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Does a better price informativeness enhance the functioning of the

commodity markets ?

Etienne Borocco ∗

January 17, 2019

Abstract

Our research question focuses on how more informatives prices aﬀect operators. Above all,

I wonder who are the winners and the losers of the lower risk generated by a higher price

informativeness. I study a two-period model with a spot market and a futures market for a

commodity. Hedgers are active on both markets and speculators trade only on the futures

market. Hedgers speculate as well on futures contracts. Information is heterogeneous. Every

group is endowed with a common signal about the net demand at maturity. First, I show the

necessity to take in account the forward curve and the prices of the output in the estimation

of the risk premium because information can modify the structural relationship. Second, I

show how more informative prices increase the elasticity of the hedging pressure to the risk

premium, exactly like when the weight of speculators increases. Last, I shed new light on the

conditions which make more precise information harmful for every agent. In this situation,

everyone loses because of a decreasing payoﬀ coming from speculation. This last eﬀect is known

as the «Hirshleifer eﬀect».

∗PSL Research University, Université Paris-Dauphine, LEDa, [SDFI], Place du Maréchal de Lattre de Tassigny,

75016 Paris.

Email: etienne.borocco@dauphine.fr

The author would like to acknowledge support from his both PhD Supervisors, Delphine Lautier and Bertrand

Villeneuve; conversations with Jérôme Mathis; remarks from Marius Zoican and David Batista Soares; and comments

from audiences in Glasgow (University of Strathclyde) and Oxford (Mathematical Institute).

1

Introduction

Commodities futures have become increasingly popular as an asset class for portfolio managers in

the ﬁrst decade of the third millenium. This process is called «ﬁnancialization» (Cheng and Xiong,

2014). In this context, governments consider information quality as a key stake to guide agents’ ex-

pectations. To witness, the G20 has launched the Agricultural Market Information System (AMIS)

in 2011. One of the aims of the AMIS is to improve information about wheat, maize, rice and

soybeans. To fulﬁll this purpose, the AMIS provides analysis, by investigation topical issues, and

forecasts of short-term supply and demand at both national and international levels.

An arising issue is who beneﬁt from this policy and also who are the losers. It is possible that

everyone’s well-being improves or at the opposite, the global welfare decreases. When everyone

loses, we call this situation a «Hirshleifer eﬀect». A more precise information can be harmful for

all the agents by destroying hedging opportunities. Less hedging is less trading which means less

business. Operators are worse oﬀ because they expect to make less money. The public disclosure

of information adds a distributive risk wich lowers the global welfare. I study how new public

information, about net demand on the spot market at maturity, impacts risk sharing. I look in

particular the consequences of the redistribution of risk sharing on the well-being of operators.

A question arising quite immediately is how the diﬀerences of information among agents aﬀects

the functions of the derivative markets. An eﬃcient market gathers the suﬃcient information in

the price which thus becomes the best estimator of the payoﬀ. The issue becomes more about the

quality of the aggregated information rather than the diﬀerences of information among operators.

In this article, we will focus on the futures only among the derivatives products because it is the

most used kind of contracts in commodities markets. Besides, the futures markets have important

economic functions. The contract prices for diﬀerent maturities will give information about the

anticipated spot prices at maturity (Lautier, 2013). This function is called price discovery. More

broadly, we can look at the functions of the commodity markets in their whole. An important

dimension is the storage. The prices have a direct inﬂuence on the storage. If the forward curve1is

upward sloping, the level of storage will be high. Because, it is proﬁtable to hold stocks to sell them

later. At the opposite, a downward sloping forward curve implies a low level of storage because the

higher spot price gives incentive to sell the commodity on the physical market immediately. The

forward curve reveals the anticipations of the market traders. The anticipations depends on the

available information heavily. The price revelation of the information is a key feature of eﬃcient

markets. A market is strongly eﬃcient if all the information is revealed including private informa-

tion.

My approach is theoretical. I apply Bayesian theory to an equilibrium model. I introduce

information in Ekeland et al. (2018). This model shows how speculation and hedging interacts

through the reciprocal feedbacks between futures and spot prices. Both are endogenous. It is a

two-period model with a spot and a futures market. On the spot market, there are spot traders

and hedgers. Hedging in this model includes storers who are naturally short and processors who

are naturally long. Storage is from the ﬁrst period to the second period. Processors buy input

for their output in the second period but they can decide to hedge it in the ﬁrst period. Thus,

the hedging pressure, which is the diﬀerence between the short and the long hedging positions,

can be net short or net long. One key result of this model is that ﬁnancialization beneﬁts to the

dominating side of hedging. Every group of agents, whatever for speculators, storers or processors,

is endowed with a common signal about the net demand at maturity. In this theoretical setting,

an eﬃcient market is deﬁned as a Fully-Revealing Rational Expectations Equilibrium (FRREE)

(Grossman, 1977). Knowing the price is equivalent to know all the private information. A unique

FRREE exists if the hedging pressure is linear. Two theorems from Grossman (1978) and Bray

(1981) are extended with a linear hedging pressure to prove the existence and the uniqueness of the

equilibrium. I show the FRREE implies the futures price is the unique predictor of the spot price.

It is a suﬃcient statistics. It means it contains all the agents need to know. In an eﬃcient market,

1The forward curves deﬁnes the prices of futures contracts according to their maturity.

2

the futures price is a biased but eﬃcient estimator of the spot price at the contract maturity. The

bias is the expected payoﬀ of speculation, which is the diﬀerence between the expected spot price

at maturity and the futures price. The bias in the futures price is called risk premium. This value

is also the income asked by speculating operators as counterparts of the risk sharing.

Moreover, noisy information makes the futures price stochastic. Thus, the diﬀerence of informa-

tion among agents implies a stochastic risk premium even in an eﬃcient market. The distribution

of the conditional risk premium, the value of the risk premium according to the information in-

cluded in the futures price, is determined by its unconditional moments. Therefore, the hedging

pressure is stochastic as well. Moreover, the sign of the basis and the spread between the futures

price of the input and the scaled forward price of the output vary with the variation of the signals.

