Forward trading in exhaustible-resource oligopoly

Transcription

1 Forward trading in exhaustible-resource oligopoly Matti Liski and Juan-Pablo Montero PRELIMINARY AND INCOMPLETE AUGUST 2007 Abstract We analyze oligopolistic exhaustible-resource depletion when firms can trade forward contracts on deliveries, a market structure prevalent in most resource commodity markets. When market interactions become arbitrarily frequent, all stocks are contracted and subgame-perfect equilibrium allocation becomes perfectly competitive. The result is in contrast with the idea that forward markets can help firms to commit to a pre-determined production plan in a dynamic game. As the resource stock is an intertemporal capacity constraint, the result sheds light on the role of capacity constraints in forward markets. (JEL classification: G13, L12, L13, L50). 1 Introduction Hotelling s theory of exhaustible-resource consumption is a building block for studies of situations where intertemporal supply is finite. Theory is used in myriad of applications which, without exceptions known to us, assume implicitly or explicitly that the commodity stock is sold in the spot market only, thereby ruling out forward trading that feature most commodity markets and markets for exhaustible-stocks in particular. Risk sharing is a reason for forward trading but, as first shown by Allaz and Vila (1993), Liski <liski@hse.fi> is at the Economics Department of the Helsinki School of Economics. Montero <jmontero@faceapuc.cl> is at the Economics Department of the Catholic University of Chile. Both authors are also Research Associates at the MIT Center for Energy and Environmental Policy Research. This research was initiated at the Center for Advanced Studies (CAS) in Oslo We thank CAS for generous support. 1

2 forward contracting has also an explanation that is unrelated to firms desire to hedge. The explanation can be described as a prisoners dilemma situation where firms must take pro-competitive forward positions to prevent competitors from taking a larger share of the final market (Allaz and Vila, 1993). In this paper we are interested in what is the strategic role of forward trading in an oligopolistic exhaustible-resource market. Resource stock held by a firm is an intertemporal capacity constraint for production. Hotelling (1931) described a simple principle for monopolistic allocation of the capacity over time: marginal value of using the capacity in different periods should be equalized in present value. Under standard assumptions, resource depletion becomes more conservative; that is, compared to the efficient path, sales are shifted to the future to increase the value of early sales. In its simplest form, the oligopolistic market follows the same principles as the monopoly, with differences in outcome analogous to those that arise in between static monopoly and oligopoly (see, for example, Lewis and Schmalensee [1980]). In a standard oligopolistic description of the exhaustible-resource market, firms are thus capacity constrained in equilibrium. In contrast with reproducible goods, forward contracting of production cannot expand what is available for consumers and it is therefore clear that the pro-competitive mechanism from the reproducible case cannot be directly copied for the nonreproducible case. Does contracting then leave oligopoly rents intact? Conclusions supporting this view can be found from the literature. 1 Our main result is in contrast with the conjecture. We characterize a subgameperfect equilibrium delivery path that coincides with the socially efficient path when the frequency with which sales can be made increases without a limit. In other words, continuous-time description of the oligopolistic resource extraction collapses to a competitive equilibrium when the resource can be sold not only in the spot but also in forward market. For intuition, consider a stock so small, or period length so large, that oneperiod demand absorbs the stock without any storage in equilibrium. Contracting plays no strategic role in such an equilibrium because the overall supply in one period is fixed. Sufficient decrease in period length, or increase in stock, leads to consumption over two periods. Now, contracting preceding spot sales plays a role, because it leads firms to race for a higher capacity share in the first period which is the more profitable market. The prisoners dilemma introduced by contracting leads to a more competitive allocation of 1 For example, Lewis and Schmalensee [1980], without explicitly studying contract markets, indicate on page 477 that the existence of perfect futures markets could facilitate the use "path strategies" (openloop strategies). Thus, alternatively put, the conjecture states that future markets could help firms to commit to production plans. 2

