European Journal of Operational Research 210 (2011) 436–447

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European Journal of Operational Research

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Interfaces with Other Disciplines

A new model for cycles in retail petrol prices

Edward Anderson ⇑Faculty of Economics and Business, University of Sydney, Australia

a r t i c l e i n f o a b s t r a c t

Article history:Received 21 September 2009Accepted 6 October 2010Available online 16 October 2010

Keywords:PricesRetail petrol price cyclesEdgeworth cycles

0377-2217/$ - see front matter � 2010 Elsevier B.V. Adoi:10.1016/j.ejor.2010.10.009

⇑ Tel.: +61 2 90367546; fax: +61 2 93137279.E-mail address: edward.anderson@sydney.edu.au

This paper explores the way that retail petrol prices may vary in a cyclical way as a result of competitivebehavior by petrol retailers when some drivers choose to fill up early when a low petrol price is available.We include a stochastic model for an individual driver’s use of petrol and consider what happens as theexpected future price of petrol is adjusted either according to observed prices or in anticipation of cyclicalbehavior. This model is different from most previous work on petrol price cycles that has focussed onEdgeworth cycles.

� 2010 Elsevier B.V. All rights reserved.

1. Introduction

Motorists in some of the large cities in Australia have becomeused to a weekly petrol price cycle (in US usage ‘gasoline’, butwe will use the word ‘petrol’ throughout this paper), with all retail-ers raising prices sharply on a Thursday. As a result, price-sensitivecustomers choose to join what may be substantial queues to fill uptheir cars on a Wednesday. Fig. 1 (taken from http://www.aip.co-m.au/pricing/retail.htm) shows the variation in prices occurringin Sydney in November 2009. Similar if less pronounced weeklyprice cycles have also been reported in Norway (Foros and Steen,2008). However, this phenomenon does not occur in most coun-tries, and it is not surprising that many Australian consumers aresuspicious of some sort of collusion taking place between petrolretailers. In 2007 the Australian Competition and Consumer Com-mission (ACCC) held a public inquiry into the price of unleadedpetrol in Australia in which the incidence of price cycles in Austra-lia was examined closely. The conclusions reached were that theseprice cycles did not indicate any lack of competition in the retailmarket for petrol.

In fact there are other retail petrol markets where price cycleson a longer and less regular basis have been observed. Wang(2005) discusses the experience in Perth, Australia, and Eckert(2003), Noel (2007) and Atkinson (2009) discuss the characteristicsof cycles that occur in Canada. There are a number of factors thathave been proposed to explain these price cycles. The most widelyaccepted explanation is that the pattern of prices are essentiallyEdgeworth cycles.

ll rights reserved.

According to this theory, firms who compete on price, observechanges in their competitors’ prices and respond by lowering theirprice to just below the competitors’ prices. This continues in analternating fashion until profits are reduced to zero, or at least nearzero. At this point there is a big incentive for one of the firms to‘relent’ and go back to a higher price, and once this happens thesecond firm will also move their price up to just below that ofthe relenting firm and the whole cycle starts again. Specificallywe suppose that there is a duopoly with Bertrand (price) competi-tion, demand is a linear function of price and that prices can onlybe set at a discrete set of points. The firms are identical with re-spect to costs. Moreover suppose that one firm can change its priceat odd numbered periods and the other in even numbered periods(which is a device to capture the idea that when firms change theirprices they commit to the new price for a minimum period oftime). This produces an alternating move dynamic. It can be ex-pected that when one firm sets a high price then the other’s bestresponse will be to undercut by the smallest increment of pricepossible. This leads over time to steadily reducing prices. Whenprofits are eroded to zero this process stops and during the follow-ing periods firms may or may not choose to raise prices again. Thefirm who raises prices is performing a type of public good: it willearn no money at the higher price and enable the other firm todo very well. But over a longer time horizon both firms will benefit.The equilibrium solution can be shown to have firms using a mixedstrategy: choosing to raise prices at the bottom of the cycle with acertain probability. Maskin and Tirole (1988b) use a Markov per-fect equilibrium applied to pairs of reaction functions to show thatthis form of Edgeworth cycle is an equilibrium provided that thediscount factor is close enough to 1.

Castanias and Johnson (1993) argue for these types of Edge-worth cycles as an explanation of the fluctuations observed in

Fig. 1. Sydney unleaded petrol prices November 2009 (Australian cents per litre).

E. Anderson / European Journal of Operational Research 210 (2011) 436–447 437

Los Angeles’ petrol prices in the period 1968–1975. In the Maskinand Tirole model an alternative equilibrium (with a constant price)also occurs. They call this a kinked demand curve equilibrium. Eck-ert (2003) considers a small adjustment in the Maskin and Tirolemodel in which the firms have different market shares when pricesare the same (though when prices are different the lower price firmstill captures the entire market). He shows how the relative sizes ofthe two firms effect the possibility of a constant price equilibriaoccurring. For cases with very different firm sizes the only equilib-ria that can occur are Edgeworth cycles. Eckert uses this observa-tion to explain why in some Canadian cities there are retail pricecycles for petrol prices, but in other cities there are not. Work byNoel (2007) gives a more detailed empirical study of Canadian pet-rol prices and shows that the observed behavior supports the the-oretical prediction from Edgeworth cycles. Specifically he observesthat in the Toronto market, which is characterized by very rapidprice cycles with an average period of about a week, the price risesoccur very quickly and are followed by slower price falls, where theprice rise is led by a large firm and the price fall is led by a smallfirm.

Eckert and West (2004) study the Vancouver area and also findsome evidence to support an Edgeworth cycle explanation. Theyfind price increases occurring on the same day across a wide geo-graphic area and being led by large firms. However they also pointout that Edgeworth cycles do not give a complete explanation ofthe behavior of this market (since there are clearly spatial effectstaking place and moreover price increases almost always occuron a Tuesday or Wednesday). Wang (2005) gives a detailed discus-sion of the Perth market and argues that firms behave very muchas would be predicted from an Edgeworth cycle. For example heshows that the leader role in a significant price increase appearsto be shared randomly amongst the large players.

Recent work by Atkinson (2009) has looked at data from petrolstations in Guelph, Ontario. This more detailed data confirms someearlier findings (that prices rise more quickly than they fall andthat price rises tend to be initiated on days between Monday andWednesday). However he also observes some phenomena thatmay require more complex explanations. A smaller number offirms than might be expected act as price leaders in raising pricesand they do so in a way that often returns prices to lower levels fora few hours after an initial price rise (Atkinson calls this behavior a‘‘sub-cycle” ). This type of more detailed discussion of the way inwhich price rises take place is also given by Wang (2008a) wholooks at data on communication and price changes arising from acartel prosecution in Australia.

Finally the work by Foros and Steen (2008) draws attention tothe way that price cycles in Norway have changed: prior to April2004, Thursday was the high price day (as occurs in the Sydneydata presented earlier) but since that date Monday has become

the high price day. It is interesting that prior to 2004 the risingphase of the price cycle was about the same length as the fallingphase, whereas more recent data shows the more typical fast riseoccurring.

Noel (2008) uses detailed numerical calculations to explore thebehavior of Edgeworth cycle equilibrium in an environment inwhich the marginal costs fluctuate and there may be some differ-entiation (so that customers have preferred retailers). He showsthat in a triopoly the Edgeworth cycles may exhibit false startbehavior that is similar to the sub-cycles observed by Atkinson.

