Transcript Topic 1: Introduction: Markets vs. Firms

Topic 6:
Static Games
Bertrand (Price) Competition
EC 3322
Semester I – 2008/2009
Yohanes E. Riyanto
EC 3322 (Industrial Organization I)
1
Introduction

In a wide variety of markets firms compete in prices




Internet access
Restaurants
Consultants
Financial services

In monopoly, setting price or quantity first makes no difference

But, in oligopoly the strategic variable matters a great deal  price
competition is much more aggressive than quantity competition
Yohanes E. Riyanto
EC 3322 (Industrial Organization I)
2
Bertrand Competition

In the Cournot model price is set by some market clearing mechanism

An alternative approach is to assume that firms compete in prices  it
leads to dramatically different results

Take a simple example

two firms producing (or selling) an identical product (mineral water or
fruits)

firms choose the prices at which they sell their products

each firm has constant marginal cost of c

inverse demand is P = A – B.Q

direct demand is Q = a – bP with a = A/B and b= 1/B
Yohanes E. Riyanto
EC 3322 (Industrial Organization I)
3
Bertrand Competition

We need the derived demand for each firm  demand conditional upon the
price charged by the other firm

Take firm 2. Assume that firm 1 has set a price of p1

if firm 2 sets a price greater than p1 she will sell nothing

if firm 2 sets a price less than p1 she gets the whole market


if firm 2 sets a price of exactly p1 consumers are indifferent between
the two firms: the market is shared, presumably 50:50
So we have the derived demand for firm 2



q2 = 0 if p2 > p1
q2 = (a – bp2)/2 if p2 = p1
q2 = a – bp2 if p2 < p1
Yohanes E. Riyanto
EC 3322 (Industrial Organization I)
4
Bertrand Competition

This can be illustrated as
follows:

Demand is discontinuous

The discontinuity in demand
carries over to profit
p2
There is a
jump at p2 = p1
p1
a - bp1
(a - bp1)/2
Yohanes E. Riyanto
EC 3322 (Industrial Organization I)
a q2
5
Bertrand Competition
Firm 2’s profit is:
Π2(p1,, p2) = 0
if p2 > p1
Π2(p1,, p2) = (p2 - c)(a - bp2)
if p2 < p1
Π2(p1,, p2) = (p2 - c)(a - bp2)/2 if p2 = pFor
1
whatever
Clearly this depends on p1.
reason!
Suppose first that firm 1 sets a “very high” price: greater
than the monopoly price of pM = (a +bc)/2b
Ac
2B
A  c aB  c a  bc
p m  A  BQ m 


2
2
2b
MR  MC  A  2 BQ  c  Q m 
Yohanes E. Riyanto
EC 3322 (Industrial Organization I)
6
Bertrand Competition
So firm 2 should just
What price
should firm 2
set?
With p1 > (aundercut
+ bc)/2b,
Firm 2’s profit looks like this:
p a bit and
At p2 = p1
all
the
Firmget
2’salmost
Profit
firm
2 ifgets
half
What
firm
1 of the
monopoly
profit
profit
pricesmonopoly
at
(a + c)/2b?
1
The monopoly
price
p2 < p1
Π2(p1,, p2) = (p2 - c)(a - bp2)
Firm 2 will only earn a
positive profit by cutting its
price to (a + bc)/2b or less
p2 = p 1
p 2 > p1
c
Yohanes E. Riyanto
(a+bc)/2b
EC 3322 (Industrial Organization I)
p1
Firm 2’s Price
7
Bertrand Competition
Now suppose that firm 1 sets a price less than (a + bc)/2b
Firm 2’s profit looks like this:
What
price
As
long as
p1 > c,
Firm 2’s Profit
Of course, firm 1
Firm
2 should
should
firmaim
2 just
will then undercut
to undercut
set now?firm 1
firm 2 and so on
p2 < p1
Then firm 2 should also
price
at c. Cutting price below cost
gains the whole market but loses
What
if firm
money
on1every customer
prices at c?
p2 = p 1
p 2 > p1
c
Yohanes E. Riyanto
p1 (a+bc)/2b
EC 3322 (Industrial Organization I)
Firm 2’s Price
8
Bertrand Competition


