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-cooperativeoligopoly 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 pricesare not sustainable in equilibrium. Hence we turn to repeated games.

Dynamic Games

Are high prices sustainable in non-cooperativerepeated 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

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

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.
  1. 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.
  1. 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.
  1. 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.

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 manypotential equilibria) we simplify matters at the outset by restricting the strategy space.

-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, high price – between monopoly and cournot levels)

= 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(Qt) . t

-Firms watch . Since they don’t know exactly Qt, they cannot tell whether low is caused by

(1)firms strategically expanding output qi above or

(2)random demand shock resulting in low demand

-Trigger Strategy: Switch to Cournot for finite T periods, if (i.e. assume lower observed price is due to firms strategically expanding output above, though this may not be the case….)

-Choice variables: qi, , T . Thus, { q*, , 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 > )

- 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 )

Let Vi(qi)= NPV of i’s profit stream if qi is used in normal periods

(issue will be, should I set qi = or set qi > ?)

probability of breakdown (i.e.) = (qi)

Vi(qi) =

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):

deviant must trade off more profit today, with higher (qi). Thus the “cartel”, on average, gets lower profit

The decision to deviate (set qi) involves selecting qi to maximise Vi(qi)

F.O.C. sets

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, or deviates and sets qi, depends on the values of { q*, , T}

There are many { q*, , 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

e.g. low , 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 (and higher the price) that can be sustained under dynamic non-cooperative oligopoly.

In making a decision about the level of the

trigger price and the length of time for reversion to Cournot T, firms trade off current profits from deviation and future losses from Cournot relative to .

For values of { q*, , 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 , 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

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:

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

If monopoly  in all states :

Gain from deviating in state of High demand 2:

Thus, for pms to be sustainable:

Benefits deviating  (discounted) costs of punishment

or, re-writing

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 < 02/3

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:

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)

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 , 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

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 (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 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

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

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

-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

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

Supply Equation

Recall, we saw in the previous topic that the general F.O.C. for firms is given as

(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:

Homogenous good, so p is same for each firm

Define market-share weighted parameter:

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:

where

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 Strepresents 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 ()
  • DM1-DM4 - structural dummies

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

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.

Summing Up: