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 an equilibrium price up, in spite of the problems of
free-riders and secret price cutting…..
In general in one-shot games – perfect information - no
reputation effects – there are big incentives to deviate and high
prices are not sustainable in equilibrium. Hence we turn to
repeated games.
1
Dynamic Games
Are high prices sustainable in non-cooperative repeated
games?
Infinite Horizon, Perfect Information, Repeated Game
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 that supports the cooperative outcome as a
non-cooperative Nash equilibrium in the repeated game:
firmi sets monopoly price in all periods if and only if, no price
set in any earlier period of the game is < the monopoly price,
otherwise, firmi sets Price = MC.
The anticipated one period gain from unilateral deviation from
high price is less than the cost of punishment forever
(competitive pricing) for certain values of  (in homework for
0.5) Thus, all firms will maximise i = tt by setting
monopoly price forever, when 0.5
2
Finite Horizon, Perfect Information, Repeated Game
Solve Finite Repeated Game in process of backward induction
Last period T: anticipated one period gain from unilateral
deviation from high price 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 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
3
Green Porter Model (1984)
Introduce uncertainty concerning the nature of demand. If
firms observe a low market price, there are two possible
stories:
1. firms are deviating from setting a low output (high price),
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.
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 ‘cheating’ is taking place.
4
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
All strategy
combinations that
form equilibria
X
X
X
X
X
Certain types of
strategy that form
equilibria
- The type of strategy used in this model is most widely used
simple “Trigger Strategy”
- the (restricted) space of strategies we use is as follows :
_
qit =
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)
_
qi
- 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
5
- 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
(2)
random demand shock resulting in low demand
qi
or
- 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)
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
6
~
P and T are given, but I decide on the level qi to set, to
maximise NPV payoffs
future earnings )
(Earnings this period + discounted
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 >
V i (qi ) 
_
q i ?)
 i (qc )
 i (qi )   i (qc )

1   1    (   T ) (qi )

~
probability of breakdown (i.e. P < P ) = (qi)
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):
7
V i (qi ) 
 i (qc )
 i (qi )   i (qc )

1   1    (   T ) (qi )
deviant must trade off more profit today, with higher (qi).
Thus the “cartel”, on average, gets lower profit
_
The decision to deviate (set qi > q i ) involves selecting qi to
maximise Vi(qi)
dV i (qi )
F.O.C. sets dq  0
i
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 , or deviates and sets qi> q i , depends on the values of { q ,
_
i
~
P , T}
*
~
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 non-cooperative dynamic oligopoly
8
~
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 .
~
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
9
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 ( p 2 )
 1 D1 ( p1 )

V   T 
( p1  c ) 
( p2  c)
2
2
2
2

t 0
D ( p )( p1  c)  D2 ( p 2 )( p 2  c)
 1 1
4(1   )
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?
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
 1m   2m 
V 

If monopoly  in all states :
 (1   )4 
10
Gain from deviating in state of High demand 2:

m
2

 2m
2

 2m
2
Thus, for pms to be sustainable:
Benefits deviating  (discounted) costs of punishment
   1m   2m 
 2m
 V  

2
 (1   )4 
or, re-writing


2 2m
  0 
3 2m   1m
Since 2m > 1m, 0 is between ½ (1=2) and 2/3 (1 = 0). 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:
1  ( p )   ( p ) 
4
1
1
2
2
max
(1   )
p1, p2
s.t .
and
 1 ( p1 )


1
4
 1 ( p1 )   2 ( p2 )
(1   )
 2 ( p2 )
 1 4  1 ( p1 )   2 ( p2 ) 

2
(1   )
2
11
Re-writing the two constraints as
1(p1) K2(p2)
2(p2) K1(p1)
where K   / (2-3)  1
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) =
K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)
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 Green-Porter, where
‘price wars’ occur in low demand periods)
12
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
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
13
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)}.

2) if   0,

n  1
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
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
14
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).
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
15
Porter (1983) A Study of Cartel Stability – the JEC 18801886
 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
- 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
16
 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
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)
17
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
Supply Equation
Recall, we saw in the previous topic that the general F.O.C. for
firms is given as
dP
P
Q  MC
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:
  it 
MRi  p1    MCi (qit )  ai qit 1
 1 
Homogenous good, so p is same for each firm
18
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:

t 
 1

MR  pt 
1


DQ
t

1 

N
where

D    ai1 1
1

i 1
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
19
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
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
20
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 < jointprofit 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
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
21
is some, not strong, historical evidence that price wars
tended to occur after unexpected demand shifts.
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.
22
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 noncooperative 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.
Mariuzzo and Walsh (2006) include Lake prices
rather than just a dummy; and allow for a
deterministic cycle as in Haltwinger and Harrington
(1991)
23
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
Halwinger-Harrington (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)
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)
24
- 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
- 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:
Marginit = 1Nvolit + 2expvolchgit+1 + 3terminalit +
4exptermchgit+1 + controls +it
Controls account for the impact of past terminal prices, past
retail prices, city and time effects
Nvol = state volume / state mean volume of retail sales in the
sample period
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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
following:
“collusive”
theory
predicts
the
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)
Predict volume changes with separate equation for each city of
the form:
Nvolt = f(past Nvolt-1) + f(past retail pt-1) month
dummies+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.
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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 Haltwinger-Harrington theory of
collusions (and with previous studies of retail gasoline)
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