In this paper we will consider a different explanation of cyclicbehavior in retail petrol markets. It is worthwhile to consider addi-tional theory in this area because there are some critical aspects ofthe usual Edgeworth explanation that may not be present inpractice.

First, many of the models discussed above have used pure pricecompetition in which a small price differential leads to the captureof the whole market, and for which there is no equilibrium withpositive firm profits (though see Noel (2008) for an exception). Itis true that petrol retailing is fiercely price competitive and it hasbeen argued that in an urban environment even price differentialsof a fraction of a cent per litre cannot be sustained in the long term.However with the time scales we are concerned with here, mea-sured in days, a high price sensitivity is appropriate rather thanan infinite one. Common sense suggests that some less price sensi-tive customers would choose to remain with a familiar or slightlymore convenient retailer even in the face of small price differences- especially if the price difference was not sustained. Wang (2008b)has obtained price and quantity data for eight stations in Perth,Australia. This enables him to estimate price elasticities whichare found to be very large; the price elasticity is nearly 20 forone petrol station. There are also clear effects related to prices atnear competitors and to previous prices at the same station.

A more important problem with the Edgeworth cycle explana-tion is that it requires a discrete set of prices and some constrainton how frequently prices are changed. Maskin and Tirole postulatean alternation of moves, or at least a minimum period of time dur-ing which a firm’s price is fixed (see Maskin and Tirole (1988a)).Each move undercuts the other player’s price by the minimumamount possible, with an incentive for making the undercuttingmoves as fast as possible. So the only way to achieve longer cyclesis to have more price points or slower moves. In the application topetrol price cycles the natural minimum change in price possible islikely to measured in fractions of a cent and is very much smallerthan the actual price changes observed. Moreover, the frequencywith which price changes can be made is often extremely rapid.Noel (2007) comments on the speed at which prices rise onceone player moves their price up. ACCC (2007) describes the ‘In-formed Sources’ price sharing service which allows subscribing

438 E. Anderson / European Journal of Operational Research 210 (2011) 436–447

retailers in Australia access to near real time information on pricesat other retailers. In order to show why the whole cycle is not com-pleted within a few hours we need to appeal to some reason for‘‘stickiness” in prices: perhaps the transaction cost involved inchanging prices, or negative consumer effects caused by changesof price during the day.

As Wang (2005) points out, the arrangements in Perth, Australiaprovide a natural experiment in which prices changes are synchro-nized and can only occur on a daily basis. However, even thoughthis is helpful to the Edgeworth explanation in limiting the speedat which price cycling takes place, it destroys an important compo-nent in the Edgeworth model, since with price changes takingplace at the same time it is hard to see why there should be analternation between the different firms as to who has price leader-ship through the cycle. In other words why should prices vary fromone day to another if each day is a ‘new start’? The Edgeworth cycleexplanation seems only to work in a market like Perth if firms havea requirement to keep the same price for more than one day at atime. Maskin and Tirole (1988b) in their discussion of endogenoustiming allow simultaneous moves but maintain a restriction thatprices, once chosen, are locked in for two periods.

In this paper we will investigate a different explanation of thephenomenon of cycles in retail petrol prices which revolves aroundthe possibility that consumers buy early when prices are low. Thisidea is related to the model proposed by Conlisk et al. (1984) forcyclic pricing by a monopolist faced with a demand for a durablegood in which new customers enter at each period, some of whomare more price sensitive than others. If customers will wait whenprices are unattractive, then there is an incentive to charge a highprice most of the time in order to obtain maximum profits fromthose willing to pay, but to have occasional periods of reducedprices in order to capture demand from more price sensitive cus-tomers. In this model consumers are well informed and can predictwhen sale periods will occur. The result is that even the less pricesensitive consumers will have a motivation to wait until a sale per-iod when it is near. From this observation we can see that the pat-tern of prices in the Conlisk et al. model has the same form as forEdgeworth cycles: prices rise to their highest level immediatelyfollowing the lowest price period, and then fall steadily towardsthe trough.

The question arises as to the circumstances in which the cyclicbehavior of this durable goods monopoly might occur in a compet-itive market for petrol. We are not the first to suggest this relation-ship (Eckert and West, Atkinson). The key component is that somecustomers are price sensitive and are captured by a low price,immediately following this the price sensitive customers are nolonger present, and the price stays high until more price sensitivecustomers accumulate when a lower price is again justified. In thepetrol market new customers arrive as old customers use up theirfuel and again need to purchase.

The strongest evidence for this behavior occurring in Australiancities is that there is substantial forward buying. In ACCC (2007)data is presented showing that about 20% of petrol is sold in capitalcities on a Tuesday when prices are usually at their lowest, andonly about 12% on a Thursday when prices are at their highest. Be-cause of the time frame involved we can only see this as a transferof demand from one day to another (rather than being due to priceelasticity of demand). With constant prices we would expect thatsales of petrol would occur in proportion to distance travelled ondifferent days.

The price sensitive customers in this market must be reacting toa difference in price between the current price and the price atwhich they would expect to purchase in the future. In most reason-able models this ‘expected future price’ against which the currentprice is compared, will itself change over time. In this paper wewill explore this behavior. We show that changes in the expected

future price can lead to price spikes occurring when the expectedfuture price falls low enough that it is no longer worthwhile forthe retailers to compete for forward purchases.

Since our aim is to provide some alternative explanations forweekly price cycles we also need to consider the nature of cus-tomer price expectations when prices have established a weeklycycle. Customer price expectations in a cyclical price environmentmay be stable as long as the expected price for tomorrow is close tothe actual price that occurred 6 days ago. We also considerwhether a well-established weekly cycle may cause retailers tothink of the entire week’s price pattern as a single decision. The lastpart of our paper considers the equilibrium price patterns that canemerge in this framework, in which a retailer takes account of theentire weekly pattern of demand when setting prices. Unfortu-nately our results in this section are less satisfactory: the cycleswe observe in this case are less significant and do not have the‘sawtooth’ pattern observed in practice.

We show that the combination of factors involving motorists’use of fuel and forward buying behavior to avoid high prices mayindeed lead to an equilibrium with cyclical variation of prices. Thisis the first time that a stochastic model of this phenomenon involv-ing forward buying of petrol has been given. Despite its limitationsit does reinforce the view that cyclical pricing behavior can emergeas a natural reaction to the competitive environment and withoutany collusion between different retailers. This work is thereforehelpful to competition authorities faced with this type of situation.

The model that we end up with is quite complex and we will de-velop the features of the model in stages. In the next section we de-fine our model in which there are a mix of motorists with differentcharacteristics. Specifically, the utility associated with delaying apurchase will vary between customers. For this model we considerthe price setting behavior in an equilibrium for a given number ofprice sensitive customers. In Section 3 we explore the behavior ofthis model when motorists use up fuel over time and move be-tween different states in a stochastic way. We show that with rea-sonable assumptions this will lead to an equilibrium in which theprice stays constant over time (after some initial oscillation). Thisassumes a fixed value of the expected future price; in Section 4we extend this model by letting the expected future price vary ina way that relates to the prices that occur in the model. Finally,in Section 5 we consider the result of allowing retailers to planahead for a weekly cycle of prices.

