Modelling Dynamic Storage Function in Commodity Markets:
Theory and Evidence
Luca Pieroni (*) and Matteo Ricciarelli (**)
Abstract
Stockholding decisions in agricultural commodity markets represent a source for strengthening the risk
management techniques related to the future markets development even if this heterogeneity and the lack
of an observable parameter summarizing stockholding decisions complicate the market analysis.
In this work, we propose a dynamic way to obtain a consistent stockholding decision rule in which the
cash and storage markets are modelled simultaneously.
We find three results: i) the definition of the marginal storage value arises directly from the optimization
program; ii) in analogy with investment theory, we use a co-state variable to model the stockholding
behaviour and iii) the reduced form of the model accounts for institutional changes. Using the U.S. corn
market data from 1973 to 2000, the empirical results, including the future price derivation, are statistically
robust and economically coherent.
JEL Classification Number: G13; L71, Q40
Keywords: Spot and Storage Markets, Convenience Yield, Inventories.
Acknowledgement: We are grateful to David Aristei for his research assistance, Federico Perali for the
useful suggestions. All errors are our own.
(*) Corresponding Author, Department of Economics- University of Perugia, Via Pascoli 20, 06123,
Perugia (Italy). Phone: +390755855280. Mail Contact: [email protected]
(**) Department of Economics and Institutions- University of Rome “Tor Vergata”, Via Columbia 2,
00133, Rome. Mail Contact: [email protected]
1. Introduction
The mechanism determining the spot price in agricultural commodity markets is
affected by the length of the production process; it is a “stylized fact” that agents
address the production plans prior to the production yield, so that the expectation
shaping mechanism is accounted for in the investment decisions. A source of
uncertainty associated with an industrial process leads the economic agents to store
inventories, even if they do have not speculative opportunities. Hence, a generalized
storage market is defined, since there is a demand for stocks of speculators, processors
and institutional agents and exists a stock supply, which reflects the willingness of the
producers of goods to release current stocks for consumption at a later time (Thurman,
1988).
Although we start our reasoning from the competitive rational expectation model
(Deaton Laroque, 1992; Chambers and Bailey, 1996) in which the state variable of the
production (or availability) is sufficient to solve the equilibrium model, we emphasize
the role played by stock levels in determining the equilibrium in the storage market. In
this paper, we follow Pindyck (1994, 2002) and use the short-run commodity market
framework to investigate the impact of a dynamic equilibrium model for prices and
stocks. In analogy to dynamic investment theory developed by Tobin (1963) and
Hayashi (1982), we make use of the discrete net change in stock (∆FS (t ) ) as a co-state
variable in order to obtain the dynamic relationship between storage and cash market.
As will be clarified in the remainder of the paper, this framework is necessary because
we generalize the competitive storage model with speculator agents in a more realistic
model invoking the co-existence of two categories of agents, processors and
speculators, which have different behaviours regarding the stockholding accumulation
activity. In fact, to the speculative arbitrage between current and expected prices, we
also include processors in the model whose inventories are accumulated for
precautionary motives related to their activity1. In the storage market, the
(unobservable) price is given by the marginal storage value (λ) which includes, as main
factor, the marginal convenience yield. Pindyck (1992) defines the marginal
1
Different from Revoredo (2001), we show that the behaviour of the agents can not be enclosed in a
unique and over-simplified decision rule concerning the stock accumulation.
2
convenience yield the value of any benefits that inventories provide, including the
ability to smooth production, avoid stock-outs and facilitate the scheduling of
production and sales2.
Our approach has several attractive elements. First, the storage depreciation does not
exclusively depend on physical factors, but is also affected by technological change and
institutional variables. The latter variable allows us to check the impact of government
policy. Second, we propose to solve the link between cash and storage markets, by
introducing a double constraint: i) the market clearing equation to obtain the spot price
equation and ii) the motion law for inventories as an expression of the marginal
convenience yield (Pindyck, 2002). Third, we introduce the dynamic investment
framework to derive consistent stockholding decision rules; thus, we account for an
endogenous risk source that is due to the uncertainty in stockholding activity. A relevant
feature of this activity is that the stock amount in a given period does not necessarily
coincide with the availability in the next period depends on the waste or physical
depreciation.
