Speculation in commodity futures markets: A simple equilibrium model

Transcription

1 Speculation in coodity futures arkets: A siple equilibriu odel Bertrand Villeneuve, Delphine Lautier, Ivar Ekeland To cite this version: Bertrand Villeneuve, Delphine Lautier, Ivar Ekeland. Speculation in coodity futures arkets: A siple equilibriu odel. séinaire Hotelling RITM ENS CACHAN, Feb 2014, Cachan, France. pp.37, <hal > HAL Id: hal Subitted on 5 Dec 2017 HAL is a ulti-disciplinary open access archive for the deposit and disseination of scientific research docuents, whether they are published or not. The docuents ay coe fro teaching and research institutions in France or abroad, or fro public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de docuents scientifiques de niveau recherche, publiés ou non, éanant des établisseents d enseigneent et de recherche français ou étrangers, des laboratoires publics ou privés.

2 Speculation in coodity futures arkets: A siple equilibriu odel Ivar Ekeland Delphine Lautier Bertrand Villeneuve February 5, 2014 Still preliinary Abstract We propose a siple and yet coprehensive equilibriu odel of the interaction between the physical and the derivative arkets of a coodity. To represent all basic econoic functions, we take three types of agents: industrial processors, inventory holders and speculators. Only the two first of the operate in the physical arket. All of the, however, ay initiate a position in the paper arket, for hedging and/or speculation purposes. First, we give the necessary and sufficient conditions on the fundaentals of this econoy for a rational expectations equilibriu to exist and we show that it is unique. Second, we propose a generalized fraework for the analysis of price relationships: the odel exhibits a surprising variety of behaviors at equilibriu which connects the noral backwardation theory and the storage theory. Third, the odel addresses the regulatory issues of speculators presence in the arket and their influence on prices. JEL Codes: D40; D81; D84; G13; Q00. 1 Introduction In the field of coodity derivative arkets, soe questions are as old as the arkets theselves, and they reain open today. Speculation is a good exaple: in his faous article about speculation and econoic activity, Kaldor 1939 wrote: Does speculation exert a price-stabilising influence, or the opposite? The ost likely answer is that it is neither, or rather that it is both siultaneously. The authors acknowledge conversations with Bruno Biais, Eugenio Bobenrieth, Alexander Guebel, Roger Guesnerie, Larry Karp, Jean-Charles Rochet, and Sophie Moinas, rearks fro Gabrielle Deange and Sven Rady, and coents fro audiences at Paris FIME Lab, Chair Finance and Sustainable Developent, ECPR, Zurich ETH, Montreal IAES, Toulouse TSE, Santiago CMM, Lyon AFFI and ISFA, Bonn Matheatical Institute, Toronto Fields Institute, Milan Bocconi University. This article is based upon work supported by the Chair Finance and Sustainable Developent and the FIME Lab. CEREMADE, Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, Paris. Eail: ekeland@cereade.dauphine.fr. DRM-Finance, Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, Paris. Eail: delphine.lautier@dauphine.fr. LEDa, Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, Paris. Eail: bertrand.villeneuve@dauphine.fr. 1

3 More than 70 years later, in June 2011, the report of the G20 FAO et al states: The debate on whether speculation stabilizes or destabilizes prices resues with renewed interest and urgency during high price episodes. [... ] More research is needed to clarify these questions and in so doing to assist regulators in their reflections about whether regulatory responses are needed and the nature and scale of those responses. Our siple perhaps the siplest possible odel of coodity trading provides insights into this question. It also proposes a way to understand how these arkets function and how the futures and spot prices are fored. Finally, it illustrates the interest of a derivative arket in ters of the welfare of the agents. In this odel, the financial arket interacts with the physical arket. There are two periods, a single coodity, a nuéraire and two arkets: the spot arket at ties t = 1 and t = 2, and the futures arket, where contracts are traded at t = 1 and settled at t = 2. The spot arket is physical there is a non negativity constraint on inventories, while the futures arket is financial shorting is allowed. There are three types of traders: inventory holders and industrial processors of the coodity, both of which operate on the two arkets, and speculators who operate on the futures arket only. All of the are utility axiizers and have ean-variance utility this choice is discussed in the presentation of the odel. There is also a price sensitive background deand or supply attributed to spot traders, which helps clear the spot arket. The sources of uncertainty are the aount of coodity produced and the deand of the spot traders at t = 2. Their realization is unknown at t = 1, but their law is coon knowledge. All decisions are taken at t = 1 conditionally on expectations about t = 2. Our ain contributions are three: existence and uniqueness of the equilibriu, extended coparative statics, and regulatory iplications. They are the consequences of the tractability of the odel. Despite nonlinear equilibriu equations, we give necessary and sufficient conditions on the fundaentals of this econoy for a rational expectations equilibriu to exist, and we show that it is unique. Moreover, it provides a unified fraework for the theory of price relations in coodity futures arkets, whereas in the literature this analysis is usually split into two strands: the storage theory and the noral backwardation theory also naed the hedging pressure theory after De Roon et al The forer focuses on the cost of storage of the underlying asset, the latter on the risk preiu. Although they are copleentary, to the best of our knowledge these two strands have reained apart up to now. We characterize the four possible equilibriu regies. While each of these four regies is siple to relate to concrete facts, we believe that our odel is the first coprehensive analysis to give explicit conditions on the fundaentals of the econoy deterining which one will actually prevail in equilibriu. We also give explicit forulas for the equilibriu prices. This enables us to characterize regies in detail and to perfor coplete and novel coparative statics. For instance, as is done in the storage theory, we can explain why there is a contango in such a case, the current basis, defined as the difference between the futures price and the current spot price, is positive or a backwardation the current basis is negative on the futures arket, and how this could change. 2