2. The model

The essential ingredient in our model is that drivers will on oc-casion choose to fill up early if they perceive the current price asunusually low (so that it may increase prior to the normal pointat which they fill up). Our starting point is to investigate the equi-librium price that occurs in a single period given a set of driverswith different characteristics.

We consider a model in which there are two petrol retailers.Each of M drivers has a car with petrol tank capacity of L + 1 units.Each day a driver uses a unit of petrol with probability r, and usesno petrol with probability (1 � r). Each driver will fill their carwhen they have only 1 unit of petrol left, but some drivers willchoose to fill early, when they have 2 units of petrol left, with aprobability that depends on the price differential between the pet-rol price on that day and their expectation of the price they willpay in the future pN.

In our basic model the retailers are profit maximizers on a sin-gle period basis, so there is no collusion and the retailers each dayplay a single stage game and set their prices at the Nash equilib-rium for that day. We assume that the retailers have full informa-tion about the distribution of car petrol holdings. (A more complex

2NX p p+ −

2XX p p+ −

2

X

Np p−

N Xp p−

Y

X

2NX p p+ −

2XX p p+ −

2

X

Np p−

N Xp p−

Y

X

Fig. 2. Region in (X,Y) plane choosing to purchase this period from retailer 1.

E. Anderson / European Journal of Operational Research 210 (2011) 436–447 439

model might involve dropping this assumption, and working onthe basis of knowledge of previous prices). For the simplest modelwe assume that prices are normalized so that variable costs for theretailers are zero. Retailers have a fixed cost of operation of K perday, but this will turn out to play no role in our analysis.

We will adopt a Hotelling type model of price competition. Wesuppose that the two retailers are at opposite ends of a line. Cus-tomers are evenly distributed along the line and go to the retailerthat maximizes their utility, allowing for the cost of travel. Thismodel gives a linear model of demand: specifically a retailer offer-ing price p against a competitor offering pX will obtain a marketshare of

0:5þ cðpX � pÞ; ð1Þ

where c is related to the cost of travel and the distance between theretailers. As c increases, corresponding to a low cost of travel, theprice competition becomes more fierce and the model approachesa form of Bertrand competition. We extend this basic Hotellingmodel to the case where customers may purchase in advance ifthe price is low enough.

Suppose that customers have a utility y which they gain frombeing able to delay their purchase. This is the simplest way to rep-resent the fact that customers will only choose to travel to a low-priced retailer and make a purchase in advance of their require-ments if the saving they make is sufficiently large. The requiredsaving here is equivalent to a utility derived from delaying pur-chase. Further suppose, for convenience, that the distribution of ybetween customers is uniform on [0,Y]. Combining this with ourprevious assumption, and assuming independence of positionand delay utility, we suppose that customers are uniformly geo-graphically distributed on the line [0,X] with a retailer at eachend of this line. We normalize distances so that there is a disutilityof 1 from travelling a distance 1, and we normalize quantities sothat filling up with petrol requires a quantity 1.

In this model the only customers who influence prices are thosewho may purchase on this day. We suppose that there are M1 cus-tomers who must purchase today and will purchase an amount L,together with a further M2 customers who may purchase today ifthis maximizes their utility and will purchase an amount L � 1.These M2 customers have an expectation that the future price ofpetrol at each retailer is pN.

We suppose the price offered by retailer 1 at location 0 is p andthe price offered by retailer 2 at location X is pX. The model isessentially symmetric: there is no difference between retailersand we can expect to find equilibrium solutions in which thetwo prices are identical. A customer at position x on the line,amongst the M1 who must purchase today, will choose retailer 1if x + p < (X � x) + pX. i.e.

x <X þ pX � p

2:

So the total demand from these customers on retailer 1 is

LM1X þ pX � p

2X: ð2Þ

Now consider a customer at position x on the line amongst theM2 who may delay and suppose this customer has a delay utility ofy. Then the customer will purchase from retailer 1 provided that

xþ p < minfX � xþ pX ; xþ pN � y;X � xþ pN � yg; ð3Þ

where the three terms in the minimization represent comparisonswith purchasing from retailer 2 today, or delaying and purchasingfrom retailer 1, or delaying and purchasing from retailer 2.

Hence we end up with a polygonal region defined in the[0,X] � [0,Y] plane determined by the three constraints

2x < X þ pX � p; y < pN � p and 2xþ y < X þ pN � p:

This is shown in Fig. 2. If pN 6 p, then the region disappears. Wewill assume that pN � p < Y so that the second constraint is activefor some choices of y. When p P pX, the third constraint does notbind, and when pX > pN then the first constraint does not bind.Otherwise all three constraints come into play. The area is given by

X2ðpN � pÞ þ HðpÞ

4;

where

HðpÞ ¼ ðpX � pÞð2pN � pX � pÞ if pN P pX > p;

¼ ðpN � pÞ2 if pX > pN > p;¼ ðpX � pÞð2pN � 2pÞ if p P pX ;

¼ 0 if p P pN :

Since we are assuming that the individual customers are uniformlydistributed over [0,X] � [0,Y], the total demand from these custom-ers at retailer 1 is

ðL� 1ÞM2ðpN � pÞ

2Yþ HðpÞ

4XY

� �: ð4Þ

The retailer profit is

PðpÞ ¼ pqðpÞ � K;

where q(p) is the total amount sold at price p given by (2) and (4).Now

dP=dp ¼ qðpÞ þ pq0ðpÞ

and

q0ðpÞ ¼ �M1L2X� ðL� 1ÞM2

12Yþ H0ðpÞ

4XY

� �:

Hence

dPdp¼ LM1

12þ pX

2X� p

X

� �þ ðL� 1ÞM2

pN

2Y� p

Yþ ðHðpÞ þ pH0ðpÞÞ

4XY

� �:

When M2 = 0, this gives a maximum profit when

12þ pX

2X� p

X¼ 0;

so p = (X + pX)/2 and there is an equilibrium solution with bothretailers setting a price of X. The Lemma below gives a descriptionof the equilibrium solutions that occur for non-zero M2.

440 E. Anderson / European Journal of Operational Research 210 (2011) 436–447

Lemma 1. When M2 > 0, there are values pA and pB in the range

XYLM1

ððL� 1ÞM2X þ LM1YÞ ;X� �

;

such that:

(a) for pN P pA there is an equilibrium at which both firms set theprice

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

q

p ¼ X þ Y þ pN

2�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðY þ pNÞ

2

4þ X2

s; ð5Þ

where Y ¼ YLM1ðL�1ÞM2

;(b) for pN 6 pB there is an equilibrium at which both firms set the

price X.

There are no other equilibrium solutions.

Proof. See Appendix A. h

It is not straightforward to write down algebraic expressions forpA and pB, but we can calculate the values numerically (see theAppendix A for more details). We write qA = pA/X, qB = pB/X andz = (YLM1)/(X(L � 1)M2). Then Fig. 3 shows the behavior of qA (thelower line) and qB (the upper line) as a function of z. In fact qA isapproximately equal to

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiz=ðzþ 2Þ

pand qB is approximately equal

toffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2z=ð2zþ 1Þ

p, and these are shown as dashed lines in the Figure.

Thus pA < pB and hence there is always at least one equilibriumsolution, and in the range (pA,pB) there are two possible equilibria.

