speculative storage in imperfectly competitive markets
TRANSCRIPT
International Journal of Industrial Organization 35 (2014) 44–59
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International Journal of Industrial Organization
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Speculative storage in imperfectly competitive markets
Sébastien Mitraille a, Henry Thille b,⁎a University of Toulouse (Toulouse Business School), Franceb Department of Economics, University of Guelph, Guelph, ON, Canada
⁎ Corresponding author.E-mail addresses: [email protected] (S. Mitra
(H. Thille).1 For example, the impact of shortage risks on some strate
the object of the European Commission (2011), who list 14Beryllium, Cobalt, Fluorspar, Gallium, Germanium, GraphitePlatinumGroupMetals, Rare earths, Tantalum, andTungstenand supply lies in the hands of a few firms or in the controltries. Smith (2009) points out that the world oil market is chply where OPEC is dominant along with speculation bycommercial” traders. Finally, non-ferrous minerals such aduced in oligopolistic conditions and trade on the London Mspeculation.
2 Frankel and Rose (2010), Kilian (2008, 2009), Kilian a
http://dx.doi.org/10.1016/j.ijindorg.2014.07.0010167-7187/© 2014 Elsevier B.V. All rights reserved.
a b s t r a c t
a r t i c l e i n f oArticle history:Received 6 February 2012Received in revised form 27 June 2014Accepted 1 July 2014Available online 10 July 2014
JEL classification:L13D43
Keywords:InventoriesCournot oligopolySpeculation
Markets for many commodities are characterized by imperfectly competitive production as well as substan-tial storage by speculators who are attracted by significant price volatility. We examine how speculativestorage affects the behavior of an oligopoly producing a commodity for which demand is subject to randomshocks. Speculators compete with consumers when purchasing the commodity and then subsequentlycompete with producers when selling their stocks, resulting in two opposing incentives: on the one hand,producers would like to increase production to capture future sales in advance by selling to speculators;while on the other hand, they would like to withhold production to deter speculation, thereby eliminatingthe additional supply from speculators in future periods.We find that the incentive to sell to speculators canbe quite strong, potentially resulting in prices sufficiently high to drive consumers from the market. Fur-thermore, these incentives are non-monotonic in the number of producers: speculative storage occursmore frequently in a relatively concentrated oligopoly than in the extremes of monopoly or perfectcompetition.
ille), [email protected]
gic commoditymarkets has beencritical rawmaterials (Antimony,, Indium, Magnesium, Niobium,) forwhich concentration is highof a very small number of coun-aracterized by oligopolistic sup-both “commercial” and “non-s nickel and aluminum are pro-etal Exchange which facilitates
nd Murphy (2014).
© 2014 Elsevier B.V. All rights reserved.
1. Introduction
Commodity market speculation and imperfect competition are twotopics that have long been of interest to economists. Despite the atten-tion given to each topic individually, little is known about market per-formance in circumstances where oligopolistic producers andcompetitive speculators interact, a topic that has recently been the sub-ject of considerable public attention.1 This attention has triggered sub-stantial academic interest investigating the relationship betweeninventories and speculative trading on commodity markets from aneconometric point of view.2 However, little is known from a theoretical
perspective on the performance of these markets in which oligopolisticproducers face the constraints that competitive speculation places onprice dynamics. In this setting, strategic considerations may be funda-mental to production decisions which ultimately affect the level of in-ventories carried in equilibrium. Understanding the consequences ofthe interaction between imperfect competition and competitive specu-lation is the issue addressed in this paper.
We analyze amodel inwhich an oligopoly produces a good over a fi-nite time horizon with demand subject to stochastic shifts. In additionto producers and consumers there are speculators who are able tostore the good between periods. When storage by these speculators ispossible, the net demand faced by producers is affected: on the onehand, speculators increase demand for the product when price is lowas they buy the good to store for future sale; on the other hand, specu-lators increase the supply of the product when price is high and theywish to sell from their inventories. Hence speculators are competitorsto producers when selling and competitors to consumers when buying,changing the effective demand faced by the producers. With imperfect-ly competitive producers, this effect of storage on net demand raises thepossibility that producers' behavior may be affected through strategicmotives to influence future demand states. It is the main purpose ofthis paper to analyze the strategic effect that speculative storage hason producer behavior.
We obtain the result that, when expected demand growth is rela-tively large, speculators may purchase the entire first period output,resulting in zero first period consumption. This occurswhen speculators
45S. Mitraille, H. Thille / International Journal of Industrial Organization 35 (2014) 44–59
value the good in the first periodmore highly than the consumers do es-sentially out-bidding them for the current production.3 This result isparticularly interesting since in much of the public discussion of recentevents in commodity markets there is a recognition, implicit or explicit,of the extreme form of competition that speculators exert to other mar-ket participants, a competition that is often viewed as detrimental byneither “ensuring an efficient, economic and regular supply to con-sumers”nor “a fair return for those investing in the industry”.4 In this re-spect, the example of developing countries struggling to satisfy theirneeds on the demand side, as emphasized by regulators or internationalagencies, is striking. In circumstances where the current demand is lowcompared to the expected future demand, which could correspond tothe recent economic boom in Asia leading to expectations of rapidgrowth in demand, the entire output may be purchased without beingconsumed and paid at the expected future price which is disconnectedfrom current demand. In that case, themarket price is “high”, the inven-tories are “large”, and the consumption is “nil”.5 Our analysis demon-strates that the likelihood that we see this outcome is increasing asproduction becomes more concentrated.
We also find that the extent to which equilibria arise in which spec-ulators carry stocks is non-monotonic in the number of firms, beinglower for the two extremes of monopoly and perfect competition andhighest for an intermediate number of firms. This result is due to twocountervailing forces in our model: first, producers have an incentiveto sell to speculators in order to capture future sales in advance; second,producers have an incentive to deter speculation in order to limit the fu-ture competition that is represented by speculative inventories. Whilethe former incentive is absent inmonopoly, but present in every oligop-olistic market where a price differential allows speculators to store, thelatter exists onlywhen the price differential is limited, so that a small re-duction of output reduces speculation. This may happen either undermonopoly, where the firm reaps the entire benefit of deterring specula-tion, or under more competitive market structures, where prices arecloser to marginal cost which limits the benefit of sales to speculators.
Modern approaches to the study of speculation in perfectly compet-itive markets were pioneered by Gustafson (1958) who was the first toformulate the problem in a dynamic programming context and high-light the difficulty for the solution of themodel caused by the restrictionthat aggregate inventories cannot be negative. Samuelson (1971)showed that the speculative storage problem could be reformulated asa social planner's problem. The theory pertaining to the rational expec-tations competitive storage model was further developed in a series ofworks including Newbery and Stiglitz (1981), Scheinkman andSchechtman (1983), Newbery (1984), Williams and Wright (1991),Deaton and Laroque (1992), andMcLaren (1999). The focus in this liter-ature is on the effects of storage on the distribution of prices caused bythemovement of production across periods resulting from randompro-duction (harvest) shocks. As aggregate inventories cannot be negative,speculators smooth prices across periods only when positive invento-ries exist. Unexpectedly high prices result in stockouts which lead to abreakdown of the price smoothing role of speculative storage.
Market power in commoditymarkets has been considered by exam-ining imperfect competition in the storage function (McLaren, 1999;Newbery, 1984; Williams and Wright, 1991), but production itself re-mains perfectly competitive in these works, hence there is no scopefor strategic considerations on the part of producers. Another common
3 This equilibrium is related to the “entitlement failure” used by Sen (1981) to explainhow famines can occur evenwhen production is not particularly low and storehouses con-tain grain. In this situation, speculators' willingness to pay for the available supply exceedsthat of consumers. This case is also relevant to the policy debate regarding anti-hoardinglaws (see Wright and Williams (1984)).
4 These objectives are part of theOPECmission in the petroleum industry, as can be readon their website, http://www.opec.org/opec_web/en/about_us/23.htm.
5 Even without the full exclusion of consumption, when both consumers and specula-tors are purchasing output, current consumption is displaced by speculative purchasesto some extent.
element of these works is a focus on agricultural commodities, forwhich the assumption of perfectly competitive production is justifiable.However, for other commodities, such asmetals, the assumption of per-fectly competitive production is harder to justify. The problem of a mo-nopoly producer facing competitive storage was touched on briefly inNewbery (1984), who demonstrated that speculation resulted in dis-continuous marginal revenue for the monopolist. Mitraille and Thille(2009) examine a monopolist producer facing competitive storageusing a model with capacity constrained speculators. They show thatboth the level and variance of prices are affected by speculation undermonopolistic production. In particular, the presence of speculatorsmay cause the monopolist to set a price higher than the monopolyone in order to eliminate competition from speculative sales in futureperiods, a result similar to that of limit pricing. In this paper we allowfor an arbitrary number of firms and impose no constraint on specula-tive capacity, which allows us to examine the consequences of specula-tion for the functioning of an oligopolistic market.
Our paper lies at the cross-roads of two streams of research in indus-trial organization: imperfect competition under demand uncertainty onthe onehand, and the literature studying strategic uses of storage on theother hand. Demand uncertaintywas examined in the seminal article byKlemperer and Meyer (1986), in which price or quantity competitioninteracts with demand uncertainty in a static game where decisionsare taken before uncertainty is revealed in order to examine the prefer-ence of firms for price versus quantity competition. In our dynamicmodel of Cournot competition, demand uncertainty can cause pro-ducers to manipulate speculation in either a pro- or anti-competitivemanner, depending on current demand relative to future demand.
The strategic use of storage has been examined in caseswhere eitherproducers or consumers perform the storage. Arvan (1985), Thille(2006), and Mitraille and Moreaux (2013) examine the former case, inwhich producers store for strategic reasons, as inventories are accumu-lated in order to gain a cost advantage. Storage by consumers is moresimilar to themodel of speculative storage that we examine in that stor-age is undertaken by competitive agents. Consumer storage with duop-oly production is examined in Anton and Das Varma (2005) and Guoand Miguel Villas-Boas (2007). Whereas Anton and Das Varma showthat consumer storage has a pro-competitive effect in a two-period du-opoly, Guo and Villas-Boas show that horizontal differentiation maymitigate this effect. Consumer storage under monopoly production isexamined by Dudine et al. (2006) who find that consumer storage canhave an anti-competitive effect as the monopolist may raise price tolimit storage. By examining a general n-firm oligopoly facing demanduncertainty, we are able to discuss under what conditions these pro-and anti-competitive effects occur. Furthermore, we show that the ef-fect of speculation on the outcome of an oligopolistic market is non-monotonic in the number of competitors.
We next provide a description of our T-periodmodel, followed by ananalysis of the equilibrium in period T-1, for which we obtain some an-alytic results due to the fact that the period T equilibrium is relativelystraightforward to obtain. We then demonstrate the robustness ofthese results by computing the equilibrium for an earlier period inwhich the influence of the final period is minimal.
