Download Strategic Interaction

Finance 510: Microeconomic
Analysis
Strategic Interaction
Recall that there is an entire spectrum of market
structures
Market Structures
Perfect Competition
Monopoly
Many firms, each with zero
One firm, with 100%
market share
market share
P = MC
Profits = 0 (Firm’s earn a
P > MC
Profits > 0 (Firm’s earn
reasonable rate of return on
invested capital
excessive rates of return
on invested capital)
NO STRATEGIC
NO STRATEGIC
INTERACTION!
INTERACTION!
Most industries, however, don’t fit the assumptions of
either perfect competition or monopoly. We call these
industries oligopolies
Oligopoly
Relatively few firms, each
with positive market share
STRATEGIC INTERACTION!
Wireless (2002)
US Beer (2001)
Music Recording (2001)
Verizon: 30%
Cingular: 22%
AT&T: 20%
Sprint PCS: 14%
Nextel: 10%
Voicestream: 6%
Anheuser-Busch: 49%
Miller: 20%
Coors: 11%
Pabst: 4%
Heineken: 3%
Universal/Polygram: 23%
Sony: 15%
EMI: 13%
Warner: 12%
BMG: 8%
Further, these market shares are not constant over time!
Airlines (1992)
Airlines (2002)
American
21
United
20
15
Delta
Northwest
Continental 11
US Air
9
14
American
19
United
17
15
Delta
Northwest
11
Continental 9
SWest
7
While the absolute ordering didn’t change, all the airlines lost
market share to Southwest.
Another trend is consolidation
Retail Gasoline (1992)
9
Shell
Chevron
8
8
8
Texaco
Exxon
Amoco
7
7
Mobil
5
5
4
4
24
Exxon/Mobil
Shell
20
BP/Amoco/Arco 18
Chev/Texaco 16
10
6
BP
Citgo
Marathon
Sun
Phillips
Retail Gasoline (2001)
7
Total/Fina/Elf
Conoco/Phillips
The key difference in oligopoly markets is that
price/sales decisions can’t be made independently
of your competitor’s decisions
Monopoly
Q  QP
Your Price (-)
Oligopoly
Q  QP, P1 ,...PN 
Your N Competitors
Prices (+)
Oligopolistic markets rely crucially on the
interactions between firms which is why we need
game theory to analyze them!
The Airline Price Wars
Suppose that American and Delta
face the given aggregate demand
for flights to NYC and that the
unit cost for the trip is $200. If
they charge the same fare, they
split the market
p
$500
$220
American
180
What will the equilibrium
be?
Q
P = $500
P = $220
P = $500
$9,000
$9,000
$3,600
$0
P = $220
$0
$3,600
$1,800
$1,800
Delta
60
The Airline Price Wars
Assume that Delta has the following beliefs about American’s
Strategy
 l  Pr P  $500
 r  Pr P  $220
Probabilities of
choosing High or
Low price
Player A’s best response will be his own set of probabilities to
maximize expected utility
pt  Pr P  $500
pb  Pr P  $220
Max pt  l ($9000)   r (0)  pb  l ($3600)   r ($1800)
pt , pb
Max pt  l ($9000)   r (0)  pb  l ($3600)   r ($1800)
pt , pb
Subject to
pt  pb  1
pt  0
pb  0
Probabilities always have to sum to one
Both Prices have a chance of being
chosen
( pt , pb ,  )  $9,000 pt l  pb  l ($3600)   r ($1800) 
  1  pt  pb   1 pt   2 pb
( pt , pb ,  )  $9,000 pt l  pb  l ($3600)   r ($1800) 
  1  pt  pb   1 pt   2 pb
First Order Necessary Conditions
9000 l    1  0
3600 l  1800 r     2  0
1  pt  pb  0
pt  0 2 pb  0
pb  0 1 pt  0
2  0 1  0
pt  0
9000 l    3600 l  1800 r
pb  0
r l 1
1  2  0
l 
1
4
r 
3
4
The Airline Price Wars
pl  1
pr  0
pt  1
pb  0
1
pl 
4
3
pr 
4
1
pt 
4
3
pb 
4
Both Randomize between
$500 and $220
pl  0
pr  1
pt  0
pb  1
Both always charge $220
Both always charge $500
Notice that prices are low most of the time!
Continuous Choice Games – Cournot Competition
p
There are two firms in an industry –
both facing an aggregate (inverse)
demand curve given by
D
Q
P  A  BQ
Aggregate
Production
Both firms have constant marginal costs equal to $C
From firm one’s perspective, the demand curve is given by
P  A  Bq1  q2    A  Bq2   Bq1
Treated as a constant by Firm One
Solving Firm One’s Profit Maximization…
TR   A  Bq2 q1  Bq 1
2
MR   A  Bq2   2Bq1  c
q1