4 Towards this analysis, we give insights into the question of the inforational content of the futures price and the price discovery function of futures arkets. As done in the noral backwardation theory, we can also copare the futures price with the expected spot price and ask whether or not there is a bias in the futures price we define the expected basis as the difference between the futures price and the expected spot price. The sign and the level of the bias depend directly on which regie prevails. For exaple, the futures price can be predicted to be lower resp. higher than the expected spot price if a synthetic index denoted by γ says that there are relatively ore resp. less storers copared to processors and that they are relatively ore resp. less risk averse. The precise thresholds depend on the nuber of speculators and their risk aversion. So the odel depicts the way futures arkets are used to reallocate risk between operators, the price to pay for such a transfer, and thus provides insights into the ain econoic function of derivative arkets: hedging. 1 Our odel allows for new types of coparative statics. For exaple, we show that when the nuber of speculators increases, say because access to the futures arket is relaxed, the volatility of spot prices at date 2 goes up. This effect sounds undesirable. Our interpretation is that speculation increases the inforativeness of prices: volatility brings ore efficiency. The echanis is quite siple. As the nuber of speculators increases, the cost of hedging decreases and deand for futures grows along with physical positions. Saller hedging costs ake storers and processors aplify the differences in their positions in response to different pieces of inforation, iplying that their arket ipact increases. This increases in turn the volatility of prices. Beyond these descriptive predictions, we use our odel to perfor a welfare analysis and to draw regulatory iplications. This question, again, is as old as derivatives arkets. Newbery 2008 suarizes well the usual yet dual appreciation of the ipact of derivatives arkets on welfare. The author akes a difference between what he calls the layan and the body of infored opinion. He explains that to the first, the association of speculative activity with volatile arkets is often taken as proof that speculators are the cause of the instability, whereas to the second, volatility creates a deand for hedging or insurance. Our odel also exhibits a dual conclusion about welfare, but it is differently stated. First, the odel allows for a clear separation between the utility of speculation and that of hedging. Then, the analysis of the ipact of an increasing nuber of speculators shows that, storers and processors, as far as their hedging activities are concerned, have opposite views on the desirability of speculators. They are useless when the positions of storers and processors atch exactly; but when one type of agents has needs higher than what the other type can supply, then the forer wants ore the latter wants less speculators because this reduces his costs of hedging. To the best of our knowledge, such an effect has never been clarified before. Literature review. Of course, the questions we have raised have been investigated before. Contrary to what is done in this paper, the literature on coodity prices however separates the question of the links between the spot and the futures prices and that of the bias in the futures 1 It is worth noticing that our odel operates even without any risk-aversion at all: if we assue that all operators or even a single one are risk-neutral, then our odel is still valid and gives the four regies described earlier. 3

5 price. The latter has been investigated first by Keynes 1930 through the theory of noral backwardation whereas the forer is usually associated to the theory of storage, initiated by Kaldor 1940, Working 1949 and Brennan The sae separation is true for the equilibriu odels developed so far. An iportant nuber of equilibriu odels of coodity prices focuses on the bias in the futures price and the risk transfer function of the derivative arket. This is the case, for exaple, of Anderson and Danthine 1983a, Anderson and Danthine 1983b, Hirshleifer 1988, Hirshleifer 1989, Guesnerie and Rochet 1993, and Acharya et al Anderson and Danthine 1983a is an iportant source of issues and odeling ideas. Copared with this work, ours is sipler the producers are not directly odeled and copletely specified. This gives us the possibility to obtain explicit forulas for the equilibriu prices and to investigate further econoics issues, like welfare for exaple. The odels developed by Hirshleifer 1988 and Hirshleifer 1989 are also inspired by Anderson and Danthine 1983a. In these papers, Hirshleifer analyzes a point which is interesting for our odel but that we leave aside: the coexistence of futures and forward arkets. Hirshleifer 1989 also asks whether or not vertical integration and futures trading can be substitute eans of diversifying risk. Let us also ention that, contrary to Anderson and Danthine 1983b, Hirshleifer 1989 and Routledge et al. 2000, we do not undertake an inter-teporal analysis in the present version of the odel. Anderson and Danthine 1983b is the inter-teporal extension of Anderson and Danthine 1983a: they allow the futures position to be revised once within the cash arket holding period. To obtain results while keeping tractable equations, the authors however ust siplify their odel so that only one category of hedger reains in the new version. When equilibriu analysis stands at the heart of all concerns which is our case, this is a strong liitation. Routledge et al give another interesting exaple of inter-teporal analysis. It is related to the literature on equilibriu odels which focuses on the current spot price and the role of inventories in the behavior of coodity prices, as in Deaton and Laroque 1992, and in Chabers and Bailey In these odels, however, there is no futures arket: arkets are coplete and there is in fact a single type of representative agent. Risk allocation being optial, these odels are not fit for the political econoy of regulatory changes. Beyond the question of the risk preiu, equilibriu odels have also been used in order to exaine the possible destabilizing effect of the presence of a futures arket and to analyze welfare issues. This is the case of Guesnerie and Rochet 1993, Newbery 1987, and Baker and Routledge As the odel proposed by Guesnerie and Rochet 1993 is devoted to the analysis of ental eductive coordination strategies, it is ore stripped down than ours. As in Newbery 1987, our explicit forulas for equilibriu prices allows for interesting coparisons depending on the presence or absence of a futures arket. Finally, contrary to Baker and Routledge 2012, we are not priarily interested in Pareto optial risk allocations: we focus instead on coparative arket perforance as easured by utilities per head. Apart fro the specific behavior of prices, the non-negativity constraint on inventories raises 4