We now have Firm 2’s best response to any price set by firm 1:

p*2 = (a + bc)/2b
if p1 > (a + bc)/2b

p*2 = p1 - “something small ()”
if c < p1 < (a + bc)/2b

p*2 = c
if p1 < c
We have a symmetric best response for firm 1

p*1 = (a + bc)/2b
if p2 > (a + bc)/2b

p*1 = p2 - “something small ()”
if c < p2 < (a + bc)/2b

p*1 = c
if p2 < c
Yohanes E. Riyanto
EC 3322 (Industrial Organization I)
9
Bertrand Competition
The best response
function for
The best response
These best response
functions
look
like
this
firm 1
function for
p2
firm 2
R1
R2
(a + bc)/2b
The Bertrand
The equilibrium
equilibrium has
isboth
with both
firms charging
firms pricing at
marginal
cost
c
c
p1
c
Yohanes E. Riyanto
(a + bc)/2b
EC 3322 (Industrial Organization I)
10
Bertrand Equilibrium

The Bertrand model shows that competition in prices gives very different
result from competition in quantities.

Since many firms seem to set prices (and not quantities) this is a challenge to
the Cournot approach

But the result is “not nice”  there are only 2 firms and yet firms charge
p=MC  Bertrand Paradox.

Two extensions can be considered


So far, firms set prices  quantities adjust  what if we have capacity
constraints?
What happen if products are differentiated?
Yohanes E. Riyanto
EC 3322 (Industrial Organization I)
11
Bertrand Equilibrium
Diaper Wars
The Kimberly-Clark Corporation, a leading diaper manufacturer, attempted to improve profits during the
economic downturn of the Summer of 2002. The company decreased the number of diapers in each pack
in order to increase the price per diaper by 5% for its Huggies brand. Kimberly-Clark’s chief executive
officer, Thomas J. Falk, expected Procter & Gamble (P&G), the second largest producer of diapers, to
respond with a similar price increase for its Pampers brand. P&G had followed Kimberly-Clark’s price
moves in the past. Kimberly-Clark and P&G had cooperated previously to increase the profits of
both firms. Cooperation is often the profit maximizing response in repeated games.
However, P&G did not respond with a cooperative price increase in this instance. P&G increased
promotional expenses to encourage retailers to cut prices on larger Pampers packs or to put up special
displays. P&G also marked its Pampers packs with “Compare,” to highlight the price difference
between brands. P&G deviated from its past cooperative strategy with Kimberly-Clark in an
attempt to increase its market share. Given the poor conditions of the market, P&G executives
believed that this one-time, non-cooperative response would maximize profits, and that any future
punishment from Kimberly-Clark would not offset the gains from improving its market position. The
conditions of the market determined the level of cooperation P&G employed.
Source: Ellison, Sarah, “In Lean Times, Big Companies Make a Grab for Market Share,” Wall Street
Journal, September 5, 2003.
Yohanes E. Riyanto
EC 3322 (Industrial Organization I)
12
Capacity Constraints

For the p = c equilibrium to arise, both firms need enough capacity to
fill all demand at p = c

But when p = c they each get only half the market

So, at the p = c equilibrium, there is huge excess capacity

So capacity constraints may affect the equilibrium

Consider an example



daily demand for product A Q = 6,000 – 60P
Suppose there are two firms: Firm 1 with daily capacity 1,000 and
Firm 2 with daily capacity 1,400, both are fixed
marginal cost for both is $10
Yohanes E. Riyanto
EC 3322 (Industrial Organization I)
13
Capacity Constraints

Is a price P = c = $10 an equilibrium?

total demand is then 5,400, well in excess of capacity

Suppose both firms set P = $10: both then have demand of 2,700

Consider Firm 1:

Normally, raising price loses some demand

but where can they go? Firm 2 is already above capacity

so some buyers will not switch from Firm 1 at the higher price


but then Firm 1 can price above MC and make profit on the buyers
who remain
so P = $10 cannot be an equilibrium
Yohanes E. Riyanto
EC 3322 (Industrial Organization I)
14
Capacity Constraints

Assume that at any price where demand is greater than capacity there
is efficient rationing.