Our main results arise directly from the continuous optimisation program and they are
summarized as follows:
*) we model two simultaneous markets - spot and storage markets maximizing the
discounted profit function and carrying out respectively a closed form for the spot price
and the marginal storage value;
**) by the marginal storage value, we can define a new state variable, q(t), as the ratio
of the marginal storage value on the replacement cost of a stored bushel; this is the tool
by which we investigate the simultaneous dynamics in spot and storage markets and
account mainly for shocks that agents consider to be transitory, whereas the effects of
permanent shocks could be questionable;
***) the empirical results are consistent with the theory. The estimation parameters in
the spot and storage markets allow us to derive the future price, verifying the statistical
robustness with the future observed prices.
2
Following the definition we relinquish the non-linearity framework to explain the price path. Even if
stock-out are virtually possible, the following arguments can be quoted: i) stock-out might occur with a
low probability; ii) stock-out can be empirically justified by looking at more disaggregated data (i.e. for a
single item or firm).
3
The rest of the paper is organized as follows. In Section 2, we present the theoretical
framework by fully characterizing the underlying U.S. corn market. We also take care to
explain the different role attributed to storage depreciation in our model. In Section 3.1,
we find the results of a dynamic optimization problem and comment on the role played
by the storage depreciation. In Section 3.2, we exploit the dynamic optimization results
to produce phase diagrams and graphs discussing the interactions among market
variables and their adjustment processes. In Section 4, we specify the econometric
model and, in particular, we model the marginal convenience yield to obtain useful
estimations. In Section 5.1, we describe the data, by checking the data generating
process of the variables, to correctly specify the econometric model. In Section 5.2, we
present the Full Information Maximum Likelihood (FIML) estimation for spot price and
the marginal convenience yield equation with a comment about the ability of the model
to match the observed data. The computations for the stocks level and future price are
also given. Finally, we make some concluding remarks.
2. The Theoretical Framework in Agricultural Commodity
Markets
The commodity prices in a setup where there is a profit maximizing agent that can carry
the goods as inventory from the current to a future period has been the main objective of
many papers in the last two decades (Deaton and Laroque, 1992, 1996; Miranda and
Rui, 1995; Ng, 1997). We propose to generalize a simultaneous competitive market, in
which the decisions of agents in the storage market are considered jointly within the
cash market. Therefore, we have that unobservable behaviour in the storage market
depends on the different aims of agents. In addition to the speculative motive, in which
the agents hold stock in order to make arbitrage by purchasing commodity and then
gambling on increasing price, we try to justify a heterogeneous behaviour with
precautionary motives by industrial consumers and public institutions: on one side,
industrial consumers reduce the cost, avoid stock-out and, generally, facilitate their own
4
production processes. On the other side, public institutions aim for a price stabilization3,
increasing the market efficiency by the reduction of volatility.
In light of these considerations, we are interested in extending the profit maximization
approach by the introduction of different variables added to internal availability
dynamics (Deaton and Laroque, 1992). By the investment theory dynamic principles
(Tobin, 1963; Hayashi, 1982), we include an equation for the marginal convenience
yield which, in the storage market, regulates the path of stockholding convenience. We
can use the discrete net change in stocks (∆FS (t ) ) as a co-state variable, which
represents the dynamic link between cash and storage markets. Consequently, we have
the motion law for the marginal storage value, extracting the spot price as a function of
the marginal net convenience yield.
In this framework, the agents “control” production and stockholding level. The profit
maximization program, with respect to the production variable determines the spot price
in the cash market. The maximization is subject to the market clearing constraint
through which the internal availability has to be equal to the demand for export and/or
consumption and for storage (Routledge, Seppi and Spatt, 2000). In this section we
analytically specify the relationships in the cash market by explicitly investigating the
fluctuations in the supply and/or demand side. Moreover, as will be shown in the next
section, market clearing constraint is a bridge between the changes in the stocks and the
dynamic storage function.
Analytically, in order to link the maximization process to the classical storage model
(Wright, 1991), we reproduce the functional physical components of market clearing
and use production (Q(t)) to find a cost function by inversion. A theoretical approach is
used to derive the supply function from behavioural relationships which describe the
interactions and expectations of the production system. The set of behavioural
relationships in the production from which we move is based on the following equation
(Chavas, Helmberger, 1996):
(
)(
Q(t ) * = HA(t ) * YD(t ) *
)
(1)
3
Another aim of public intervention could be to limit the public stockholding, which is very expensive to
maintain over the time.