2. Model
Consider the market for a homogeneous product consisting of pro-ducers, consumers, and speculators. We wish to focus on the effects ofspeculation alone, so we assume that all agents are risk-neutral. Thereare n producers who compete in quantities over T periods. Let δ ∈(0, 1] be the agents' common discount factor. In each period t, firm i ∈{1, …, n} chooses the quantity to produce, qti ∈ ℝ+. All firms face thesame quadratic cost function of c
2 qit2. Convex production costs are not
crucial to our results, however, as we discuss below, speculators' behav-ior can lead to non-concavities in producer revenue, which can be mit-igated by assuming sufficiently convex production costs. We assume
46 S. Mitraille, H. Thille / International Journal of Industrial Organization 35 (2014) 44–59
that firms are unable to store their production between the twoperiods.6
In order for speculation to be potentially profitable, we require someprice variation over time. We ensure this by having demand vary ran-domly across periods,7 which also implies that storage is potentially so-cially valuable in this model as it allows producers to smooth theirconvex production costs over time. In each period, consumers' aggre-gate demand for the product is given by,
D at ; ptð Þ ¼ max at−pt ;0½ � ð1Þ
where at is drawn from a uniform distribution on [0, A] for each t. Com-binedwith sufficient cost convexity, the assumption of a uniform distri-bution for the demand uncertainty allows us to compute closed-formsolutions for the equilibrium quantities in period T-1. Random changesin αt may be interpreted as random shocks affecting the distribution ofincome in the population of consumers from period to period, modify-ing the willingness to pay for the product sold by firms and stored byspeculators. The distribution of the intercept of market demand is com-mon knowledge to both producers and speculators who learn the cur-rent state of demand, at, prior to any production or storage decisionsin a period. Information is consequently symmetric between producersand speculators.
We denote the aggregate quantity produced in period t by Qt, andthe aggregate quantity produced by all firms but i by Qt
−i, where Qt =∑ i = 1
n qti and Qt
−i = ∑ j = 1,j ≠ in qt
j. Producer i's payoff,Πi, is
Πi ¼ E0XTt¼1
δt ptqit−
c2qit
2h i; ð2Þ
where Et denotes the expectation operator with regard to the randomdemand intercept conditional on information at time t.
We model speculation in the sameway as is done in the competitivespeculative storage literature (Deaton and Laroque, 1992, 1996;Newbery, 1984). In each period a large number of risk-neutral, price-taking agents exist who have access to a storage technology and faceno borrowing constraints. The storage technology allows speculators tostore the commodity at a unit cost ofw between periods, so the expectedreturn to storing a unit of the commodity between periods t and t+1 isδ(Et[pt + 1]−w)− pt. We use Xt to denote aggregate speculative sales inperiod t (purchases have Xt b 0) and Ht to denote beginning of periodstocks. Consequently, inventory dynamics follow Ht + 1 = Ht − Xt andwe assume that inventories unsold at the end of period T can be disposedof at no cost. To ensure that speculation is at least potentially profitable,we assume that consumers' expected valuation exceeds the unit cost ofstorage, A/2 ≥ w. In addition, since it plays an important role in themodel, we define pt
e(Ht + 1) = Et[pt + 1] to represent the expectednext-period price when stocks of Ht + 1 are carried into the next period.
The timing of moves within a period is simply the standard Cournottiming with demand adjusted for net speculative supply: producers ob-serve at andHt and then set their output, qti. Price adjusts to clear themar-ket with demand now equal to the sum of consumer demand andspeculators' net demand. We search for the subgame-perfect Nash equi-librium to this game with rational expectations on the part of all agents.In any period t = 1, 2, …, T, Vi,t(Ht, at) will denote the present value ofthe expected payoff to producer i from t to T when producers play theirNash equilibrium strategies, and Vi,t
e (Ht + 1) = Et[Vi,t + 1(Ht + 1, at + 1)]will denote the expected continuation payoff to producer iwhen specula-tors carry Ht + 1 into the next period.
6 Clearly it would bemore realistic to allow firms to also store, however that would adda substantial degree of complexity to the analysis potentially clouding the effects of inde-pendent speculators. See Arvan (1985), Thille (2006), and Mitraille and Moreaux (2013)for analyses of producer storage under imperfect competition.
7 Having the source of variation be indemandmeans that for a sufficiently large numberof producers speculation will not occur as price will tend to equality in the two periods.
In order to complete the specification of producer payoffs, we nowwork out the effect of speculation on the net demand faced by pro-ducers. The aggregate behavior of these speculators ensures that pt ≥δ(pte(Ht + 1) − w) with a stockout occurring if the inequality is strict.The non-negativity constraint on aggregate speculative inventories im-plies that speculators' aggregate behavior satisfies the complementaritycondition
Ht−Xt≥0; pt−δ pe Htþ1� �
−w� �
≥0;
Ht−Xtð Þ pt−δ pe Htþ1� �
−w� �� � ¼ 0:
ð3Þ
Either no inventories are carried and the return to storage is nega-tive, or inventories are carried and the return to storage is zero. Wewill use Xt∗(Qt) to denote the equilibrium storage undertakenwhen pro-ducers sell Qt in aggregate. The market clearing price, Pt(Qt), must besuch that, given Qt, the total of consumer and speculative purchases sat-isfy
Dt at ; Pt Qtð Þð Þ−X�t Qtð Þ ¼ Qt : ð4Þ
FromEq. (3), there is a threshold level of aggregate output,whichwedenote Qt
L, below which pt N δ(pte(0) − w), as speculators cannot carrynegative inventories. This threshold is the level of output which leadsto zero return to speculation when there is a stockout:
at−Ht−QLt ¼ δ pet 0ð Þ−w
� �: ð5Þ
For Qt b QtL the slope of net demand coincides with that of consumer
demand, while for Qt N QtL the slope of net demand differs from that of
consumer demand due to the dependence of speculative sales on aggre-gate output. Note that it is possible that Qt
L b 0 in which case a stockoutcannot occur.
Another threshold for aggregate output is possible due to the com-petition that speculators pose to consumers when they purchase thegood for storage. Speculators' purchase the good when the price is rela-tively low. To the extent that a lowprice results froma low at, speculatorand consumer demands are negatively correlated in this situation, in-troducing the possibility that speculators squeeze consumers out ofthe market. The exclusion of consumers will occur if speculators valueproducers' output more highly than consumers do: δ(pte(Ht + Qt) −w) N at. Consequently, there is another threshold output, which we de-note Q̂ t, for which only speculators buy ifQtbQ̂ t and consumers buy andspeculators carry inventories if Qt NQ̂ t . This threshold is determined bythe level of aggregate output that just extinguishes consumer demand:
at ¼ δ pet Ht þ Q̂ t
� �−w
� �: ð6Þ
As with QtL, the slope of the demand faced by producers changes dis-
continuously at Q̂ t. It is important to note that only one of QtL and Q̂ t can
be positive as long as pte() is a decreasing function.8
The final aspect of the behavior of speculators that needs to be deter-mined is thequantity that theybuyor sellwhenaggregate output exceedsthe relevant threshold, Qt
L or Q̂ t . This quantity can be determined implic-itly by the relationship between Pt and pt
e(Ht + 1) that must hold. We de-note speculative sales in this case as eXt Qtð Þ, which is the solution in X to
at−X−Qt ¼ δ pet Ht−Xð Þ−w� �
: ð7Þ
Applying the Implicit Function Theorem to Eq. (7) shows that eXt Qtð Þis strictly decreasing as long as pte is decreasing, which is verified in thefollowing sections. The higher the aggregate output, the smaller the
8 If both thresholds were positive, both Eqs. (5) and (6) must hold. The left hand side ofEq. (6) is clearly higher than that of Eq. (5) while the right hand side of Eq. (6) is lowerthan that of Eq. (5) if the expected price function is decreasing in future stocks, so bothEqs. (5) and (6) cannot hold simultaneously.
47S. Mitraille, H. Thille / International Journal of Industrial Organization 35 (2014) 44–59
market clearing price is and the larger speculative purchases are (in ab-solute value), inducing larger final period inventories. Speculative salesare then given by
X�t Qtð Þ ¼
Ht if Qt≤QLt and QL
t N0−Qt if Qt≤Q̂ t and Q̂ t N0eXt Qtð Þ otherwise:
8><>: ð8Þ
As only one ofQtL andQ̂ t can be positive, only one of thefirst two con-
ditions in Eq. (8) is possible for a given (Ht, at). The quantities QtL and eXt
Qtð Þ are consistent with what is found in the competitive storage liter-ature. Compared to that literature however, the existence of Q̂ t isnovel, as in our setting the possibility that consumers do not purchasemust be considered. Studying this possibility is particularly importantas the incentives of oligopolists to produce when consumer demand islow enough are entirely driven by the future expected price paid byspeculators and not the value of consumers' current demand.
Given the behavior of speculators in Eq. (8), we can now state the in-verse demand net of speculation that producers face:
Pt Qtð Þ ¼at−Ht−Qt if QT−1≤QL
t and QLt N0
δ pet Htþ1 þ Q� �
t−w� �
if QT−1≤Q̂ t and Q̂ t N0at−eXt Qtð Þ−Qt otherwise:
8><>: ð9Þ
With the behavior of speculators determined by Eqs. (8) and (9), pay-offs to producers can be specified entirely in terms of output. Themargin-al payoff to a firm in any period will be discontinuous at an output thatresults in aggregate output of Qt
L or Q̂ t , so the nature of the equilibriumin period t depends on where aggregate output falls in relation to thesethresholds. First, a stockout may occur in which speculators sell their en-tire stocks and consumers buy the total of speculative and producer sales.Wewill denote this type of equilibriumwith a C. Second, consumersmaybuy nothing and speculators purchase the entire output of firms, whichwe will denote with an S. Third, consumers may make some purchasesand speculators carry inventory into the following period, denoted witha CS. Finally, there is the possibility that producers deter speculators byproducing exactly Qt
L in aggregate, which we will denote with an L.It is not possible to find the equilibrium analytically for the general T-
period game due to the difficulty in determining the expected price func-tion.9 In our game, the discontinuities inmarginal profit also contribute tothe difficulty in finding closed-form solutions. However, we are able toderive analytic results for the equilibrium in period T-1 which we turnto in the next section. Following that, we verify that the types of equilibriathat we find in period T-1 exist also in earlier periods by solving the gamewith numerical approximation techniques in Section 4.
