Download Dynamic Price Competition in Homogenous Products

Dynamic Price Competition in Homogenous
Products
Chicago Tradition on Cartels:
Friedman 1973, Newsweek on the OPEC
Cartel: Cartels involve setting a price in which
it would be optimal for somebody to deviate by
secret price cutting. Thus, all cartels are
unstable, including OPEC, and so there is no
need to worry…….
However, this fails to recognise that there are
some mechanisms under non-cooperative
oligopoly models which can hold a equilibrium
price up, in spite of the problems of free-riders
and secret price cutting…..
1
Static Model – One-Shot Game
 an industry is faced with a sum of marginal
cost schedules and a demand schedule.
 Under perfect competition, P0 = MC.
S = MC
P1
P2
P0
Exess Market
Capacity
D
 Industry agrees P1> P0, knowing the
demand, and bargaining among the firms
about the quotas each can have
 tendency for an individual to offer P2< P1,
steal customers from other firms and
operate at full capacity (for the firm –
market wide excess capacity at p2 is still
positive), thus taking more than his quota.
 With perfect information, other firms
notice it’s quota has been stolen, and so
that cheating has been going on
 One-Shot game – perfect information - no
reputation effects – big incentive to deviate
– high price not sustainable in equilibrium
2
Dynamic Games
Are high prices sustainable?
Infinite Horizon,
Repeated Game
Perfect
Information,
firmi payoff is discounted sum of profits:
i = tt
where discount factor is 0 <  < 1 (higher
values give greater weight to the future)
Strategy for firmi in repeated game maps prices
set by all players in periods 1……T-1 into Pit
for firmi in period t
“Trigger Strategy”: I am either going to do
this, or that; and there is something called the
trigger that shoots me from this to that.
3
Trigger strategy that supports the cooperative
outcome as a non-cooperative Nash
equilibrium in the repeated game:
firmi sets monopoly price P0 in all periods if
and only if, no price set in any earlier period of
the game is < P0. Otherwise, firmi sets P = MC.
The anticipated one period gain from unilateral
deviation from high price P0 is less than the
cost of punishment forever (competitive
pricing) for certain values of  (0.5)
Thus, all firms will maximise i = tt by
setting monopoly price forever, when 0.5
Finite
Horizon,
Repeated Game
Perfect
Information,
firmi payoff is discounted sum of profits:
i = tt
firmi sets high P0 in all periods if and only if,
no price set in any earlier period of the game is
< P0. Otherwise, firmi sets P = MC.
4
Solve Finite Repeated Game in process of
backward induction
Last period T: anticipated one period gain from
unilateral deviation from high price P0 brings
about no future punishment. Thus, incentive
for representative firm to deviate. All firms
thus deviate in final period T, and P = MC
Second Last Period T-1: Treat as last period.
Perfect information, so all know P = MC in last
period T. Thus, anticipated one period gain
from unilateral deviation from high price P0 at
T-1 brings about no additional future
punishment. Thus, incentive for representative
firm to deviate. All firms thus deviate in final
period T-1, and P = MC
Similar for each preceding period
So First Period, T=1, we have all firms
deviating and setting P = MC
Co-operative prices are not sustainable in
Finite Repeated Games with Perfect
Information
5
Green Porter Model (1984)
More realistic assumption of uncertainty – do
not exactly know what demand is.
If firms observe a low market price, there are
two possible stories:
1. firms are deviating from setting a low
output (high price), or
2. actual demand is low (quotas are set based
on anticipated demand).
Which is true?
Could try to use an observable market signal to
distinguish between 1) and 2). However, the
environment in which firms operate is actually
quite complicated.
6
Underlying Assumptions of the Green Porter
Model:
1.Market is stable over time i.e. fluctuations
in the demand curve are described by a
stationary stochastic process.
2. In this model, there is no route around the
signal extraction problem, so ‘cheating’
(firms expanding output) can not be
distinguished from a low demand.
3.All information is public, except ‘own
output’. So if some other player is
producing above its quota, then that is not
detectable by other parties.
4.NB: The information which is used to
police the arrangement is imperfectly
correlated with actions i.e. by observing
market price – which is only imperfectly
correlated with whether or not there is
‘cheating’.
7
Structure of the Green Porter Model:
Is given by the structure of the market……
- n firms
- homogenous goods
- i(qi, P) is current profit per period of firm i;
P = price and qi = sales
- As in many models (because there are many
potential equilibria) we simplify matters at
the outset by restricting the strategy space.
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
All strategy
combinations that
form equilibria
X
Certain types of
strategy that form
equilibria
- The type of strategy used in this model is
most widely used simple “Trigger Strategy”
8
- the (restricted) space of strategies we use is
as follows :
_
_
qit = q i if t is ‘normal’ (low output q i , high
price – between monopoly and
cournot levels)
c
= q i if t is ‘revisionary’ (higher output,
lower price i.e. Cournot)
- assume Cournot behaviour in revisionary
periods
- the focus of interest is qi - what level of
output will profit maximising firms select?