6 another issue. Epirical facts indeed testify that there is ore than a non-negativity constraint in coodity arkets: the level of inventories never falls to zero, leaving thus unexploited soe supposedly profitable arbitrage opportunities. The concept of a convenience yield associated with inventories, initially developed by Kaldor 1940 and Brennan 1958 is generally used to explain such a phenoenon, which has been regularly confired, on an epirical point of view, since Working In their odel, Routledge et al introduce a convenience yield in the for of an ebedded tiing option associated with physical stocks. Contrary to these authors, we do not take into account the presence of a convenience yield in our analysis. While this would probably constitute an interesting iproveent of our work, it is hardly copatible with a two-period odel. Recent attepts to test equilibriu odels ust also be entioned, as they are rare. The tests undertaken by Acharya et al could be used as in fruitful source of inspiration for further developents. As far the analysis of the risk preiu is concerned, the epirical tests perfored by Hailton and Wu 2012 and Szyanowska et al. forth., as well as the siulations proposed by Bessebinder and Leon 2002 are other possible directions. 2 The odel This is a two-period odel. There is one coodity, a nuéraire, and two arkets: the spot arket at ties t = 1 and t = 2, and a futures arket, in which contracts are traded at t = 1 and settled at t = 2. It is iportant to note that short positions are allowed on the futures arket. When an agent sells resp. buys futures contracts, his position is short resp. long, and the aount f he holds is negative resp. positive. On the spot arket, short positions are not allowed. In other words, the futures arket is financial, while the spot arkets are physical. There are three types of traders. Processors P, or industrial users, who use the coodity to produce other goods which they sell to consuers. Because of the inertia of their own production process, and/or because all their production is sold forward, they decide at t = 1 how uch to produce at t = 2. They cannot store the coodity, so they have to buy all of their input on the spot arket at t = 2. They also trade on the futures arket. Storers I for inventory, who have storage capacity, and who can use it to buy the coodity at t = 1 and release it at t = 2. They trade on the spot arket at t = 1 and at t = 2. They also operate on the futures arket. Speculators S, or oney anagers, who use the coodity price as a source of risk, to ake a profit on the basis of their positions in futures contracts. They do not trade on the spot arket. In addition, we think of these arkets as operating in a partial equilibriu fraework: in the background, there are other users of the coodity, and producers as well. These additional agents 2 For a recent and exhaustive study on this question, see for exaple Syeonidis et al

7 will be referred to as spot traders, and their global effect will be described by a deand function. At tie t = 1, the deand is µ 1 P 1, and it is µ 2 P 2 at tie t = 2. P t is the spot price at tie t and the deand can be either positive or negative; the superscript indicates a rando variable. All decisions are taken at tie t = 1, conditionally on the inforation available for t = 2. The tiing is as follows: For t = 1, the coodity is in total supply ω 1, the spot arket and the futures arket open. On the spot arket, there are spot traders and storers on the deand side, the price is P 1. On the futures arkets, the processors, the storers and the speculators all initiate a position, and the price is F. Note that the storers have to decide siultaneously how uch to buy on the spot arket and what position to take on the futures arket. For t = 2, the coodity is in total supply ω 2, to which one has to add the inventory carried by the storers fro t = 1, and the spot arket opens. The processors and the spot traders are on the deand side, and the price is P 2. The futures contracts are then settled at that price, eaning that every contract brings a financial result of P 2 F. There are N P processors, N S speculators, N I storage copanies I for inventories. We assue that all agents except the spot traders are risk averse inter-teporal utility axiizers. To take their decisions at tie t = 1, they need to know the distribution of the spot price P 2 at t = 2. We will show that, under ean-variance specifications of the utilities, there is a unique price syste P 1, F, P 2 such that all three arkets clear. Uncertainty is odeled by a probability space Ω, A, P. Both ω 2, µ 2 and P 2 are rando variables on Ω, A, P. At tie t = 1, their realizations are unknown, but their distributions are coon knowledge. Before we proceed, soe clarifications are in order. Production of the coodity is inelastic: the quantities ω 1 and ω 2 which reach the spot arkets at ties t = 1 and t = 2 are exogenous to the odel. Traders know ω 1 and µ 1, and share the sae priors about ω 2 and µ 2. A negative spot deand can be understood as extra spot supply: if for instance P 1 > µ 1 /, then the spot price at tie t = 1 is so high that additional eans of production becoe profitable, and the global econoy provides additional quantities to the spot arket. The nuber µ 1 deand when P 1 = 0 is the level at which the econoy saturates: to induce spot traders to deand quantities larger than µ 1, one would have to pay the, that is, offer negative price P 1 < 0 for the coodity. The sae reark applies to tie t = 2. We separate the roles of the industrial user and the inventory holder, whereas in reality industrial users ay also hold inventory. It will be apparent in the sequel that this separation need not be as strict, and that the odel would accoodate agents of ixed types. In all cases, agents who trade on the physical arkets would also trade on the financial arket for 6