3. Equilibrium in T-1
In this sectionwe present results that characterize the equilibrium inperiod T-1 by deriving closed-form representations of the expectedprice function and equilibrium quantities. We start by presenting theequilibrium outcome for period T and then derive speculators' optimalbehavior and the implications for producers' marginal profit. We thenanalyze the equilibrium in period T-1
3.1. Period T equilibrium
Since there is no salvage value to inventories held at the end of T, spec-ulatorswill sell the quantity they have stored as long as PT N 0. In this caseaggregate speculative sales are equal to XT = HT. If on the other hand the
9 This is true even for the perfectly competitive case as has been known since Gustafson(1958).
aggregate quantity stored by speculators is such that the market price isequal to zero once HT is sold, then the aggregate speculative sales cannotexceed the maximal quantity consumers are ready to buy given the ag-gregate production of firms, aT − QT. In that case, aggregate speculativesales are simply XT = aT − QT and the price is zero. Consequently, the in-verse demand faced by producers in the final period is
PT QTð Þ ¼ max aT−HT−QT ;0f g: ð10Þ
Given the inverse demand net of speculative sales, PT(QT), each pro-ducer chooses the quantity qT
i to maximize its second period profit, πiT ¼
PT QTð ÞqiT−c2q
iT2, with respect to the quantity produced qT
i . The unique andsymmetric pure strategy Nash equilibrium for the final period subgame isgiven by the following functions of the period T state, (HT, aT):
q�T HT ; aTð Þ ¼ 1n
1−βð Þmax aT−HT ;0½ �;Q�T HT ; aTð Þ ¼ 1−βð Þmax aT−HT ;0½ �;
ð11Þ
p�T HT ; aTð Þ ¼ βmax aT−HT ;0½ �; ð12Þ
X�T HT ; aTð Þ ¼ min HT ; aT½ �; ð13Þ
with the final period value given by equilibrium profit:
VT HT ; aTð Þ ¼ π�T HT ; aTð Þ ¼ γ max aT−HT ;0½ �ð Þ2: ð14Þ
where β=(1+ c)/(n+1+ c) and γ=(2+ c)/2(n+1+ c)2 both be-longing to (0,1). For a given c, β reflects the impact of competition on themarket price andγ reflects the impact of competition on eachfirm's profitin period T. The larger the number of producers the closer β and γ are tozero. The expected price function pT − 1
e (HT) can now be defined10:
pe HTð Þ≡ ET−1 PT½ jHT � ¼Z A
HT
β a−HTð Þ=A da ¼ βA=2ð Þ 1−HT=Að Þ2 ð15Þ
3.2. Implications for producer marginal profit
Before determining the equilibrium in period T-1 we need to under-stand the implications of speculators' behavior, as given in Eqs. (8) and(9), on producers' marginal profit. It is possible to do this analyticallysince we have closed-form solutions for QL, Q̂ , and eX QT−1ð Þ .11 LetΠi(qT − 1
i , QT − 1−i ) denote the total profit to firm i over the last two pe-
riods when it produces qT − 1i and the other firms produce qT − 1
−i in ag-gregate while incorporating the period T equilibrium:
Πi qiT−1;Q−iT−1
� �¼ PT−1 QT−1ð ÞqiT−1−
c2qiT−1
2
þδET−1 VT HT−1−X�T−1 QT−1ð Þ; aT
� �� � ð16Þ
withQT− 1= qT− 1i +QT− 1
−i . As speculators' position,XT− 1∗ (QT− 1), and
inverse demand, PT − 1(QT − 1), are continuous12 and piecewise differ-entiable, individual profit is also continuous and piecewise differentia-ble. The marginal profit of a firm choosing a level of output qT − 1
i ,where it exists, is equal to
Πiq qiT−1;Q
−iT−1
� �¼ ∂PT−1
∂QT−1qiT−1 þ PT−1 QT−1ð Þ−c qiT−1
þ2δγβ
dX�T−1
dQT−1pe HT−1−X�
T−1 QT−1ð Þ� �ð17Þ
10 For the rest of this section, we drop the T-1 subscript on variables where there is noconfusion.11 These are presented in Lemma 1 of Appendix B.12 Continuity follows from the definitions of QL, Q̂ , and eX QT−1ð Þ.
48 S. Mitraille, H. Thille / International Journal of Industrial Organization 35 (2014) 44–59
where the last term is from Lemma 2 of Appendix B. Marginal profit isdiscontinuous at the thresholds QL and Q̂ for two reasons: fromEq. (9), the marginal effect of output on price is discontinuous; and,from (8), the marginal effect on expected future profit is discontinuousthrough the effects on future inventories. When a stockout occurs, 0 ≤QT − 1 ≤ QL, no inventories are carried into the final period so expectedperiod T profit is not affected by output and period T-1marginal profit issimply that of a Cournotmodelwith consumer demand reduced byHT-1.We denote marginal profit in this case by Πq
C(qT − 1i , QT − 1
−i ). When in-ventories are carried into the final period, both price and future inven-tories are affected by a firm's choice of output in a manner thatdepends on whether consumers purchase the good or not. In the casethat consumers are excluded from purchasing in T-1, 0≤QT−1≤Q̂ , wedenotemarginal profit asΠq
S(qT− 1i ,QT− 1
−i ),whilewhen consumers pur-chase and inventories are carried into the final period we denote mar-ginal profit as Πq
CS(qT − 1i , QT − 1
−i ).13
In order to find values of (HT − 1, aT − 1) for which a unique symmet-ric equilibrium exists, we work under an assumption that guaranteesthat marginal profit is strictly decreasing apart from the points atwhich aggregate outputmeets the thresholds QT − 1
L and Q̂ T−1. Whereasthis property is trivially satisfied when a stockout occurs, it is not auto-matically the case when speculators carry inventories to the final peri-od. A sufficient condition for marginal profit to be strictly decreasingin the S and CS cases is that production cost be sufficiently convex:
Assumption 1. (Cost convexity). c ≥ δβ
This assumption ensures that when there is an equilibrium candi-date of the S or CS type, it is the unique one of that type. The only poten-tial problems for the existence or uniqueness of equilibrium are thenassociated with the discontinuities in marginal payoffs at QT − 1
L andQ̂ T−1 . Note that, since β is declining in n, Assumption 1 is moreconstraining for more concentrated market structures. This impliesthat, if Assumption 1 holds for n = 1, it will hold for all n. We cannow summarize the characteristics of period T-1 marginal profit in T-1with the following proposition:
Proposition 1. The marginal profit of producer i, Πqi (qT − 1
i , QT − 1−i ),
(i) jumps up at the output level such that consumers start to buy theproduct, qiT−1 ¼ Q̂−Q−i
T−1 , whenever this output level exists and ispositive;
(ii) jumps up or down at the output level such that speculators start tobuy the product, qT − 1
i = QL − QT − 1−i , whenever this output level
exists and is positive.
Furthermore, under Assumption 1, marginal profit for firm i is strictlydecreasing when inventories are carried into the final period.
Proof. See Appendix A.
Proposition 1 has important consequences for the nature of equilib-rium. First, as is well known, the possibility of upward jumpingmargin-al profitmay generatemultiple local solutions to thefirm's optimizationproblem. Second, the possibility of an downward jump at qT − 1
i =QL−QT − 1− i suggests that an equilibrium may exist in which all producers
choose output levels such that the industry output is equal to QL. Inthis case a stockout is induced and the first period price is equal toδ(βA/2 − w). As Mitraille and Thille (2009) showed in this context foramonopoly, this limit outcome, L, is a possibility in an oligopoly as well.
In addition, Proposition 1 guarantees that a local solution to the firstorder condition is unique in the interior of the regions in which C, S, andCS are possible outcomes. To establish the existence of an equilibrium,we then need only check that no profitable deviation to another typeof equilibrium (across the QL or Q̂ thresholds) exists.
13 The precise expressions for ΠqC, Πq
S, and ΠqCS are given by Eqs. (67), (65), and (66) in
Appendix B.
3.3. Equilibrium
The equilibrium quantities for each of the different types of equilibriaare now determined, beginningwith the one in which consumers are ex-cluded from the market. The consumer exclusion equilibrium, S, occurswhen speculators' marginal valuation of the T-1 output is higher thanaT-1. We establish the existence of an S equilibrium in the followingproposition:
Proposition 2. For aT-1, HT-1, and n sufficiently small, an S equilibriumexists in which output for an individual firm is given by
qS ¼δ β−2γ þ β
n
� �A−HT−1ð Þ þ c
nA−
ffiffiffiffiΔ
p
δ β−2γ þ 2βnð Þn
with
Δ ¼ − δβn
A−HT−1ð Þ þ cnA
2þ 4δ
β−2γ2
þ βn
A δwþ c
nA−HT−1ð Þ
� �
Proof. See Appendix A.
The S equilibrium occurs in circumstances where current demand,aT-1, and existing inventories,HT-1, are low compared to expected futuredemand, and only occurs for relatively concentrated market structures,which is due to the strategic incentive to increase T-1 sales by selling tospeculators.
Turning now to the other types of equilibria, the existence of C, CS,and L has been demonstrated in the previous literature for monopolisticand perfectly competitive production. In the case of monopoly produc-tion,Mitraille and Thille (2009) show that all three of these are possible,while the large literature with competitive production shows that bothC and CS occur in that case. Not surprisingly, these equilibria may alsooccur in our oligopoly model and we present the conditions underwhich they obtain for period T-1 in Appendix B. Here we present theequilibrium output for each of these cases in period T-1.
The smoothing equilibrium candidate, CS, with individual firm outputof qCS, has both speculators carrying inventories to the finalperiod and consumers buying the product in period T-1. It obtains whenΠq
CS(qCS, (n − 1)qCS) = 0 and we show in Appendix B that output isgiven by
qCS ¼1n
aT−1−HT−1 þ δwþ A−A − K2
3K1þ Z�
−δβ
A2
− K2
3K1þ Z�
2 ð18Þ
where Z⁎, K1, and K2 are functions of aT-1, HT-1 and model parameters.The output levels for the equilibria inwhich speculators do not carry
stocks into the final period are much simpler to obtain. In the stockoutequilibrium, C, speculators sell all inventories HT-1 they own at thebeginning of period T-1. Equilibrium quantities are defined by Πq
C(qC,(n− 1)qC) = 0, which is simply the Cournot outcome when consumerdemand is reduced by speculative inventories HT-1. Individual produceroutput is then
qC ¼ 1n
1−βð Þmax aT−1−HT−1;0½ �: ð19Þ
Finally, speculation may be deterred. As Proposition 1 states, mar-ginal profit may be jumping down at an aggregate output level equaltoQT− 1
L . It is possible thatfirms choose output levels lower than qCS, en-suring that speculators sell all inventories in period T-1. When specula-tion deterrence occurs, individual firm production is
qL ¼1n
aT−1−HT−1−δβA2þ δw
; ð20Þ
Fig. 1. Equilibria in period T-1.
49S. Mitraille, H. Thille / International Journal of Industrial Organization 35 (2014) 44–59
and in this case the equilibrium price is pt − 1 = δβA/2 − δw, which isthe minimum price consistent with zero inventories being carried tothe final period.