3 the capacity; in the limit where a given overall stock is sold arbitrarily frequently, the allocation is competitive. Our strategy of exposition is to start with the above two-period illustration. While helpful in explaining the basic mechanism, the extensive form of the two-period model is misleading, because firmsshouldbeabletochoosehowlongthemarketinteraction lasts in equilibrium. For example, firm i may respond to firm j s heavy contracting in period t by avoiding own contracting at t and allocating more capacity to less contracted period t +1instead. This difference in extensive form is an important difference to the basic Allaz and Vila model where firms are trapped to face the prisoners dilemma in a particular spot market. In section 3, we set up the general version of the model where deliveries and future contract positions are chosen on a period-by-period basis depending on current physical stocks and positions inherited from the past. We first characterize a discrete-time version that we solve by backward induction. We describe the dynamic contract coverage, which shows that contract positions are altered for all future dates in each forward market interaction. Then, we solve the continuous-time limit of the discrete model to obtain the main result. It is important to emphasize that we have not ruled out asymmetric equilibria and equilibria that can be sustained by non-stationary strategies in the continuous-time limit. These possibilities and discussed in concluding remarks. This research is related to three strands of literature. First, this work is closely related to the basic exhaustible-resource theory under oligopolistic market structure. This literature has focused on developing less restrictive production strategies for firms (from path to decision rule strategies) 2 and also on including more realistic extraction cost structure (towards stock-dependent costs). 3 None of the papers in this literature explicitly consider the effect of forward trading on equilibrium. Second, there is a recent literature on organization of trade in dynamic oligopolistic competition under capacity constraints (Dudey [1992], Biglaiser-Vettas [2005], Bhaskar [2006]). These papers focus on dynamic price competition and also on the efficiency losses and changes in division of surplus 2 Loury (1986), Polansky (1992), and Lewis-Schmalensee (1980) use path strategies; Salo-Tahvonen (2001), for example, use decision-rule strategies. 3 Salo and Tahvonen (2001) solve their model with stock-dependent costs, so that the overall amount of the resource used is endogenously determined in equilibrium. In this sense, the resource is only economically exhausted. In our model, the resource is physically exhausted as the cost of using it is independent of the stock level. Econonomic exhaustion is arguably a more reasonable concept; we leave it open for future research how replacing physical capacity with economic capacity would alter the contracting incentives. 3

4 caused by strategic buyers. We depart from this literature by assuming non-strategic but forward looking buyers, and we consider quantity competition in two dimensions (spot and forward markets). Third, there is a literature on forward trading starting with Allaz and Vila (1993) who analyze a static Cournot market.mahencandsalanie(2005)show that price competition can reverse the implication; forward trading leads to less competition under price competition. Liski and Montero (2006) show that forward contracting by a firm can be seen as strategic investment in firm s own production, which explains the dependence of implications on the form of competition; 4 it is clear that our current model would produce different results under price competition. Liski and Montero also develop a repeated interaction model of forward contracting, and this modeling approach is also used in the current paper. 2 Two-period illustration The implications of forward contracting for the equilibrium path of a depletable-stock market can be best explained by firstconsideringasimpleexampleoftwoperiodsand then extending the analysis to the general case in which the number of periods is endogenously determined. This section will also serve to introduce the notation and key assumptions that we will use throughout the paper. 2.1 Notation and assumptions Consider two symmetric firms (i =1, 2), each holding a stock of a perfectly storable homogenous good, denoted by s i = s j = s, tobesoldintwoperiods(t =1, 2). To avoid uninteresting cases, we assume throughout this section that the "capacity constraint" represented by s i is binding for both firms, i.e., no stock is left unused at the end of the second period. 5 There are no production (or extraction) costs other than the shadow cost of not being able to sell tomorrow what is sold today. Firms attend the spot market in t =1, 2 by simultaneously choosing quantities qt i and q j t. For technical simplicity, we assume that the spot price at t, which is denoted by p s t, 4 Selling forward contracts is a tough investment in the sense that it lowers the rival s profit allelse equal; thus, the strategic-investment models of Bulow et al. (1985) and Fudenberg and Tirole (1984) predict that firms over-invest in forwards (i.e., go short) when they compete in quantities and underinvest in forwards (i.e., sell fewer forwards or go long) when they compete in prices. 5 In the general case, where the number of periods is endogenously determined, the stock left for the last two periods is always consistent with exhaustion in the last period. 4