8 two separate purposes: hedging their risk, and aking additional profits. In the sequel, we will see how their positions reflect this dual purpose. Note also that the speculators would typically use their position on the futures arket as part of a diversified portfolio; our odel does not take this into account. We also suppose that there is a perfect convergence of the basis at the expiration of the futures contract. Thus, at tie t = 2, the position on the futures arkets is settled at the price P 2 then prevailing on the spot arket. For the sake of siplicity, we set the risk-free interest rate to 0. In what follows, as we exaine an REE rational expectation equilibriu, we look at two necessary conditions for such an equilibriu to appear: the axiization of the agent s utility, conditionally on their price expectations, and arket clearing. 3 Optial positions and arket clearing 3.1 Utilities All agents have ean-variance utilities. For all of the, a profit π brings utility: E[ π] 1 2 α ivar[ π] 1 where α i is the risk aversion paraeter of a type i individual. Beside their atheatical tractability, there are good econoic reasons for using ean-variance utilities. They are not of von Neuann-Morgenstern type, i.e. forula 1 cannot be put in the for Eu X for soe function u. However, Mean-variance utilities capture well the behavior of firs operating under risk constraints. The capital asset-pricing odel CAPM in finance, for instance, consists in axiizing E[ R] under the constraint Var[ R] ν, where R is the return on the portfolio, which is equivalent to axiizing E[ R] λvar[ R], where λ is the Lagrange ultiplier. In financial arkets, as in coodities arkets, agents are ostly firs, not individuals, and they have risk constraints iposed on the fro inside anagers controlling traders and fro outside regulators controlling the fir. This is what forula 1 captures. For the sake of siplicity, we have kept the variance as a easure of risk, but we expect that our results could be extended to ore sophisticated ones coherent risk easures, at the cost of atheatical coplications. 3.2 Profit axiization Speculator. the r.v.: For the speculator, the profit resulting fro a position in the futures arket f S is π S f S = f S P 2 F, 7

9 and the optial position is: f S = E[ P 2 ] F α S Var[ P 2 ]. 2 This position is purely speculative. It depends ainly on the level and on the sign of the bias in the futures price. The speculator goes long whenever he thinks that the expected spot price is higher than the futures price. Otherwise he goes short. Finally, he is all the ore inclined to take a position as his risk aversion and volatility of the underlying asset are low. Storer. The storer can hold any non-negative inventory. However, storage is costly: holding a quantity x between t = 1 and t = 2 costs 1 2 Cx2. Paraeters C cost of storage and α I risk aversion characterize the storer. He has to decide how uch inventory to buy at t = 1, if any, and what position to take in the futures arket, if any. If he buys x 0 on the spot arket at t = 1, resells it on the spot arket at t = 2, and takes a position f I on the futures arket, the resulting profit is the r.v.: π I x, f I = x P 2 P 1 + f I P 2 F 1 2 Cx2. The optial position on the physical arket is: x = 1 C ax{f P 1, 0}. 3 The storer holds inventories if the futures price is higher than the current spot price. position is the only one, in the odel, that directly links the spot and the futures prices. This is consistent with the theory of storage and, ore precisely, its analysis of contango and the inforational role of futures prices. The optial position on the futures arket is: This position can be decoposed into two eleents. This f I = E[ P 2 ] F α I Var[ P 2 ] x. 4 First, a negative position x, which siply hedges the physical position: the storer sells futures contracts in order to protect hiself against a decrease in the spot price. Second, a speculative position, structurally identical to that of the speculator, which reflects the storer s risk aversion and his expectations about the relative level of the futures and the expected spot prices. Processor. The processor decides at tie t = 1 how uch input y to buy at t = 2, and which position f P to take on the futures arket. The revenue fro sales at date t = 2 is y β 2 y2 Z, where Z is our convention for the forward price of the output, and the other factor reflects decreasing arginal revenue. Due to these forward sales of the production, this revenue is known at tie t = 1. 8

10 The resulting profit is the r.v.: π P y, f P = y β 2 y2 Z y P 2 + f P P 2 F. An easy coputation then gives his optial decisions, naely: y = 1 ax{z F, 0}, βz 5 fp = E[ P 2 ] F α P Var[ P 2 ] + y. 6 The futures arket is also used by the processor to plan his production, all the ore so if the price of his input F is below that of his output Z. The position on the futures arket, again, can be decoposed into two eleents. First, a positive position y, which hedges the position on the physical arket: the processor goes long on futures contracts in order to protect hiself against an increase in the spot price. Then, a speculative position reflecting the processor s risk aversion and his expectations about the level of the expected basis. Rearks on optial positions. In this fraework, all agents have the possibility to undertake speculative operations. After having hedged 100 percent of their physical positions, they adjust this position according to their expectations. The separation of the physical and the futures decisions was derived by Danthine As shown by Anderson and Danthine 1983a, this property does not hold if the final good price is stochastic, unless a second futures arket for the final good is introduced. Note also that in the scenario where there is no futures arket, as depicted in appendix E, the quantities hold on the physical arket necessarily have a speculative diension. As we shall see, this separation result is very convenient for equilibriu analysis. This is one of the reasons why we choose, for the processor, not to introduce uncertainty on the output price and/or on the quantities produced. Although we assue that all individuals are identical in each category of agents, ore subtle assuptions could be retained without uch coplication. For exaple, reark that if the storers had different technologies, say, storer i with i = 1,..., N I had technology C i, then, instead of N I C ax{f P 1, 0}, total inventories would be i 1/C i ax{f P 1, 0}. In other words, storers are easily aggregated. In the following, when relevant, we shall use the index n I representing a synthetic nuber of storage units, and per-unit inventories X defined by: n I := { NI /C if storers are identical, i 1/C i X := ax{f P 1, 0}. otherwise, Siilarly, if processors had different technologies, say, processor i with i = 1,..., N P had 9