3.4. The non-monotonic effects of market structure
Wenowdemonstrate that the incentive to sell to speculators is non-monotonic in the number of firms by examining two specific cases: i)the incentive to sell to speculators when they are the only source of de-mand, and ii) the existence of the L equilibrium. To examine the incen-tive to sell to speculators in isolation, it is useful to examine the Sequilibrium described in Proposition 2 with HT-1 = aT-1 = 014
and zero cost of production, c = 0, in order to examine the marginalprofit of an individual firm when aggregate production is zero. This al-lows us to emphasize how the incentive to sell to speculators varieswith the number of firms while abstracting from any otherincentive to produce.15 In this case, demand in T-1 can only comefrom speculators, who buy any output produced as long as PT-1 is nottoo high. To characterize the incentive for producers to sell to specula-tors in this setting, consider themarginal profit faced by a firmwhen in-dustry output is in fact zero. In this case Πq
S takes the relatively simpleform16:
ΠSq 0;0ð Þ ¼ δ β−2γð ÞA=2−wð Þ ð21Þ
Note that for c = 0, β− 2γ= (n− 1)/(n + 1)2, which is zero for amonopolist: even though speculators would be willing to buy the com-modity at a positive price, the monopolist will not supply them withoutput. This is not the case for an oligopoly. For n N 1,Πq
S(0, 0) is strictlypositive if the cost of storage is low enough and firms will want toproduce positive output even with zero consumer demand. In addition,this effect is non-monotonic in n: (n− 1)/(n+1)2 reaches amaximumfor n = 3 and stays above 1/9 (its value when n = 2) for n between 2and 5. It decreases to 0 as n goes to infinity. In this simple setting, thisnon-monotonicity is due to the strategic effect of competitive storage:by selling to speculators in T-1, firms capture some of the period Texpected demand. This effect is absent under monopoly, wherethere is no competition for future demand, and under perfect competi-tion, where the strategic effect is absent. To summarize this result wehave
Proposition 3. For aT-1 and HT-1 close to zero and c= 0, a firm's marginalprofit when industry output is zero,Πq
S(0, 0), is larger under oligopoly thanunder either monopoly or perfectly competitive market structures.
For c N 0 this strategicmotive to sell to speculators is balanced by theincentive to smooth production costs and the net effect of market struc-ture on the equilibrium outcome depends on the relative strength ofthese two incentives.
We can also see the non-monotonic effect of the number of firms byexamining the conditions under which the limit equilibrium, L, occurs.When QL N 0, the L equilibrium will exist if there is a feasible aT-1 −HT-1 for which marginal profit is downward-jumping at QL − (n − 1)qL: Πq
C(qT − 1L , (n − 1)qT − 1
L ) ≥ 0 and ΠqCS(qT − 1
L , (n − 1)qT − 1L ) ≤ 0.
From the expressions for ΠqC and Πq
CS given in Appendix B, these twoconditions are satisfied simultaneously if
w≥ A 1þ cð Þ n−1ð Þ2 nþ 1þ cð Þ2 : ð22Þ
14 The argument that follows does not rely onHT-1 and aT-1 being zero, just that they arelow enough that Proposition 2 applies.15 Clearly c = 0 violates Assumption 1, but we are not attempting to compute the equi-librium in this particular case, rather we only want to isolate the incentive to sell tospeculators.16 See Eq. (65) in Appendix B for the general form for Πq
S.
Clearly, formonopoly the right hand side of Eq. (22) is zero and thereis a set of values for aT-1 − HT-1 in which an L equilibrium exists for anyw N 0. Furthermore, the right hand side of Eq. (22) is non-monotonic inn, increasing at n = 1 and then decreasing to zero for large values of n,which means that it is less likely to be satisfied for market structuresconsisting of an intermediate number of firms. As the limit equilibriumrepresents thewillingness of firms to deter speculation,we can summa-rize this result as:
Proposition 4. The possibility that an equilibrium exists in which specula-tive storage is deterred is non-monotonic in n.
With these two simple cases, we have shown that the strategic incen-tive to sell to speculators is non-monotonic in the number of firms. Thegeneral relationship between market structure and speculation in periodT-1 equilibriumalso depends on the desire to smooth production costs, sothe net effect is difficult to discern in general.We now turn to a computedexample to illustrate this relationship for a particular set of parametervalues.
3.5. Example
Although we have explicit solutions for output in each type of equi-librium, the profit comparison required to rule out deviations to anotherequilibrium type is too complex to perform analytically. However, it isstraightforward to compute them for a particular example, using theclosed-form expressions for output determined above. We set A = 20,δ = 0.95, w = 0.2, and choose c = 0.6 which satisfies Assumption 1for n ≥ 1. Consequently, we are certain that marginal profit for firm i isdecreasing apart from the discontinuities at QL − QT − 1
−i or Q̂−Q−iT−1.
We find the T-1 equilibrium as described above for n = 1, 2, …, 80 andfor 500 equally spaced values for aT-1∈[0,20]. The proportion of occur-rences of each type of equilibrium for each value of n is illustrated inFig. 1 for zero initial stocks,17 HT-1 = 0.
The C equilibrium is the unique equilibrium for large first period de-mand and for large n, with more than 78 firms speculation is blockadedand only the C equilibrium is possible. The CS equilibrium occurs for alarge set of parameters and is the unique equilibrium over a substantialrange. There are a large number of situations for which we have multi-ple equilibria of either C or CS. An equilibrium of the CS type existsmore
17 We are most interested in what happens with zero initial stocks as this case allows adirect comparison to the Cournot equilibrium in the absence of storage.
Fig. 2. The effects of speculative storage on expected price and welfare relative to that in the absence of speculation.
50 S. Mitraille, H. Thille / International Journal of Industrial Organization 35 (2014) 44–59
than 50% of the time for a wide range of n, which implies that specula-tors are carrying inventory even though demand is expected to fall.
The consumer exclusion equilibrium (S) occurs over a significantrange of stateswhen themarket is not very competitive.When it occurs,it is usually the unique outcome, although there are a relatively smallnumber of cases in which either S or CS can be the equilibrium. Thelimit equilibrium (L) occurs for a small number of demand stateswhen there is one firm or when there are 76–78 firms.
The proportion of demand states for which the CS equilibrium ob-tains in Fig. 1 is non-monotonic in the number of firms. This is trueeven when we consider only the situations in which CS is the uniqueequilibrium. To examine the implications of non-monotonicity in thetype of equilibrium for expected price and welfare, we compute specu-lative purchases, expected price and welfare for n = 1, …, 80 and plotthe results in Fig. 2. The effect on prices and welfare is computed asthe difference between the equilibrium values with speculation and
Fig. 3. Equilibria in period T-1: Increased storage cost.
those without, expressed as a proportion of the latter. We see that thelevel of speculative purchases is non-monotonic in the number offirms, peaking at around 10 firms and slowly declining to zero at over70 firms. Expected period T price mirrors the level of stocks carriedand clearly speculation has the effect of reducing period T prices. How-ever, the effect on T-1 price is dependent onmarket structure: with rel-atively few firms speculation causes an increase in expected T-1 price,due largely to the relatively high prices when the equilibrium is S.With more firms competing, however, the effect of speculation is to re-duce expected T-1 price as the S equilibrium occurs only rarely. The im-plications of these effects on welfare are shown in the lower right panelin Fig. 2 in which we plot the effects of speculation on the discountedsum of expected welfare (consumer surplus plus producer profit)against the number of producers. Again we see a non-monotonic effect:welfare increases for relatively concentrated market structures, but ac-tually falls for relatively unconcentrated ones. The magnitude of thewelfare effects of speculation in this example is not large (at mostunder 2% of expected welfare in the absence of speculation) but thereis potential for substantial redistribution between consumers and pro-ducers within periods as seen by the effects on the expected T-1 price,which can be as high as 15%.
To see the effects of higher storage costs and to demonstrate that thelimit equilibrium need not be as rare as we see in Fig. 1, in Fig. 3 we plotthe equilibria that obtain with a value ofw=1.1 and all other parame-ters the same as in Fig. 1. Not surprisingly, equilibria with storage occurmuch less frequently and the stockout equilibrium is the unique out-come for n N 12. The L equilibrium occurs somewhat more frequentlyfor the monopoly and when there are 10 to 12 firms. The higher costof storage makes deterrence of speculators easier to attain whileretaining the non-monotonicity discussed above.
4. Equilibrium for t b T-1
Explicit expressions for output in the T-1 equilibrium were possibledue to the relatively simple equilibrium in period T. In particular, spec-ulators' behaviorwas quite simple in that they sell their stocks as long aspT N 0, allowing us to compute the expectation of the period T pricewithlittle difficulty. In periods prior to T-1 however, we are not able to derive
Fig. 4. Equilibria in period T-10.
51S. Mitraille, H. Thille / International Journal of Industrial Organization 35 (2014) 44–59
closed-form solutions for pte and Vte due to more complex behavior by
speculators in the following period. Consequently, we turn to numericalapproximation methods to find approximate equilibria to the game forperiods T-2 and earlier in order to demonstrate the analytic resultsestablished for period T-1 continue to obtain for t b T-1.
We apply the collocation method,18 using cubic splines to approxi-mate the expected price and value functions, denoting these approxi-mations ρt and vi,t. We start with the period T solution presentedabove and iterate backwards. Given a vector of m values for Ht-1 be-tween 0 and Hmax, H ¼ 0;H1;…;Hmax
� �we compute the vectors pet
and Vet which are the expected price and value19 associated with future
stocks given by each element of H, i.e., for k = 1, 2,…, m,
petk ¼Z A
0p�tþ1 a;Hk
� �=Ada ð23Þ
and
Vetk ¼
Z A
0V�tþ1 a;Hk
� �=Ada ð24Þ
with pt + 1∗ (a, H) and Vt + 1
∗ (a, H) the equilibrium price and value func-tions in period t+ 1 given state (a,H) and period t+ 1 approximationsρt + 1 and vt + 1. Our approximation to pt
e(H), ρt(H), is found by fitting acubic spline to the H and pet vectors. Similarly, we fit a cubic spline to Hand Ve
t to generate our approximation to Vte, vt(H).
In summary, to find the solution in any period t0:
Step 0 Compute the period T equilibrium price and value, p�T a;Hk� �
andV�T a;Hk� �
, k = 1, … m, using Eqs. (12) and (14). Set t = T-1.Step 1 For each k = 1, …, m compute
petk ¼Z A
0p�tþ1 a;Hk
� ��ρtþ1;νtþ1Þ=Ada; ð25Þ
Vetk ¼
Z A
0V�tþ1 a;Hk
� ��ρtþ1;νtþ1Þ=Ada ð26Þ
Step 2 Fit a cubic spline to H;pet� �
and H;Vet
� �to form ρt(H) and vt(H)
Step 3 While t N t0, decrement t by one and return to Step 1.
When computing the equilibrium for period t+1an equilibrium se-lection is required for the cases in which multiple equilibria occur. Weassume producers play C when both C and CS are possible and CSwhen both CS and S are possible.20
In order to analyze a representative period to check the robustnessof the analytic results for the period T-1 equilibrium, we choose avalue for t0 for which precedes T sufficiently for there to be little varia-tion in the expected price function from period to period. In particular,we iterate until there is nomore than a 0.1% variation inρt Hk
� �for each
k between periods t0 + 1 and t0. The resulting t0 depends on n, so wepresent results for the earliest t0 over n=1,…, 80, which for the param-eters used in our example gives t0 = T-10 when n = 3.
