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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».
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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
µ1mP1. 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 ˜µ2mP2. 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 εjN(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= (yβ
2y2)ZyP2+fP(P2F)(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],fPR
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(ZF, 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(P2P1) + fI(P2F)1
2Cx2(8)
C is the parameter for the quadratic storage cost function.
The program of the storer is thus:
max
xR+,fIRUI=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(FP1,0), 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 := nIXnPY(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[P2F|FI]
αIVar1[P2|FI]X
C(15)
fP=E1[P2F|FP]
αPVar1[P2|FP]+Y
β Z (16)
fS=E1[P2F|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 nIXin the first period to sell it in the next period.
The processors buy the quantity nPYon 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
+µ1mP1
| {z }
spot demand
(18)
˜ω2
|{z}
spot supply
+nIX
|{z }
storage out
=nPY
| {z }
processors demand
+ ˜µ2mP2
| {z }
spot demand
(19)
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I write the clearing equation of the spot market in the second period (19) in function of the hedging
pressure defined in (14):
P2=˜
ξ2HP
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]
m1
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(˜
ξ2H P )
F=Pj={I,P,S }NjΨj,ξE1[ξ2|F ,sj]
mΥξ1
mφ HP
(31)
Such as the following defined variables are:
nP:= NP
β,Z
nI:= NI
C
X:= max(FP1,0)
Y:= max(ZF, 0)
HP := nIXnPY
(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: F0,P10and P20almost surely.
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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 :
mP1nIX=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(˜
ξ2nIX+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.
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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(P2F) + 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
hjDj,fjREUj[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 (hj0) 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 (hj0) 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(P2F)hjP2+Hj(hj)with hj0. I make this choice in our model for clarity’s
sake. A processor has a quantity of inputs y0with a gross profit function (G(y) = yβ
2y2). It
would have been equivalent to assume y0with a gross profit function such as (G(y) = yβ
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=µ1ω1+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+ (µ1ω1)nI
m+n