11 technology β i, then total input deand would be i 1/β iz ax{z F, 0} instead of N P βz ax{z F, 0}. Thus, when relevant, we shall use the index n P representing a synthetic nuber of processing units, and per-unit deand Y defined by: 3.3 Market clearing n P := The spot arket at tie 1. N P βz 1 Z i 1 β i Y := ax{z F, 0}. if processors are identical, otherwise, On the supply side we have the harvest ω 1. On the other side we have the inventory n I X bought by the storers, and the deand of the spot traders. Market clearing requires: ω 1 = n I X + µ 1 P 1, hence: P 1 = 1 µ 1 ω 1 + n I X. 7 The spot arket at tie 2. We have, on the supply side, the harvest ω 2, and the inventory n I X sold by the storers; on the other side, the input n P Y bought by the processors and the deand of the spot traders. The arket clearing condition is: ω 2 + n I X = n P Y + µ 2 P 2, with X and Y as above. We get: P 2 = 1 µ 2 ω 2 n I X + n P Y. 8 The futures arket. Market clearing requires: N S f S + N P f P + N I f I = 0. Replacing the f i by their values, we get: F = E[ P 2 ] N P α P Var[ P 2 ] + N I α I + N S α S n I X n P Y 9 Reark that if different agents of the sae type K K = P, I, S had different risk aversions α Kj for j = 1,..., N K, then we would see j 1/α Kj instead of N K /α K in Equation 9. This is an illustration of a ore general fact: we su up the inverse of the risk aversions of all agents to represent the inverse of the overall or arket risk aversion. See our synthetic index γ below. 10

12 Suary and definition The equations characterizing the equilibriu result fro the optial choices on the physical arket, Equations 3 and 5, the clearing of the spot arket at dates 1 and 2 Equations 7 and 8, as well as the clearing of the futures arket 9: X = ax{f P 1, 0} 3 Y = ax{z F, 0} 5 P 1 = 1 µ 1 ω 1 + n I X 7 P 2 = 1 µ 2 ω 2 n I X + n P Y 8 F = E[ P 2 ] Var[ P 2 ] N P α P + N I α I + N S α S HP 9 where the Hedging Pressure, HP, represents the unbalance of the paper arket and is defined as follows: HP := n I X n P Y. 10 Equation 9 gives a foral expression for the bias in the futures price, which confirs and refines the findings of Anderson and Danthine 1983a. It shows indeed that the bias depends priarily on fundaental econoic structures storage and production costs ebedded in the hedging pressure and the nuber of operators, secondarily on subjective paraeters agents risk aversions, and thirdly on the volatility of the underlying asset. Note also that the sign of the bias depends only on the sign of HP, which of course is endogenous. As the risk aversion of the operators only influences the speculative part of the futures position, it does not ipact the sign of the bias, at least in this partial equilibriu equation. Finally, when HP = 0, there is no bias in the futures price, and the risk transfer function of arkets is entirely undertaken between hedgers, because their positions on the futures arket are opposite and atching exactly. Thus the absence of bias is not exclusively the consequence of risk neutrality but ay have other structural causes. 4 Existence and uniqueness of the equilibriu 4.1 Notations We first focus on the equilibriu of the spot arkets and set: ξ 1 := µ 1 ω 1, ξ 2 := µ 2 ω 2, ξ 2 := E[ µ 2 ω 2 ], Note that ξ 1, ξ 2 and ξ 2 represent scarcity, i.e. excess basic deand with respect to basic supply, and reeber that the distribution of µ 2 ω 2 is coon knowledge. In what follows, it turns out that only the expectation and the variance of this quantity will be needed. We also assue that Var[ ξ 2 ] > 0. Thus there is uncertainty on the future availability of the coodity, and it is 11

13 the only source of uncertainty in the odel. In addition, reark that fro 8, we can easily derive the expectation and the variance of P 2 : E[ P 2 ] = 1 ξ 2 n I X + n P Y, 8E Var[ P 2 ] = Var[ ξ 2 ] 2. 8V Finally, we introduce the following notation, where is the price sensitivity of the deand: γ := N P α P Var[ ξ 2 ] + N I α I + N S α S This paraeter encodes a lot of inforation about the arket structure and will be very helpful in what follows. We have 1 γ +. If one of the agents, the processor for instance, is riskneutral, then α P = 0, so that γ = 1. If all the agents are pure arbitrageurs, so that α K = + for all K, then γ = +. In what follows, we assue that α P, α I and α S all are non-zero nubers, a restriction that is readily lifted. 4.2 Definitions Definition 1. An equilibriu is a faily X, Y, P 1, F, P 2 such that processors, storers and speculators act as price-takers, all arkets clear, and all prices are non-negative in all states of the world: X 0, Y 0 11 P 1 0, F 0 12 P 2 ω 0 for alost every ω Ω. 13 We will need an interediary notion, where positivity of P 2 is no longer required: Definition 2. A quasi-equilibriu is a faily X, Y, P 1, F, P 2 such that all prices, except possibly P 2, are non-negative, processors, storers and speculators act as price-takers and all arkets clear. Technically speaking, a quasi-equilibriu is a faily X, Y, P 1, F, P 2 R 4 + L 0 Ω, A, P such that Equations 3, 5, 7, 8 and 9 are satisfied. We now give two existence and uniqueness results, the first one for quasi-equilibria and the second one for equilibria. 12