- Market Demand: P = P(Qt) . t

- Firms watch P . Since they don’t know

exactly Qt, they cannot tell whether low P is
caused by
(1) firms strategically expanding output qi
_
above q i or
(2) random demand shock resulting in low
demand
9
- Trigger Strategy: Switch to Cournot for finite

~
T periods, if P < P (i.e. assume lower
observed price is due to firms strategically
_
expanding output above q i , though this may
not be the case….)
~
*
~
- Choice variables: qi, P , T . Thus, { q , P , T}
is an equilibrium satisfying the condition that
no firm wishes to deviate (i.e. Nash
Equilibrium)
- Let the discount factor be 0 <  < 1 (higher
values of  give greater weight to future
profit)
10
Mechanism:
Calculate the Payoff (NPV of future profits)
from deviating, when everyone plays the above
strategies
Need to worry about different future paths
pricing may take …..
_
Even if no-one else deviates (expands qi > q i )
- I might deviate and trigger Cournot
revisionary period for T finite periods
- I might not deviate, but low demand
triggers Cournot for T finite periods
~
and T are given, but I decide on the level qi
to set, to maximise NPV payoffs (Earnings
this period + discounted future earnings )
P
Let Vi(qi)= NPV of i’s profit stream if qi is
used in normal periods
_
_
(issue will be, should I set qi = q i or set qi > q i ?)
V i (qi ) 
 i (qc )
 i (qi )   i (qc )

1   1    (   T ) (qi )
probability of breakdown (i.e. P < P ) = (qi)

~
11
Vi(qi) satisfies: Vi(qi) =
T t i

 i (qi )   (1   (qi ))V (qi )   (qi )    (qc )   TV i (qi )
 t 1

i



no trigger( probability (1 ( qi ))
earn V i ( qi ) ( discounted ) from now on



trigger( with probability  ( qi ))
earn cournot ( discounted t ) profits for T periods
V i ( qi ) ( discounted T ) from when go back to normal
Discounted future earnings =
discounted payoffs if no Cournot reversion
(with probability 1 - (qi))
+ discounted Cournot payoffs for T periods if
breakdown (with probability (qi))
+ discounted payoffs from the point that we go
back to normal
Solving for Vi(qi), we obtain (as written earlier):
 (q )
 i (qi )   i (qc )
V i (qi )  i c 
1   1    (   T ) (qi )
deviant must trade off more profit today, with
higher (qi). Thus the “cartel”, on average,
gets lower profit
12
_
The decision to deviate (set qi > q i ) involves
selecting qi to maximise Vi(qi)
F.O.C. sets
dV i (qi )
0
dqi
The optimal q* chosen takes into account: the
expected gains from one period deviation of qi
by one unit, and the expected costs of
triggering breakdown.
Note that the expected costs of deviating
include the loss of profit for T periods (cournot
strategy played)
And the fact that a breakdown may happen by
accident (low demand), and thus the future
benefit you forego is one you only get
sometimes – this reduces the expected cost
Whether the maximising firm sets the co_
operative output level q i , or deviates and sets
_
*
~
qi> q i , depends on the values of { q , P , T}
13
*
~
There are many { q , P , T} triples that form a
Nash Equilibrium, such that the F.O.C.
requirement holds so that no firm will want to
deviate along the equilibrium path of the game
i.e low output (and high price) is feasible and
no-one wants to deviate
under noncooperative dynamic oligopoly
~
e.g. low P , needs high T to be a sufficient
deterrent to deviation…..
The more severe the punishment (longer T
and/or more competitive behaviour during
revisionary period) and the greater the
weighting given to future profits (), the lower
_
the output q (and higher the price) that can be
sustained under dynamic non-cooperative
oligopoly.
In making a decision about the level of the
~
trigger price P and the length of time for
reversion to Cournot T, firms trade off current
profits from deviation and future losses from
_
Cournot relative to q i .
14
*
~
For values of { q , P , T}, there is a Nash
Equilibrium – nobody finds it worthwhile to
deviate from low output (high price), and this
is common knowledge.