5
(
The produced quantity stems from the interaction between the harvested area HA(t ) *
(
)
)
and the yield of that area YD(t ) * . The equation (1) overcomes the over-simplifying
i.i.d. hypothesis proposed by Deaton and Laroque (1992). To relax the i.i.d. hypothesis,
we take the logarithm in (1) and separate the harvested area and yield (Chambers and
Bailey, 1996). We introduce the following structural relationships to explain the
underlying processes for a harvested area:
(
)
HA(t ) = ln HA(t ) * = β 0 + β 1 AP (t ) + b(t )
(2)
where APt is the extension of the cultivated corn area in terms of planted area; the APt
variable is assumed to be linearly linked to the current prices of the same commodity
( Pm (t ) ) and its substitute ( Ps (t ) ) according to the above equation:
AP (t ) = ln AP (t ) * = κ 0 + κ 1 Pm(t ) + κ 2 Ps (t ) + k (t )
(3)
Substituting (3) for (2), we get an explicative relationship for the harvested area4:
HA(t ) = β 0 + β 1κ 0 + β 1κ 1 Pm(t ) + β 1κ 2 Ps(t ) + β 1 k (t ) + b(t )
(4)
The other explanatory variables for the supply equation are in the yield equation
specified as follows:
YD(t ) = ln YD(t ) * = ω 0 + ω 1 PR (t ) + ω 2TH (t ) + ω 3Wage (t ) + w(t )
(5)
where (PR(t ) ) is an average precipitation index during the flowering stage, (TH (t ) )
represents the effects of technological modifications in agricultural production, whereas
(Wage(t ))
is a measure of the unit cost in labour input. The supply equation is then
completed by a set of technical and economic variables that evaluate the potential
production. We find:
Q (t ) = ln Q (t ) * = α * + β 1*Wage (t ) + β 2* Pm(t ) + β 3* Ps (t ) + β 4* PR (t ) + β 5*TH (t ) + e(t ) * (6)
where the next equalities has been set:
4
Such an equation takes some features from Nerlove ‘s model (1989) and is introduces them to capture
decision process of the farmer in allocating the available areas. This improves the over-simplifications of
the Nerlove model which also involve other variables. The myopic behaviour implied in this equation is
not in contrast with the rationality environment of the paper. In fact, the rationality comes from the
stockholding strategies adopted by the agents.
6
α * = β 0 + β 1κ 0 + ω 0
β 1* = ω 3 , β 2* = β 1κ 1 , β 3* = β 1κ 2 , β 4* = ω1 , β 5* = β 1κ 3
e(t ) * = β1 k (t ) + b(t ) + w(t )
Equation (6) is a well-done synthesis of two different processes of agricultural
commodity production. On one hand, we find into (6) some features from the allocation
process of the farmer, such as the current commodity or its substitute prices, the cost of
production and the technology index; on the other hand, it is worth noting that in this
production equation the i.i.d. hypothesis survives. Deaton and Laroque (1992) justify
the hypothesis through some arguments related to the influence of the weather. As in Ng
and Ruge-Murcia (1997), this factor is explicitly modelled in (6) thanks to the
introduction of some weather-dependent variables, i.e. precipitation index.
Some important relationships are derived by net-export demand considering separately
the export and import demands of the commodity5. We provide the equation describing
the export demand:
X (t ) = ψ 0 + ψ 1 EXC (t ) + e(t )
(7)
We do not model the imports, M(t), because it is negligible with respect to the exports
for many U.S. commodity markets, such as the corn market. Therefore, the net-export
equation coincides with the export demand equation and we can write the following
behavioural relationship:
NX t = X (t ) − M (t ) = ψ 0 + ψ 1 EXC (t ) + e (t )
(8)
On the demand side, building up the model is relatively straightforward. A commonly
adopted framework is to assume that consumption does not react to the expected price,
but is much more sensitive to the current price. This hypothesis is due to the peculiarity
of agricultural commodities for which the expectations driving the setup of production
5
Export demand X(t) is represented by a behavioural equation with the goal of analyzing the relationship of this
variable with the world corn price, Pw(t). We add an equation taking into account the world demand and world
supply excesses (Wd(t) and Ws(t)) and we add a relationship where the world corn price is dependent on the exchange
rate, EXC(t). The equations are reported below:
Ws(t ) = λ1 + λ2 Pw(t ) + s (t )
Wd (t ) = λ0 − λ3 Pw(t ) + d (t )
X (t ) = Wd (t ) − Ws(t ) + x(t )
Pw(t ) = v0 + v1EXC (t ) + p (t )
Solving for Xt in the set of equations, we find the (9).