14 4.3 Quasi-equilibriu Theore 1. There is a quasi-equilibriu if and only if ξ 1, ξ 2 verifies: and then it is unique. ξ 2 n P γz if ξ 1 0, 14 ξ 2 n P γz + n I + n P γ ξ 1 n I if n I Z ξ 1 0, 15 ξ 2 + n Iγ ξ 1 n I if ξ 1 n I Z, 16 Proof. To prove this theore, we begin by substituting Equation 8E in Equation 9. We get: F γn P Y n I X = ξ We now have two equations, 7 and 17 for P 1 and F. Replacing X and Y by their values, given by 3 and 5, we get a syste of two nonlinear equations in two variables: { P1 n I ax{f P 1, 0} = ξ 1, 18 F + γ n I ax{f P 1, 0} n P ax{z F, 0} = ξ Reark that if we can solve this syste with P 1 > 0 and F > 0, we get P 2 fro 8. So the proble is reduced to solving 19 and 18. Consider the apping ϕ : R 2 + R 2 defined by: ϕp 1, F = P 1 n I ax{f P 1, 0} F + γ n I ax{f P 1, 0} n P ax{z F, 0}. In R 2 +, take P 1 as the horizontal coordinate and F as the vertical one, as depicted by Figure 1. There are four regions, separated by the straight lines F = P 1 and F = Z: Region 1, where F > P 1 and F < Z. In this region, both X and Y are positive. Region 2, where F > P 1 and F > Z. In this region, X > 0 and Y = 0. Region 3, where F < P 1 and F > Z. In this region, X = 0 and Y = 0. Region 4, where F < P 1 and F < Z. In this region, X = 0 and Y > 0 Moreover, in the regions where X > 0, we have X = F P 1 and in the regions where Y > 0, we have Y = Z F. So, in each region, the apping is linear, and it is obviously continuous across the boundaries. To prove the theore, we have to show that the syste 19 and 18 has a unique solution. It can be rewritten as: ξ1 ϕp 1, F =, ξ 2 13

15 F X * >0 2 3 Y * =0 X * =0 Y * =0 A 1 X * >0 Y * >0 M 4 X * =0 Y * >0 45 O P 1 Figure 1: Physical and financial decisions in space P 1, F : the 4 regions. and it has a unique solution if and only if the right-hand side belongs to the iage of F, which is depicted by Figure 2, and developed in appendix A. This leads to the conclusion of the proof. 4.4 Equilibriu A quasi-equilibriu is an equilibriu if P 2 is positive in all futures states of the world. Clearly, if soe realizations of ξ 2 are sufficiently low, there will be no equilibriu: states of extree abundance are inconsistent with positive prices. By equation 8, the exact condition is: inf{ ξ 2 } n I X n P Y Theore 2. An equilibriu obtains if and only if ξ 1, ξ 2 satisfies 14, 15, 16, plus the additional condition: inf{ ξ 2 } n ξ2 I ξ 1 n P + n I Z ξ 2 + n I + n I + n P γ + n I n P γ ξ2 ξ 1 and it is then unique. inf{ ξ 2 } n I + n I 1 + γ in Region 1, in Region 2, inf{ ξ 2 } 0 in Region 3, inf{ ξ 2 } n P Z ξ 2 + n P γ The proof of this theore is given in appendix A. in Region 4. 14

16 ξ 2 2 X * >0 Y * =0 ϕ A 3 X * =0 Y * =0 1 X * >0 Y * >0 ϕm 4 X * =0 Y * >0 1 ϕo Figure 2: Physical and financial decisions in space ξ 1, ξ 2 : the 4 regions. To be coplete, we ust ask a copleentary question: for a given ξ 1, ξ 2, is there a distribution of ξ 2 such that an equilibriu exists? Theore 3. If ξ 1, ξ 2 supports a unique quasi-equilibriu under the ters of Theore 1, there is a distribution of ξ 2 supporting an equilibriu if and only if ξ 2 n I ξ n I n P Z + γ 1n P + n I γ + γ 1n P ξ 2 n I ξ 1 + n I γ ξ 2 in Region 1, in Region 2, 0 in Region 3, ξ 2 n P Z + γ 1n P in Region 4. The proof can be found in appendix A. These additional constraints also have two characteristic points, which are denoted by convention ϕ O and ϕ A see appendix A. The whole set of constraints is pictured in Figure 3. For a low ξ 1, the constraint P 1 0 atters ore excessive abundance at t = 1 ust be avoided, whereas for a low ξ 2, the constraint P 2 0 is the ost restrictive one excessive abundance at t = 2 should be avoided. Reark that, in the absence of a futures arket see appendix E, the existence conditions for an equilibriu are restricted: the four regions, in this scenario, are included in those of the basic scenario. 15

17 ϕ A ξ 2 ϕ ' A ξ 1 Ψ ϕ ' O ϕo Figure 3: Theore 3 s existence conditions in space ξ 1, ξ 2 : zoo on region 1. 5 Equilibriu analysis In this section we analyze the equilibriu in two steps. First, we exaine the four regions depicted in Figure 1. They correspond to very different types of interactions between the physical and the financial arkets. Second, we turn to Figure 2 on which we read directly the ipact of initial net scarcity ξ 1 and expected net scarcity ξ Prices, physical and financial positions A first general coent on Figure 1 is that in Regions 1 and 2 where X > 0, the futures arket is in contango: F > P 1. Inventories are positive and they can be used for inter-teporal arbitrages. In Regions 3 and 4, there is no inventory X = 0 and the arket is in backwardation: F < P 1. These configurations are fully consistent with the theory of storage. The other eaningful coparison concerns F and E[ P 2 ]. Fro Equation 9, we know that hedging pressure HP gives the sign and agnitude of E[ P 2 ] F, i.e. the price paid by the hedgers to transfer their risk in the futures arket. The analysis of the four possible regions, with a focus on Region 1 it is the only one where all operators are active and it gathers two iportant subcases, enables us to unfold the reasons for the classical conjecture: backwardation on the expected basis, i.e. F < E[ P 2 ]. More interestingly, we show why the reverse inequality is also plausible, as entioned by several epirical studies. 3 The equation HP = 0 cuts Region 1 into two parts, 1U and 1L. It passes through M as can be 3 For extensive analyses of the bias in a large nuber of coodity arkets, see for exaple Faa and French 1987, Kat and Ooen 2007 and Gorton et al