Now suppose that pN is high enough for the lower price equilib-rium given by (5) to occur. If we define

a ¼ LM1

ðL� 1ÞM2;

so Y ¼ Ya, then

dpda¼ Y

21� Y þ pNffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðY þ pNÞ2 þ 4X2

q0B@

1CA > 0:

Hence, as the number of customers in M2 increases, a decreases andthe equilibrium price will drop. This provides a potential explana-tion for cycles of prices. When there is a low price then many driv-ers with a low amount of fuel fill up. In the next period there arefewer drivers in this state, M2 is reduced and the price rises again.But if this is to be a route to explain price cycles then we need tounderstand the behavior over time of the proportion of drivers withdifferent amounts of fuel: this is the topic of the next section.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0z

Fig. 3. qA and qB as functions of z.

3. Equilibrium price with constant usage pattern

Our discussion has been motivated by the search for anexplanation of cyclical pricing behavior. However, we will showin this section that, when the pattern of usage is constant fromone day to the next, and the expected future price is alsoheld constant, then prices may oscillate but will approach anequilibrium.

Our model of petrol usage is stochastic - we suppose that anindividual driver uses petrol in a way that varies from day today, but the probabilities involved are held constant. We supposethat the fuel in an individual vehicle is divided into portions, sothat a vehicle is in state i if has between (i � 1)/L and i/L of fuelat the beginning of the day. In the absence of forward buyingfor petrol, we suppose that the state of a car follows a Markovchain. We think of this as a model in which most cars are driveneach day, but possibly not far enough to make a transition to thenext lowest state. As far as an individual car is concerned thismeans that the Markov assumption will not hold, but it is a rea-sonable approach to the modelling of the whole population ofcars.

We shall suppose that only cars in state 1 consider the possibil-ity of filling in advance if the price is low enough. Hence even if acar in state 1 is not driven far enough to be forced to fill up (trans-fer to state L), if the price is low enough it will do so. Also cars thattransfer into state 1 consider filling up if the price is low enough.From our discussion in the previous section we can assume thatboth retailers set the same price p, and write b = h(pN � p) for theprobability that a car fills up in advance from state 1. Notice thatb will vary as the numbers of cars in different states varies. Inthe event that the car in state i is not filled up through this pricesensitive demand it moves to state i � 1 with probability r, witha car in state 1 moving to state L with probability r. Otherwisethe car remains in the same state.

Thus the car follows a Markov chain with an L � L time-varyingtransition matrix given by

P ¼ ðPijÞ ¼

ð1� rÞð1� bÞ 0 0 . . . 0 rþ bð1� rÞ

rð1� bÞ ð1� rÞ 0 . . . 0 rb

0 r ð1� rÞ . . . 0 0

. ..

0 0 0 . . . r 1� r

0BBBBBBBBBBB@

1CCCCCCCCCCCA:

In the case that there is a finite number M of cars, we can re-move the time varying characteristic of this system by consideringa system with states (n1,n2, . . . ,nL) where ni is the number of cars instate i and

PL1ni ¼ M. In this case the price is given as a function of

n1 and n2 and hence all transition probabilities are determined bythe state (probabilities are given by binomial distribution typeexpressions). Hence this is a finite irreducible Markov chain. More-over because all transitions are possible including no change at allin the state from one period to the next, it is easy to see that thiscannot have periodic behavior. Hence there is a single equilibriumdistribution on the states and we will not observe cyclic pricebehavior.

We will observe price fluctuations arising from fluctuations inthe numbers n1 and n2. However the price becomes nearly constantfor large enough M. In fact in the limit of large M we obtain a deter-ministic model in which we replace numbers ni with the propor-tion qi of cars in state i, so that qi = ni/M. Finding an equilibriumin this model will determine the equilibrium price in the stochasticmodel for large M.

0.51

0.52

0.53

0.54

0.55

0.56

0.57

0.58

0.59

1 6 11 16 21 26 31 36Day

Price

Fig. 4. Price evolution for Example 1.

E. Anderson / European Journal of Operational Research 210 (2011) 436–447 441

The dynamics in this model are as follows:

q1ðt þ 1Þ ¼ ð1� rÞð1� bÞq1ðtÞ þ rð1� bÞq2ðtÞ;qiðt þ 1Þ ¼ ð1� rÞqiðtÞ þ rqiþ1ðtÞ; i ¼ 2; . . . ; L� 1;qLðt þ 1Þ ¼ ðr þ bð1� rÞÞq1ðtÞ þ rbq2ðtÞ þ ð1� rÞqLðtÞ:

In order to make the algebra a little simpler we will take pN = X.Thus we have

pðtÞ ¼ 12

YaðtÞ þ 3X �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðYaðtÞ þ XÞ2 þ 4X2

q� �;

where

aðtÞ ¼ LM1

ðL� 1ÞM2¼ Lrq1ðtÞðL� 1Þðð1� rÞq1ðtÞ þ rq2ðtÞÞ

:

In this case we have h = 1/(2Y) since, from (4), the number of cus-tomers who bring forward their purchase is a proportion

bðtÞ ¼ ðX � pðtÞÞ2Y

:

At an equilibrium we can drop the t dependence in both p and b,and the proportions, qi, i = 1,2, . . . ,L satisfy the equations

q1 ¼ ð1� rÞð1� bÞq1 þ rð1� bÞq2;

qi ¼ ð1� rÞqi þ rqiþ1; i ¼ 2; . . . ; L� 1:

We can solve these recursions and after normalizing so thatPqi ¼ 1, we obtain

q1 ¼rð1� bÞ

rð1� bÞ þ ðL� 1Þðr þ b� rbÞ ;

qi ¼ðr þ b� rbÞ

rð1� bÞ þ ðL� 1Þðr þ b� rbÞ ; i ¼ 2; . . . ; L:

So we have

a ¼ Lr2ð1� bÞðL� 1Þðð1� rÞrð1� bÞ þ rðr þ b� rbÞÞ ¼

Lrð1� bÞðL� 1Þ :

Substituting for b we obtain

a ¼ Lrð2Y � ðX � pÞÞðL� 1Þ2Y

:

Now the expression for p gives

ðYaþ XÞ2 þ 4X2 � ðYaþ 3X � 2pÞ2 ¼ 0: ð6Þ

Hence substituting for a, and after some algebra, we obtain the fol-lowing quadratic for p:

ð2þ Lr � 2LÞp2 þ ð6LX � 6X � 2LXr þ 2LYrÞpþ 2X2 � 2LXYr � 2LX2 þ LX2r ¼ 0:

When p = X the right hand side simplifies to 2X2(L � 1) > 0 andwhen p = 0 the right hand side is

X2ð2þ Lr � 2LÞ � 2LXYr;

which is negative, since L > 2/(2 � r). Hence there is exactly one rootfor p in the range (0,X).

We have demonstrated the existence of a constant price equi-librium. The Markov behavior for individual motorists ensures thatthere is a mixing between the motorists holding different amountsof petrol, rather than cohorts of motorists moving through the sys-tem together. The end result is that a cyclical price behavior cannotbe maintained and we find the system settles down to the constantprice equilibrium after some initial oscillations of price. This isillustrated in the following numerical example.