Buyers with the highest willingness to pay are served first.

Then we can derive residual demand.

Assume P = $60

total demand = 2,400 = total capacity

so Firm 1 gets 1,000 units


residual demand to Firm 2 with efficient rationing is Q = 5000 –
60P or P = 83.33 – Q/60 in inverse form.
marginal revenue is then MR = 83.33 – Q/30
Yohanes E. Riyanto
EC 3322 (Industrial Organization I)
15
Capacity Constraints (Efficient-Rationing Rule)
Efficient rationing rule: consumers with highest
willingness to pay are served first.
Price
100
83.33
residual
demand
1000 units
60
1400 units
(firm 2)
1000 units
(firm 1)
2400 units
Yohanes E. Riyanto
EC 3322 (Industrial Organization I)
Quantity
16
Capacity Constraints

Residual demand and MR:
Price

Suppose that Firm 2 sets P = $60.
Does it want to change?



since MR > MC Firm 2 does
not want to raise price and lose
buyers
since QR = 1,400 Firm 2 is at
capacity and does not want to
reduce price
$83.33
Demand
$60
MR
$36.66
$10
MC
1,400
Quantity
Same logic applies to Firm 1 so P = $60 is a Nash equilibrium
for this game.
Yohanes E. Riyanto
EC 3322 (Industrial Organization I)
17
Capacity Constraints

Logic is quite general

firms are unlikely to choose sufficient capacity to serve the
whole market when price equals marginal cost




since they get only a fraction in equilibrium
so capacity of each firm is less than needed to serve the whole
market
but then there is no incentive to cut price to marginal cost
So we avoid the Bertrand Paradox when firms are capacity
constrained
Yohanes E. Riyanto
EC 3322 (Industrial Organization I)
18
Product Differentiation

Original analysis also assumes that firms offer homogeneous
products

Creates incentives for firms to differentiate their products

to generate consumer loyalty

do not lose all demand when they price above their rivals


keep the “most loyal”
We will discuss this when we cover the product differentiation
topic.
Yohanes E. Riyanto
EC 3322 (Industrial Organization I)
19
Topic 7:
Sequential Move Games
Stackelberg Competition
EC 3322
Semester I – 2008/2009
Yohanes E. Riyanto
EC 3322 (Industrial Organization I)
20
Introduction

In a wide variety of markets firms compete sequentially




One firm makes a move.

new product

advertising
Second firms sees this move and responds.
These are dynamic games.

May create a first-mover advantage

or may give a second-mover advantage

May also allow early mover to preempt the market
Can generate very different equilibria from simultaneous move games
Yohanes E. Riyanto
EC 3322 (Industrial Organization I)
21
Sequential Move Game
X
terrorist
fly to X
pilot
fly to the
original
destinati
on Y
bomb
-1,-1
not bomb
1,1
bomb
-1,-1
not bomb
Pilot-Terrorist
Game
2, 0
Pilot
Y
BB’
-1 , -1
-1 , -1
BN’
-1 , -1
0,2
NB’
1,1
-1 , -1
NN’
1,1
0,2
Terrorist
Nash-Equilibria: (X,NB’); (Y,BN’); (Y,NN’)
Yohanes E. Riyanto
EC 3322 (Industrial Organization I)
22
Sequential Move Game

Thus, there are multiple pure strategy NE. This greatly reduce our
ability to generate predictions from the game.

We need another solution concept that can narrow down the set of
NE outcomes into a smaller set of outcomes.

We need to eliminate NE that involves non-credible threat
(unreasonable).

From the example: terrorist’ strategy that involves bomb threat is not
credible, because once his information set is reached he will never
carried out the threat.

Thus, we need to be able to eliminate (X, NB’) and (Y, BN’).
Yohanes E. Riyanto
EC 3322 (Industrial Organization I)
23
Sequential Move Game

Refinement:  Subgame Perfect Nash Equilibrium: A Strategy
profile is said to be a subgame perfect Nash equilibrium if it
specifies a Nash Equilibrium in every subgame of the original
game.