5 is given by the linear inverse demand function p s t = p s (qt i + q j t )=a (qt i + q j t ). Firms are also free to simultaneously buy or sell forward contracts that call for delivery of the good at any of the spot markets that follow; therefore, quantities qt i and q j t must cover both spot-market sales and forward obligations, if any. For each of the two periods we let the forward market open before the spot market, so in t =1firms can take forward positions for both the spot market at t =1and the one at t =2,whereasint =2firms can only take forward positions for the remaining spot market. 6 Forward transactions by firm i at period t =1for the first and second spot markets are denoted by f1,1 i and f1,2, i respectively. Similarly, forward contracting at t =2is denoted by f2,2. i We adopt the convention that f i > 0 when firm i is selling forward contracts (i.e., taking a short position) and f i < 0 when is buying forwards (i.e., taking a long position). 7 We further assume that forward positions are observable and the delivery of contracts is enforceable. 8 For clarity, it may be useful to think of forward contracts as physical delivery commitments, although the results do not depend on this, i.e., contracts can be purely financial (see Liski and Montero [2006]). Note that while position f1,1 i calls for delivery of the good at t =1, position f1,2 i does not need be equal to the actual delivery at t =2since the forward market at t =2allows the firm to change its overall position for the spot market at t =2. For example, firm i can nullify its overall forward position at t =2(i.e., f1,2 i + f2,2 i =0) by buying/selling f2,2 i = f1,2. i Theforwardpriceatt for a delivery at τ t is denoted by p f t,τ. 2.2 Equilibrium To facilitate the exposition, suppose for a moment that f1,2 i = f2,2 i =0,sothatfirms sell forwards only for the first spot market. Our idea is to show that the equilibrium outcome derived under this assumption is the equilibrium outcome also when f1,2 i and f2,2 i are unconstrained. Since the deliveries in the first period determine what is left to be 6 Again, this restriction is relevant only in this two-period example; in the general case firms face no final spot market for which positions can be taken. 7 In equilibrium, the possibility of taking a long position is not used since forward positions can be interpreted as strategic investments in firm s own production, and these investments will be positive (i.e., positions will be short) as long as firms compete in quantities. However, it is important to allow for this possibility, because otherwise firms might be able to commit to aggressive behavior in some future spot market by the fact the positions cannot be adjusted downwards. 8 The contract market is modelled as in Allaz and Vila (1993). Demand for contracts comes from speculators who share the same information as the producers and therefore there will be no arbitrage profits. 5

6 sold in the second period, i.e., q2 i = s i q1 i (recall that the size of the stocks constraints firms actions, so there are no strategic decisions at t =2), we just need to concentrate on theforwardandspotsubgamesatperiodt =1. Following backward induction principles, we first consider the spot subgame at t =1. Given the forward contract commitments f1,1 i and f1,1, j firm i s present-value payoff from sales in the two spot markets is given by π s,i 1 = p s (q i 1 + q j 1)(q i 1 f i 1,1)+δp s (q i 2 + q j 2)q i 2 where δ<1is the discount factor. Since the firm has already pocketed the revenue from forward contracts, it is selling only q1 i f1,1 i to the spot market at t =1. Because of the capacity constraint s i = q1 i + q2, i the subgame that starts at the spot market in t =1reduces to a static (Nash-Cournot) game of simultaneous choice of q1 i and q1. j Firm s i best response to q j 1 (and q2)satisfies j the intertemporal optimization principle that discounted marginal revenues should be equalized across periods, that is a 2q i 1 q j 1 + f i 1,1 = δ(a 2q i 2 q j 2) (1) Solving, we obtain the (subgame perfect) equilibrium allocation q i 1(f i 1,1,f j 1,1) = a(1 δ)+3δsi +2f i 1,1 f j 1,1 3(1 + δ) (2) q2(f i 1,1,f i 1,1) j = a(1 δ)+3si 2f1,1 i + f j 1,1. (3) 3(1 + δ) Before moving to the forward subgame, it is useful to see how the contract coverage affects the intensity of the spot competition. If firms sign no contracts, i.e., f1,1 i = f j 1,1 =0, we obtain the pure-spot oligopoly equilibrium. Unlike the perfectly competitive equilibrium where spot prices rise at the rate of interest (i.e., p 1 = δp 2), 9 in a pure-spot oligopoly spot prices rise at rate strictly lower than the interest rate: p 1 > δp 2, p 1 > p 1 and p 2 <p 2. As can be seen from (1), this derives directly from the equilibrium condition that marginal revenues go up at the rate of interest. In other words, oligopolists move away from competitive pricing by shifting production from the present to the future. When firms go short in the forward market, f1,1 i > 0 and f j 1,1 > 0, spotmarkets become more competitive in that firms are credibly committing more production to the 9 The perfectly competitive allocations are q 1 =[a(1 δ)+δ(si + s j )]/(1 + δ) and q 2 = s q 1. 6