7
plans make most of the consumption plans inelastic. Therefore, processors relinquish a
forward-looking perspective, choosing an optimal consumption period by period
(Revoredo, 2001). The consumption equation is specified as follows:
Con (t ) = Con + α 1 Pm (t ) + u (t )
(9)
In such a specification, we assume that consumption fluctuates around a stable level
(Con ) and deviations from that value are time justified by factors that depend upon
either internal or external demand for the commodity. Moreover, it is worth noting that
current price is a factor which simultaneously influences internal and storage demand
and, hence, including stocks in this function seems to be redundant. Once consumption
and net exports are defined, we can collapse these definitions into the Demand for corn:
D(t ) = Con(t ) + NX (t ) = d 0 + d1 Pm(t ) + d 2 EXC (t ) + vt
(10)
where:
d 0 = Con + ψ 0
d1 = α 1
(11)
d2 = ψ1
vt = u t + et
This variable is an indicator of the (internal and external) demand coming from the cash
market.
The model is completed by the market clearing condition, which establishes the balance
between the internal availability (Deaton and Laroque, 1992) and demand components:
i) the cash market demand, D(t), and ii) the storage market component, FS(t).
The market clearing constraint is given by:
Q(t ) + (1 − δ )FS (t − 1) = D(t ) + FS (t )
(12)
where δ represents the depreciation of the stock when we go from one period to another
and/or directly linked to inefficiencies which arise contemporaneously with the
carryover activity. Market clearing constraint ensures that the current production and
incoming inventories (i.e. internal availability) balance time by time internal/external
consumption and demand for storage. In previous works (Deaton and Laroque, 1992;
Routledge, Seppi and Spatt, 2000), δ has been considered to be a measure of the
physical storage depreciation or the waste related to spoilage in the agricultural
8
commodity market. Therefore, it is crucially dependent upon technological and natural
variables. The former affects the storage transformation rate through an improvement in
storage mechanism and techniques, while the latter are related to the specific nature of
the agricultural commodity. We can find commodities which are very storable while
other commodities are subject to an intense physical depreciation. In crop field
commodities (corn, soybean, wheat), the physical depreciation rate is not large but it
cannot be neglected. Indeed, the assumption of a time-invariant δ seems to be
unrealistic and prevents us from searching for the dynamics which drive stockholding
decisions. In contrast to Deaton and Laroque (1992), this work attempts to model a
time-variant storage depreciation rate. In the theoretical model, the institutional variable
is endogenously considered, conditioning the equilibrium of the state variables in spot
and storage markets. The relationship underlying the storage depreciation rate adds to
physical factors also the institutional variables. For example, since the 70s, the U.S.
Government has provided the farmers with a loan rate6, the stockholding decision has
therefore been influenced. In contrast to the government stock accumulation, the loan
rate (Loan(t)) is an indirect way to fine tune the allocation process that comes out from
the market. In this sense, we can say that policy intervention could be linked to the
depreciation rate, because the different technologies in private and public stockholding
functions may lead to different levels in the depreciation rate when the policies change.
Once equipped with this theoretical background, we can specify the depreciation rate as
a deterministic function related to either technological or institutional factors:
⎡
⎤
⎣
⎦
δ = δ ⎢ STH (t ), IF (t )⎥
(13)
where STH(t) and IF(t) are respectively the storage technology and the institutional
factors. In particular, δ can be limited by the development of storage technology,
whereas the institutional factors are modelled by the inclusion of the loan rate which is
essentially a proxy of government intervention in the agricultural commodity markets.
6
The loan rate is the unitary price (per bushel, per quintal or per pound) at which the Government loans
money to farmers in order to let them store the product at the end of the period, giving them the
opportunity to sell it in the next period.
9
3. Stockholding Decision in a Dynamic Perspective
In this Section, we derive spot price and marginal storage value equations directly from
the optimization program. In doing so, we introduce into the theoretical framework
presented in Section 2, the dual relationship between production (Q(t)) and cost function
(C(t)) and the economic hypotheses about the market structure. We generally know that:
C (t ) = C [Q(t )]
(14)
and, consequently:
∂C
= C ′[Q(t )] = mc(t )
∂Q(t )
(15)
where mc(t) is the direct marginal cost of production.