For the entire game, the NE are (X,NB’); (Y,BN’); (Y,NN’)

For the two subgames:
terrorist
bomb
not
bomb

-1,-1
terrorist
1,1
bomb
not
bomb
-1,-1
2,0
Hence, (X,NB’); (Y,BN’); are not SPE. The terrorist will always
choose Not Bomb (NN)
Yohanes E. Riyanto
EC 3322 (Industrial Organization I)
24
Subgame Perfect Equilibrium

To get the SPE  “backward induction” method (‘look ahead reason
back’)

Analyze a game from back to front (from information sets at the end
of the tree to information sets at the beginning). At each information
set, one eliminates strategies that are dominated, given the terminal
nodes that can be reached.
1
3, 8
1
7, 9
1, 2
1
2, 1
10, 4
2
1
2
0, 5
1
4, 0
8, 3
Yohanes E. Riyanto
EC 3322 (Industrial Organization I)
25
Stackelberg Competition

Let’s interpret first in terms of Cournot

Firms choose outputs sequentially


Leader sets output first, and the choice is observed by the follower.

Follower then sets output upon observing the leader’s choice.
The firm moving first has a leadership advantage

It can anticipate the follower’s actions

can therefore manipulate the follower

For this to work the leader must be able to commit to its choice of output

Strategic commitment has value
Yohanes E. Riyanto
EC 3322 (Industrial Organization I)
26
Stackelberg Competition
t=1
firm 1 choosing
its optimal
quantity (q*1) to
maximize its
profit.
Yohanes E. Riyanto
t=2
Time Period
firm 2 observes the
optimal quantity
choice of of firm 1
(q*1) and sets its
optimal quantity
(q*2(q1))
EC 3322 (Industrial Organization I)
27
Stackelberg Competition

Assume that there are two firms with identical products

As in our earlier Cournot example, let demand be:

P = A – B.Q = A – B(q1 + q2)

Marginal cost for for each firm is c

Firm 1 is the market leader and chooses q1

In doing so it can anticipate firm 2’s actions. So consider firm 2.
Demand for firm 2 is:


P = (A – Bq1) – Bq2
Marginal revenue therefore is:

MR2 = (A - Bq1) – 2Bq2
Yohanes E. Riyanto
EC 3322 (Industrial Organization I)
28
Stackelberg Competition
Equate
This is marginal
firm 2’s revenue
with
marginal cost
MR2 = (A - Bq1) – 2Bq2 best
response
q2
function
MC = c
But firm 1 knows
Firm 1 knows that
this is how firm 2
 q*2 = (A - c)/2B what
- q1/2q2 is going
to be
will reactSotofirm
firm11’s
can
Demand for firm 1 is:
output
choicefirm 2’s
anticipate
P = (A - Bq2) – Bq1
(A – c)/2B
From
earlier example we know reaction
P = (A - Bq*2) – Bqthat
this is the monopoly output. This is an
1
important
result. The Stackelberg
leader
P = (A - (A-c)/2) – Bq
1/2
S
Equate
marginal
revenue
(A – c)/4B
chooses
the
output as a monopolist would.
 P = (A + c)/2 – Bq1/2 with same
marginal cost
R2
But
2 isequation
not excluded from the market
Marginal revenue for firm
1firm
is: this
Solve
q1
MR1 = (A + c)/2 - Bq1 for output q1
(A
–
c)/B
(A – c)/2B
(A + c)/2 – Bq1 = c
 q*1 = (A – c)/2B
Yohanes E. Riyanto
 q*2 = (A – c)4B
EC 3322 (Industrial Organization I)
29
Stackelberg Competition
Aggregate output is 3(A-c)/4B
So the equilibrium price is (A+3c)/4
Leadership benefits
Firm
1’s best
the leader
firm 1response
but
harms
Leadership
the follower
benefits
function
is
“like”
consumers
firm
but
Compare
this with
firm2 2’s
reduces
aggregate
the
Cournot
profits
equilibrium
q2
(A-c)/B
Firm 1’s profit is (A-c)2/8B
R1
Firm 2’s profit is (A-c)2/16B
We know that the Cournot
equilibrium is:
qC1 = qC2 = (A-c)/3B
(A-c)/2B
C
(A-c)/3B
The Cournot price is (A+c)/3
S
(A-c)/4B
R2
Profit to each firm is (A-c)2/9B
(A-c)/3B (A-c)/2B
Yohanes E. Riyanto
EC 3322 (Industrial Organization I)
(A-c)/ B
q1
30
A Comparison of Oligopoly Equilibria
Output
Price
Firm
Industry
Monopoly
360
360
Cournot Duopoly
240
Stackelberg Duopoly
Leader
Follower
Competitive Market
(P=MC)
Profit
Consumer
Surplus
Firm
Industry
64
129.6
129.6
64.8
480
52
57.6
115.2
115.2
540
46
97.2
145.8
0
259.2
360
180
64.8
32.4
720
28
0
Market demand function: P = 1 – 0.001Q and MC=0.28
(linear demand and constant MC)
Yohanes E. Riyanto
EC 3322 (Industrial Organization I)
31
A Comparison of Oligopoly Equilibria
2
57.6
32.4
Cournot
Stackelberg
Monopoly
Bertrand
& Competitive
Solution
Yohanes E. Riyanto
57.6 64.8
EC 3322 (Industrial Organization I)
129.6
1
32
Stackelberg and Commitment