7 present. Another way to explain this is by looking at (1): contracts increase firms marginal revenues making them to behave more aggressively in the spot markets. In fact, if f1,1 i = f j 1,1 = a(1 δ)/2, the competitive solution is implemented. Conversely, as firms go long in the forward market, i.e., f1,1,f i j 1,1 < 0, spot markets become less competitive. And if f1,1 i = f j 1,1 = a(1 δ)/4, the monopoly solution is implemented. 10 Obviously, in equilibrium firms do not trade any arbitrary amount of forwards. Firms, speculators and consumers are assumed to have rational expectations in that they correctly anticipate the effect of forward contracting on the spot market equilibrium. Thus, in deciding how many contracts to buy/sell in the forward market at t = 1, firm i evaluates the following payoff π i 1 = p f 1,1f i 1,1 + π s,i 1 (f i 1,1,f j 1,1) where π s,i 1 (f i 1,1,f j 1,1) are the spot (subgame-perfect) profits. Rearranging terms, firm i s overall profits as a function of f i 1,1and f j 1,1 canbewrittenas π i 1 =(p f 1,1 p s 1)f i 1,1 + p s 1q i 1(f i 1,1,f j 1,1)+δp s 2q i 2(f i 1,1,f j 1,1) where p s t = p s (qt(f i 1,1,f i 1,1)+q j j t (f1,1,f i 1,1)) j for t =1, 2. As in Allaz and Vila (1993), the arbitrage payoff (p f 1,1 p s 1)f1,1 i is zero since speculators and/or consumers share the same information as producers and thus p f 1,1 = p s 1.Therefore,firms are left with the (capacity constrained) Cournot profit from the two periods, p s 1q1 i + δp s 2q2. i Using the symmetry of the problem (s i = s j ) and linear demand, equilibrium forward sales become f1,1 i = f j 1,1 = a (1 δ) 5 implying the equilibrium deliveries 11 q i 1 = q i 2 = µ 1 6 a(1 δ)+3δsi 3(1 + δ) 5 (4) 1 3(1 + δ) ( 6 5 a(1 δ)+3si ). (5) Themereopportunityoftrading forwardhas created a prisoner s dilemma for the two firms bringing them closer to competitive pricing. Forward trading makes both firms worse off relative to the case in which they stay away from the forward market. However, if firm j does not trade any forwards, then firm i hasalltheincentivestomakeforward 10 The monopoly allocations per firm are q m 1 =[a(1 δ)+2δ(s i + s j )]/2(1 + δ) and q m 2 = s q m Note that since δ<1, f i 1,1 <q i 1. 7

8 sales (i.e., f1,1 i > 0) as a way to allocate a larger fraction of its total stock s i to the first period,whichisthemostprofitable of the two (recall that p 1 >δp 2). In the reproducible commodity (Cournot) game of Allaz and Vila (1993), forward trading allows a firm to capture Stackelberg profits given that the other firm has not sold any forwards by credibly committing in advance a larger quantity of production. In our depletable-stock game, forward trading allows a firm to capture Stackelberg profits by committing a larger fraction of its overall stock to the first period. But in either case, both firms face the same incentives, so both end up selling forwards. This is the pro-competitive effect of forward contracts; first documented by Allaz and Vila (1993). Let us now relax the assumption that contract positions can only be taken for the firstspotmarket(i.e.,f1,1, i f1,2 i and f2,2 i are unconstrained). Proposition 1 Equilibrium deliveries are given by (4) and (5), and equibrium forward positions satisfy f1,1 i δf1,2 i = a (1 δ) 5 For the proof of the proposition, let us work backwards and consider the last spot subgame (t =2): firms can only sell what is left of the stock so there are no decisions to make, other than meeting delivery commitments and putting the rest to the spot market. The same capacity constraint dictates behavior at the forward subgame at t =2. Selling contracts at this point cannot change delivery allocations and thus f2,2 i =0. 12 Consider then the first spot subgame (t =1), where the delivery allocation is still open. Given what has been contracted for the two periods, the condition equalizing present-value marginal revenues must hold, a 2q i 1 q j 1 + f i 1,1 = δ(a 2q i 2 q j 2 + f i 1,2), or a 2q i 1 q j 1 + f i 1,1 δf i 1,2 = δ(a 2q i 2 q j 2) Therefore, the payoff-relevant variables in the forward subgame are not the individual positions f1,1 i and f1,2 i but the composite position f1,1 i δf1,2. i By the same backward induction arguments laid out before, in equilibrium firms will choose f1,1 i and δf1,2 i as to satisfy f1,1 i δf1,2 i = a(1 δ)/5, which essentially leads to the same equilibrium level of contracting found earlier. It is irrelevant how firms transact in the contract market as long as their overall position satisfies f1,1 i δf1,2 i = a(1 δ)/5 (and, of course, f1,1 i q1 i and f1,2 i q2,where i 12 Technically speaking f2,2 i and f j 2,2 do not need to be zero; they can be any ammount that satisfy 0 f2,2 i + f j 2,2 + f 1,2 i + f j 1,2 si 2 + sj 2 (check this). 8