Example 1. If we take L = 5, r = 0.8 and X = Y = 1 and suppose thatthe system starts with equal numbers of cars in each of the fivestates then the initial value of a is 1 and the initial price is given by

pð1Þ ¼ 12

1þ 3�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ 1Þ2 þ 4

q� �¼ 0:58579:

Then the price evolves in the way shown in Fig. 3. The finalequilibrium price is 0.55051. The equilibrium distribution of theproportion of cars in the five states is q1 = 0.15505, q2 = q3 =q4 = q5 = 0.21124.

4. Price cycles when price expectations vary

The previous example shows that when price expectations areconstant and usage patterns do not vary then any oscillations inprice will die out, and long-lasting price cycles will not occur. How-ever it is clear that the model should reflect changing expectationsof price in some way. In Example 1 the price expectation pN re-mains at 1, whereas actual prices quickly drop to around 0.5.

It is reasonable therefore to assume that current prices will af-fect the expected future price pN. There are various ways in whichwe could model this. In Example 2 we suppose that an exponentialsmoothing model operates in which the expected future price isupdated on the basis of the observed price.

Example 2. In this example we repeat the assumptions of Example1, but we assume that the new price expectation is given by acombination of the old price expectation and the actual price lastperiod:

pNðt þ 1Þ ¼ cpðtÞ þ ð1� cÞpNðtÞ;

where c 2 (0,1) is the smoothing constant.Fig. 4 shows the way that the price changes when c = 0.2. It is

necessary to decide which equilibrium will be chosen when pN isbetween pA and pB. We take the low price equilibrium where thisexists. This corresponds to keeping prices, low following low priceson the previous day, unless one of the retail firms can gain from anincrease in price. The alternative of resetting prices to X at the ear-liest possible point in the cycle, requires more coordination be-tween the players, since they make a change in the price beforethis is ‘forced’. On the other hand, as soon as a lower price equilib-rium is possible this will be chosen in preference to that with priceX. As in Example 1 we take L = 5, r = 0.8 and X = Y = 1 and supposethat the system starts with equal numbers of cars in each of thefive states.

Example 2 has a period of 7 days, but this is a result of the par-ticular parameters we have chosen. For example, a higher value of

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1 6 11 16 21 26 31 36 41Day

Price

Fig. 6. Price cycles for Example 3.

442 E. Anderson / European Journal of Operational Research 210 (2011) 436–447

c will result in a shorter cycle. During the initial period (days 1 to10) the price is slowly driven lower: the actual price drops to0.4389 (with the expected price also being driven lower). At theend of this period the ratio a = LM1/((L � 1)M2) (which is equal toY here) has dropped to 0.7054. The lowest price expectation atwhich the low price equilibrium is sustainable is pA, which is equalto 0.511 for this value of a. At period 11 the expected price dropsbelow this threshold and the equilibrium jumps up to a price of1. This single high price lifts the following period’s expected priceto a value above the threshold, so the low price equilibrium isagain chosen. At this point the cycle repeats.

At the same time as prices are being changed the number ofmotorists in different states also varies, but with quite a complexpattern. On the day after the high price occurs (call this day 1 ofthe cycle) there are relatively fewer motorists in state L = 5, sincethere has been no buying forward. This smaller cohort mostlymove to state 4 on day 2 of the cycle, and this process repeats withthe smallest cohort arriving at state 1on day 5 of the cycle At thesame time there is a large cohort which fills up the day after thehigh price and is in state L = 5 on day 2 of the cycle. As the largercohort moves down through the states, on each day 20% of itsmembers do not change state and by day 5 of the cycle there is arelatively even spread of motorists among the states 2–5.

Observe that in this example the expected future price is heav-ily influenced by the last few days. It is a reduction in price fromthe previous day that is necessary in order to persuade motoriststo buy forward. Within each cycle this effect keep on driving priceslower, until such time as it is no longer worthwhile for the retailersto encourage forward buying.

However once a regular pattern of behavior is established itwould seem odd for motorists to ignore the price cycle in formingexpectations of the future price. In this case it makes more sense topredict the future price by anticipating a repeat of the previousprice behavior. This type of calculation for expected future pricesturns out to make it harder to generate a cyclic pattern of prices,which at first sight is surprising. It is necessary to be careful inthe way that price expectations are set if we want to observe reg-ular price cycles.

Example 3. In Fig. 5 below, the solid line shows the pattern ofprice behavior with the same arrangements as Example 2 butwhere the expected future price on day j of the cycle in cycle t + 1 isgiven by

pNðj; t þ 1Þ ¼ cXT�1

i¼1

wipðtT þ jþ iÞ þwT pðtT þ jÞ" #

þ ð1� cÞpNðj; tÞ;

so that w1 is the weight given to the price on the following day(assuming that the cycle of prices repeats), w2 is the weight given

0

0.2

0.4

0.6

0.8

1

1.2

1 6 11 16

Price

Fig. 5. Price cycles

to the price two days a head and so on. The special treatmentfor wT arises because the price on day j of the cycle is not availableto use as a proxy for the price T days ahead. In this exampleT = 7 and we have used a weight vector w with values(0.3,0.25,0.15,0.1,0.1,0.1,0)0. The value of c is set to 0.6.

In Fig. 6 we have also shown as a dashed line what happenswhen we use the weight vector (1,0,0,0,0,0,0)0 corresponding toan estimate of the future price given by what we expect for the fol-lowing day of the week. This suggests (and other experiments con-firm) that with this pattern of weights it is not possible for aweekly pattern of prices to be retained. This is true no matter whatstarting point we use and no matter how long the simulation isrun. In this case the fact that drivers have an expectation of weeklyprice movements destroys the very weekly cycles that give rise tothat expectation. More specifically we can say that weekly pricecycles are more likely to be stable when the drivers do not usethe anticipated price tomorrow in deciding whether to buy for-ward (but instead take an expectation based on an average overwhat they expect over the next few days).

Example 4. Now we look in more detail at what happens in anenvironment in which the pattern of usage varies during the cycle.In this arrangement we have the same model as before except thatthe probability r is now a function of the day of the week. To modelthe variations in petrol usage through days of the week we use dataon vehicle movements collected at random points in Sydney todevelop a profile of kilometers travelled and assume that petrolusage is proportional to this. Table 1 shows the vehicle movementdata collected from 43 data points randomly selected from NSWGovernment data for Sydney in 2005. Here fj is the proportion of allvehicle movements recorded on day j.

21 26 31 36 41Day

for Example 2.

Table 1Vehicle movements by day of week.

Mon. Tue. Wed. Thu. Fri. Sat. Sun.

Vehicle movements 26,729 27,483 27,862 28,437 28,997 24,270 20,346fj 0.145 0.149 0.151 0.154 0.157 0.132 0.111

E. Anderson / European Journal of Operational Research 210 (2011) 436–447 443

We set the value of r on day j of the cycle to be kfj with k chosenso that the average r value is 0.8 as in the previous examples. Inthis arrangement we find that the highest price day is Friday (ingeneral working with other examples of usage patterns suggeststhat the high price occurs on the day before a drop in usage takesplace). Fig. 7 shows the cycle of prices that develops with all otherdata the same as for the example shown in Fig. 6. When there areweekly cycles in usage the weekly pattern of price changes is morereadily established and it is no longer necessary to choose theweights wi so carefully.