It modiﬁes randomly the coeﬃcients of the linear relationship of the spot price at maturity on the

futures price. In this framework, an increasing weight of the speculators decreases the cost of risk

sharing but does not improve the precision of the signals. I deﬁne here the precision as the inverse

of the variance.

Finally, I show that the higher price informativeness about the net demand at maturity in-

creases the elasticity of the risk-bearing capacity exactly like ﬁnancialization does. The cost of risk

sharing decreases. In an eﬃcient market, an additional signal improves the precision of the suﬃ-

cient statistics revealed by the futures price in the ﬁrst period. Therefore, the conditional variance

of the spot price decreases in the second period. This lower volatility means a less risky investment

so an incentive to speculate. The operators who are risk-bearers accept at a lesser cost. In other

words, they get a smaller risk premium. The eﬀect on welfare is ambiguous. The utility arising

from speculation decreases because of the diminishing risk premium coming from speculation but

increases with the higher price precision. Nonetheless, we show that when the price precision is

low, an additional signal can increase global welfare by rising utilities both from speculation and

hedging. When the prices are very noisy, the AMIS improves the global welfare by increasing the

expected proﬁt of speculation. At the opposite, this policy may generate a «Hirshleifer eﬀect»

because a more precise information is harmful when the expected gains of risk-sharing decrease

such as the utility of every operator decreases.

The paper is organized as follow. Section 1 reviews the literature about information aggregation

mainly in the commodity markets. Section 2 describes the settings of the model. The main equa-

tions are in the section 3. Section 4 characterizes the equilibrium showing its existence and its

uniqueness. Section 5 show how information makes stochastic the asset pricing. Last, section 6

discussed the policy implications of this article.

1 Literature review

The equilibrium model of Ekeland et al. (2018) is a production economy where there is a feedback

between the spot and futures prices which are both endogenous. Therefore, the expectation of the

spot price at maturity is endogenous too. Therefore, we can study how information impacts pro-

duction and speculation decisions through the interaction of prices and expectations. I show a more

informative futures price decreases the absolute value of the risk premium as ﬁnancialization does.

Ekeland et al. (2018) shows a lower absolute risk premium favors the dominating side of hedging.

For example, if the hedging pressure is net short, short hedging will become less expensive and will

increase with a lower positive risk premium. Nonetheless, we have an additional eﬀect on welfare in

comparison to Ekeland et al. (2018). A more precise information decreases the conditional variance

meaning speculation is less risky. This eﬀect is opposite to the decline of the payoﬀ highlighted

by Ekeland et al. (2018). If the eﬀect of the disminishing risk is stronger, everyone wins. At the

opposite, everyone can loose because the declining payoﬀ of speculation is a burden on the welfare

of every operator. Otherwise, we are on the situation highlighted by Ekeland et al. (2018) with a

decreasing payoﬀ hurting speculators and the dominated of hedging but which does not oﬀset the

hedging gains of the dominating side.

Our approach is complementary of Sockin and Xiong (2015) which tackles informational fric-

3

tions which oﬀset the cost eﬀect on price. In their model, information is about the strength of

the global economy. They show how the macroeconomic aspects can shape the asset pricing of the

futures contracts. While in the ours, information is about the supply side, more precisely about

the net demand of the commodity at maturity. I focus about microeconomic aspects while the

approach of Sockin and Xiong (2015) is more macroeconomic. This is the reason why we say our

both approaches are complementaries. Both are models with a rational-expectations equilibrium

(REE) under asymmetric information as deﬁned by Grossman (1981). The agents have rational

expectations, i.e traders know how the economy works. They gather all the available information

and they can compute the state of the economy through the set of prices directly. I thus make

the assumption that the traders use the correct model. Sockin and Xiong (2015) highlight the

importance of informational frictions. Each good producer observes a private signal about a com-

mon productivity factor. The authors have a macroeconomic approach. The productivity shock of

end-users is a macroeconomic factor. Informed agents convey their information about the macroe-

conomic situation to the commodity prices. This informational eﬀect can be so strong that it can

oﬀset the cost eﬀect meaning there is a commodity demand increasing with the spot price. For

more realism, they refuse normal distribution for the parameters. Their variables are log-normal

and at the end, the informational eﬀect can oﬀset fundamental values. Their key message is that

speculators have an indirect eﬀect on commodity supply and demand through the feedback of fu-

tures price. The latter can impact the commodity demand and the spot price. If the informational

eﬀect oﬀsets the cost eﬀect, a rising futures price will counter-intuitively decreases the basis and

thus the storage level.

Although our model is a pure extension of Ekeland et al. (2018), it has a similar structure to

Goldstein and Yang (2017a). The producers of the authors’ model are equivalent to the storers in

our article. They have the same maximization program (their linear cost parameter plays the same

role than the spot price in the period 1 for storers). Moreover, when the futures price increases

under the inﬂuence of speculators, farmers or storers get incentives to increase the supply in the

next period. Therefore, a higher futures price drives down the spot price in contango. At last, their

model rely on a strong assumption. Speculators have a perfect information about the commod-

ity’s demand shock but they bring noise because of their security portfolio hedging in Goldstein

and Yang (2017a). Therefore, an increasing weight of speculators has an ambiguous eﬀect on the

price informativeness and the risk premium. A similar eﬀect is generated when there are strategic

complementarities in the acquisition of information.

A key feature of our model is the common error for each group. It can be interpreted as a biased

consensus. More speculators with no new information bring liquidity only, consistent with Chinn

and Coibion (2014). The literature highlighted the existence of an optimistic biased consensus

among analysts (Knill et al., 2006). They noticed that speculators can get information about oil

and gas producers through analysts. The issue is their forecast of corporate earnings are often too

optimistic. In their empirical analysis, "a measure of aggregate earnings surprise for the industry"

is used as a proxy for information asymmetry. The later is considered as proportional to the former.