A  Bq 2   c

2B
In Game Theory Lingo, this is Firm One’s Best Response
Function To Firm 2
 Ac  1
q1  
  q2
 2B  2
q2
 Ac


 B 
Note that this is the optimal
output for a monopolist!
 Ac


 2B 
q1
Further, if Firm two produces
 Ac


 B 
q2
 Ac


 B 
It drives price down to MC
P  A  BQ
 Ac 
P  A  B
c
 B 
 Ac


 2B 
q1
The game is symmetric with respect to Firm two…
 Ac  1
q1  
  q2
 2B  2
q2
 Ac


 B 
 Ac  1
q2  
  q1
 2B  2
Firm 1
 Ac


 2B 
Firm 2
 Ac


 2B 
 Ac


 B 
q1
1 Ac 
q1  q  

3 B 
*
2  Ac 
Q  q1  q  

3 B 
*
2
q2
*
*
2
1  Ac  2 Ac   Ac 

 


2 B  3 B   B 
Firm 1
Competitive
Output
Monopoly
Output
q 2*
There exists a unique Nash
equilibrium
Firm 2
*
q1
q1
A numerical example…
Suppose that the market demand for computer chips (Q is in millions)
is given by
P  120  20Q
Intel and Cyrix are both competing in the market and have a
marginal cost of $20.
1  120  20  5
q q  
   1.67 M
3  20  3
*
I
*
C
P  120  20(3.33)  $53.33
Had this market been serviced instead by a monopoly,
P  120  20Q
MC  $20
Q*  2.5M
P  120  20(2.5)  $70
dQ P
1  70 

 
  1.4
dP Q
20  2.5 
MC
p
 1
1  
 
$20
$70 
1 

1 

 1.4 
With competing duopolies
P  120  20q2   20q1  86.6  20q1
MC  $20
Q*  1.67 M
P  86.6  20(1.67)  $53.33
MC
p
 1
1  
 
dQ P
1  53.33 

 
  1.6
dP Qi
20  1.67 
$20
$53.33 
1 

1 

 1.6 
One more point…
Monopoly
Duopoly
Q*  2.5M
P  $70
Q*  1.67 M
P  $53.33
  ($70  $20)2.5  $125
  ($53  20)1.67  $55
If both firms agreed to produce 1.25M chips (half the monopoly output),
they could split the monopoly profits ($62.5 apiece). Why don’t these
firms collude?
Suppose we increase the number of firms…
N
P  A  BQ  A  B  qi
i 1
Demand facing firm i is given by (MC = c)


P  A  B  j i q j  Bq i
Qi
 Ac  1
q1  
  Qi
 2B  2
Firm i’s best response to its N-1 competitors is given by
 Ac  1
qi  
  Qi
 2B  2
Further, we know that all firms
produce the same level of output.
Qi  ( N  1)qi
Solving for price and quantity, we get
Ac
qi 
( N  1) B
N A  c
Q
( N  1) B
A
 N 
P

c
N 1  N 1
Expanding the number of firms in an oligopoly
Ac
qi 
( N  1) B
N A  c
Q
( N  1) B
A
 N 
P

c
N 1  N 1
Note that as the number of firms increases:
Output approaches the perfectly competitive
level of production
Price approaches marginal cost.
Lets go back to the previous example…
Recall, we had an aggregate demand for computer chips and a constant
marginal cost of production.
P  120  20Q
MC  $20
q*  1.67 M
Q  2q  3.33
P  $53.33
  $56
CS = (.5)(120 – 53)(3.33) = $112
p
$112
$53
D
What would it be worth to consumers
to add another firm to the industry?
3.33
Q
With three firms in the market…
P  120  20Q
MC  $20
CS = (.5)(120 – 45)(3.75) = $140
q*  1.25M
Q  3q  3.75
P  $45
  $31
p
$140
$45
D
A 25% increase in CS!!
3.75
Q
Increasing Competition
6
80
70
5
60
4
50
3
40
30
2
20
1
10
Num ber of Firm s
Firm Sales
Industry Sales
Price
97
93
89
85
81
77
73
69
65
61
57
53
49
45
41
37
33
29
25
21
17
13
9
0
5
1
0
Increasing Competition
300
250
200
150
100
50
Num ber of Firm s
Consumer Surplus
Firm Profit
Industry Profit
97
93
89
85
81
77
73
69
65
61
57
53
49
45
41
37
33
29
25
21
17
13
9
5
1
0
Now, suppose that there were annual fixed costs
equal to $10
P  120  20Q
MC  $20
How many firms can this industry support?
Ac
qi 
( N  1) B
A
 N 
P