~
Does this mean, if players observe P < P , they
can agree to ignore it, knowing that it can’t
have been caused by a deviant?
No – since removing the punishment does not
discipline the firms, and so you get unilateral
incentives to deviate…..
Green Porter Predictions
1. Interprets Revisionary Periods as ‘Price
Wars’
2. ‘Price Wars’ should occur sometimes
3. ‘Price wars’ should happen when demand
is low.
4. Firms should not cheat (in equilibrium,
‘price wars’ happen only due to demand
shocks)
5. output during normal periods exceeds the
monopoly level, but is lower than Cournot
15
Rotemberg-Saloner (1986)
 Homogenous Goods
 2 price setting symmetric firms, infinite
game
 No uncertainty
 i.i.d. Demand shock: at each period can be
low (D1(p)) or high (D2(p)) with probability
½ . Assume D2(p) > D1(p)  p
 In each period, state of demand is known
before choose p
Thus, discounted profits of each firm from
any two prices is given by:

1 D2 ( p2 )
 1 D1 ( p1 )

V   
( p1  c) 
( p2  c ) 
2 2
2 2

t 0
D ( p )( p  c)  D2 ( p2 )( p2  c)
 1 1 1
4(1   )
T
Can we enforce a (non-cooperative)
agreement? Is there {p1*,p2*}in which deviating
from a price ps when demand is in state s is not
privately optimal?
16
Assume firms set pm in each state of demand
Assume maximal punishment : if observe
p<pm: p=mc and zero profits forever
Incentives to deviate? Highest in high demand
period, so consider these….
Maintain high prices if profit from deviating <
profit from cooperating
If monopoly  in all states :
 1m   2m 
V 

(
1


)
4


Gain from deviating in state of High demand 2:
m
m


 2m  2  2
2
2
Thus, for pms to be sustainable:
Benefits deviating  (discounted) costs of
punishment
  1m   2m 
 2m
 V  

2
(
1


)
4


or, re-writing
17
2 2m
  0  m
3 2   1m
Since 2m > 1m, 0 is between ½ (1=0) and 2/3
(1 = 2). i.e. cooperative outcomes are
sustainable in non-cooperative oligopoly where
  0 such that ½ < 0 < 2/3
When demand is high, the temptation to
undercut is important. The punishment is an
average of high and low profit (so less severe
than if high demand were to persist with
certainty)
What if  between ½ and 0? Can not support
monopoly prices in high demand periods. Then
choose (p1,p2) to max firms expected payoffs
subject to the incentive (no undercutting)
constraints:
18
1
max p1, p2
s.t.
and
4
 1 ( p1 )   2 ( p2 )
(1   )
 1 ( p1 )


1
4
 1 ( p1 )   2 ( p2 )
(1   )
 2 ( p2 )  14  1 ( p1 )   2 ( p2 )