Their results show a "large degree of information asymmetry" on the futures markets for oil and

gas. Moreover, they remind that the errors are not distributed identically and independently among

speculators. The analysts’ forecasts can be biased in the same direction. An explanation with ra-

tional agents has been brought (Lim, 2001). Analysts in an uncertain information environment and

who are reliant on the management access as primary source are more likely to make optimistic

bias forecasts about the companies’ earnings. Nonetheless, the signal in our model is not about

earnings but the net demand in the next period. It is hard to say if forecasts about net demands

are biased. For example, the forecasts errors of the U.S Department of Agriculture (USDA) about

harvests of have been associated mostly with structural changes. There is no evidence that they

are biased systematically toward leniency or pessimism (Isengildina-Massa et al., 2013).

Some speculators can be informed. For example, Hau (2001) studies the electronic trading

system Xetra of the German Security Exchange which provides data source on the equity trades of

756 professional traders located in 23 diﬀerent cities and eight European countries. He showed that

4

traders located outside Germany in non-German-speaking cities show lower proprietary trading

proﬁt in comparison to local German traders. In commodity markets, there is the same kind of

concentration in cities like Geneva, Singapore or Houston. Thus, we can suppose that informed

speculators exist. This the point of view of Khoury and Martel (1989) who made the assumption

that the speculators are more informed than the hedgers:

« These speculators can exploit economies of scale and of specialization in order to have

access to a continuous ﬂow of additional relevant information concerning future supply

and demand conditions in the spot ,and futures markets at a much lower cost than

hedgers (Khoury and Martel 1985, 1986) [...] Moral hazard and/or the loss of lucrative

trading opportunities by speculators hamper the transfer of the needed information

to hedgers; and the cost (in terms of time and money) to hedgers of acquiring this

information on their own generally exceeds the beneﬁt to them. »

Khoury and Martel (1989) show how information asymmetry can lead to a positive storage level

when the futures price is lower than the spot price. If the basis is negative, stockholders expect

the futures price to decrease enough such as the proﬁt on the futures hedge exceeds the loss from

the storage activity. This eﬀect is reinforced when there is a high discount rate and a low cost of

storage. Therefore, they provide an alternative explanation to convenience yield, which is the ben-

eﬁt associated with holding a physical good Kaldor (1939). When the forward curve is decreasing,

the market is said in "backwardation". Nonetheless, Khoury and Martel (1989) remain quite ad

hoc and their alternative explanation did not replace the convenience yield to modelize storage in

a backwardated market. I do not get a similar result in our model.

Others authors have made diﬀerent assumptions. The involved agents in the physical commod-

ity trading as the storers or the processors can exploit their information for speculation (Cheng and

Xiong, 2014). For example, their knowledge of the local physical market allow them to exploit in-

formation frictions. Therefore, they get informational advantages that they can exploit. According

to Vives (2010), the informed speculators are the producers while the processors are uninformed

hedgers:

« The private information of producers cannot help the production decisions because it

comes too late, but allows them to speculate in the futures market where uninformed

speculators (market makers) and other hedgers operate. This will tend to diminish

the hedging eﬀectiveness of the futures market and consequently diminish the output

of risk-adverse producers (since they will be able to hedge less of their production).

The adverse selection is aggravated with more precise information. Adverse selection

is eliminated if the signal received by producers is made public. However, more public

information may decreases production because it destroys the insurance opportunities

».

The last eﬀect is the "Hirshleifer eﬀect". When all the information is released, it can lead to a

no-trade situation such the utility of the agents can decrease. Insurance opportunities are created

by the risk sharing between the agents. New information modiﬁes risk sharing and thus how op-

erators trade among them on the futures market. Operators can be impacted negatively by this

redistribution of risks, Therefore, information release adds a distributive risk to the technological

risk. Hirshleifer (1971) shows information has no social value in a pure exchange economy. There-

fore, agents in a pure exchange economy with random endowment can be hurt. Better information

decreases the amount of risk to share. Thus there is less trading on the risk-sharing market. This

phenomenon occurs in ﬁnancial markets (Goldstein and Yang, 2017b). If agents trade less goods

between them, it means they rather tend to consume their endowments. Thus, the new allocation

of risk becomes Pareto inferior to the one with no information. Schlee (2001) shows that one suﬃ-

cient condition, for the better information to be Pareto inferior in a pure exchange economy, is that

«all agents are risk averse and the economy has a representative agent who satisﬁes the expected

utility hypothesis with a concave diﬀerentiable von Neumann-Morgenstern utility function.» In this

5

case, the concavity of the utility function in beliefs make the agents dislike information in a pure

exchange economy. In our model all the agents have a CARA utility function which satisﬁes the

criterion for a representative agent. This is equivalent to the equilibrium prices reﬂecting a kind

of average of the risk aversions and of the conditional variances of each agent according to their

information set and preferences (Lintner, 1969).

Nonetheless, our equilibrium is not an endowment economy. Storers can transfer an amount of

commodities from a period to an other. Our model is a production economy because storers can

carry one unit from the ﬁrst period to the ﬁnal one after. A better information can help producers

to make better decisions about their output level (Eckwert and Zilcha, 2001). I get two contrary

eﬀects: the decrease of the risk-sharing business which harms operators while the improvement of

production decisions can improve welfare. Therefore, information can increase or decrease agents’

well-being. Prior to signals release, traders do not know in which direction prices will move. Sul-

ganik and Zilcha (1996) study an equilibrium with competitive risk-averse ﬁrms which hedge the

foreign exchange risk of their production they sell abroad. They show that more information is

not always beneﬁcial on futures markets for foreign exchange. The futures price can change in a

disadvantageous direction for traders, speciﬁcally when there is a positive risk premium.

The other issue is the bias in the futures price which is also called the «risk premium». If the

equilibrium is fully revealing, the futures price can be a suﬃcient statistics meaning that it in-

cludes all the necessary information to get the best estimate. However, it can be a biased statistics.