18 F A 1U Δ n I X * n P Y * >0 n I X * n P Y * <0 1L 2 3 M 4 45 O P 1 Figure 4: Physical and financial decisions in space P 1, F zoo on Region 1. seen in Figure 4. This frontier can be rewritten as: : n I F P 1 n P Z F = Along the line, there is no bias in the futures price, and the risk is exchanged between hedgers: storers and producers have perfectly atching positions and they insure each other. Above, HP > 0 and F < E[ P 2 ]. This concerns the upper part of Region 1 Subregion 1U and Region 2. The net hedging position is short and speculators in long position are indispensable to the clearing of the futures arket. In order to induce their participation, there ust be a profitable bias between the futures price and the expected spot price. This backwardation on the expected basis corresponds to the situation depicted by Keynes 1930 as the noral backwardation theory. Below, HP < 0 and F > E[ P 2 ]. This concerns the lower part of Region 1 Subregion 1L and Region 4. The net hedging position is long and the speculators ust be short, which requires that expected spot price be lower than the futures price. Table 1 suarizes for each region the relationships between the prices and the physical and financial positions. Attentive scrutiny of the table shows very contrasted regies. For exaple, in Region 2, we have siultaneously a contango on the current basis and a backwardation on the expected basis or a positive bias. In short, P 1 < F < E[ P 2 ]. In Region 3, in the absence of hedging of any sort, the futures arket is dorant, and this is no bias on the expected 17

19 2 P 1 < F F < E[ P 2 ] F > Z X > 0 f S > 0 Y = 0 1U P 1 < F F < E[ P 2 ] F < Z X > 0 f S > 0 Y > 0 P 1 < F F = E[ P 2 ] F < Z X > 0 f S = 0 Y > 0 1L P 1 < F F > E[ P 2 ] F < Z X > 0 f S < 0 Y > 0 4 P 1 > F F > E[ P 2 ] F < Z X = 0 f S < 0 Y > 0 3 P 1 > F F = E[ P 2 ] F > Z X = 0 f S = 0 Y = 0 Table 1: Relationships between prices, physical and financial positions. basis. Region 4 is the opposite of Region 2: the arket is in backwardation and, as X = 0, the net hedging position is long, the net speculative position is short and the bias is negative. In short, P 1 > F > E[ P 2 ]. 5.2 Supply shocks In Figure 2, ξ 1, ξ 2 easure scarcity, not abundance: ξ 1 is the extent to which current production ω 1 fails short of the deand of spot traders, and ξ 2 is the expected extent to which future production will fall short of the deand of spot traders. Assue that no arket is open before ξ 1 is realized, and that ξ 1 brings no news about ξ 2. We can take ξ 2 as fixed, and see what happens on equilibriu variables, depending on the value of ξ 1. To fix ideas, suppose that we expect a oderate scarcity at date 2 ξ 2 = ξ 2 in Figure 5. In the case of a low ξ 1 abundance in period 1, we are in Subregion 1U. If ξ 1 is bigger, we are in Subregion 1L, and if ξ 1 is even bigger, the equilibriu is in Region 4. The interpretation is straightforward. If period 1 experiences abundance Subregion 1U, there is assive storage: the current price is low and expected profits are attractive, since a future scarcity is expected. Storers need ore hedging than processors, first because inventories are high, second because the expected release of stocks reduces the processors needs. Thus, there is a positive bias in the futures price and speculators have a long position. For a less arked abundance Subregion 1L, storage is ore liited. The storers hedging needs diinish while that of the processors increases. So the net hedging position is long, the bias in the futures price becoes negative and the speculators have a short position. If the coodity is even scarcer Region 4, there is no storage, only the processors are active and they hedge their positions. This exaple illustrates quite siply why, when there is a contango on the current basis, we 18

20 ξ 2 2 ϕ A 3 1U Δ ϕm 4 ξ 2 = ξ 2 1L 1 ϕo Figure 5: Physical and financial decisions in space ξ 1, ξ 2 zoo on Region 1. can have either an expected backwardation or an expected contango. 6 The ipact of speculation The ipact of speculation can be studied in two ways. First, the difference between having and not having speculators. This is the approach taken in particular by Newbery We propose results in this vein in Appendix E. Suary of existing and new results to be written. Second, one can analyze the effect of increasing speculation. One ay think either of a relaxed access to the futures arkets, or of a sudden rise in risk appetite. We can translate these changes as an increase in the nuber N S, or as a decrease of the risk aversion α S, or even via the decrease of the risk aversion of any of the other actors α I or α P. The key observation is that all these possible causes ipact the synthetic index γ in the sae way: it decreases. Indeed, N S, α P, α I and α S and Var[ ξ 2 ] appear only through the single paraeter γ. This suggests us a siple strategy to perfor the coparative statics of speculation. In the following, we ean by increased speculation any of the above entioned causes decreasing γ. This expression is used for convenience, but the reader should keep in ind that this eans several slightly different things. Subsection 6.1 shows the ipact on prices and quantities; Subsection 6.2 prolongs with detailed welfare analysis and the political econoy of speculation. 19