c
N 1  N 1
 i  ( P  $20)qi  $10  0
Solve for N
With a fixed cost of $10, this industry can support 7 Firms
140
130
120
110
100
90
80
70
60
50
40
30
20
10
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
The previous analysis was with identical firms.
 Ac  1
q1  
  q2
 2B  2
q2
 Ac  1
q2  
  q1
 2B  2
Firm 1
Suppose Firm 2’s marginal costs
are greater than Firm 1’s….
q 2*
Firm 2
*
1
q
q1
Suppose Firm 2’s marginal costs
are greater than Firm 1’s….
q2
 A  c1  1
q1  
  q2
 2B  2
 A  c2  1
q2  
  q1
 2B  2
c2  c1
Firm 1
q 2*
Firm 2
*
1
q
q1
Firm 2’s market share drops
 A  c2  2c1 
q1  

3B


 A  c1  2c2 
*

+ q2  
3B


*
 2 A  c1  c2  
Q

3B


As long as average industry
costs are the same as the
identical firm case
c1  c2
c
2
Industry output and price are
unaffected!
Note, however, that production is undertaken in an inefficient manner!
With constant marginal costs, the firm with the lower cost
should be supplying the entire market!!
Market Concentration and Profitibility
N
P  A  B  qi
Industry Demand
i 1
P  ci si

P

 H 


P  c  10,000 

P

The Lerner index for Firm i is related
to Firm i’s market share and the
elasticity of industry demand
The Average Lerner index for the
industry is related to the HHI and the
elasticity of industry demand
The previous analysis (Cournot Competition) considered quantity as the
strategic variable. Bertrand competition uses price as the strategic
variable.
p
Should it matter?
P*
D
Q*
P  A  BQ
Q
Just as before, we have an industry
demand curve and two competing
duopolists – both with marginal cost
equal to c.
Q  a  bP
P  A  BQ
A
1
a
b
B
B
Cournot Case
Bertrand Case
p1
 A  Bq2 
p1
p2
D
q1
D
q1
Price competition creates a discontinuity in each firm’s demand curve –
this, in turn creates a discontinuity in profits
if p1  p2
0



 a  bp1 
 1  p1 , p2   ( p1  c)
 if p1  p2
 2 


 ( p  c)( a  bp ) if p  p
1
1
2
 1
As in the cournot case, we need to find firm one’s best response
profit maximizing response) to every possible price set by firm 2.
(i.e.
Firm One’s Best Response Function
Case #1: Firm 2 sets a price above the pure monopoly price:
p2  pm
p1  pm
Case #2: Firm 2 sets a price between the monopoly price and marginal cost
pm  p2  c
p1  p2  
Case #3: Firm 2 sets a price below marginal cost
c  p2
p1  p2
Case #4: Firm 2 sets a price equal to marginal cost
c  p2
p1  p2  c
What’s the Nash equilibrium of this game?
Bertrand Equilibrium: It only takes two firm’s in the
market to drive prices to marginal cost and profits to
zero!
However, the Bertrand equilibrium makes some very restricting
assumptions…
Firms are producing identical products (i.e. perfect
substitutes)
Firms are not capacity constrained
An example…capacity constraints
Consider two theatres located side by side. Each theatre’s marginal
cost is constant at $10. Both face an aggregate demand for movies
equal to
Q  6,000  60 P
Each theatre has the capacity to handle 2,000 customers per day.
What will the equilibrium be in this case?
Q  6,000  60 P
If both firms set a price equal to $10
(Marginal cost), then market demand is
5,400 (well above total capacity = 2,000)
Note: The Bertrand Equilibrium (P = MC) relies on each firm
having the ability to make a credible threat:
“If you set a price above marginal cost, I will
undercut you and steal all your customers!”
4,000  6,000  60P
P  $33.33
At a price of $33, market demand is 4,000 and both firms operate at capacity
Imperfect Substitutes
Recall our previous model that included travel time in the
purchase price of a product
Length = 1
Customer
Firm 1
x
Distance to Store
~
p  p  tx
Consumers
places a value V
on the product
Travel Costs
Dollar Price
Imperfect Substitutes
Now, suppose that there are two competitors in the market –
operating at the two sides of town
Customer
Firm 2
1 x
Firm 1
x
The “Marginal Consumer” is indifferent between the two competitors.
~
~
V  p1  tx  V  p2  t (1  x)
We can solve for the “location” of this customer to get a demand curve
Imperfect Substitutes
p2  p1  t
x
2t
Customer
Firm 2
1 x
 p2  p1  t 
D1  
N
2t