9 q1 i and q2 i are the equilibrium quantities given by (4) and (5), respectively). For example, firm j can fully contract its period-two deliveries (i.e., f j 1,2 = q2) j and simultaneously take a short position in period-one spot market equal to f1,1 i = a(1 δ)/5+δq2. j Firm i, on the other hand, might just take a short position in period-one equal to f1,1 i = a(1 δ)/5, or alternatively, go long in period-two in an amount equal to f1,2 i = a(1 δ)/5δ. In understanding the implications of forward contracting on the equilibrium quantities and prices, an important lesson from this two-period example is that we do not need to consider the full set of forward contracting possibilities. We can restrict attention to one-period-ahead contracting. The reason for this is quite simple. The strategic role of forward contracting is to allow firms to credibly reallocate (beyond the pure-spot allocation) their total stock across periods. And one-period-ahead contracting is sufficient for such strategic channel to be fully operational. More interestingly, we shall see that restricting attention to one-period-ahead contracting is not specific tothetwo-period setting but extends to the general model as well. 2.3 Lessons and questions Let us now discuss what we can learn from the two-period model. The two periods illustrate that despite the capacity constraint, contracting introduces a prisoners dilemma type of situation to firms and thereby enhances competition. While mechanism towards more competition is different from the reproducible-good case, the basic idea in Allaz- Vila (1993) seems to apply here. The question then is if the lesson is the same as in Allaz and Vila? To answer this we need to first update what is incomplete in Allaz and Vila. In the original reproducible case, the extensive form of the model is such that all forward markets open before any spot delivery takes place. This timing implies that firms are trapped to face the prisoners dilemma in a single spot market as many times as there are forward market openings. Obviously, this extensive form is critical to the result that forward markets enhance competition. In a more realistic formulation, where the contract and spot markets open repeatedly one after the other, Liski and Montero (2006) show that forward markets enlarge the set of collusive profits that history-dependent strategies can sustain. The extensive form in Liski and Montero is more realistic also when there are intertemporal capacity constraints. Thus, it is not reasonable to assume that all contracting takes place before stock consumption begins; contracts should be traded as stock depletion progresses. This opens up possibilities that are not present in the two-period model. First, 9

10 like in Liski-Montero (2006), nonstationary strategies are now possible. 13 Second, firms are not by definition trapped to deliver their stocks in some given periods but, rather, free to open new spot markets as a response to heavy contracting by other firms. Therefore, in the true stock depletion equilibrium with contracting, the time period of consumption is endogenously determined. To illustrate this latter effect, consider the following variant of the basic Allaz and Vila equilibrium: forward market opens n times before the spot but firms take turns in the forward market. It is straightforward to show that the first-mover advantage disappears when n, and the equilibrium converges to perfect competition as in the original model. In the exhaustible-resource model, we conjecture that the first-mover in the contract market contracts his entire capacity to early markets, eliminating other firms incentives to enter contracting (contracting can no longer influence other firms behavior). 3 The model We now build upon the framework introduced in the two-period example above and proceed to the general model. Our plan is to introduce the model firstindiscretetimeso that the extensive form of the game becomes clear, and then by letting the period length vanish we characterize the continuous time version. The continuous time limit identifies the most competitive sales path of given resource stocks, in the sense that there is no apriori restriction on firms possibilities to trade forward contracts. Continuous time allows also relative simple description of equilibrium dynamics. Throughout the rest of the paper, we assume that the demand is linear and that there are two symmetrical firms. 3.1 Discrete-time model The extensive form can be described as follows. Periods run from zero to infinity, and each period has to stages as in the two-period setting, that is, in each period forward contracting stage is followed by spot market. In any given period t, the spot market opens with contract commitments made at earlier dates 0, 1, 2,.., t 1 and also at the forward market t. For firm i, wedenotethecommitmentsmadepriortot for market t 13 Given that the stocks are finite and choke price above marginal costs (=zero), one is tempted to conclude that the model can be solved only by backward induction. However, exhaustible-resource model is similar to Coasian durable-good monopoly (consumer stock replaces the role of the resource stock), and for the Coasian model we know that two firms can sustain monopoly profits even in the gap case. Forthisreason,thesameoutcomecannotberuled out in the exhaustible resource setting. 10