It is crucial that the leader can commit to its output choice



without such commitment firm 2 should ignore any stated intent by
firm 1 to produce (A – c)/2B units
the only equilibrium would be the Cournot equilibrium
So how to commit?



prior reputation
investment in additional capacity
place the stated output on the market

Given such a commitment, the timing of decisions matters

But is moving first always better than following?

Consider price competition
Yohanes E. Riyanto
EC 3322 (Industrial Organization I)
33
Stackelberg and Price Competition

With price competition matters are different

first-mover does not have an advantage

suppose products are identical

suppose first-mover commits to a price greater than marginal
cost

the second-mover will undercut this price and take the market

so the only equilibrium is P = MC

identical to simultaneous game
Yohanes E. Riyanto
EC 3322 (Industrial Organization I)
34
Application: Advertising & Competition

The game (firm 1 and 2)
t=1
t=2
Time Period
firm 1 chooses
advertising level
(a) in order to
enhance demand


firm 1 and 2
compete in a
Cournot fashion
(choosing quantity
Level)
Firm 2 observes the
choice of a of firm
1
The market demand faced by the two firms p  a  q1  q2
Firms produce at zero costs, but firm 1 incurs advertising costs of
Yohanes E. Riyanto
EC 3322 (Industrial Organization I)
2a 3
81
35
Application: Advertising & Competition

(Start from t=2): Solve the Cournot best response function of the
two firms at the end of the game (t=2), taking the advertising level
determined in t=1 as given.
2a 3
1  a  q1  q2 q1 
81

Derive the f.o.c. w.r.t. q1 and solve for q1, we get the best response
function.
q*1  f q2  

a  q2 
2
Similarly derive the best response fu. For firm 2.
 2  a  q1  q2 q2

a  q1 
q 2  f q1  
*
2
Yohanes E. Riyanto
EC 3322 (Industrial Organization I)
36
Application: Advertising & Competition

*
*
The Cournot Nash equilibrium can be obtained. q 1  q 2 

The equilibrium price is then, p  a 



a
3
a a a
 
3 3 3
Hence, firm 1’s profit function as a function of a is,
2a 3
1  a  q1  q2 q1 
81
2
2a 3
a
1    
81
3
(Now at t=1): Firm 1 chooses its advertisement level (a) to maximize its
profit.
1 2a 6a 2
f .o.c.


0
a*  3
a
9
81
The SPE strategy profile is
a
a
a*  3 q1a  
q2 a  
3
3
Yohanes E. Riyanto
EC 3322 (Industrial Organization I)
37