Moosa and Al-Loughani (1994) tested both unbiasedness and eﬃciency of the futures price of WTI

crude oil for the period from January 1986 to July 1990. Their time series are monthly. They ﬁnd

a bias when they regress monthly spot returns on monthly futures returns. Moreover, they ﬁnd

autocorrelated residuals which mean that the futures price is not an eﬃcient forecaster. Last, they

ﬁnd a time varying risk premium which they ﬁt adequately with a GARCH-M(1,1) process. Chinn

and Coibion (2014) run the tests for four diﬀerent types of commodity prices: energy, agricultural

products, precious metals, and base metals. For energy, they include petroleum, natural gas, gaso-

line, and heating oil. Corn, soybeans, and wheat are the three agricultural commodities in their

sample. For precious metals, they consider gold and silver while their set of base metals consists

of aluminum, copper, lead, nickel, and tin. Their monthly data cover the period from 1990 to

2012. They ﬁnd evidence of unbiasedness for energy but not for other commodities. However, they

ﬁnd that the increase of liquidity did not reduce bias. It did not improve eﬃciency neither. They

noticed an increased comovement among commodity futures and that since the 2000s, the basis

tended to lose forecasting power, meaning the predictive content of commodity prices declined.

Vives (2010) elaborated a two-period model of asymmetric information in a futures market.

There are on one hand speculators who are whether informed or uninformed. The informed specu-

lators have a signal on the future spot price at the next period. In the Vives model, the spot price

is exogenous to the futures price. On an other hand, there are hedgers who are informed about an

individual endowment shock. This endowment is the quantity that the hedger, who is a processor,

will sell on the manufactured good markets. The model is quite complicated to solve with strong

limitations such as the exogeneity of the spot price and of the quantities of manufactured goods.

The issue is the same for Perrakis and Khoury (1998) where their spot price is also exogenous in

their asymmetric information model. In their model, the spot price is a martingale process. Thus,

they test their model on three commodities markets of the Winnipeg Commodities Exchange’s

(WCE), the Canadian futures market of commodities, between 1982 and 1994. At this time, the

WCE was the only futures market for canola and barley. As a benchmark, they tested their model

on the wheat market. Wheat is exchanged in Chicago too. They found non signiﬁcant results

for wheat markets but they got a signiﬁcantly Fully-Revealing Rational Expectations Equilibrium

(FRREE) for the barley and canola markets. The authors acknowledge that the assumption of an

exogenous price is reasonable for small markets like canola and barley at the time of the WCE.

However, it seems less consistent for ﬁnancialized markets. The activities of hedger present on

both markets may have consequences on the spot market. It is true in particular for storers. If the

futures price is high, they will buy more spot. Therefore, the demand increases on the spot market

6

so the spot price increases.

I show that it is possible to have a very tractable model with endogenous spot prices and quan-

tities of manufactured goods with an asymmetry on the net spot demand. The traders have a

diverse information. Private information is revealed to traders through signals which are known

only by the recipients. Each group of operators is endowed with a common signal. Thus, ﬁnancial

traders get their own signal common to their whole group. They include this information in their

set of knowledge to take their positions. Therefore, they inﬂuence, through their positions, the

future price which does aﬀect the hedging pressure and thus the stored quantities and the proces-

sors’ demand. I observe a «Hirshleifer eﬀect» in some cases which is quite similar to Sulganik

and Zilcha (1996), with prices moving in a disadvantageous direction for traders. Nonetheless, this

eﬀect exists for both positive and negative risk premium.

2 The settings of the model

I extend the two-period rational expectations equilibrium model of Ekeland et al. (2018) to in-

troduce the additional feature of information asymmetry à la Grossman (1977). Therefore, the

description of the model is largely inspired by the paper of the core model (Ekeland et al., 2018).

The model is based on three periods. In t=0, the markets are not open yet and there is no infor-

mation. I call this period of the market ex ante. The unconditional moments are computed.

There is one commodity, a numéraire, and two markets: the spot market at times t = 1 and t = 2

and a futures market in which contracts are traded at t = 1 and settled at t = 2. The model allows

for short positions on the futures market. When an agent sells (resp. buys) futures contracts,

her position is short (resp. long), and the amount of futures contracts she holds is negative (resp.

positive). On the spot market, short positions are not allowed. There is a binding nonnegative

constraint on inventories. In other words, the futures market is ﬁnancial, while the spot market is

physical. There are three kinds of operators which make intertemporal decisions:

•Storers or inventory holders (I) have storage capacity and can use this capacity to buy the

commodity at t = 1 and release it at t = 2. They trade on the spot market at t = 1 and at

t = 2. The storers also operate on the futures market. Thus they can hedge the sale of their

inventories at the second period on the futures market at the ﬁrst period. They are naturally

long on the spot market.

•Processors (P), or industrial users, use the commodity to produce other goods that they sell

to consumers. Because of the inertia of their own production process and/or because all of

their production is sold forward, they decide at t = 1 how much to produce at t = 2. They

cannot store the commodity, so they have to buy all of their input on the spot market at t =

2. They also trade on the futures market. Thus they can hedge the purchase of their inputs

at the second period on the futures market at the ﬁrst period. They are naturally committed

to buy on the spot market.

•Speculators (S), or money managers, use the commodity price as a source of risk to make

a proﬁt out of their positions in futures contracts. They do not trade on the spot market.

Speculators play a role of liquidity providers in the futures market (Vives, 2010). They share

risk with hedgers. Market-making or risk-bearing is a source of beneﬁts.

There is a weight (Nj)j∈{I,P,S}for each of the groups described above. I assume that all agents

(except the spot traders) are risk averse, inter-temporal utility maximizers. They make their

decisions at time t = 1 according to their expectations for time t = 2. Spot traders do not

participate in the futures market. For small businesses like farms, learning futures trading and

transaction costs can be a signiﬁcant deterrent to trade futures contracts (Hirshleifer, 1988). Thus,

some operators in the spot market renounce to participate to the futures market.