11 by Ft i (the aggregate position for market t ) and contract sales made at t by (ft,τ) i τ t. Thus, the contract coverage of firm i at spot market t is Ft i + ft,t. i Wedefine the state at period t forward subgame as I t =(s i t,s j t, F i t, F j t) where F i t =(Ft i,ft+1,f i t+2, i...) denotes aggregate positions that firm i is holding for all future dates at t. The state at period t spot subgame is then (I t, ft i, f j t ) whereweadoptthe notation ft i =(ft,t,f i t,t+1,f i t,t+2, i...) to denote what firm i contracted at period t forward market opening. We are interested in equilibria where strategies depend on the current state only and therefore look for forward-contracting strategies that are functions of the form ft i = f i (I t ). That is, given what is the state at period t forward subgame, this vector-valued function determines the forward transactions made for all periods τ t at period t. Similarly, we look for spot market strategies of the form q i t = q i (I t, f i t, f j t ). That is, deliveries to market t depend on the remaining stocks, positions inherited from previous periods, and contracting made at period t. In this framework, we can derive the following equilibrium delivery rule. Proposition 2 Let T be the last period of consumption in equilibrium, and k =1,..., T the number of periods to go to T. Then, equilibrium delivery per firm at period t = T k is given by q i T k = q j T k = {a 3 [P k i=1 δi 1 kδ k ][1 + T k 3+2(T k) ]+δk s i T k} 1 P k. (6) i=0 δi We develop (6) by backward induction from T, but it may be helpful to see first the accounting of profits at some arbitrary t from the equilibrium path. Let V i (I t ) denote firm i s equilibrium payoff in the beginning of period t when state is I t. 14 In the forward subgame, firm i s best response defines V i (I t )=max{p s t(q i ft i t ft,t i Ft i )+ P p f t,τft,τ i + δv i (I t+1 )} (7) τ=t 14 In equilibrium, payoffs depend only on state I t and we do not need to keep track on calendar time. Note, however, that equation (6) expresses deliveries for some given T. In expressions below, it should be understood V already contains information about the number of market openings needed for stock depletion. 11

12 given f j t and that firm correctly anticipates productions in the next spot stage, qt i = q i (I t, ft i, f j t ) and q j t = q j (I t, ft i, f j t ), andp s t = p s (I t, ft, i f j t ). Note that since there is no arbitrage profit, p s t = p f t,t will hold for all t, and thus the payoff along the equilibrium path will satisfy V i (I t )=p s t(qt i Ft i )+δv i (I t+1 ), (8) where quantities and prices are given by equilibrium strategies. Effectively, we are finding contracting profiles (ft i, f j t ) starting with F0 i = F j 0 =0and generating the above value such that no shot-deviations are profitable. In each spot subgame, contracting is taken as given and best-response deliveries are chosen: V i (I t, ft i )=max{p s t(q i qt i t ft,t i Ft i )+δv i (I t+1 )}. (9) We can now use this structure to derive (6) by progressing inductively backwards from T. We do this here only for three periods (t =0, 1, 2) asthisissufficient for revealing the equilibrium contracting pattern. Having solved the two-period model, we know that deliveries in period 1 spot subgame satisfy q1(i i 1,f1,1,f i 1,1) j =[a(1 δ)+3δs i 1 +2(f0,1 i + f1,1) i (f j 0,1 + f1,1)] j 1 3(1 + δ). (10) That is, in this stage firm takes the existing contract coverage to spot market 1 as given, f0,1 i + f1,1, i and equalizes marginal revenues over the remaining two periods as explained for the two-period model. In the forward subgame at t =1previous contracting (f0, i f0) j from period t =0is given, and, in view of (8), best-response f1 i = f1,1 i is given by 15 max{p s 1(q i f1,1 i 1 f0,1)+δp i s 2q2} i where p s 2 q2 i = V i (I 2 ) is the last period value when q2 i = s i 2 = s i 1 q1 i for i =1, 2. Solving and imposing symmetry gives the best-response as depending on previous contracting by the two firms: f1,1 i = 1 5 [a(1 δ) f 0,1 i f0,1]. j The overall coverage at the outset of spot market t =1, which is the relevant determinant of deliveries in view of (10), is therefore f i 0,1 + f i 1,1 = 1 5 [a(1 δ)+4f i 0,1 f j 0,1]. (11) 15 Note that for reasons already explained for the two period model, we can ignore contracting for the very last spot market (t =2), as the capacity at the outset of the very last period is fixed. 12