5. Price cycles with non-myopic retailers

Up to this point we have modelled the players as myopic ornearly so. For the customers there is just a single expected futureprice. It would make more sense for customers to have an estimateof the price tomorrow, the price for the day after tomorrow, etc.The Conlisk et al. paper includes this kind of thinking to create areducing price as we move toward a sale period.

For the retailers, the model outlined above assumes no foresightat all. Retailers take no account of the impact on their future saleswhen setting today’s price. In this section we incorporate non-myopic behavior. Our results are limited but it is valuable to con-sider the different modelling choices that are required. The firstquestion to resolve is the nature of the decisions made by the play-ers and the information that they possess. One possible approach isto make no assumption of a weekly cycle and consider a Markovperfect equilibrium approach, in which retailers set prices in eachperiod and the customers’ expectations of future prices are givenon the basis of perfect foresight. Because of the interactions be-tween prices and the numbers of customers in various states, webelieve a full treatment of this is analytically intractable and itseems very hard to find even a numerical approach to the solutionof this game.

Instead of this approach we will take an easier alternative. Thediscussion of previous sections gives some indication of how aweekly cycle of prices might occur. We now ask the followingquestion: What weekly pattern of prices would be an equilibriumfor the retailers if the entire set of prices is set just once and thereis full information about this weekly pattern of prices.

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1 6 11 16 2D

Price

Fig. 7. Price cycles when usage

It is clear that more price sensitive customers (with a low valueof delay utility y) are more likely to buy petrol early. This will leadto these customers being preferentially selected out at some pointsin the weekly cycle. The retailer should take account of this and sothe distribution of customers in [0,Y] can no longer be assumed tobe uniform.

First, suppose that there is a weekly cycle of prices and bothretailers set the price pk on day k. There is then an equilibrium dis-tribution of customers in different states on different days of theweek, and we set nk(i) to be the number of customers in state ion day k. We let

hk ¼ pNðkÞ � pk ¼Xkþi

wipkþi � pk; k ¼ 1;2; . . . ;7;

where pn = pn�7 for n > 7.In order to track the number of customers in different states

according to their y values, we define Nk(i,v) to be the number ofcustomers in state i on day k with y > v, so nk(i) = Nk(i,0).

On day k all the customers with y values lower than hk thateither (a) start in state 1 or (b) move from state 2, will choose tofill up early and move to state L, so

Nkþ1ð1;vÞ ¼ Nkð2;maxðhk;vÞÞrk þ Nkð1;maxðhk;vÞÞð1� rkÞ; ð7ÞNkþ1ði;vÞ ¼ Nkðiþ 1; vÞrk þ Nkði;vÞð1� rkÞ; i ¼ 2;4; . . . ; L� 1;

ð8ÞNkþ1ðL;vÞ ¼ NkðL;vÞð1� rkÞ þ Nkð1; vÞ

� Nkð1;maxðhk;vÞÞð1� rkÞ þ rkðNkð2; vÞ� Nkð2;maxðhk;vÞÞÞ: ð9Þ

Here we have N8(�, �) = N1(�, �). These recursions can be solved forv = 0 and v = hk for those hk > 0. The Nk(i,hj) values need to be scaledso thatX

i

Nkði; hjÞ ¼ðY � hjÞ

YM; k ¼ 1;2; . . . ;7; ð10Þ

which follows from the uniform distribution of Y values.On day k there will be (1/2)nk(1)rk customers who run out of

petrol and buy L units of petrol from retailer 1. In addition on dayswhen hk > 0 there will be

ð1=2Þðnkð1Þ � Nkð1; hkÞ þ rkðnkð2Þ � Nkð2; hkÞÞÞ;

1 26 31 36 41ay

rates vary during the week.

444 E. Anderson / European Journal of Operational Research 210 (2011) 436–447

customers who choose to fill early with L � 1 units of petrol. Hencewe can calculate the total profit made by each retailer:

P ¼X7

k¼1

pk

2nkð1Þrk þ _Ikðnkð1Þ � Nkð1; hkÞ þ rkðnkð2Þ � Nkð2; hkÞÞÞh i

;

where Ik = 1 if hk > 0 and is zero otherwise.To find an equilibrium we need to consider what happens to

the retailer’s profit when one retailer changes its price and theother does not. Our assumption is that there is a single set ofweekly prices, and hence a change in price on any day will bereplicated in future weeks. We need to decide what happens tocustomer price expectations if the two retailers have differentprices. We model this by taking an average of the two pricesand in a similar way we take the current price to be the averageof that at the two retailers. So that if retailer l uses a price pðlÞk onday k, then

hj ¼X

i

wi ljþipð1Þjþi þ ð1� ljþiÞp

ð2Þjþi

� �� ljp

ð1Þj þ ð1� ljÞp

ð2Þj

� �;

where pðlÞn ¼ pðlÞn�7 for n > 7 and

lj ¼X þ pð2Þj � pð1Þj

2X;

Mon Tue Wed Thu Fri Sat Sun

Equilibrium 1pk 1.0042 0.9950 1.0039 1.0029 1.0021 1.0021 0.9865hk �0.00522 0.00651 �0.00363 �0.00346 �0.00404 �0.00496 0.01481

Equilibrium 2pk 0.9956 1.0039 0.9967 1.0036 1.0029 0.9925 1.0052hk 0.00536 �0.00430 0.00440 �0.00418 �0.00407 0.00889 �0.00611

0.975

0.98

0.985

0.99

0.995

1

1.005

1.01

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Fig. 8. Equilibrium for Example 5 showing two weeks of prices.

which represents the probability that a customer is in the part ofthe line segment that chooses retailer 1. Hence

@hj

@pð1Þl

¼ wl�j lj �pð1Þj � pð2Þj

2X

!; j – l and

@hj

@pð1Þj

¼ ðw7 � 1Þ lj �pð1Þj � pð2Þj

2X

!:

Observe that if there is a change in price for retailer 1 then the hvalues will change and this will change the Nk values. However ifthe ordering of the h values is unchanged, then the recursions(7)–(9) are also unaltered. Hence we can track the effect of changesthrough these recursions and the fact that, from (10),

Xi

@Nkði; hjÞ,@pl ¼ �

MY@hj=@pl and

Xi

@nkðiÞ,@pl ¼ 0:

We have used a program in GAMS with CoinIpopt as the solver tosearch for an equilibrium solution for specific examples. This searchis carried out by considering all possible orders of the alphas andlooking for a solution which is a local equilibrium. Candidate equi-librium solutions are checked by letting one player choose its pricesat the potential equilibrium, while the other player maximizes itsprofit considering all possible orders of the alpha values separately.