Further, the futures and spot markets operate in a sort of partial equilibrium framework: in

the background, there are other sellers of the commodity, and processors as well. These additional

7

agents are referred to as spot traders, and their global eﬀect can be described by a demand function.

I will use the notation "˜" for the realized values of the random variables at the period 2. All traders

make their decisions at time t = 1, conditionally on the information available for t = 2. The timing

is as follows:

•For t = 1, the spot and the futures markets are open. Spot traders supply ω1and demand

µ1−mP1. The spot price is P1, futures price is Fand mis the elasticity of the spot demand.

•For t = 2, the spot market is open and the futures contract are settled. Spot traders supply

˜ω2and demand ˜µ2−mP2. The spot price is P2. The futures contracts are then settled.

I assume that there is a perfect convergence of the basis at the expiration of the futures

contract. Thus, at time t = 2, the position on the futures market is settled at price P2that

is prevailing on the spot market.

I assume that (µ2, ω2)is a vector of normal variables and cov(µ2, ω2) = 0, then µ2and ω2are

independent. ˜

ξ2= ˜µ2−˜ω2is the realized value of the net exogenous demand realized at period 2.

For simplicity’s sake, we will say net demand. The normality of variables allows the existence of a

linear equilibria with information asymmetry. My model follows a linear-normal setting.

I distinguish private and public information (Tang, 2014). Private information is content which

is known only by a share of the population of operators. In our model, each group (speculators,

storers and processors) is endowed with a signal which is common to every of its members. A signal

which is known only by a speciﬁc group is thus considered as private. At time t = 1, operators

receive a signal (sj)j∈{I,P,S }common to the group which they belong. This signal is unbiased such

as:

∀j∈ {I, P, S}, sj=˜

ξ2+εjwith εj∼N(0, σ2

j)(1)

I could assume groups receive information about the random spot supply or demand but it does

not change the outcome of the model. In a rational expectations equilibrium (REE) à la Grossman

(1977), owning an asset brings a payoﬀ in the last period. The main unknown factor is what is

random. An important point is our REE relies on a linear-normal model Vives (2010). The latter

implies the payoﬀs are linear-quadratic. The random parameters and the signals follow a normal

distribution. Therefore, the conditional expectations are aﬃne so they are the sum of a linear

combination of the informative variables (signals and prices) and a constant (the unconditional

expectation). A linear combination of normal variables just gives an other variable. Agents antici-

pate the random factor of the payoﬀ which follows a normal distribution. Whatever, the factor is

a sum of normal variables or not, there is no diﬀerence, because we still end with a unique normal

variable which agents estimate.

The assumption of a common group signal allows us to separate the issues of liquidity and infor-

mation. Indeed, a new agent does not bring a new information necessarily. Operators can get their

information from common sources like forecasts from institutions. Moreover, some components of

the net demand may be impossible to uncover (Stein, 1987). I make the assumption that there

is no information cost. It allows us to study the fully-revealing rational expectations equilibrium

(FRREE) which is not implementable with information cost.

Before we proceed, some clariﬁcations are in order :

•Production of the commodity is inelastic: the quantities ω1and ˜ω2that reach the spot

market at times t = 1 and t = 2 are exogenous to the model. Operators know ω1and µ1

, and share the same prior about ˜ω2and ˜µ2. The operators making intertemporal decisions

(storers, processors and speculators) update their decision according to their information

set. The latter includes the signal received by the operator according to one’s group and

public information at time t = 1. I deﬁne public information as content known by the whole

population of operators. Everyone on the market knows prices. The last ones are endogenous

variables which are the results of clearing equations. Prices are the outcome of the positions

of the agents based on their information. Thus, operators can infer the private information

8

of the other agents from prices. Therefore, we can write the information set ((Fj)j∈{I,P,S })

such as :

∀j∈ {I, P, S},Fj= (sj, F, P1)(2)

•A negative spot demand equals extra spot supply. If for instance P1>µ1

m, then the spot price

at time t = 1 is so high that additional means of production become proﬁtable, and the global

economy provides additional quantities to the spot market. The number µ1(demand when

P1= 0) is the level at which the economy saturates that induces spot traders to demand

quantities larger than µ1,that is, the traders oﬀer a negative price P1< 0 for the commodity.

The same situation occurs at time t = 2.

•I set the risk-free interest rate at 0.

3 Main equations

In this section, we describe the main equations of the model. In subsection 3.1, we explain the

utilities and proﬁt functions of the industrial hedgers. From them, we derive optimal positions.

Then, subsection 3.2 computes the market-clearing equations. All the equations are taken directly

from Ekeland et al. (2018). I add an information set to for every group of traders.

3.1 Industrial hedging

Hedgers make two choices at t = 1. First, they choose the amount of commodities they will use

for their economic ativities. Second, they determine their positions on the futures market.

Subsubsection 3.1.1 shows how processors hedge, then subsubsection 3.1.1 details the hedging of

the storers and 3.1.3 derives the hedging pressures from the hedgers positions computed previously.

3.1.1 Processor’s hedging

The processor seeks to hedge the quantity of input y bought in the second period on the spot

market at the price P2. Z is a constant which depends of the output price.

The realized proﬁt function at time t=2 is:

πP= (y−β

2y2)Z−yP2+fP(P2−F)(3)

βis the parameter of the quadratic function of production. fPis the position of the processor on

the futures market.The processor’s utility is mean variance. Its maximization program is thus:

max

y∈[0,1

β],fP∈R

UP=E1[πP|FP]−αP

2Var1[πP|FP](4)

Such as FP= (sP, F, P1).αPand sPrespectively are the risk aversion and the signal of the

processor. Therefore, the processor’s optimal decisions (f∗

P, y∗)are:

y∗=Y∗

β Z (5)

f∗

P=y∗+E1[P2|FP]−F

αPVar1[P2|FP)(6)

Such as Y∗= max(Z−F, 0) which is gross payoﬀ from the arbitrage to hedge inputs.