13 We can now move on to the first period. In the spot subgame at t =0, firms take contracts from the first contracting stage (f0, i f0) j as given and solve max{p s 0(q i q0 i 0 f0,0)+δp i s 1q1 i + δ 2 p s 2q2} i =max{p s 0(q i q0 i 0 f0,0)+δv i i (I 1 )}, understanding how (11) together with (10) determine the continuation value V i (I 1 ).We find that firm i s best response in delivered quantities satisfies q0(i i 0, f0, i f0)={a(1 j + δ 2δ 2 )+3δ 2 s i 0 +2H0 i H0} j 1 3(1 + δ + δ 2 ). where H0 i =(1+δ)f0,0 i δ 2 f0,1. i We have now enough material to solve the final step, namely the first forward subgame where firm i chooses f0 i as a best response to f j 0 to solve Solving yields max p s 0q i {f0,0 i,f 0,1 i } 0 + δv i (I 1 ) f0,0 i = f j 0,0 = a (1 + δ 2δ 2 ) + a δ 2 (1 δ) 5 (1 + δ) 7 (1 + δ) f0,1 i = f j 0,1 = a (1 δ) 7 H0 i = H j 0 = a 5 (1 + δ 2δ2 ). Now, using these we can find expressions for equilibrium deliveries q0 i and q1 i (and q2 i = s i 1 q1). i Direct substitution of terms shows that deliveries are given by the expression (6) when T =3and k =1, 2. It is easy, while tedious, to verify that this same pattern emerges whatever the assumed number periods T needed for stock depletion. 16 To understand the economics of deliveries, but it proves useful to rewrite (6) as qt i k = q j T k = {a 3 [P k i=1 δi 1 kδ k ]+δ k s i T k} 1 P k i=0 H(T,k) + P k (12) δi i=0 δi where H(T,k)= a 3 [P k i=1 δi 1 kδ k T k ] 3+2(T k). Without forward markets, H(T,k)=0and the delivery per firm equals the path obtained in pure spot sale equilibrium. Term H(T,k) thus expresses directly how contracting 16 Upon request we can provide details on how to calculate the term H0 i directly for each period, without explicitly solving the precise pattern for future contracing. This helps in solving multiperiod examples. Besides verifying the pattern, little additional insight is gained from the mechanics. 13

14 increases supplies, compared to pure spot equilibrium, in a given period that is preceded by T k forward market openings and followed by k periods of deliveries. Term T k 3+2(T k) in H(T,k) indicates how many times firms face the prisoners dilemma from contracting, and term a 3 [P k i=1 δi 1 kδ k ] in H(T,k) weights the importance of the competitive pressure by taking into account what fraction of the remaining supply is at stake in the current market. Note that the marginal revenue at the spot stage of period T k is MR T k = a 2q i q j + H(T,k) so that H(T,k) takes the role of the contract coverage of the static model. Term H(T,k) measures the dynamic contract coverage, as it is a function of all previous and future contracting, and it is a measure of contracting level in this market. 3.2 Continuous-time model In standard exhaustible-resource theory, period length is not essential for understanding the economics of sales over time. 17 Hotelling s arbitrage condition, which just requires choosing sales to achieve the market rate-of-return for the underlying asset, has the same economics meaning whatever the frequency of market openings. Here, with forward contracting, frequency of transactions directly influences the profitability of holding the good. When period length is sufficiently large, any given initial holdings are consumed in just two periods in equilibrium, and the firms face the prisoners dilemma from contracting only once. Depletion of the same holdings require increasingly many periods if the period length becomes shorter; in the limit, the two-period model is transformed into a continuous time version. In the latter, after any positive interval of time, firms face the prisoners dilemma arbitrarily many times, and so it becomes reasonable to expect that the equilibrium becomes competitive. It proves useful to demonstrate first how the period length can be incorporated into the standard spot sale equilibrium. Let denote the period length and assume it takes 17 Exceptions: Stokey-Reinganum... 14

15 three periods to exhaust the initial holdings in equilibrium. To be concrete, conditions a 2q0 i q j 0 = δ(a 2q1 i q1), j a 2q1 i q j 1 = δ(a 2q2 i q2), j (q0 i + q1 i + q2) i = s i 0, for i = 1, 2 must hold equilibrium. delivery: The conditions lead to the following first-period More generally, q i 0 = q j 0 = a 3 {(1 + δ 2δ2 )+δ 2 s i 0 } 1 (1 + δ + δ 2 ). q i T k = q j t k = {a 3 [P k i=1 δi 1 kδ k ]+δ k si T k } 1 P k i=0 δi is the period T k delivery when it takes T periods to deplete the commodity stocks. It thus clear that period length only scales the stock size in the expression for deliveries, when the number of periods is taken as given. 18 But this same conclusion holds for deliveries in the contracting equilibrium: dynamic contract coverage, H(T,k) in (12), depends only on the number of times market opens before and after T k, but not on how short or long these openings are. Therefore, we can immediately write (12) as follows foragivenperiodlength: qt i k = q j T k = {a 3 [P k i=1 δi 1 kδ k ]+δ k si T k } 1 H(T,k) P k + P k (13) i=0 δi i=0 δi Let τ denote the real time used for consumption of stocks, and let r be the continuous time discount rate. Proposition 3 As 0, subgame-perfect equilibrium deliveries per firm approach the socially efficient deliveries at any given t>0. Proof. Let τ = T be the real time associated with T discrete steps of size. Since T = k +1we can replace k = τ 1 and δ = e r in (13) to write deliveries as a depending on the assumed real time and period length. Note that given t>0 is preceded by t/ forward markets so that T k 18 The period lenght will influence the real time needed for depletion, as will become clear shortly, but we first derive expressions that hold for a given real time. 15