21 6.1 Speculators ipact on prices and quantities Increased speculation has equilibriu effects, so that the causal relationships between speculation, prices and quantities ust be used with care. We propose however a sequence of concoitant theoretical facts that are easy to understand and eorize. This subsection ust be seen as a big proposition, the proofs being given in Appendix D.1. The equilibriu analysis assues that ξ 1 is known when arkets open, naely at date 1. But for the observer, ξ 1, or rather ξ 1, can be seen as rando at a previous stage, say date 0. In order to analyze variance, we consider now that prices are functions of two rando factors ξ 1 and ξ 2, with ξ i = E[ ξ i ] i = 1, 2. We assue that the two factors are independent. We focus in particular on Var 0 [ ] instead of Var 1 [ ξ 1 ], as was done iplicitly up to now. The subscript gives the date at which the statistics are calculated; given the absence of abiguity in the sequel, the subscript will be dropped. The analysis of an increase of speculation is studied as a decrease of γ. A topical case is an increase of N S but other causes via the above entioned paraeters are also worth considering. We study first Regions 2 and 1U, where E[ P 2 ] > F, and where the physical agents are sellers in aggregate in period 2 they prefer the sure as of date 1 F to the rando P 2. References below are to the coluns of Table 2. Reark that in the table the absolute value E[ P 2 ] F gives the equilibriu cost of risk coverage to physical agents. This cost is the starting point of our econoic analysis. Let s see first prices and quantities in level. Increasing speculation increases in fact the overall capacity to absorb risk. In our copetitive setting, this eans that hedging becoes cheaper: the expected argin E[ P 2 ] F > 0 decreases see colun E[ P 2 ] F. As risk anageent becoes cheaper for storers, they increase their inventories, whatever the shock observed in period 1 see colun X. For the processors, hedging was a double win: it was reducing risk and was profitable. The rent or subsidy they were receiving is diinished by an increased speculation; thus they reduce their takes see colun Ỹ. Increased inventories eans an increased deand in period 1, thus a price increase see coluns P 1. Quite logically, the effect is a lower price in period 2, due to the extra units drawn fro inventories ssee colun P 2. Let s now turn to the variances. The decrease in the hedging cost enables storers to be ore reactive to first-period prices, so that overall their opportunistic purchases attenuate even ore production or deand shocks on prices: the covariance of inventories and price is negative and it increases in absolute value with speculation. This explains the lower variance of P 1 see colun Var[ P 1 ]. 20

22 The consequence of the previous effect is that there is ore variance of the quantity of the coodity delivered in period 2. This adds noise to the current shocks and thus the variance of P 2 increases. See colun Var[ P 2 ]. F and P 2 get closer as speculation increases see coluns F and P 2. This convergence eans that their variances have the sae sense of variation with respect to speculation. see coluns Var[ F ] and Var[ P 2 ]. Whenever E[ P 2 ] < F, the effects are siilar but reverse. Processors are the agents who need the ore speculators, and they increase they position as speculation increases. Storers in contrast lose on the rent they draw fro being structurally contrarians. E[ P 2 ] F F X Ỹ P1 P2 Var[ F ] Var[ P 1 ] Var[ P 2 ] 2 0 1U 1L 4 0 } E[ P 2 ] F > 0 } F E[ P 2 ] > 0 3 F = E[ P2 ] Table 2: Ipact of speculators on prices and quantities Speculation, prices and quantities in suary. Table 2 shows that Regions 2 and 4 can be viewed as ere subcases of Subregions 1U and 1L. Reark for exaple that P 1 decreases in Subregion 1L whereas it is constant in Region 4. This is due to the fact that the storers are active in Subregion 1L but not in Region 4, and underlines how iportant the stocks are for the functioning of a coodity arket. Inventories indeed appear as the transission channel for shocks in the space between the paper and the physical arkets and in the tie between dates 1 and 2. For this is through inventories that a shock appearing in the paper arket i.e. the rise of speculation ipacts the level and variances of the physical quantities and the prices. This result is close to the analysis of Newbery As far as the level of the different variables is concerned, our odel shows that the ipact of an increase of speculation depends, in the end, on which side of the hedging deand doinates. The physical quantities, for exaple, increase for the operators benefiting fro lower hedging costs whereas they decrease for the others. This aplifies the difference in the positions of the operators and consequently their arket ipact. The analysis of the variances is less straightforward. The ost siple effect is the ipact on Var[ F], which always diinishes under the pressure of a ore intense speculative activity provided that there are stocks in the econoy. As regards to the spot prices, an increase of speculation has a stabilizing effect at tie 1 and a destabilizing one at tie 2. The latter result however ight 21

23 be odified in a three-period odel, where the quantities at tie 2 would be influenced by the futures price of a contract expiring at tie 3. It could also be changed if the price of the output, Z, could be adjusted as an answer to a shock. Up to now, indeed, there is nothing in the odel that could absorb a shock at tie 2. This version of the odel illustrates the fact that financial arkets ay destabilize the underlying arkets, though the ter is inappropriate since it only refers to a statistical property. Of course a higher price volatility doesn t ean a lower welfare, quite the contrary: ore volatility eans that prices are ore effective/inforative signals. The ipact of arkets on prices volatilities is often a naïve aspect of welfare analysis. We will go further on this point in the next subsection. Note finally that Appendix E copletes this analysis with coparisons based on another scenario where the futures arket is closed. 6.2 Speculators ipact on utilities In this section, we express the equilibriu indirect utilities of the various types of agents, and we copute their sensitivities with respect to the paraeters, in particular the nuber of speculators. We proceed in two steps. First, we copute the indirect utilities as functions of equilibriu prices P 1 and F. Second, we copute the elasticities of P 1 and F to deduce the elasticities of the indirect utilities. We restrict ourselves to the richer case, i.e. Region 1, where all agents are active. For the sake of siplicity, we return to an analysis where ξ 1 is known when arkets open. Recall that then we have F < Z and P 1 < F. The speculators indirect utility is given by: U S = f SE[ P 2 ] F 1 2 α Sf 2 S Var[ P 2 ], where we have to substitute the value of f S and and Var[ P 2 ], which leads to: U S = E[ P 2 2 ] F 2α S Var[ ξ 2 ] The storers indirect utility is given by: U I = x + f I E[ P 2 ] x P 1 f I F 1 2 Cx α Ix + f I 2 Var[ P 2 ], where we substitute the values of f I, x and Var[ P 2 ]: U I = E[ P 2 2 ] F 2α I Var[ ξ 2 ] 2 + X 2 2C