The futures market is also used by the processor to plan his or her production. If the price of the

input F is below the margin per input Z, the processors will produce. The position on the futures

market can be decomposed into two elements: a hedging component y∗(the processor goes long on

9

futures contracts in order to protect himself against an increase in the spot price) and a speculative

one: E1[P2|FP]−F

αPVar1[P2|FP)(7)

Processors use their position on the futures market to speculate. The speculative component is

positive (resp. negative) if the expected spot price is higher (resp. lower) than the futures price.

Therefore, the overall position of processors is diﬀerent of the amount of inputs they need to hedge

if futures price is not equal to its expected payoﬀ. The separation of the physical and the futures

decisions is consistent with Danthine (1978).

3.1.2 Storer’s hedging

The storers buy units of input x in the ﬁrst period to sell them in the second period.

The storer gets the following realized proﬁt at time t=2:

πI=x(P2−P1) + fI(P2−F)−1

2Cx2(8)

C is the parameter for the quadratic storage cost function.

The program of the storer is thus:

max

x∈R+,fI∈RUI=E1[πI|FI]−αI

2Var1[πI|FI](9)

Such as FI= (sI, F, P1).αIand sIrespectively are the risk aversion and the signal of the storer.

I deﬁne also the optimal hedge position x∗:

x∗=X∗

C(10)

Such as X∗= max(F−P1,0), which is the gross payoﬀ of the contango arbitrage, excluding the

storage cost. If the futures price is higher than the spot price in the ﬁrst period, the storer will

store an amount of the commodity to sell it at the futures price.

The optimal futures position of the storer is:

f∗

I=−x∗

|{z}

hedging

+E1[P2|FI]−F

αIVar1[P2|FI]

| {z }

speculation

(11)

The separation between hedging and speculation is still veriﬁed. First, they hedge 100 percent

of their physical positions, then they adjust this position according to their expectations.

3.1.3 Hedging pressure

The hedging decisions described above are independent of the private signal, which is consistent

with Danthine (1978). The private signal has a direct consequence on the conditional expectation

of the spot price and indirectly by its precision on the risk-adjusted information advantage. While

hedging decisions are both independent of the degree of risk aversion and of the risk aversion.

The reason is that the futures price is certain. Therefore, the Danthine separation of hedging

and speculation decisions implies the hedging issue is solved in a certain environment, while the

speculative position is decided in an uncertain one. The storers compare the future price to the

spot price in the period 1. The processors do the same between the futures price and the forward

price of their output. The interests of the two categories are opposite. If the future price increases,

the storers have a stronger incentive to increase their hedge while the processors would wish to

10

decrease their one.

I will use synthetic weights of processing units (nP) and storing units (nI) when it is relevant:

nP:= NP

β Z (12)

nI:= NI

C(13)

The hedging pressure (or the unbalance of hedging positions) is represented by

HP := nIX∗−nPY∗(14)

It is important to notice the hedging pressure is a weighted sum of the hedging positions. They are

not adjusted by the speculative components of the futures positions of the hedgers. Therefore, the

hedging pressure is public information because it relies on the prices which are known by everyone.

3.2 Clearing of the markets

All the agents have a mean-variance program and the received signal is the same for all the agents

of the same group. For example, all the processors have the same signal. Likewise storers and spec-

ulators have an other signal which is identical for all the population in their group. Subsubsection

3.2.1 lists the optimal positions for the diﬀerent groups on the futures markets. Subsubsection 3.2.2

shows the spot clearing conditions and subsubsection 3.2.3 the futures clearing condition. Then we

derive the system of the market clearing conditions in subsubsection 4.1.1.

3.2.1 The optimal positions on the futures markets

The traders are endowed with the information of their group. The set of their information includes

the price and the private signal common to all the group members. The speculators do not hedge so

their position is limited to a speculating component. According to the hedging positions described

by (5) and (10), we get:

fI=E1[P2−F|FI]

αIVar1[P2|FI]−X∗

C(15)

fP=E1[P2−F|FP]

αPVar1[P2|FP]+Y∗

β Z (16)

fS=E1[P2−F|FS]

αSVar1[P2|FS](17)

Fjand αjfor j=I, P, S respectively stands for the information set which is unbiased and for the

risk aversion. Obviously, speculators have a speculating position only and not a hedging position.

I notice, the conditional moments of operators are included in the futures positions and not in the

spot ones. Therefore, the information goes ﬁrst through the futures price.

3.2.2 The clearing of the spot market

On the spot market, there is a physical constraint on the market-clearing condition. Only positive

quantities are allowed. Thus, the supply has to be equal to the demand to clear the spot market.

At the ﬁrst period, the supply and the demand of the spot traders are known, respectively µ1and

ω1. At the period 2, the random supply ( ˜ω2) and demand of the spots traders (˜µ2) are not known

in the ﬁrst period. The storers buy a quantity nIX∗in the ﬁrst period to sell it in the next period.

The processors buy the quantity nPY∗on the spot market at period 2 that they hedged in the

previous period. So we can derive the market-clearing conditions for both periods:

ω1

|{z}

spot supply

=nIX∗

|{z }

storage in

+µ1−mP1

| {z }

spot demand

(18)

˜ω2

|{z}

spot supply

+nIX∗

|{z }

storage out

=nPY∗

| {z }

processors demand

+ ˜µ2−mP2

| {z }

spot demand

(19)

11

I write the clearing equation of the spot market in the second period (19) in function of the hedging

pressure deﬁned in (14):

P2=˜

ξ2−HP

m(20)

Such as ˜

ξ2= ˜µ2−˜ω2which is the random exogenous net demand. I deduce the conditional moments:

E1[P2|Fj] = E1[ξ2|Fj]−HP

m(21)

Var1[P2|Fj] = Var1[ξ2|Fj]

m2(22)

The hedging pressure (H P ) and the storage level (nIX∗) are functions of the futures and spot

prices (respectively Fand P1) according to the deﬁnition (14) and the clearing condition (18).