In the remainder of the paper, our empirical analysis will concern the U.S. Corn Market,
hence it is reasonable to assume market competitiveness. In light of this, we find that
the equilibrium spot market condition is provided by:
mc (t ) = Pm (t )
3.1. The Determinants of Stockholding Decision
In stockholding commodity markets, the uncertainty feature plays a relevant role since it
can delay the agent decisions in order to wait for new information to arrive about
prices, costs and other market conditions (Pindyck, 1991). The exogenous production
level includes the uncertainty into the opportunity to postpone the sale. Analogously to
the Net Present Value model (hereafter, NPV) decision principle, an agent decides to
invest or not by the comparison of the discounted benefit flows and the cost of the
investment7. This principle has been pervasively used to investigate the stockholding
decisions; many researchers (among others, Revoredo, 2001) argue that the markets
record positive storage only if the discounted expected price for the next period is
greater than all the costs which an individual has to suffer in order to store the
7
Generally, the NPV principle does not provide dynamically consistent decision rule whether agents face
with irreversibility and uncertainty on investments (Pindyck, 1991). However, the irreversibility feature in
the commodity storage markets does not explicitly arise. Indeed, the irreversibility is ruled out from
commodity storage either for the low depreciation affecting the stored commodity from one period to
another or for the intensive development of commodity exchange markets which increase the investment
liquidity.
10
commodity, i.e. physical storage cost including the current price. Even if the inclusion of
the current price can represent an attempt to model the opportunity cost and, hence, it
could improve the econometric performance of this model with respect to the
investment neoclassical models (Pindyck, 1991), this strategy is not enough to explain
the stylized facts affecting both price and stock series (in a wide set of different
commodities) and does not provide satisfactory results as pointed out by Ng and RugeMurcia (1997).
In this section, we characterize a different approach to carry out optimal and
dynamically consistent stockholding decision rule which can be obtained from dynamic
programming. The representative agent maximizes the profit function under the market
clearing condition. The objective function to be maximized is given by:
∞
Π (FS (t ), Q(t ) ) = ∫ e − IRt [Pm(t ) D(t ) − TC (t )]
(16)
0
where TC is the total cost function. Following Pindyck (2002), we specify the TC
through the separation of three different components: i) total production costs; ii)
marketing costs, i.e. the cost of re-planning production to avoid stock-out and facilitate
deliveries and iii) storage cost.
Total production costs take on the following quadratic form:
C (Q (t ) ) = (c 0 + η (t ) )Q (t ) +
1
c1Q (t ) 2 + c 2 Ctd (t )Q (t )
2
(17)
in which η(t) is a shock affecting on the amount of fixed costs, i.e. it captures the
technological shock in the cost function. Ctd(t) is the per unit production cost including
the cost for wage, irrigation and fertilizers.
In our model, in order to describe the complementary behaviour of the processors, we
term marketing cost ,Φ(.), all the costs that an agent has to suffer whenever he decides
to re-plan production, avoid stock-out, facilitate deliveries. This cost function typically
reflects a precautionary behaviour which characterizes the activity of processors with
inelastic goods. The marketing cost function [Φ (FS (t ), Pm(t ) ) ] has the following
property:
φ (t ) = −
∂Φ
∂FS (t )
(18)
11
i.e. the marginal convenience yield (φ) (hereafter, convenience yield), which is the
derivative of the marketing cost function with respect to the stock level with a change in
the sign.
The storage cost function is assumed to be linear in the stock level:
SC (FS (t ) ) = kFS (t )
(19)
where k is the cost of carry which is assumed to be invariant over time. The cost of
carry (k) is a portion of the total storage cost and includes the physical storage cost (the
per unit cost of the warehouse) and the foregone interest.
Therefore, collecting all the total cost function components, we can define:
TC (t ) = C (Q(t ) ) + Φ(FS (t ), Pm(t ) ) + SC (FS (t ) )
(20)
The maximization problem of the economic agent is:
∞
max ∫ e − IRt {Pm(t )D(t ) − TC (t )}dt
(21)
0
This is subject to two different constraints: the market clearing condition and the stock
accumulation equation. The market clearing condition has already been presented
among the structural equations, whereas the stock accumulation process is the
following, where for brevity, we omit specifying the relationship for δ assumed in the
above section:
FS& (t ) = FS (t ) − (1 − δ )FS (t − 1) − δFS (t )
(22)
Such a formalisation8 of the motion law for the stock level is worth looking at in depth.
It tells us that the change in stocks is given by: i) the stockholding level at time t (FS(t))
which results from the agents’ decision rule; ii) the net stock availability (i.e. initial
condition) for period t ((1-d)FS(t-1)) and iii) the wasted stock (δFS(t)) related to
inefficiencies in the stock carryover, i.e. amount of commodity which is ruined and/or
wasted during the carryover9.
8
The motion law can also be re-written by aggregating some terms in the right hand side. In particular,
defining ∆FS=FS(t)-(1-δ)FS(t-1), we get: FS& (t ) = ∆FS (t ) − δFS (t ) . However, in a continuous time setup,
the use of ∆(.) operator can generate some misunderstandings.
