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178.307 Markets, Firms and Consumers
Lecture 11: Competition
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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
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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.
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Cournot Model
See tutorial exercises
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Betrand Duopoly
02
0))(1())(1(
))((
cbppa
cpbppap
bqpacpcqqp
bppaq
ji
ijii
i
jiiiiii
jii
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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.
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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.
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Stackelberg Game
02
2
1
2
)(
2)(
)(
11
1
1112111
11122111
121
122
1121111
bqca
q
cqqqb
cabbqaq
cqqqbRbqaq
qb
caqRq
cqqbqbqacqPq
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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.
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Simple Spatial Competition
Hotelling Model (1929)– See tutorial
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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
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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.
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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
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Solution
F
Dn
F
Dn
F
Dn
n
DF
2
1
24
04
2
2
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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
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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.
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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
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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…
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Profit of Bank
2
1
1
01
)(
n
D
nnD
nnrrD
nrrD
nrr
iD
iD
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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.
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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.
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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
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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.
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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).