Moreover, the spot price at maturity (P2) varies negatively with the hedging pressure according to

(20). I already highlight a feedback from the futures price to the spot prices at both periods. If

information varies, the futures positions are modiﬁed. Thus, the futures price varies which change

the hedging pressure and the spot price at t = 1 and t = 2.

3.2.3 The clearing of the futures market

For the futures market, we get the following market-clearing condition:

X

j={I,P,S}

Njfj= 0 (23)

According to the agents’ positions given by (15),(16) and (17), the clearing equation (23) becomes:

NI(E1[P2|FI]−F

αIVar1[P2|FI]−X∗

C) + NP(E1[P2|FP]−F

αPVar1[P2|FP]+Y∗

β Z ) + NS

E1[P2|FS]−F

αSVar1[P2|FS]= 0

⇔X

j={I,P,S}

Nj

E1[P2|Fj]−F

αjVar1[P2|Fj]−H P = 0

I get the futures price:

F=1

ΥP

(X

j={I,P,S}

Njψj,pE1[P2|Fj]−HP)(24)

Such as:

ΥP:= X

j={I,P,S}

NjΨj,p (25)

ψj,p := (αjVar1[P2|Fj])−1, j =I , P, S (26)

Ψj,P is the inverse of the product of the risk aversion and of the conditional variance of the price

to the signal of the agent i. It is the risk-adjusted information advantage (Vives, 2010). Higher it

is, higher is the speculative position which is equal to the spread between the conditional expected

spot price times the informational advantage. ΥPis related to market depth because it impacts

the sensitivity of the futures price to the hedging pressure (HP).

If we inject the conditional expectations (21) and variances (22) of net demand in the equation of

the futures price according to the moments of the spot price at maturity (24), we get:

F=1

ΥξX

j={I,P,S}

Njψj,ξ

E1[˜

ξ2|Fj]

m−1

mφ HP (27)

12

Such as :

Υξ:= X

j={I,P,S}

Njψj,ξ (28)

ψj,ξ := (αjVar1[ξ2|Fj])−1, j =I , P, S (29)

φ:= 1 + 1

mΥξ

(30)

The sensitivity (φ) of the demand to the hedging pressure (Ekeland et al., 2018) is given by (30).

When the hedging pressure (HP changes by one unit, the futures price moves by φ

m. The market

depth, which we deﬁne as the inverse of the impact of the hedging pressure on the futures price

(Vives, 2010), can be measured thus by m

φ. The futures market is deep if a variation of the hedging

pressure is absorbed with a limited impact on the moves of the futures price. Higher is Υξ, lower is

the sensitivity of the price to the hedging pressure. When Υξleans toward inﬁnite, the sensitivity

of the futures price to the hedging pressure declines toward one and so the market depth tends to

m.

4 Characterization of the equilibrium

Above, we have deﬁned a suﬃcient condition for the existence of a rational-expectation equilibrium.

Now, we deﬁne the necessary conditions.

4.1 Solving the equilibrium

4.1.1 System to solve

Eventually, we have the market-clearing equations (Ekeland et al., 2018). They correspond to the

equilibrium on the spot and on the future markets. On the spot, the supply is equal to the demand.

The short positions are forbidden. While on the futures market which is ﬁnancial, the equilibrium

is met when the sum of the positions is null. Therefore, we get:

P1=1

m(ξ1+nIX∗)

P2=1

m(˜

ξ2−H P )

F=Pj={I,P,S }NjΨj,ξE1[ξ2|F ,sj]

mΥξ−1

mφ HP

(31)

Such as the following deﬁned variables are:

nP:= NP

β,Z

nI:= NI

C

X∗:= max(F−P1,0)

Y∗:= max(Z−F, 0)

HP := nIX∗−nPY∗

(32)

4.1.2 A piecewise linear equilibrium

I can deﬁne the equilibrium from the market-clearing conditions (31).

Deﬁnition 4.1 (Equilibrium) An equilibrium is a family of quantities and prices (X∗, Y ∗, P1, F, P2)

such that:

1. The nonnegativity constraint of quantities is fulﬁl led : (X∗, Y ∗)∈R2

+

2. Prices are nonnegative: F≥0,P1≥0and P2≥0almost surely.

13

3. Each agent, of a group j={I, P, S}, relies on one’s information set which is composed of a

private signal (sj=˜

ξ2+j) and public information which includes the futures price F and

the spot price P1. Therefore, we get the following information set :

∀j∈ {I, P, S},Fj= (sj, F, P1)(33)

4. The following market-clearing conditions for the spot and futures markets in the ﬁrst period

is fulﬁlled :

mP1−nIX∗=ξ1

mF +φHP =Pj={I,P,S }NjΨj,ξE1[ξ2|Fj]

Υξ

(34)

5. The following condition for the spot market at maturity is fulﬁlled:

P2=1

m(˜

ξ2−nIX∗+nPY∗)(35)

There is an unique equilibrium for each sub-region. I get 4 regions (Ekeland et al., 2018):

•The region 1 where F > P1and Z > F so both kind of industrialist are hedging.

•The region 2 where F > P1and Z < F so storers are hedging only.

•The region 3 where F < P1and Z < F so no one is hedging.

•The region 4 where F < P1and Z > F so processors are hedging only.

Figure 1: Physical and ﬁnancial decisions in space (P1, F ): the four regions deﬁned by Ekeland

et al. (2018)

I will show that this equilibrium is fully revealing through the futures price. It means that all

the private information is revealed through the futures price. More precisely, the futures price is a

suﬃcient statistics of all the private signals.

4.2 A fully-revealing equilibrium based on the futures price

4.2.1 The characteristics of a FRREE

A fully revealing rational expectations equilibrium (FRREE) means that the information from

the private signals to make the best estimate of the payoﬀ is revealed by the prices (Grossman,

1976). Mathematically, the prices are a suﬃcient statistics of signals. Therefore, knowing prices is

equivalent to know all the signals. I will show that the futures price is a suﬃcient statistics alone.

Thus, the futures price is an eﬃcient est fulﬁlls its function of price discovery.