The equilibrium type that occurs in period T-10 with HT-10 = 0 is il-lustrated in Fig. 4. The qualitative nature of Fig. 4 is similar to that ofFig. 1, illustrating that the analytic results obtained for period T-1 are ro-bust to situations with a more distant time horizon. One notable differ-ence between the two figures is that, while the set of circumstances inwhich the CS equilibrium obtains is similar, there is an increase in the
18 Judd (1998), Ch. 11provides a description of themethod,whichhe applies to the com-petitive storage problem in Ch. 17.4.19 We look for symmetric equilibria only, so the expected value function is identical foreach firm.20 As with finitely repeated games, for which Benoit and Krishna (1985) show that mul-tiple stage game equilibria can result in players able to sustain a large range of outcomes,other subgame perfect equilibria may arise here due to the potential multiplicity of equi-libria in any period prior to T.
number of states inwhich CS is unique relative to themultiple equilibriawith C. Essentially, the set of states for which the stockout equilibrium,C, is possible has shrunk. The reason for this is that in T-1 sales to spec-ulators are guaranteed to be re-sold in the following period, while in T-10 there is a possibility that theywill not be re-sold, effectively loweringthe cost to producers of allowing speculators to carry inventories. Thiseffect reduces the incentives for firms to trigger a stockout. For similarreasons, the frequency of the consumer exclusion equilibrium, S, is larg-er thanwas the case in Fig. 1:firms aremorewilling to allow speculativestorage resulting in a higher price which is more likely to cause con-sumers to withdraw from purchasing output.
5. Conclusion
Our results indicate that speculation storage is more likely to occurunder an oligopoly market structure than under the extremes of monop-oly and perfect competition. Furthermore, speculation canhave a dramat-ic effect on the nature of the equilibrium in a non-cooperative setting. Ofparticular interest are the results that consumers may be excluded frompurchasing in a particular period and that the degree of speculative stor-age can vary non-monotonically with market structure. We demonstrat-ed with an example that this can cause non-monotonic effects on pricesand welfare. Although the net effect on total welfare is modest, thereare significant distributional effects of speculation between producersand consumers as well as over time and these effects vary with marketstructure. As many commodity markets feature oligopolistic production,our results suggest that the prevalence of concern about speculationmay be warranted.
Appendix A. Proofs
A.1. Proposition 1
Proof. (i) For 0 b qiT−1 b Q̂−Q−iT−1, pT-1 N aT-1 and consumer demand
is zero, while if qiT−1NQ̂−Q−iT−1, pT-1 b aT-1 and both con-
sumer and speculative demand is positive. Consequently,the individual firm's first period inverse demand flattensas the threshold Q̂−Q−i
T−1 is crossed, which results in thefirm's profit being less sensitive to increases in output forqT − 1i larger than Q̂−Q−i
T−1 than for smaller levels of out-put. Hence, marginal profit must take an upward jump atqiT−1 ¼ Q̂−Q−i
T−1.(ii) In this case the discontinuity in marginal profit occurs at
qT − 1i = QL − QT − 1
− i . A stockout occurs and only
52 S. Mitraille, H. Thille / International Journal of Industrial Organization 35 (2014) 44–59
consumers buy the product when the output of firm i islower than this value, and consequently the marginalprofit at the left of the discontinuity is
ΠCq QL−Q−i
T−1;Q−iT−1
� �¼ aT−1−HT−1− 2þ cð ÞQL þ 1þ cð ÞQ−i
T−1:
ð27Þ
SinceHT=0 is also equal to 0 to the right of qT − 1i =QL−QT − 1
−i , themarginal profit at the right of the discontinuity is
ΠCSq QL−Q−i
T−1;Q−iT−1
� �¼ aT−1−HT−1−QL
− 1þ c− 11þ δβ
QL−Q−i
T−1
� �−
2γ aT−1−HT−1−QL þ δw� �
β 1þ δβð Þ :
ð28ÞSimplifying the last term with aT-1 − HT-1 − QL + δw= δβA/2 from
the definition of QL, the difference across the discontinuity is equal to
ΠCSq QL−Q−i
T−1;Q−iT−1
� �−ΠC
q QL−Q−iT−1;Q
−iT−1
� �¼ QL−Q−i
T−1−δγA1þ δβ
:
ð29Þ
Clearly QL − QT − 1−i N 0 for the discontinuity to be relevant, but this
differencemay be larger or smaller than δγA and consequently themar-ginal profit may have upward or downward jumping at QL − QT − 1
i .For the second part of Proposition 1, differentiating Eq. (65) yields
ΠSqq qiT−1;Q
−iT−1
� �¼ δ 1−2
γβ
dpe HTð ÞdHT
−δβ 1−HT
A
þ δβ
qiT−1
A−cqiT−1;
ð30Þ
where we have used HT = HT-1 + QT-1 in S. From Eq. (15) we have
dpe HTð ÞdHT
¼ −β 1−HT
A
; ð31Þ
so we have
ΠSqq ¼ −2δ β−γð Þ 1−HT
A
þ δβ
qiT−1
A−c: ð32Þ
The first term in Eq. (32) is negative since β N γ. By Assumption 1, δβb c, so δβ qiT−1
A bc since qT − 1i b A must be true. Consequently, Πqq
S b 0.The same analysis can be performed forΠq
CS(qT − 1i , QT − 1
−i ). Differen-tiating Eq. (66) with respect to qT − 1
i yields
ΠCSqq ¼ − 1−2
γβ
deXdQT−1
!δdpe HTð ÞdHT
deXdQT−1
− 1þ deXdQT−1
!
þ 2γβδpe HT−1−eX QT−1ð Þ� � d2eX
dQ2T−1
−qT−1d2eX
dQ2T−1
−c: ð33Þ
Considering the first term in Eq. (33), dpe HTð ÞdHT
b0 as established above,deX
dQT−1b0 from Lemma 1, and γ b β, so the first term is negative. From
Lemma 1, deXdQT−1
is smaller than one in absolute value, so the second termin Eq. (33) is also negative. Also from Lemma 1, we have
d2eXdQ2
T−1
¼ − δβ=A
1þ δβ 1− HT−1−eX QT−1ð Þ� �
=A� �3≤0;
ð34Þ
so the third term in Eq. (33) is negative. Finally, under Assumption 1, wehave
δβqiT−1=A
1þ δβ1− HT−1−eX QT−1ð Þ� �
=A 3 bδβ≤c: ð35Þ
since qT − 1i b A. Consequently,−qiT−1
d2eXdQ2
T−1−c≤0 andΠqq
CS b 0. □
A.2. Proposition 2
Proof. To prove existence, we note that a symmetric S equilibrium ex-ists if the following conditions are satisfied:
I. Q̂ N0.II. A solution, qS, satisfying Πq
S(qS, (n − 1)qS) = 0 exists with qS∈0; Q̂=n� �
. By Proposition 1, a firm's marginal profit is decreasingin its own output, which results in unique and continuous bestresponses. Consequently, qS∈ 0; Q̂=n
� �if
(a) A producer's best response to its competitors' producingzero output is to produce positive output. This is ensured if
ΠSq 0;0ð ÞN0; ð36Þ
(b) A producer's best response to its competitors each produc-ing Q̂=n is to produce less than Q̂=n. This is ensured if
ΠSq Q̂=n; n−1ð ÞQ̂=n� �
b0: ð37Þ
III. Producers do not wish to deviate by increasing output to inducesales to consumers. Proposition 1 establishes thatmarginal profitjumps upward at q ¼ Q̂− n−1ð ÞQ̂=n, so there may a profitabledeviation into theCS region. A sufficient (but not necessary) con-dition to rule such a deviation out is
ΠCSq Q̂− n−1ð ÞqS; n−1ð ÞqS� �
b0: ð38Þ
We examine the conditions under which each of these conditionsholds in turn.
First, condition I is satisfied if
aT−1b 1−HT−1
A
2 δβ2
A−δw: ð39Þ
As the right hand side of this inequality is decreasing in both HT − 1
and n (since β is decreasing in n), Eq. (39) holds for small but positivevalues of aT-1, HT-1, and n.
Condition IIa requires
β−2γð ÞA2
1−HT−1
A
2−wN0 ð40Þ
or
HT−1bA 1−ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2wA β−2γð Þ
s !ð41Þ
which is independent of aT-1. As β-2γ is decreasing in n, the right handside of Eq. (41) is decreasing in n, and so is more likely to hold forsmall HT-1 and small n.
Condition IIb requires
δ β−2γð ÞA2
1−HT−1 þ Q̂A
!2
−δw− δβ 1−HT−1 þ Q̂A
!þ c
!Q̂nb0:
ð42Þ
Since the final term on the left hand side of Eq. (42) is negative, itsuffices to demonstrate that the first two terms sum to a negative quan-
tity. From the definition ofQ̂ in Lemma1,wehave1− HT−1þQ̂A ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaT−1þδwδβA=2
q,
so the first two terms of Eq. (42) will be negative if
β−2γð Þ aT−1 þ δwδβ
−wb0; ð43Þ
53S. Mitraille, H. Thille / International Journal of Industrial Organization 35 (2014) 44–59
or
aT−1b2δγ
β−2γw: ð44Þ
Since the right hand side of Eq. (44) is strictly positive, there arevalues of aT-1 N 0 that satisfy Eq. (44). Consequently there are positivevalues of aT-1 for which Eq. (42) and hence Eq. (37) are satisfied. Notethat since 2γ
β−2γ ¼ 2þc1þcð Þ nþ1þcð Þ− 2þcð Þ, the right hand side of Eq. (44) is de-
creasing in n, so the more concentrated the market is, the larger therange of aT-1 for which Eq. (44) holds.
Using Eqs. (66) and (15), condition III requires
aT−1b 1þ deXT−1
dQT−1þ c
!Q̂− n−1ð ÞqS� �
þ 2γ aT−1 þ δwð Þ
β 1þ δβ 1−HT−1 þ Q̂A
! !ð45Þ
where we have used eX Q̂� �
¼ −Q̂ . Since −1b deXT−1dQT−1
b 0 from Lemma 1and Q̂ N n−1ð ÞqS in a feasible S equilibrium, the right hand side ofEq. (45) is strictly positive. Consequently, Eq. (45) holds for a sufficient-ly small, but positive value for aT-1.
To demonstrate that all four conditions can be satisfied simulta-neously, note that inequalities (39), (44), and (45) define a subset ofaT-1 bounded between zero and a valuewhich is positive for sufficientlysmall HT-1 and n. In addition, Eq. (41) holds for sufficiently small valuesof HT-1 and n. Hence, the set of values of aT-1, HT-1 and n which satisfiesconditions I–III is non-empty, and an S equilibrium exists in this game.
To determine the equilibrium level of output, we find qS that solvesΠq
S(qS, (n − 1)qS) = 0. From Eqs. (65) and (15) this becomes
δ β−2γð ÞA2
1−HT−1 þ QT−1
A
2−δw−δβ 1−HT−1 þ QT−1
A
qS−cqS ¼ 0;
ð46Þ
which, using HTS = HT − 1 + QT − 1
S , becomes
δ β−2γð ÞA2
1−HST
A
!2
−δw−δβ 1−HST
A
!HS
T−HT−1
n−c
HST−HT−1
n¼ 0:
ð47Þ
Making the substitution HST ¼ A−A 1−HS
TA
� �for the last two occur-
rences ofHTS in Eq. (47) allows us to express the solution as a second de-
gree polynomial in ZT ≡ 1− HSTA , which is the probability that demand
exceeds period T inventories:
δ β−2γð ÞA2Z2T−δw−δβZT
A 1−ZTð Þ−HT−1
n−c
A 1−ZTð Þ−HT−1
n¼ 0: ð48Þ
Collecting terms, this can be written as K1ZT2 + K2ZT + K3 = 0 with
K1 ¼ δβ−2γ
2þ β
n
A
K2 ¼ − δβn
A−HT−1ð Þ þ cnA
K3 ¼ −δw− cn
A−HT−1ð Þ
ð49Þ
Note that K1 is positive and K2 is negative,21 so the discriminant, K22−4K1K3, is positive. There are two solutions to consider, and as ZT is aprobability, we must keep the one between 0 and 1. The solutions are
21 For K3 to be positive, it is necessary that A-HT-1 b 0, that is it is necessary that invento-riesHT-1 exceed themaximal demand A; but this would force the price to 0 in period T andmakes any increase in inventories in period T-1 not profitable anyway, a contradiction tothe case we are examining.