9
For example, some units of a storable commodity could be crashed and/or get wet during transportation
from the market to the warehouse and, in turn, partially affect the agents’ stockholding choice.
12
In order to solve the maximization problem, we employ the Hamiltonian technique. The
Hamiltonian for this task is:
⎧⎡
1
2
⎪⎢ Pm(t )(Q(t ) + FS (t − 1) − FS (t ) ) − (c0 + η (t ))Q (t ) − 2 c1Q(t )
⎪
H = e − IRt ⎨⎢
c Ctd (t )Q(t ) − Φ[FS (t )] − kFS (t )
⎪⎣ 2
⎪⎩λ (t )[FS (t ) − (1 − δ )FS (t − 1) − δFS (t )]
⎤ ⎫
−⎥ ⎪
+
⎥ ⎪⎬
⎦ ⎪
⎪⎭
(23)
The first order conditions with respect to the production and stock level yield are:
∂H
= 0 ⇒ Pm(t ) = c0 + c1Q(t ) + c 2 Ctd (t ) + η (t )
∂Q(t )
∂H
Pm(t ) − [φ (t ) − k ]
= 0 ⇒ λ (t ) =
∂FS (t )
1 − δ (STH (t ), Loan(t ) )
(24)
The F.O.C. with respect to the stockholding level provides an expression for the
marginal storage value and is strictly linked to the spread between the current market
price and the marginal net convenience yield. It is worth noticing that the storage
depreciation, as defined in (13), affects the marginal storage value. If this parameter is
close to one, the storage depreciation to increases the marginal storage value, because it
includes a risk premium due to the probability of losing a fraction of the stored
commodity. Investigating the meaning of the storage depreciation rate in terms of
marginal storage value, we set consider price and marginal convenience yield to be
fixed at their equilibrium values:
Pm(t ) = Pm
(25)
φ (t ) = φ = k
and, hence, (24) for λ(t) simplifies into:
λ=
Pm
1−δ
(26)
In Graph 1, we plot the behaviour of the equilibrium marginal storage value with a
varying storage depreciation rate. It is graphically worth noting that if the storage
depreciation rate is zero, then the equilibrium marginal storage value equals the
equilibrium value for price, i.e. with a perfect storage technology the equilibrium
marginal storage value coincides with the equilibrium price. There is no place for a risk
premium due to the imperfection in the storage mechanism. If the storage depreciation
13
rate approaches one (the upper bound of its domain), the equilibrium marginal storage
value tends to infinity, i.e. the value of a unitary increase in stockholding level is very
high given an inefficient storage technique. We have seen in the previous section that
the stored level of the commodity can be affected by institutional factors. The
stockholding choice can be influenced by the indirect taxes, interest policy and/or by the
role (directly) played by government’s stock on the overall inventory accumulation.
Hence, government can drive the agents to obtain a better allocation.
The F.O.C. with respect to production can be further manipulated by replacing the
definition for Q(t) which comes from the market clearing equation and involves the
structural representations of the internal/external demand. This sequence of
substitutions gives us:
Pm(t ) = δ 0 + δ 1 ∆FS (t ) + δ 2 EXC (t ) + δ 3 Ctd (t ) + ξ (t )
(27)
where we have set the following equalities:
δ0 =
c0
c3
c1
c1
cd
;δ 2 = 1 1 ;δ 3 =
+
d 0 ;δ 1 =
1 − c1α 1 1 − c1α 1
1 − c1α 1
1 − c1α 1
1 − c1α 1
ξ (t ) =
1
[η (t ) + c1v(t )]
1 − c1α 1
(28)
Therefore, the equilibrium price is intimately related to the change in stocks and to a
pair of exogenous variables such as the exchange rate and unit cost of production.
The co-state condition provides the motion law for the marginal value of stocks, i.e.
λ(t).
−
[
]
∂H
d
=
λ (t )e − IRt ⇒ λ& (t ) = (1 + IR)λ (t ) − Pm(t )
∂∆FS (t ) dt
(29)
Recalling Tobin (1969), we can define a ratio on which the stockholding decision rule
can be grounded. This relevant parameter is:
q(t ) =
λ (t )
(30)
Pm(t )
where the denominator is the replacement cost of a stored unit of corn which, under our
assumptions, equals the spot market price. Then, market records a positive storage if
q(t) is greater than one, whereas market decreases its stocks with q(t)<1. The q(t)
14
function links the marginal storage value to the current price, including it in
stockholding functions. Moreover, it is worth noting that the optimal stockholding
decision rule summarized into q(t) depends upon various parameters that come from
market environment (Pindyck, 1991). Finally, we can conclude that the stockholding
decisions are made by comparing the marginal storage value carried out in the profit
maximization. This ratio is a powerful tool in surveying the stockholding behaviour of
farmers in the storage market.