16 in H(T,k) is replaced with t/. Fixingbothτ and 0 <t<τ, we can take the limit of (13) as 0: qt i = a (e r(τ t) 1 r(τ t)) rs i t + 2 e r(τ t) 1 e r(τ t) 1. (14) Consider then socially optimal deliveries starting with overall stock s i t + s j t = s t.denote the socially optimal total delivery by qt at some t 0 t0 τ.itmustsatisfy a q t 0 =(a q t )e r(t0 t), because prices grow at the rate of interest over the depletion period t 0 t<τ.solving for qt = q(q t,t,t 0 ) and using the exhaustion condition 0 yields Z τ t q(q t,t,t 0 )dt 0 = s t, q t = a (er(τ t) 1 r(τ t)) e r(τ t) 1 + rs t e r(τ t) 1. (15) Thus, equilibrium delivery per firm at each (s i t,s j t) given by (14) is equal to one half of the total socially efficient delivery qt at s i t + s j t = s t. 4 Concluding remarks We conclude by discussing the next steps in this research. First, it seems likely that there are asymmetric contracting equilibria that leave firms with some oligopoly rents. Suppose firm i contractsallofitsstocktotheveryearlymarketssothatthisfirm is effectively leaving its holding in competitive hands; in the extreme limit, firm i can contract more per period than the market can absorb so that buyers will store the stock to subsequent periods. Firm j 0 s best response in the forward market is obviously not to match i s contracting, but rather to reallocate spot deliveries away from markets where the competitive stocks are sold. Indeed, if i s stock is in competitive hands, j s best response has already been described by Salant (1976): firm j is now a dominant firm playing against a fringe of small stockholders. Thus, by committing its entire stock through the forward market, firm i receives what competitive agents receive on aggregate and firm j receives what the dominant firm receives in the Salant equilibrium. It is interesting to note that i s payoff is larger because it sells each unit of its stock at weakly higher prices than firm j. In view of this observation, we believe that there are asymmetric equilibria that leave some oligopoly rents to both firms. 16

17 Second, it is well known that two strategic agents can sustain collusion even in situations where backward induction equilibria are competitive. This was demonstrated by Gul (1987) in connection with durable-good oligopoly (gap case). In exhaustible-resource markets, nonstationary and collusive strategies can be expected to work when the consumption period is infinite, for example, due to stock-dependent extraction costs. Based on Gul s result, we cannot rule out collusive trigger strategies even though there exists a finite-horizon backward-induction equilibrium in our setting. If such collusive equilibria exist, forward market should make collusion easier as the equilibrium after deviation is more competitive than without forward markets. This line of reasoning implies that the general result of Liski-Montero (2006) holds also when firms are capacity constrained. On the other hand, if capacity constraints rule out collusive outcomes, the pro-competitive effect of contracting is quite extreme. For these reasons, it is important to analyze if the Gul s result can hold in the exhaustible resource market. Third, price competition in the spot market implies that forward trading reduces competition in the traditional reproducible-good market (see Mahenc-Salanie [2005]). It would be interesting to learn whether this anti-competitive effect helps firms to move from the oligopolistic equilibrium to monopoly in the exhaustible-resource market. Finally, we have assumed forward-looking but nonstrategic consumers throughout this paper. Strategic large buyers can extract part of the producers rent, and it is not clear how contracting of deliveries might affect the equilibrium. These questions are left open for future research. References [1] Allaz, B., and J.-L. Vila (1993), Cournot competition, forward markets and efficiency, Journal of Economic Theory 59, [2] Bhaskar, V. (2006), Dynamic Countervailing Power: Public vs Private Monitoring, working paper. [3] Biglaiser, G., and Vettas, N. (2005), Dynamic price competition with capacity constraints and strategic buyers, working paper. [4] Bulow, J.I., J.D. Geanakoplos, and P.D. Kemplerer (1985), Multimarket oligopoly: strategic substitute and complements, Journal of Political Economy 93,

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