A problem arises with examples in which all the customers areprice sensitive (having some finite delay utility). One strategy forthe retailers is to set a very high price on day k in such a way thaton the previous day hk�1 > 1. This will imply that all the eligible

customers will fill up early and no cars require petrol on day k. Thismeans that the normal equilibrium considerations on day k, as thetwo retailers compete for customers, do not occur with the resultthat there is nothing to stop very high prices occurring on thatday (leading to high prices on other days as well). This type ofbehavior is removed if there is some mechanism to avoid all theprice sensitive customers buying no matter how high the price dif-ferential. We have used a model in which only a proportion k of po-tential forward buying customers choose to do so. This leads tomore complex recursions:

Nkþ1ð1;vÞ ¼ kNkð2;maxðhk;vÞÞrk þ ð1� kÞNkð2;vÞrk

þ kNkð1;maxðhk;vÞÞð1� rkÞ þ ð1� kÞNkð1;vÞð1� rkÞ;Nkþ1ði;vÞ ¼ Nkðiþ 1;vÞrk þNkði;vÞð1� rkÞ; i ¼ 2;4; . . . ; L� 1;Nkþ1ðL;vÞ ¼ NkðL;vÞð1� rkÞ þNkð1;vÞðrk þ kð1� rkÞÞ

� kNkð1;maxðhk;vÞÞð1� rkÞ þ rkkðNkð2;vÞ�Nkð2;maxðhk;vÞÞÞ:

Example 5. Our final example uses the same weight vector asExample 3 and the variations in r values given in Example 4. Wetake a k value of 0.9. We have carried out an extensive search tofind equilibrium prices for this example. The table below show theprices and h values for just two possible equilibria amongst many.

The profit for the first equilibrium is 280.027, the profit for thesecond is 280.018. Fig. 8 shows the pattern of prices displayed overa two week period for the first equilibrium. The exact shape of theweekly pattern is influenced by the weight vector w used indetermining pN, by L and the rk values, which are all relativelyarbitrary choices made for this example.

Observe that both these solutions show little variation in pricefrom one day to another and in addition have significantly higherprices than the myopic equilibria we have considered earlier. Thehigher prices arise since competition on one day to win customersreduces the number of customers available on the following day.

E. Anderson / European Journal of Operational Research 210 (2011) 436–447 445

Since the non-myopic price model allows for the future impacts ofprices, it is relatively less competitive. The amount of variation inprice can be influenced by the problem parameters - for exampleit is higher for smaller values of Y. We find price variations of be-tween 1% and 8%, depending on the size of Y. In none of the exam-ples we have considered has the price variation from peak to troughreached the degree of variation observed in Australian markets.

6. Conclusions

This paper discusses a new explanation for price cycles for retailpetrol that does not use Edgeworth cycles, but instead arises fromthe interaction of customer expectations of future prices and profitmaximization by competing retailers. In our model price expecta-tions vary through the cycle and low price expectations have the ef-fect at a certain point of making it no longer worthwhile for retailersto try to entice forward buying amongst customers - at which pointa higher price occurs. All of this happens within an environment inwhich the number of customers who may consider forward buyingvaries from day to day as petrol is used (modelled through a dis-crete state approximation of the amount of petrol in each car withprobabilities for transitions between the states). We note that thismodel does not link the cyclical behavior to the specification ofthe grid of prices, as occurs with Edgeworth cycles. Moreover thetime period of the cycle is somewhat endogenous. Though we as-sume that prices are changed at the beginning of each day, anddo not change at any other time, the actual length of the cycle de-pends on the rate at which fuel is used by motorists.

There are two factors which determine the nature of the cycles.First, the distribution of motorists in respect to the petrol they havein their vehicles, and second, the expectations that motorists formabout the savings to be obtained by buying before it is necessary.Looking at the distribution of petrol holdings suggests that thehighest prices occur on the day following the lowest prices - whenthe fewest number of price sensitive customers are present. How-ever Example 1 illustrates how this effect on its own is not enoughto maintain a pattern of price cycles. More generally the models wehave developed suggest that this behavior is likely to be dominatedby the second issue related to customer expectations. Future priceexpectations by the motorists acts to determine the current price:when the expected future price is lower, then the current price alsoneeds to be set lower in order to encourage forward buying bymotorists.

When price cycles are not well established the price expectationwill probably be formed by considering what has happened overthe last few days. This can lead to slowly decreasing prices till apoint is reached when it is no longer worthwhile for retailers topursue the price sensitive customers and the price jumps up toits highest level. So price cycles are likely to occur and in the rightcircumstances they can be weekly cycles. This is illustrated inExample 2.

We might expect that well-established expectations of a cycleof a certain length will serve to make the cyclic behavior more pro-nounced. However we find that this is not the case; especially ifdrivers use the same amount of petrol on each day of the week.Once price cycles are well-established we would expect customersto use this to predict the future price. This will make it harder toreach a stable equilibrium pattern as is illustrated in Example 3.

The existence of a stable weekly cycle with customer expecta-tions determined by this has the effect of making prices loweston the day after a high price, with prices then increasing slowlyduring the cycle, since the expectations of future prices are lowestat the beginning of the cycle and increase slowly as the high priceday gets closer. The intuition here is that retailers do not needto set low prices immediately before a high price day as price

sensitive customers will not need such a low price to make itworthwhile to fill up early. A general tendency for increasing pricesduring the cycle is also present with petrol usage that varies overthe days of the week, and this is illustrated in Example 4.

In summary our model suggests that stable weekly price cyclesare only likely to occur when petrol usage also shows a weekly pat-tern. Moreover our model predicts that the slow rise and sharp fallobserved in many actual markets could be changed once a weeklypattern is well-established. This is an interesting finding for bothregulators and participants in these markets.

We have also explored a second version of our model which as-sumes that a weekly cycle has been established and is taken by allparticipants as given. In this model we suppose a much greater de-gree of foresight on the part of retailers, who decide on a fixedweekly pattern of prices in advance. We use numerical techniquesto seek an equilibrium in these weekly price patterns. In this mod-el, illustrated in Example 5, we found that there are a variety ofequilibria that may occur. Our experiments indicate that price cy-cles usually have more than one peak in price as well as havingonly a small amount of variation from one day to the next.

Our work is the first to model the influence of forward buying ina petrol market and has shown that cyclical price patterns may begenerated through the mechanism we have discussed, but thatthere is a critical role for the way that consumers’ expectationsof future prices are set. Our models do not easily replicate the pat-tern of prices observed in practice and work remains to be doneusing more realistic models both for petrol usage and for themotorists decision on whether or not to buy forward.

Appendix A

Proof of Lemma 1. First consider the optimal choice of p giventhat pX = X. The retailer profit is

PðpÞ ¼ pM2

2YðpN � pÞ þ HðpÞ

2X

� �þ p M1

2X � p2X

� �� K:

Thus

dP=dp ¼ M2

2YH0ðpÞp

2Xþ pN � 2p

� �þM1

XðX � pÞ:

We want to identify the circumstances when p = X is optimal. WhenpN P X, consider reducing p from a value X. Then we haveH(p) = (X � p)(2pN � X � p) and the derivative as p approaches Xfrom below is H0(p) = �2(pN � X). Thus

dP=dp ¼ M2

2Y2ðX � pNÞp

2Xþ pN � 2p

� �þM1

XðX � pÞ ¼ �M2X

2Y

and the profit is increased by reducing p from its value of X. Hencep = Xcannot be optimal when pN P X, so consider the case thatpN < X.