Invoking again the equilibrium in (27) and evaluating q(t) in that point we obtain an
instrument for investigating the equilibrium stockholding decisions:
q=
λ
Pm
=
1
1− δ
(31)
Given q(t) definition and the equilibrium conditions, without the price uncertainty the
stockholding decisions are made by only looking at the storage depreciation rate. This is
due primarily to the lack of price uncertainty and therefore, the risk of stockholding
decision is included in the depreciation rate.
3.2. Dynamics
The goal of this section is to show dynamic behaviour of agents in making stockholding
decisions and to emphasize the impact of this decision on the other market variables.
Moreover, we can shed some light on the short run adjustment process started up by a
shock in one of the exogenous variables involved in the model. Before estimating the
model, a qualitative representation of the model implications will be given. In doing so,
we derive the isocline for q(t) in (∆FS(t), q(t))-space. Computing the derivative with
respect to time in (30) and using the co-state condition in the place of the time
derivative of the marginal storage value, we have the following structure:
⎡
P& m(t ) ⎤
q& (t ) = q(t ) ⎢(1 + IR) Pm(t ) −
⎥ −1
Pm(t ) ⎦
⎣
(32)
Setting q& (t ) = 0 and solving for q(t), we get:
15
q(t ) =
1
(33)
⎡
P& m(t ) ⎤
+
−
(
1
)
(
)
IR
Pm
t
⎢
⎥
Pm(t ) ⎦
⎣
Recalling (27), we obtain a relationship that links q(t) to the changes in stocks through
the equilibrium price. Panel a) of Graph 1 shows the equilibrium in this space: the
intersection of the isocline for q(t) and the vertical axis (i.e. the locus where ∆FS(t)=0).
Independent of the sign of c1, we do not find a stable branch. This means that the
adjustment process only takes place along the isocline: whenever we are out of the
isocline, the adjustment paths diverge from equilibrium. Therefore, faced with a shock
to the exogenous variables, the adjustment path jumps from the old to the new isocline,
in order to move along it to the new equilibrium point. Panel a) of Graph 1 was drawn
based on the hypotheses that c1 in the cost function is greater than zero. We are
interested in explaining the reaction of market variables, i.e. the price, to a shock on the
exchange rate. An increase in the exchange rate yields a downward shift of isocline for
q(t) in panel a) of Graph 1. An increasing exchange rate cuts the external demand and,
in turn, generates expectations of a supply excess. The role of expectations is
remarkable: the adjustment process differs if the shock in the exchange rate is conceived
as transitory or persistent. If transitory, the shock is expected to vanish over time and
therefore the stockholding behaviour is affected to a limited degree. If persistent, the
shock impinged on the exchange rate determines a changes the stockholding behaviour.
We have considered a case in which the shocked variable (i.e. the exchange rate) is not
expected to revert to its ex-ante shock value. However, the mechanism in stockholding
accumulation affects the interaction of processors and speculators. The initial decrease
of stocks is essentially due to speculators: the expected supply excess causes an
expected decrease in spot price, leading them to sell out inventories in order to obtain
stock gain. The flexible reaction of speculators also induces processors to partially
postpone the stock purchase. The difference is that the activity of processors cannot
totally stop the purchase of the commodity because they have to maintain their
production/transformation process. Hence, the co-existence of processors and
speculators results in a temporary excess supply, which has to be re-absorbed. The
action of processors contributes greatly to restore the equilibrium, but their behaviour is
16
sticky. Finally, the action of processors increase in the marginal convenience yield due
to precautionary motives. It is reasonable to assume that if stock level is low, then the
net convenience yield is increased by the precautionary instance, whereas a small
marginal convenience yield is compatible with high stock levels. The increasing
marginal convenience yield, combined with a price effect, which will be explained in
the remainder of the section, re-absorbs the stock variation and lowers the value of q(t).
Similar arguments can be developed even if the shocks do not directly hit an exogenous
variable of the model. Macroeconomic shocks, asymmetric change in policy (i.e.
concerning only one economic sector) or shocks of production could affect the corn
storage behaviour. However, greater care has to be given to evaluate the different levels
of persistence of the shock on the market variables.