2
(1   )
2
Re-writing the two constraints as
1(p1) K2(p2)
2(p2) K1(p1)
where K  (2-3)
Intuitively, second constraint is the binding one
(high demand). So for any p1, choose p2 to
maximise subject to it.
But this solution gives an objective function which
increases in p1 up to p1m , so set
p1 = p1m and
then choose p2 subject to 2(p2) = 1(p1m)
So charge p1m in low demand, and p2< p2m in high
demand (note: this does not mean necessarily price
levels in one period are higher compared to another –
depends on what the demand function and thus what
monopoly price is each period)
19
Rotemberg and Saloner:
 No uncertainty
 Rational comparison of gains from deviating to
losses of punishment
 Harder to support monopoly pricing in good
times than bad, since incentive to deviate is
higher (i.i.d. assumption important here –
assumes good times not known to be followed
by even better times….)
 Consistent with ‘Countercyclical Pricing’
 Revisions in prices interpreted as ‘Price War’
 ‘Price Wars’ occur in Booms (unlike GreenPorter, where ‘price wars’ occur in low demand
periods)
20
Haltwinger and Harrington (1991)
Replace i.i.d. demand shifts with predictable
demand movements (e.g. business cycle,
seasonal fluctuations …)
Thus, different periods differ in returns to
deviating (as with Rotemberg and Saloner)
But here, since different periods have different
futures, they also differ with respect to the loss
due to punishment.
 Homogenous Goods
 n Price setting symmetric firms, infinite
game
 Deterministic Demand Cycles
 Demand curves increase (at every p) until
^
t,
and then decrease until cycle is
complete.
 Maximal Punishment: if firm deviates, then
we get reversion to zero profits forever
21
Firms sustain max joint profits subject to the
constraint that the price path is supportable
by a subgame perfect equilibrium. Thus,
punishment must > value of deviation for
each period
P(t ) 

   p( )  c D( p( )); / n


t

t 1
n  1 p(t )  c D( p(t )); t / n  D(t )
t here refers to the period in the cycle
discounted future loss from deviating in period
t from the cooperative price path (foregone
future higher profits)  one time gain from
deviating
The equilibrium they derive depends on the
value of 
 ^ 
1) if    ,1 (i.e  is large enough), firms
maintain pm forever, and whether this is pro- or
counter-cyclical depends on the form of the
sequence of {D(p,t)}.
22
 n  1
2) if   0, n  i.e.  is low enough, then the
p = c forever (can not sustain cooperative
prices. Note that, for a high enough value of n
this is actually a likely event)
3) there is a range of  values where we will
only not maintain monopoly outcomes at one
point in the cycle, and that point is after the
peak. i.e. the point at which cooperative
outcomes can not be maintained is always
when demand is falling
If lowered  further, then there would exist
many more such points over the cycle where
cooperative outcomes could not be sustained
However,
for the same level of demand, the point when
demand is falling will always loose the
ability to maintain cooperative outcomes
faster than the point at which demand is
rising
23
Two forces at work
Higher demand makes it more profitable to
cheat
Falling demand makes punishment from
deviating smaller
Thus, it is when demand is high and falling that
monopoly prices can not be maintained
Note :
 This is all relative to the monopoly price,
which in turn depends on how demand
curves shift over the cycle (e.g. if they
become more elastic when demand grows,
prices will be countercyclical…)
 When prices fall < pm, this does not mean
profits fall (no price war or punishment in
this sense).
24
Haltwinger and Harrington
 Current price depends on current demand
and on expectations of future demand
 Gain to deviating from established pricing
rule varies over the cycle, and is highest
when demand is strongest
 (discounted) loss from deviation varies
over the cycle, and is lowest when demand
is anticipated to be falling in immediate
future
 for the same level of demand, prices will
always be lower during periods of falling
demand than during rising demand.
 Thus, it is possible that prices may be
procyclical
during
booms,
and
countercyclical during recessions
25
Porter (1983) A Study of Cartel Stability –
the JEC 1880-1886
 JEC – a railroads freight cartel controlling
eastbound freight from Chicago (preceded
Sherman Act 1890, and so was explicit).
 Cartel took weekly stock of sales
 Cartel reported official prices and market
share quotas weekly in the “Chicago
Railway Review”
 However, clearing arrangements allowed
 Market demand highly variable (some 70%
of annual business was undertaken by
steamships when Lakes openend), so actual
market shares depended on actual prices
(could be different from official rate) and
the realisation of the demand shock
 Porter (1983) believed there was an internal
enforcement mechanism, which was a
variant of a trigger price strategy, used by
the JEC to maintain collusion
26
- We observe price and quantity movements
over time. Are they due to (exogenous)
shifts in the demand and cost functions? Or
are they due to price wars?
- Porters Main Objective: Establish the
existence of price wars
JEC gathered and disseminated weekly
information to member firms
 TQG – total quantity of grain recorded as
shipped by JEC members – varies
dramatically over period
 GR - index of grain rate prices of the JEC
 PO
- dummy variable = 1 when the
“Railway Review” reported that a price
war was occurring (though conflicts with
other indices of when a price war was
occurring that were available for that
period)
 PN – Porters estimate of when there was a
price war
27
Various changes in industry structure over
the period
 2 entrants to the railroad industry
 1 exit from the cartel
 opening and closing of alternative means
of transport (the Great Lakes)
 various seasonal effects
Thus, assumptions behind the ‘repeated game’
are suspect. Paper allows for the change in
structure to cause exogenous changes in the
various cartel prices (but only prices in
punishment phases)
Porter’s Model:
Demand Equation
ln Qt = 0 + 1 ln Pt + 2 Lt + 1t
Lakes is the main outside option
Lt = 1 Great Lakes open to shipping (all seasons,
save Winter)
= 0 Otherwise
28
Supply Equation
Recall, we saw in the previous topic that the
general F.O.C. for firms is given as
dP
P
Q  MCi
dQ
(where  = 0 for competitive industry;  = 1 for
collusive industry;  = 1/N for cournot industry)
N firms, asymmetric with respect to costs
ci(qi) = aiqit + Fi
i = 1,….,N
Thus, Marginal Revenue for firm i:
 1   it 
  MCi (qit )  ai qit 1
MRi  p
 1 
Homogenous good, so p is same for each firm
29
Define market-share weighted parameter:
N
 t   it sit
i 1
Conduct is allowed to vary over time (this is
the essence of the Green-Porter model – varies
between normal and revisionary behaviour).
Adding up MR condition over the N firms, and
solving for the quantities, we obtain the
industry marginal revenue conditions:
 1  t 
  DQt 1
MR  pt 
 1 
N
where