In this case the profit has no term from M2 while p > pN andhence in this region p = X is optimal. Finally we consider p < pN.Then H(p) = (pN � p)2 and

dP=dp¼M2

2Y�ðpN �pÞp

XþpN �2p

� �þM1

XðX�pÞ

¼ 12XY

�pM2pN þp2M2þXM2pN �2XpM2þ2M1XY �2pYM1

� �:

Thus dP/dp = 0 when

�pM2pN þ p2M2 þ XM2pN � 2XpM2 þ 2M1XY � 2pYM1 ¼ 0;

or

p2 � pðpN þ 2X þ 2YÞ þ XpN þ 2XY ¼ 0;

where Y ¼ YM1=M2. This has solution

446 E. Anderson / European Journal of Operational Research 210 (2011) 436–447

p ¼ X þ Y þ pN

2�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX2 þ Y þ pN

2

� �2r

:

Pis a cubic in p with a positive coefficient of p3 so the maximum va-lue of P occurs at the lower root of dP/dp = 0. Finally we can calcu-late the value of P with this price p (given pX = X). We have

PðpÞ ¼ pM2

2YðpN � pÞ þ ðpN � pÞ2

2X

!þ p M1

2X � p2X

� �� K

and this is to be compared with P(X) = M1(X/2) � K. Hence the solu-tion with p = X is optimal if

M1X2� pM2

2YðpN � pÞ þ ðpN � pÞ2

2X

!� p M1

2X � p2X

� �> 0

for each choice of p < pN. i.e. if

2YðX � pÞ2 � pðpN � pÞð2X þ pN � pÞ > 0: ð11Þ

Consider the derivative of this with respect to pN it is

�pð2X þ 2ðpN � pÞÞ;

which is negative for p < pN. Hence if the solution with p = X is opti-mal, then it is still optimal for pN made smaller. This establishes theexistence of pB as claimed in the lemma. However the algebra in-volved in calculating a closed form expression for the value of pB

is complex and we will not go through this.Notice that when pN = X the condition (11) for the equilibrium

to exist is

ðX � pÞð2XY � 2pY þ p2 � 3XpÞ > 0:

When p = X � d then this inequality is

dð�2X2 þ Xdþ d2 þ 2ydÞ > 0;

which clearly fails when d is small and positive. Hence pB < X.Now we consider the possibility of an equilibrium at which

both players choose a price below X. We assume thatpN > max(pX,p). Then if pX > p we have H(p) = (pX � p)(2pN � pX � p)and H0(p) = 2p � 2pN, so

HðpÞ þ pH0ðpÞ ¼ ðpX � pÞð2pN � pX � pÞ þ 2p2 � 2ppN ;

dPdp¼ M1

2þM1pX

2XþM2pN

2Y� p

M2

YþM1

X

� �

þ M2

4XYð2pNpX þ 3p2 � p2

X � 4pNpÞ:

Observe that dPdp is continuous as p crosses pX. For a symmetric equi-

librium at p = pX we need

M1

2þM2pN

2YþM1p

2X� p

M2

YþM1

X

� �þ M2

2XYðp2 � pNpÞ ¼ 0;

or

Xyþ XpN � pð2X þ YÞ þ pðp� pNÞ ¼ 0:

This is a quadratic in p with solution

p ¼ X þ Y þ pN

2� 1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðY þ pNÞ

2 þ 4X2q

(since we assume that p < pN, we must choose the lower of the twopossible roots).

Consider how p varies with pN. We have

dpdpN¼ 1

2� ðY þ pNÞ

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðY þ pNÞ

2 þ 4X2q 2 ð0;1=2Þ: ð12Þ

Observe that when

pN ¼ p� ¼ XY=ðX þ YÞ;

we have

p ¼ 2XðX þ yÞ þ yðX þ yÞ þ Xy2ðX þ yÞ � 1

2ðX þ yÞ

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðy2 þ 2XyÞ2 þ 4X2ðX þ yÞ2

q

¼ 2X2 þ 4Xyþ y2

2ðX þ yÞ � 2Xyþ 2X2 þ y2

2ðX þ yÞ ¼ p�:

Thus this equilibrium can only occur when pN > p* since we requirepN > p.

For a maximum we need d2P/dp26 0.

d2P=dp2 ¼ �M1

X�M2

Y� M2

4XYð2H0ðpÞ þ pH00ðpÞÞ:

Now, when pX > p, then H00(p) = 2 and so

d2P=dp2 ¼ �M1

X�M2

Y� M2

4XYð6p� 4pNÞ ð13Þ

and when p > pX, then H0 0(p) = 4 and so

d2P=dp2 ¼ �M1

X�M2

Y� M2

4XYð12p� 4pX � 4pNÞ: ð14Þ

When p = pX observe that the expression (13) is larger than (14)and so the second derivative requirement for an optimum can betranslated to

2Y þ 2X þ ð3p� 2pNÞP 0:

We can substitute in the expression for p and after some algebrathis inequality becomes

44X2 þ 70XY � 10XpN þ 23Y2 � 10YpN � p2N P 0: ð15Þ

Notice that this inequality will fail to hold when X and Y are veryclose to zero. In practice however we can expect that pN, which isthe expected price in the future, is not far away from the equilib-rium price of X that we calculated when M2 = 0. For any value ofpN that is less than 3X we can easily check that the inequality(15) will hold.

The next step is to check that this really is an equilibrium, andthat there is no advantage in one of the players choosing a value ofp higher than pN; foregoing all the price sensitive demand, butcapturing the demand from the non-price sensitive customers at ahigher price. In this case the best choice of price is (X + p)/2 and theprofit available from this strategy is M1(X + p)2/(8X) � K where p isgiven by (5). This has to be compared with

P ¼ pM2

2YðpN � pÞ þ ðpN � pÞ2

2X

!þ p M1

12

� �� K:

The condition that the symmetric equilibrium exists is that.

pM2

2YðpN � pÞ þ ðpN � pÞ2

2X

!þ p M1

12

� ��M1

ðX þ pÞ2

8X> 0;

which can be simplified to

GðpNÞ ¼ 4pðpN � pÞð2X þ pN � pÞ � YðX � pÞ2 > 0; ð16Þ

where p is given by (5). Then

dGdpN¼ 8ppN � 8p2 þ 8Xp

þ dpdpN

4p2N � 16Xpþ 8XpN þ 12p2 � 16ppN þ 2YX � 2Yp

� �¼ 8pðpN � pÞ þ 8Xp 1� 2

dpdpN

� �

þ dpdpN

4ðpN � pÞ2 þ 8p2 þ ð8pN þ 2YÞðX � pÞ� �

:

E. Anderson / European Journal of Operational Research 210 (2011) 436–447 447

From (12) and since p is less than both pN and X we can deduce that Gis increasing in pN. So there is some lowest value of pN at which (16)holds, which we call pA. Moreover we have Gðp�Þ ¼ �YðX � p�Þ2 < 0and hence pA > p*. h

To evaluate qA = pA/X, qB = pB/X in terms of z = (YLM1)/(X(L � 1)M2), we can note that qB is the maximum value forwhich

Gðq; qBÞ ¼ 2zð1� qÞ2 � qðqB � qÞð2þ qB � qÞP 0 ð17Þ

for every q < qB. The expression (17) follows from (11) in the proofof Lemma 1.In fact for given qB and z, the minimum value of G(q,qB)in the range we are interested in occurs at

q� ¼ 23ðzþ qB þ 1Þ � 1

3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4z2 þ 8zqB � 4zþ q2

B þ 2qB þ 4q

:

Then qB is given by solving G(q*,qB) = 0.Moreover, again using the analysis given in the proof of Lemma

1 we can show that qA is given by the solution to

zð1� qÞ2 � 4qðqA � qÞð2þ qA � qÞ ¼ 0;

when

q ¼ 1þ zþ qA

2� 1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðzþ qAÞ

2 þ 4q

:

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