Z0T ¼ −K2−
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiK2ð Þ2−4K1K3
q =2K1 and Z}
T ¼ −K2 þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiK2ð Þ2−4K1K3
q =2K1, where ZT′ is negative.22 Therefore we have
Z}T ¼ 1−HS
T
A¼
δβn
A−HT−1ð Þ− cnAþ
ffiffiffiffiΔ
p
δ β−2γ þ 2β=nð ÞA ð50Þ
with
Δ ¼ − δβn
A−HT−1ð Þ þ cnA
2
þ 4δβ−2γ
2þ β
n
A δwþ c
nA−HT−1ð Þ
� �ð51Þ
The aggregate quantity produced is then equal to
QST−1 ¼ HS
T−HT−1 ¼ A−A 1−HST
A
!−HT−1
¼ A−HT−1−δβn
A−HT−1ð Þ− cnAþ
ffiffiffiffiΔ
p
δ β−2γ þ 2βn
¼ δ β−2γ þ β
n
� �A−HT−1ð Þ þ c
nA−
ffiffiffiffiΔ
p
δ β−2γ þ 2βnð Þ
ð52Þ
and qT − 1S =QT − 1
S /n completes the proof. □
Appendix B. T-1 equilibrium details
In this appendix we provide detailed derivations of equilibriumquantities for period T-1. We first examine speculators' behavior andthen present producers marginal profit and finally derive the equilibri-um output for each type of equilibria.
B.1. Speculator behavior
The quantities that characterize speculators' behavior is summarizedin the following lemma:
Lemma 1. There exist unique QL, Q̂ , and eX QT−1ð Þ given by
QL ¼ aT−1−HT−1−δβA2þ δw;
Q̂ ¼ A 1−ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaT−1 þ δwδβA=2
s !−HT−1; and
eXT−1 QT−1ð Þ ¼ HT−1 þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 2δβð ÞA2 þ 2 aT−1−HT−1 þ δw−QT−1ð ÞδβA
q− 1þ δβð ÞA
δβ
with the properties that
(i) deXdQT−1
¼ − 1
1þδ β 1− HT−1−eX QT−1ð Þ� �
=A� �b0
(ii) Q̂ andQL cannot be both positive: either one is positive and the othernegative, or both are negative.
Proof. The expression (and uniqueness) of QL is straightforward fromusing Eq. (15) in Eq. (5):
QL ¼ aT−1−HT−1−δ βA=2−wð Þ: ð53Þ
For Q̂ , using Eq. (15) in Eq. (6) we have
δβA2
1−HT−1 þ Q̂ T−1
A
!2
−δw−aT−1 ¼ 0 ð54Þ
22 Indeed eitherK2 N 0 and ZT′ is obviously negative, or K2≤ 0 but in that case asK3 b 0 thesquare root of the discriminant exceeds −K2, that is ZT′ b 0.
54 S. Mitraille, H. Thille / International Journal of Industrial Organization 35 (2014) 44–59
giving immediately
Q̂ T−1 ¼ A 1−ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaT−1 þ δwδβA=2
s !−HT−1: ð55Þ
To determine eXT−1 QT−1ð Þ, using Eq. (15) in Eq. (7) yields
aT−1−X−QT−1 þ δw−δβA2
1−HT−1−XA
2¼ 0 ð56Þ
Adding the difference HT-1 − HT-1 to the left-hand side and usingHT = HT-1 − X we can express this as
aT−1−HT−1−QT−1 þ δwþ A−A 1−HT
A
−δβ
A2
1−HT
A
2¼ 0 ð57Þ
which is equivalent, denoting ZT ¼ 1− HTA , to solve in ZT:
aT−1−HT−1−QT−1 þ δwþ A−AZT−δβA2Z2T ¼ 0: ð58Þ
The discriminant of Eq. (58) can be written as Δ = (1 + 2δβ)A2
+ 2(aT − 1 − HT − 1 + δw − QT − 1)δβA. If QT−1baT−1−HT−1 þ δwþ1þ2δβδβ
A2, then Δ N 0 and the only potentially positive root is
ZT ¼ffiffiffiffiΔ
p−A
δβA
¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 2δβð ÞA2 þ 2 aT−1−HT−1 þ δw−QT−1ð ÞδβA
q−A
δβA; ð59Þ
and speculators position X solves
1−HT−1−XA
¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 2δβð ÞA2 þ 2 aT−1−HT−1 þ δw−QT−1ð ÞδβA
q−A
δβA
or
eXT−1 QT−1ð Þ ¼ HT−1
þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 2δβð ÞA2 þ 2 aT−1−HT−1 þ δw−QT−1ð ÞδβA
q− 1þ δβð ÞA
δβ
ð60Þ
If QT−1≥aT−1−HT−1 þ δwþ 1þ2δβδβ
A2, then Δ b 0 and no real roots
exist for Eq. (58); moreover, the left-hand side of Eq. (58) is negativefor all ZT, meaning that the first period price is lower than the expectedsecond period price net of the cost of storage: speculators would preferto buy an infinite amount in that case.
Property (i) results application of the Implicit Function Theorem toEq. (7) or via direct differentiation of Eq. (60).
To prove property (ii), we need to study the sign of each threshold.From Eq. (53),
QL≥0 ⇔ aT−1≥HT−1 þ δβA=2−δw; ð61Þ
and from Eq. (55),
Q̂ T−1≥0 ⇔ aT−1≤ 1−HT−1
A
2 δβ2
A−δw: ð62Þ
For HT-1 = 0, the right-hand sides of Eqs. (61) and (62) are equal,so clearly only one of QL and Q̂ is be positive. For HT-1 N 0, note thatthe right hand side of Eq. (61) is increasing in HT-1 while that ofEq. (62) is decreasing in H T-1. Consequently, there are only threepossibilities: i) QL N 0 and Q̂b0, ii) QL b 0 and Q̂ N0, and iii) QL b 0and Qb0. □
B.2. Producer marginal profits
The effects of period T-1 output choice on expected period T profit issummarized in the following lemma:
Lemma 2. When speculators carry stocks into period T, HT N 0, aproducer's expected period T profit is decreasing in aggregate T-1 produc-tion, QT-1:
∂ET−1 π�T½ �
∂QT−1¼ 2
γβdX�
T−1
dQT−1pe HT−1−X�
T−1 QT−1ð Þ� �≤0
where dXT − 1∗ /dQT − 1 is equal to−1when speculators buy the entire pe-
riod T-1 production, or to deX=dQT−1∈ −1;0ð Þ when both consumers andspeculators buy in period T-1.
Proof. Using Eq. (14), expected second period profit is
ET−1 π�T HT ; aTð Þ� � ¼ γ
Z A
HT
a−HTð Þ2A
da ¼ γA2
31−HT
A
3ð63Þ
As HT = HT − 1 − XT − 1∗ (QT − 1), we have
∂ET−1 π�T½ �
∂QT−1¼ γA − ∂HT
∂QT−1
1−HT
A
2¼ 2γ
∂X�T−1
∂QT−1
pe HTð Þβ
ð64Þ
where A 1−HT
A
� �2has been replaced by 2pe(HT)/β using Eq. (15). □
We now present themarginal profit of a producer's choice of outputin T-1 depending on which regime of C, S, or CS obtains. When aT − 1 −HT − 1 ≤ δ(βA/2 − w) speculators always carry stocks and consumersare excluded from the market when total output is lower than Q̂ . Themarginal profit of firm i for an output level qiT−1≤Q̂−Q−i
T−1 is equal to
ΠSq qiT−1;Q
−iT−1
� �≡ δ 1−2
γβ
pe HT−1 þ QT−1ð Þ−δw
−δβ 1−HT−1 þ QT−1
A
qiT−1−cqiT−1;
ð65Þ
where we use an S superscript to denote a case in which speculatorspurchase the entire output. IfqiT−1≥Q̂−Q−i
T−1 marginal profit is equal to
ΠCSq qiT−1;Q
−iT−1
� �≡ δ 1þ 2
γβ
deXdQT−1
!pe HT−1−eX QT−1ð Þ� �
−δw
− 1þ deXdQT−1
!qiT−1−cqiT−1;
ð66Þ
where we use a CS superscript to denote the case in which both con-sumers purchase and speculators carry stocks into the final period.
When aT − 1 − HT − 1 ≥ δ(βA/2 − w) consumers always purchasepositive quantities and a stockout occurs when total output is lowerthan QL. The marginal profit of firm i for an output level qT − 1
i ≤ QL −QT − 1−i is equal to
ΠCq qiT−1;Q
−iT−1
� �≡ aT−1−HT−1−2qiT−1−Q−i
T−1−cqiT−1; ð67Þ
where a C superscript denotes the case in which consumers purchasethe entire T-1 output plus speculative stocks. Finally, if qT − 1
i ≥ QL −QT − 1−i then the marginal profit is again given by Eq. (66).