D    ai1 1
1

i 1
30
The implied Supply Equation is therefore:
ln pt = -ln (1+t/1) + ln D + ( -1)ln Qt
We identify t by putting on some structure
about how it varies.
Porter assumes there are only two regimes: one
that is collusive, and one that is a price war
He estimates the following:
ln pt = 0 + 1 ln Qt +2 St + 3 It + 2t
0 + 1 log Qt +2 St represents the price in
punishment periods
St = set of market structure dummies that
accommodate entry/exit
It = dummy = 1 during collusive regime
31
Theory predicts:  higher during collusive
regime, and therefore 3 should be positive
(since 1 is negative)
When the It are known, identification is as in
Bresnahan
When It not known, they are estimated using a
straight maximum likelihood
Data and Results:
 GR - $/100 pounds shipped (average of
self-reported prices
 TQG – total quantity of grain shipped
 PO – cheating dummy = 1 if collusion is
reported by Railway Review (not really
used)
^
 PN – estimated cheating dummy ( I t )
 DM1-DM4 - structural dummies
32
Table 3: Results
Collusion Dummies indicate collusive price
40% - 50% higher than price in the
punishment phase
TSLS: IV procedure where Porter
instruments for GR and TQG. Cooperative
prices > prices in punishment phase
BUT, Porter reports that these cooperative
prices < joint-profit maximising prices (when
absolute value of elasticity should = 1)
Does this imply that cost of maintaining a
collusion too high? Or at least, too high when
environment varied from period to period?
Lakes: Dummy = 1 when one could ship on
Great Lakes
33
Figure 1: GR, PO, and PN series
 Punishment phase does correspond to
price wars, but price wars seem to vary in
duration and magnitude
 Revisions to price wars happened more
regularly in later periods after the new
entry (and hence, when there are more
cartel members)
 Model implies that price wars should
occur when there is unanticipated low
realised demand. Porter does not find this
in the demand errors. Could be due to
several missing variables from demand
system that may have dominated the
behaviour of those errors and known to
the agents at the time (not to the
econometrician today – eg price of
freighter traffic on the Great Lakes).
There is some, not strong, historical
evidence that price wars tended to occur
after unexpected demand shifts.
34
Summing Up:
1. Green and Porter (1984) prediction that
price wars should occur sometimes.
This is tested by Porter (1983) - the
paper seems to document the existence
of an omitted variable on the supply
side, which he interprets as “price
wars”
2. However, he does not model what
drives it. There is no explanation of
why price wars start or how long they
last (vary in duration and magnitude)
3. Green and Porter (1984) prediction that
price wars should happen when
demand is low.
Porter (1983) regresses price war
occurrence on indicator variables and
finds nothing. As mentioned above,
the power of this test is low due to lack
of data.
35
4. Porter (1983) allows for change in
industry structure (entry and exit) – but
(i) does not tackle the issue of how
much the existence and success of a
cartel induces change in structure and
(ii) assimilates the two new entrants
into the cartel without much of a fight
5. Green and Porter (1984) prediction that
in a non-cooperative oligopoly, firms
should not cheat – in equilibrium price
wars occur only due to demand
shocks.
This is not tested by Porter (1984).
Ellison (1994) considers this.
6. Walsh and Whelan (2004) include
Lake prices rather than just a dummy;
and allow for a deterministic cycle as
in Haltwinger and Harrington (1991)
36
Borenstein and Shephard (1996) – Dynamic
Pricing in Retail Gasoline Markets
Not a study of an established cartel
Objective: Demonstrate Collusion AND