14

Deﬁnition 4.2 (Fully-Revealing Rational Equilibrium) With Sas the set of the private sig-

nals such as S={(sI, sP, sS)},P2as the spot price at maturity and FM= (F, P1)the market

information set which includes the prices in the ﬁrst period, a FRREE exists if the two following

conditions hold (Bray, 1981):

E1[P2|FM] = E1[P2|S](36)

Var1[P2|FM] = Var1[P2|S](37)

The private signals are normal variables, as explained in section 2, thus (36) implies (37).

The conditions described by Bray (1981) have an economic meaning. They disregard their private

signal to look at the information included in the prices only. A FRREE implies the same conditional

moments because the agents get the same information. Therefore, the speculative positions vary

according to the risk aversion only. In a FRREE, if we compute the ratio of two speculative

positions, the result will be a ratio of the risk aversion.

Nonetheless, we will reduce the market information set to the futures price for two motives.

First, the expectations are included in the speculative positions which are traded directly on the

futures market. Second, the expectations are included in the spot price when the latter is a function

of the futures price. This condition is met only when the market is in contango (F > P1).

First, we will show that a FRREE through the futures price exists and then it is unique.

4.2.2 Existence of a FRREE

The concept of an artiﬁcial economy was ﬁrst coined by Grossman (1978) and then generalized by

Bray (1981). In the artiﬁcial economy, the traders pool their priors before trading. It means the

whole information in the economy is common knowledge. This situation is equivalent to a FRREE

because a suﬃcient statistics is equivalent to own all the information. Therefore, if an equilibrium

exists in an artiﬁcial economy, it means that the FRREE exists too. I will assume an artiﬁcial

economy with K groups and I the set for all the information. I will demonstrate the theorem of

existence below. I have chosen the name of the "Bray-Grossman theorem" in tribute to the two

authors quote above.

First, we show a lemma of separation between the speculative and the hedging positions, short or

long. This lemma allows us to prove the existence of the FRREE with a hedging pressure. I call it

the Danthine lemma in tribute of Danthine (1978) which ﬁrst establishes a separation result for a

long hedging position.

Lemma 4.1 (Danthine Separation) I call a proﬁt function Danthine-separable if the futures

position, derived from the optimization of the expected utility, is a sum of the speculative and the

hedging positions. A proﬁt function such as πj=fj(P2−F) + hjP2+Hj(hj)of agent j with a

mean-variance utility, such as EUj[πj|πj] = E1[πj|Fj]−αj

2Var1[πj|Fj], is Danthine-Separable if

Hj(hj)is continuously diﬀerentiable (class C1) and does not depend of any conditional moment or

random variable. The function Hjis the impact of the hedgers’ economic activity on one’s proﬁt.

fjis a futures position and hjis a hedging position. P2is the spot price at maturity and Fis the

futures price.

Proof.

The program of the agent j is:

max

hj∈Dj,fj∈REUj[πj|Fj] = E1[πj|Fj]−αj

2Var1[πj|Fj](38)

I get the following ﬁrst-order conditions:

EUj[πj|Fj]

∂ fj

= 0 (39)

EUj[πj|Fj]

∂ hj

= 0 (40)

(41)

15

These conditions give respectively:

fj=E[P2|Fj]−F

αjVar[P2|Fj]−hj(42)

H0(hj) = αjVar[P2|Fj](fj+hj)−E[P2|Fj](43)

If we substitute the expression of fjin (42) to (43), we get :

H0(hj) = F(44)

Hjis a C1function so there is a solution to the equation above. This solution does not depend

of any moment or random variable. Therefore, we show there is a separation between speculation

and hedging.

End of proof.

If the agent is naturally a short hedger, the hedging position is positive (hj≥0) and Hj(hj)

is a cost function. F is the price rewarding a long position. The long hedger increases her long

hedge until the marginal cost is equal to the price. In our model, the storer has a storage level (x)

associated to a cost function C(x) = P1x+C

2x2, which is increasing and convex. If at the opposite,

the agent is naturally a long hedger, the hedging position is negative (hj≤0) and Hj(hj)is a

revenue. Therefore, a long hedger increases her marginal revenue until it is equal to F, which acts

as a constant marginal price. In the situation of a long hedger, it is equivalent to assume a proﬁt

function πj=fj(P2−F)−hjP2+Hj(hj)with hj≥0. I make this choice in our model for clarity’s

sake. A processor has a quantity of inputs y≥0with a gross proﬁt function (G(y) = y−β

2y2). It

would have been equivalent to assume y≤0with a gross proﬁt function such as (G(y) = −y−β

2y2).

Now, we can separate speculative and hedging positions. Therefore, we can show the existence

of a FRREE in a futures market where a hedging pressure exists. The latter needs to be linear in

the futures price to get an equilibrium in the linear-normal setting.

Proposition 4.2 In an eﬃcient market, the relationship between the futures price and the hedging

pressure is linear.

First, let us consider the region 1 where F > P1and Z > F .

According to the spot price at the ﬁrst period given by the market-clearing conditions (31), the

spot price is a linear function of the futures price when the storers are active (nI>0):

P1=µ1−ω1+nIF

m+nI

(45)

The futures price is itself a function of the suﬃcient statistics. The spot price does not reveal

more information than the futures price because it is a function of the suﬃcient statistics through

the futures price. I notice, the spot price does not include any information when the storers are

inactive (nI= 0) because the spot price is not a function of the futures price. The information is

conveyed from the futures market to the spot market in contango but is not when the market is

backwardated. Nonetheless, both prices are simultaneous so the transfer is instantaneous. If the

market is in contango, there is price discovery on both markets while there is only information in

the futures one in case of backwardation.

I consider the following linear hedging pressure of the futures price:

HP =−γ0+γ1F(46)

According to the hedging pressure given by the market-clearing conditions (31) we get in the region

1:

γ1=mnI

m+nI

+nP(47)

γ0=nPZ+ (µ1−ω1)nI

m+n