55S. Mitraille, H. Thille / International Journal of Industrial Organization 35 (2014) 44–59
B.3. Derivation of qCS
Output in a CS equilibrium obtains when ΠqCS(qCS, (n − 1)qCS) = 0
and is detailed in the following proposition:
Proposition 5. In a smoothing equilibrium, a firm individual output isequal to
qCS ¼1n
aT−1−HT−1 þ δwþ A−A − K2
3K1þ Z�
−δβ
A2
− K2
3K1þ Z�
2
where Z⁎ is the solution23 to Z3 + EZ+ F= 0 for which K23K1
≤Z�≤1þ K23K1
,and where
E ¼ 1K1
K3−K22
3K1
!; F ¼ 1
K1K4 þ 2
K32
27K21
−K3K2
3K1
!;
with K1 ¼ δ 1þ cð Þδβ A2;
K2 ¼ δ 1þ cð Þ þ δβ 1þ 2cð Þ−2nδγð ÞA2;
K3 ¼ −δ 1þ cð Þδwþ c− 1þ cð Þδβð ÞA− 1þ cð Þδβ aT−1−HT−1ð ÞK4 ¼ − nþ cð Þδw−cA−c aT−1−HT−1ð Þ:
Proof. In a symmetric CS equilibrium output is determined by the solu-tion to the first-order condition Πq
CS(qT − 1CS , (n − 1)qT − 1
CS ) = 0. SincepT−1 ¼ aT−1−eXT−1 QT−1ð Þ−QT−1 ¼ δ pe HTð Þ−wð Þ in CS with pe(HT)given by Eq. (15), this reduces to
δβA2
1−HT
A
2−δw− 1þ deXT−1
dQT−1
!qiT−1−cqiT−1−
2γδβA2
1−HT
A
� �2β 1þ δβ 1−HT
Að Þð Þ ¼ 0:
ð68Þ
Using ZT = 1− HT/A, we use Eq. (58) that expresses aggregate pro-duction as
QT−1 ¼ aT−1−HT−1 þ δwþ A−AZT−δβA2Z2T ; ð69Þ
which can be used with Lemma 1 to get
deXT−1
dQT−1¼ −1
1þ δβ 1−F HT−1−eXT−1
� �� � ¼ −11þ δβZT
: ð70Þ
We can now express the zero marginal payoff condition as
δβA2Z2T−δw− 1þ c− 1
1þ δβZT
qCST−1−
2γδβA2Z2T
β 1þ δβZTð Þ ¼ 0: ð71Þ
Since qT − 1CS =QT− 1
CS /n in a symmetric equilibrium, using Eq. (69)wehave
δβA2Z2T−δw− 1þ c− 1
1þ δβZT
aT−1−HT−1 þ δwþ A−AZT−δβA2Z2T
n
−2δγ
A2Z2T
1þ δβZT¼ 0;
ð72Þ
23 The specific closed form solution to this cubic equation is presented in the proof forbrevity.
which is a cubic equation in ZT that we write as K1ZT3 + K2ZT
2 + K3ZT+K4 = 0, with
K1 ¼ nþ 1þ cð Þ δβð Þ2 A2¼ δ 1þ cð Þδβ A
2K2 ¼ nδβ
A2−2nδγ
A2þ δβc
A2þ A 1þ cð Þδβ
ð73Þ
¼ δβ nþ cþ 2 1þ cð Þð Þ−2nδγð ÞA2
¼ δ 1þ cð Þ þ δβ 1þ 2cð Þ−2nδγð ÞA2
ð74Þ
K3 ¼ −nδwδβ þ cA− 1þ cð Þδβ aT−1−HT−1 þ δwþ Að Þ¼ −δ 1þ cð Þδwþ c− 1þ cð Þδβð ÞA− 1þ cð Þδβ aT−1−HT−1ð Þ ð75Þ
K4 ¼ −nδw−c aT−1−HT−1 þ δwþ Að Þ¼ − nþ cð Þδw−cA−c aT−1−HT−1ð Þ ð76Þ
To determine the solution, we can first change the variable ZT ¼ Y−K23K1
. This first change leads to the depressed cubic equation
Y3 þ 1K1
K3−K22
3K1
!Y þ 1
K1K4 þ 2
K32
27K21
−K3K2
3K1
!¼ 0 ð77Þ
Then, defining
E ¼ 1K1
K3−K2ð Þ23K1
!ð78Þ
F ¼ 1K1
K4 þ 2K2ð Þ3
27 K1ð Þ2 −K3K2
3K1
!; ð79Þ
the cubic equation we need to solve is:
Y3 þ EY þ F ¼ 0: ð80Þ
The following definition states the solution to a depressed cubicequation
Definition 1. [CRC (1996)] Let P Zð Þ ¼ Z3 þ E � Z þ F be a third degreepolynomial. The three solutions {Zk}k = 0,1,2 to P Zð Þ ¼ 0 are:
1. if 27F2 + 4E3 ≥ 0, given by the real quantity:
Z0 ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi− F
2þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiF2
4þ E3
27
s3
vuut þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi− F
2−
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiF2
4þ E3
27
s3
vuut;
and the two other roots are complex,2. if 27 F2 + 4E3 b 0, given by the quantities:
Zk ¼ 2
ffiffiffiffiffiffiffiffiffi− E
3
rcos
13arccos
3F2E
ffiffiffiffiffiffiffiffiffi−3
E
r−k
2π3
! !for k ¼ 0;1;2;
which are all real.
FromDefinition 1, depending on the sign of 27E3+4 F2, there can beup to three real solutions to this equation. As we are searching for aprobability, we select the solution Y⁎which verifies the fact thatZT ¼ Y�
− K23K1
belongs to [0,1], that is K23K1
≤Y�≤1þ K23K1
. □
56 S. Mitraille, H. Thille / International Journal of Industrial Organization 35 (2014) 44–59
B.4. Characterization of T-1 equilibrium
In this appendix we determine the conditions governing the exis-tence of the alternative types of equilibria in terms of values of (HT − 1,aT − 1). As a complete characterization is not possible analytically due tothe difficulty in comparing deviation profits when marginal profit isupward-jumping at QL or Q̂ , in this situation we demonstrate how theoptimal deviation output is calculated when computing the results forthe example in the text. There are three broad situations to examine de-pending on the signs of QL and Q̂ which we consider in turn.
B.4.1. QL, Q̂b0The case in which both QL and Q̂ are negative corresponds to a situ-
ation in which qCS is the unique symmetric Nash equilibrium to thegame. This happens if:
1−HT−1
A
2 δβ2
A−δw≤aT−1≤HT−1 þ δβA=2−δw ð81Þ
As themarginal profit is continuous and decreasing in the individualquantity, there are no global deviations to consider in this case.
B.4.2. QL N 0Consider now the case inwhichQL is positive and Q̂ is negative. From
previous analysis, this occurs if aT − 1 ≥ HT − 1 + δβA/2 − δw. In thiscase, three different equilibrium candidates must be considered. More-over let us distinguish two sub-cases, δβ A
2−δw≥0 and δβ A2−δwb0.
In the case where δβ A2−δwb0, storing is never profitable for specu-
lators: themaximal return net of storage cost they can obtain in the lastperiod is negative. Consequently a stockout in period T-1 always occurswhen δβ A
2−δwb0, and depending on the sign of aT-1 − HT-1 the priceand the quantity produced in period T-1 are positive or nil, and the equi-librium output is equal to
qC ¼ max1n
1−βð Þ aT−1−HT−1ð Þ;0�
ð82Þ
Note that theNP equilibrium inwhich no production takes place canoccur if simultaneously the return of speculation is negative, δβ A
2−δwb
0, and the initial inventories are too large compared to potential de-mand, aT-1 − HT-1 b 0.
In the case where δβ A2−δw≥0, the marginal profit may jump up or
down at an aggregate output level equal toQL. However aswe arework-ing under the assumption that aT−1−HT−1−δβ A
2 þ δw≥0, the NP equi-librium cannot be an outcome. There are three possible equilibria wemust examine: C, CS, and L. Let us start with the L equilibrium, forwhich each firm' marginal profit is downward jumping.
B.4.2.1. L equilibrium. A limit equilibrium L exists in period T-1when themarginal profit jumps down at the individual output
qL ¼QL
T−1
n¼ aT−1−HT−1−δβA=2þ δw
n; ð83Þ
that is when
ΠCq qLT−1; n−1ð ÞqLT−1
� �≥0
⇔ aT−1−HT−1≤δA2−δ
wβ
ð84Þ
and
ΠCSq qLT−1; n−1ð ÞqLT−1
� �≤0
⇔ aT−1−HT−1≥δ 1þ cð Þ þ nþ cδβ þ c 1þ δβð Þ δβ
A2−δw
− δγnA
δβ þ c 1þ δβð Þð85Þ
These two conditions are the exact conditions underwhich an L equi-librium exists, and as themarginal profit is piecewise decreasing no indi-vidual deviation from L exists when these conditions are satisfied.
B.4.2.2. C equilibrium. A C equilibrium will exist if
max ΠCq QL
T−1− n−1ð ÞqCT−1; n−1ð ÞqCT−1
� �;ΠCS
q QLT−1− n−1ð ÞqCT−1; n−1ð ÞqCT−1
� �n o≤0
ð86Þ
These conditions can be expressed as a function of themodel param-eters only, using the expressions of the marginal profits and of thequantities:
ΠCq QL− n−1ð ÞqC ; n−1ð ÞqC� �
≤0
⇔ aT−1−HT−1≥2þ cð Þn δβA=2−δwð Þ1þ cð Þ 1þ n−1ð Þβð Þ
ð87Þ
and
ΠCSq QL− n−1ð ÞqC ; n−1ð ÞqC� �
≤0
⇔ aT−1−HT−1≥δβ þ 1þ δβð Þ 1þ cð Þð Þn δβ
A2−δw
−δnγA
1þ δβð Þ 1þ cð Þ−1ð Þ 1þ n−1ð Þβð Þ
ð88Þ
If marginal profit is upward jumping at QL − (n− 1)qT − 1C we need
to check that there is no profitable deviation in which a producer trig-gers speculative storage.We showhere how this deviation is computed.Denote qdCS(qC) the solution in q toΠq
CS(q, (n− 1)qT − 1C ) = 0. Following
the proof of Proposition 5,marginal profit can be expressed as a functionof the level of inventories Hd
CS(qC), and the individual quantity qdCS(qC),
as:
δβA2
1−HCSd qCð ÞA
!2
−δw− 1þ c− 1
1þ δβ 1−HCSd qCð ÞA
� �0@ 1AqCSd qCð Þ
−2γδβ
A2
1−HCSd qCð ÞA
2
β 1þ δβ 1−HCSd qCð ÞA
� �� � ¼ 0:
ð89Þ
SinceQdCS(qC)= qd
CS(qC)+ (n− 1)qCmust satisfy the arbitrage equa-tion of speculatorswhich links the aggregate quantity toHd
CS(qC), the de-viation quantity must be such that
qCSd qCð Þ ¼ aT−1−HT−1 þ δwþ A−A 1−HCSd qCð ÞA
!
−δβA2
1−HCSd qCð ÞA
!2
− n−1ð ÞqC
¼ n− n−1ð Þ 1−βð Þn
aT−1−HT−1ð Þ þ δwþ A−A 1−HCSd qCð ÞA
!
−δβA2
1−HCSd qCð ÞA
!2
ð90Þ
This expression can be plugged into marginal profit to give a third
degree polynomial into 1− HCSd qCð ÞA ¼ ZCS
d qCð Þ, with coefficients K1d(qC),
K1d(qC), K3
d(qC) and K4d(qC) to replace the coefficients K1, K2, K3 and K4 de-
fined in Proposition 5. This polynomial can be solved using Definition 1and expressions E and F from Proposition 5. Existence of a C equilibriumrequires that producing qdCS(qC) results in lower profit that is obtained inthe C equilibrium.