Examine its Form - is pricing of retail gasoline
consistent with predictions of HalwingerHarrington (1991) type models?
Looks for reduced form implications that are
consistent with the data (1986-1992)
Haltwinger-Harrington: harder to support
collusive prices if, all else equal, future
demand is lower 
1.
Collusive margins will respond to
anticipated changes in cost and demand
2.
Controlling for current demand, margins
will respond positively to expected
increase
in
near-term
demand
(punishments are likely more effective so
can support a higher price)
37
3.
Controlling for current input prices,
margins will respond negatively to
expected increase in input prices
(punishments are less effective and we
can’t support higher prices)
Retail gasoline:
- differentiated product market (mostly by
location)
- known seasonal changes in demand and
input prices (primary input is wholesale
gasoline)
- many ‘related’ firms in each market, which
doubt whether the joint profit maximising price
can be sustained (without side payments
between firms, which is illegal)
- Data are by city (so abstract from intracity
competition)
- Figure 2 shows seasonal in quantities –
shows distinct seasonal pattern, so there are
periods when future demand is expected to be
higher than current demand and vice-versa
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- Figure 1 shows seasonal in price - terminal
price is the closest they have to a wholesale
price, so margins are roughly proportional to
the difference between the terminal price and
the retail price (only roughly, as there are
different types of contracts between retailers
and suppliers so margins can depend on the
nature of the vertical contract).
- (Note that the terminal price series is much
more erratic than the quantity series, making it
difficult to see a seasonal in the margins. This
is the market with OPEC - various political
and collusive considerations are important in
determining the terminal price)
Basic Equation:
Margin = 1Nvol + 2expvolchg
3exptermchg + controls +
+
Controls account for the impact of past
terminal prices, past retail prices, city and time
effects
39
Nvol = state volume / state mean volume of
retail sales in the sample period
Assumes (absent incentives for collusion) retail
price would be a distributed lag of past
terminal and retail prices about an equilibrium
determined by volume and city effects.
Haltwinger-Harrington
predicts the following:
“collusive”
theory
2 > 0 (if anticipated demand , punishments
are likely more effective so can support a
higher price)
AND
3 < 0 (if anticipated terminal prices ,
punishments are less effective and we can’t
support higher prices)
The data: average monthly prices in ~ 60
cities over 5 year period (1986-1992)
40
Predict volume changes with separate equation
for each city of the form:
Nvolt = f(past Nvol) + monthdummies+f(time)
High fit (0.80 – 0.95) mainly due to the
seasonal
Predict terminal prices similarly - city-by-city
regression as a function of month, past terminal
p and past crude prices. Fit is only 0.3 – 0.6.
Terminal or input prices vary in a much less
predictable way than volume.
41
Table 2: Results (correcting for endogeneity)
 2 > 0 AND 3 < 0
 Margins (not price) are increasing in
quantity sold (Nvolt), and by about the
same amount as margins increase with
expected volume changes
Note: average margin =
(retail – terminal price = ~ 10.6 cents) / (average
terminal pricet = ~ 73 cents pergallon)
 Numbers not very large (effect of a one
deviation change in the expected volume on
margin, calculated at the mean, is about
0.26 cents – and similar for impact of
terminal price change) but they are
significant
 Results consistent with HaltwingerHarrington theory of collusions (and with
previous studies of retail gasoline)
42