Firm 1
x
 p1  p2  t 
D2  (1  x) N  
N
2t


Both firms have a marginal cost equal to c
 p2  p1  t 
1  ( p1  c)
N
2t


 p1  p2  t 
 2  ( p2  c)
N
2t


 p2  t  c 
p1  

2


 p1  t  c 
p2  

2


Each firm needs to choose
price to maximize profits
conditional on the other
firm’s choice of price.
Bertrand Equilibrium with imperfect substitutes
p2
Firm 1
Firm 2
t c
t c


 2 
t c


 2 
t c
p1
Cournot vs Bertrand
Suppose that Firm two‘s costs increase. What happens in each
case?
Bertrand
p2
Cournot
q2
Firm 1
Firm 1
Firm 2
Firm 2
p1
q1
Cournot vs Bertrand
Suppose that Firm two‘s costs increase. What happens in each
case?
Cournot (Quantity Competition): Competition is very
aggressive
Firm One responds to firm B’s cost increases by
expanding production and increasing market share\
Best response strategies are strategic substitutes
Bertrand (Price Competition): Competition is very passive
Firm One responds to firm B’s cost increases by
increasing price and maintaining market share
Best response strategies are strategic complements
Stackelberg leadership – Quantity Competition
In the previous example, firms made price/quantity decisions
simultaneously. Suppose we relax that and allow one firm to choose
first.
P  A  BQ
Both firms have a marginal cost equal to c
Firm A chooses
its output first
Firm B chooses
its output second
Market Price is
determined
Firm B has observed Firm A’s output decision and faces the
residual demand curve:
P   A  Bq A   Bq B
TR   A  Bq A qB  Bq
2
B
MR   A  Bq A   2Bq B  c
A  c qA
qB 

 qB q A 
2B
2
Knowing Firm B’s response, Firm A can now maximize its
profits:
P   A  Bq B   Bq A
A  c qA
qB 

2B
2

A  c  Bq A
P

2
2
2

A  c q A Bq
TR 

A
2

A  c
MR 
 Bq
2
2
A
c
qA

A  c

2B
Monopoly Output
qA

A  c

2B
Ac
qB 
4B
Essentially, Firm B acts as a monopoly
in the “Secondary” market (i.e. after A
has chosen). Firm B earns lower
profits!
A  c qA
qB 

2B
2
1  Ac 


2 B 
Monopoly
Output
3 A  c 
q A  qB 
4B
2 Ac 


3 B 
3 Ac   Ac 



4 B   B 
Cournot
Output
Stackelberg
Output
Competitive
Output
Sequential Bertrand Competition
With identical products, we get the same result as before (P =
MC). However, lets reconsider the imperfect substitute case.

p2  

p1  t  c 

2


p1  

p2  t  c 

2

We already derived each
firm’s best response
functions
Now, suppose that Firm 1 gets to set its price first (taking into account
firm 2’s response)
 p2  p1  t 
1  ( p1  c)
N
2t


Sequential Bertrand Competition
Take the derivative
and set equal to
zero to maximize
profits
 c  3t  p1 
1  ( p1  c)
N
4t


3t
p1  c 
2
5t
 p1  t  c 
p2  
c
2
4


Note that prices are higher than under the simultaneous move
example!!
Sequential Bertrand Competition
3t
p1  c 
2
3
D1  N
8
5
D2  N
8
5t
 p1  t  c 
p2  
c
2
4


18
1 
Nt
32
25
2 
Nt
32
In the simultaneous move game, Firm A and B charged the
same price, split the market, and earned equal profits.
Here, there is a second mover advantage!!
Cournot vs Bertrand: Stackelberg Games
Cournot (Quantity Competition):

Firm One has a first mover advantage – it gains
market share and earns higher profits. Firm B loses
market share and earns lower profits

Total industry output increases (price decreases)
Bertrand (Price Competition):
Firm Two has a second mover advantage – it charges a
lower price (relative to firm one), gains market share and
increases profits.
Overall, production drops, prices rise, and both firms
increase profits.