57S. Mitraille, H. Thille / International Journal of Industrial Organization 35 (2014) 44–59
B.4.2.3. CS equilibrium. A speculation equilibrium CS exists in period T-1whenΠq
CS(qCS, (n− 1)qCS) = 0, and when no individual deviation to anaggregate output level implying no inventories to be carried over exists.A sufficient condition for this is
min ΠCq QL− n−1ð ÞqCS; n−1ð ÞqCS� �
;ΠCSq QL− n−1ð ÞqCS; n−1ð ÞqCS� �n o
≥ 0
ð91Þ
that is:
ΠCq QL− n−1ð ÞqCS; n−1ð ÞqCS� �
≤0⇔ aT−1−HT−1≤n 2þ cð Þ1þ c
δβA=2−δwð Þ
þ n−1ð Þ δwþ A−A − K2
3K1þ Z�
−δβ
A2
− K2
3K1þ Z�2
ð92Þ
and
ΠCSq QL− n−1ð ÞqCS; n−1ð ÞqCS� �
≥0
⇔ aT−1−HT−1≤nþ cþ δ 1þ cð Þð Þ δβ
A2−δw
−δγnA
1þ cð Þ 1þ δβð Þ−1
þ n−1n
δwþ A−A − K2
3K1þ Z�
−δβ
A2
− K2
3K1þ Z�
2 ð93Þ
where Z⁎ is the solution given in Definition 1.If marginal profit is upward jumping at (n− 1)qT − 1
CS − QL we needto check that there is no profitable deviation in which a producer trig-gers a stockout. We show here how this deviation is computed. The de-viation qd
C(qCS) is the solution in q to ΠqC(q, (n − 1)qCS) = 0, which is
equal to
qCd qCSð Þ ¼ aT−1−HT−1
2þ c− n−1
2þ cqCS; ð94Þ
where qCS has been obtained in Proposition 5. Existence of a CS equilib-rium requires profit under this deviation be lower than that in the CSequilibrium.
B.4.3. Q̂ N0The case in which QL is negative and Q̂ is positive occurs if aT − 1 ≤
(1 − HT − 1/A)2δβA/2 − δw. In this case, three different equilibriumcandidate must be considered: S, CS, and NP equilibrium. Let usstart with the latter.
B.4.3.1. NP equilibrium.An equilibrium involving no productionNP existsin period T-1 when the marginal profit is negative Πq
S(0, 0) ≤ 0:
ΠSq 0;0ð Þ ¼ δ β−2γð ÞA
21−HT−1
A
2−δw≤0 ð95Þ
Note that when this condition holds, we can also check that ΠCSq
0; Q̂T−1
� �≤0, which guarantees that there is not a CS equilibrium:
ΠCSq 0; Q̂� �
¼ δβA2
1−HT−1 þ Q̂A
!2
−δw−2δγ
A2
1−HT−1 þ Q̂A
!
1þ δβ 1−HT−1 þ Q̂A
! ≤0
ð96Þ
B.4.3.2. S equilibrium. An S equilibrium exists in period T-1 whenΠqS(qS,
(n − 1)qS) = 0. As the marginal profit is upward jumping at Q̂ T−1, asufficient condition for this equilibrium to exist is
ΠCSq Q̂− n−1ð ÞqS; n−1ð ÞqS� �
≤0
⇔ δβA2−δw− 1þ c− 1
1þ δβ
A 1− ffip aT−1 þ δw
δβA=2
−HT−1
−n−1n
δ β−2γ þ βn
� �A−HT−1ð Þ þ c
nA−
ffiffiffiffiΔ
p
δ β−2γ þ 2βnð Þ
!− δγA
1þ δβ≤ 0
ð97Þ
where Δ is determined in Proposition 2.If (97) fails,wemust check that noprofitable deviation exists that in-
volves a firm inducing sales to consumers. The deviation qdCS(qS) can be
determined in the same way as was qdCS(qC). It is the solution in q toΠq
CS(q, (n− 1)qS)= 0, and is obtained by linking the aggregate quantityqdCS(qS) + (n− 1)qS to Hd
CS(qS) through the arbitrage equation of specu-lators (98) which defines the quantity as a function of speculative in-ventories:
qCSd qSð Þ þ n−1ð ÞqS ¼ aT−1−HT−1 þ δw
þA−A 1−HCSd qSð ÞA
!−δβ
A2
1−HCSd qSð ÞA
!2
:
ð98Þ
Using the same method as in the proof of Proposition 5, marginalprofit evaluated at this quantity results in a third degree polynomial inZ≡ 1− HCS
d qSð ÞA , the solution of which is used to determine the optimal devi-
ation quantity. The S equilibrium exists if such a deviation yields lowerprofit than is obtained in the S equilibrium.
B.4.3.3. CS equilibrium.A CS equilibriumexists in period T-1whenΠqCS(q-
CS, (n− 1)qCS)=0 andwhen no profitable individual deviation exists toan aggregate output level which leads to an exclusion of consumers.Marginal profit is upward jumping at Q̂ , so a sufficient condition forthis equilibrium to exist is
ΠSq Q̂− n−1ð ÞqCS; n−1ð ÞqCS� �
≥ 0⇔δ β−2γð ÞA2aT−1 þ δwδβA=2
−δw
− δβ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaT−1 þ δwδβA=2
sþ c
! A 1−
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaT−1 þ δwδβA=2
s !−HT−1
−n−1n
aT−1−HT−1 þ δwþ A−A Z�− K2
3K1
−δβA2
Z�− K2
3K1
2!!
≥0
ð99Þ
To check that no profitable deviation exists in which a producercauses consumers to cease buying the product, we compute qd
S(qCS),the solution to Πq
S(qdS, (n − 1)qCS) = 0. This deviation quantity solves
HSd qCSð Þ ¼ HT−1 þ n−1ð ÞqCS þ qSd qCSð Þ; ð100Þ
or
qSd qCSð Þ ¼ A−A 1−HSd qCSð ÞA
!−HT−1− n−1ð ÞqCS: ð101Þ
Using the same method as in the proof of Proposition 2, marginalprofit evaluated at this quantity results in a second degree polynomialin Z ≡ 1− HCS
d qSð ÞA , the solution of which determines the optimal deviation
quantity. The CS equilibrium exists if such a deviation does not yieldhigher profits than obtains in the CS equilibrium.
Fig. 5. Expected price function, ρt(H), t = T − 10,…, T.
58 S. Mitraille, H. Thille / International Journal of Industrial Organization 35 (2014) 44–59
Appendix C. Numerical solution method
In this appendix we provide some addition details about the numer-ical method used to compute the equilibrium for periods prior to T-1.
C.1. Algorithm
We describe here the algorithm for the solution of the T-periodmodel, which is computed with Python code that utilizes numericalroutines from NumPy and SciPy24 (available from the authors on re-quest). The solution is computed recursively beginning from the analyt-ic period T solution described in the paper:
1 Starting value: ρT-1(H) is given by Eq. (15) and vT-1(H) by integrating(14).
2 For periods T-1 to T-k, ρT-1(H) and vT-1(H) are determined by fittingcubic splines to pt
e(Ht + 1) and eVt Htþ1ð Þ as follows:- Generate expected price and value for each element of a pre-determined vector of next-period stocks at which to perform the
approximation,25 Hk ¼ 0;H1k ;H
2k ;…;H
mk
� �.
- The expected price and value are computed by numerical inte-gration of the Nash equilibrium price and value function in pe-
riod T-k + 1 for each Hjk . This integration is performed by
approximating the uniform distribution for at on a uniformgrid over [0,A] with 300 values, each with probability 1/300.
The expected values for eachHjk are collected in the vectors ep ¼
p0;…; pm� �
and eV ¼ V0;…;Vm� �
.
- The SciPy routine interpolate. UnivariateSpline is used to fit
cubic splines to the H; ep� �and H; eV� �
data, resulting in the ap-
proximations ρT − k() and νT − k().3 Nash equilibrium calculation: The period t Nash equilibrium is com-
putedusing the approximations pt+ 1e and eVtþ1whichwere computed
in the previous step. In commuting this equilibrium numerical rou-tines are required to find eXk , Q̂k , and qk
CS. We use the SciPy routineoptimize.brentq for these calculations. An equilibrium selection ruleis required for those situations in which multiple equilibria occur.We impose that the C equilibrium is playedwhen both C and CS equi-libria exist, and the CS equilibrium is played when both the S and CSequilibria exist. This rule minimizes the amount of storage that oc-curs. For any (a,Hk) the equilibrium is computed as follows- Compute Qk
L and Q̂ k.- If Qk
L b 0 and Q̂kb0 the equilibrium is either CS or NP.- If Qk
L the equilibrium is C, L, or CS.- If Q̂ kN0, the equilibrium is S, CS, or NP.- In each case, when marginal profit is upward-jumping check thatno profitable deviation exists: e.g. for an equilibrium candidate C,if marginal profit is positive at the level of output that induces anaggregate quantity of Qk
L (ΠqCS(Qk
L − (n − 1)qkC, (n − 1)qkC) N 0),compute the optimal deviation for a firm qd/CS N Qk
L − (n − 1)qkC.If profit from the deviation exceeds that in the C equilibrium can-didate, the C equilibrium is ruled out.
To illustrate that the expected price function has “stabilized” by pe-riod T-10,we plot ρt(H), t= T− 10,…, T in Fig. 5 for the case n=3. Boththe level and slope of ρt(H) change very little from T-10 to T-9, particu-larly for Ht =0, the case of interest in Fig. 4. We plot the expected valuefunction for each period in Fig. 6 which shows that by T-10, it is increas-ing in a parallel fashion. The slope of the expected value function is es-sentially stabilized at Ht = 0 by T-10.
24 Jones et al. (2001).25 The vector of interpolation nodes,Hk is allowed to vary with k in order to ensure thatthe maximal level of stocks at which we perform the interpolation,H
mk , is large enough to
cause s. By doing this, we ensure that equilibrium inventories remain within the approx-imation interval.
C.2. Accuracy
Wenowprovide some evidence of the accuracy of our numerical so-lution. We do this by first examining how accurate the method is atcomputing the equilibrium in T-1 which we can compare with the ana-lytic solution. Second, we examine the approximation residuals for theT-10 solution.
In order to examine the accuracy of ourmethod at replicating thepe-riod T-1 solution, we can compute the interpolated expected price andvalue functions, ρT-1(HT) and vT-1(HT) and compare them with the ana-lytic solutions give by Eq. (15) and the integration of Eq. (14). For theparameters given in the paper andHm=A=20, the difference betweenthe interpolated expected price and value functions an the actual is ofthe order 10−15 for any order of approximation,m, which is not surpris-ing since the expected period T price and value are quadratic functionsofHT. Consequently, the solution using the interpolated approach yieldsthe same results as reported in Fig. 1.
To judge the accuracy of the approximation for period prior to T-1, weexamine the approximation residuals which are computed by comparingthe interpolated expected price function, ρt(H) to the “actual” expectedprice on a finer grid than was used to generate the approximation. The“actual” expected price is computed by integrating pt + 1(at + 1, H) overat + 1 given ρt + 1(H). We use a grid five times finer that the one used
Fig. 6. Expected value function, νt(H), t = T − 10, …, T.
Fig. 7. Period T-10 approximation residuals for expected price function (left) and expected value function (right).
59S. Mitraille, H. Thille / International Journal of Industrial Organization 35 (2014) 44–59
to compute Fig. 4 and plot the residuals in Fig. 7. We were unable to gethigher accuracywith this algorithmwhich seems to be related to difficul-tywith numerical integration of the price and value functions. Since equi-librium output is discontinuous at points where the type of equilibriumchanges when marginal profit is upward jumping, equilibrium price andprofit are also discontinuous in at and Ht.
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