD

i

iD

iD

iDi

Di

rr

rrr

nr

rrr

nrrD

The solution here requires the use of the product rule

Simplify

2

2

2

21

11

11

iD

iD

iDi

D

iD

iD

iD

iD

rrr

nrr

rrr

n

rr

Only one solution is possible, if all banks charge the same interest rates…

Profit of Bank

2

1

1

01

)(

n

D

nnD

nnrrD

nrrD

nrr

iD

iD

Free Competition Output

F

Dn

F

Dn

DFnn

DF

e

2

22

Free competition leads to too many banks This provides scope for regulation Note that decreasing ‘r’ (e.g. by reserve requirement)

has no effect.

Predatory Pricing

A firm sets prices below cost, in an attempt to drive competitors out of the market.

– It hopes to recoup losses after the competitors have been driven out.

– It does so by exploiting market power after the exit of these other firms.

It is difficult to distinguish aggressive pricing in a competitive market from predatory pricing.

Predatory pricing is usually regarded as illegal.

Theoretical Work

Selten (1978) began with a chain-store game.

Accomodating Firm 2 weakly dominates Fighting for Firm 1.

Firm 2’s type is unknown. If W, it leaves, if T, it stays.

Firm 2

T W

Firm 1

Fight -1, -1 a, 0

Acc. 0, b a, 0

Hence

Firm 1 may wish to adopt predatory in an infinite length game.

The per-period payoff (-1)q +(1-q)a > 0

The Firm has to be patient (does not discount future too much).

In a finite length game, Firm 1 always accomodates in the last period.

Backward induction then implies it will always accommodate.

Predatory pricing requires games of infinite length.

Conclusion

Predatory Pricing is not as common as some people believe.

Conditions depend on asymmetry of information.

Predatory firm has to have better information on each firm’s costs.

Experiments on theory (Issac and Smith) confirm these aspects.

– Predatory pricing does not occur with complete information.

– With incomplete information, some players do slash prices to signal toughness (reputation effect).