In panel b) of Graph 1, the dynamic interaction which has just been explained is
transposed in the (∆FS(t), Pm(t))-space. Panel b) helps one to understand the
mechanism which allows the new equilibrium point to be achieved with a lower value
for q(t). In order to clarify the effect of an exchange rate shock on the price level, it is
helpful to carry out a relationship between corn price and the change in stocks. In doing
so, we rearrange the expression for Pm(t) in the first stage of maximization. The
relationship linking corn price and change in stocks takes the following linear form:
Pm(t ) = K + δ 1 ∆FS (t ) + ε (t )
(34)
where K includes all the exogenous variables evaluated at their equilibrium levels and
the error term. In other words:
K = K ( EXC , Ctd )
(35)
In this space, the equilibrium price is located on the intersection between the price-stock
line and the vertical axis, i.e. the equilibrium price is achieved only when the change in
stocks is zero as shown in panel a). As the exchange rate increases, the price-change in
stocks shifts up and identifies a higher equilibrium price (Pm2). The adjustment path can
be interpreted as the interaction among speculators and processors. An increase in the
exchange rate results in current and expected supply excesses and, in turn, a
depreciation of the commodity. Such a situation leads speculators to sell out inventories
and decrease the price thereby increasing the availability in the market. The system
moves along the line from Pm0 to D. At point D, the interaction among speculators and
17
processors takes place in the following manner: i) processors have a positive demand
for the commodity in order to supply their production/transformation process; ii)
speculators know that processors are buying and that processor demand will cause
prices to increase. For this reason (i.e. the spread between the expected and the current
price becomes positive), speculators also have a positive demand for corn. The action of
speculators strengthens that of processors: the price jumps from D to E. Such an abrupt
jump in the adjustment mechanism is sparked by the massive demand due to speculative
opportunities. Hence, the reaction of the speculators to the pressure applied on the
system availability by the processors is instantaneous, whereas the processors behaviour
is much more regular and is essentially driven by their underlying production activity.
For this reason, the movement from E to the final equilibrium point (Pm2) is only
justified by the processor’s activity.
Now, we can explain more extensively the factors that reduce the equilibrium value of
q(t). As shown in panel (a) the lower q(t) is not only due to the increased convenience
yield, but also to the increasing equilibrium price which is established in panel b), i.e. in
the cash market. Thus, it is straightforward to track down the connection between
storage and cash market. The analytical derivation of the explicit functional relationship
linking cash and storage market exploits the motion law that arises from the co-state
condition (29), i.e. for the marginal storage value. Hence, we can depict the isocline
λ& (t ) = 0 in the (Pm(t), φ(t)) space. Replacing λ(t) with the second equation contained in
(24) and setting λ& (t ) = 0 , we see an explicit relationship of the variation of the marginal
storage value in terms of price and convenience yield. Then, solving for Pm(t) yields:
Pm(t ) = −
1 + IR
[φ (t ) − k ]
IR + δ
(36)
The relationship plotted in panel c) is that between storage and cash market: it expresses
the spot price in terms of net convenience yield. Continuing with the example of a
shock on the exchange rate, the effect of such a shock causes an upward shift on the line
in panel c). An increase in the exchange rate induces a positive change in either price or
net convenience yield and a parallel shift of this straight line. The marginal net
convenience yield determined by projecting the equilibrium spot price through the
isocline for the marginal storage value is the equilibrium marginal convenience yield,
18
once the adjustment has fully taken place. In fact, the isocline for the marginal storage
value depicted in this panel ensures that the equilibrium stockholding is unchanged.
Panel d) shows the relationship between the decisional stockholding parameter and
convenience yield. The peculiar shape of this relationship is worthy of commenting:
stockholding decisions are made by looking at q(t). Given a value for the net marginal
convenience yield, we can see the compatibility of two different values of q(t). In other
words, a given positive value of net marginal convenience yield can justify either
increasing (q(t)>1) or decreasing (q(t)<1) stockholding mechanisms. Therefore,
observing the marginal net convenience yield is not enough to say anything about
agents stockholding decisions of agents.
4. Econometric specification
The empirical tests in the U.S. corn market are specified using in both the price equation
as specified in (27) and the convenience yield equation (φ). To derive the econometric
form of last equation, in analogy to the total production cost function introduced in
Section 2, we consider a quadratic equation for the total marketing cost function (Φ):
Φ( Pm(t ), FS (t ), T (t )) = c0 + [− a 0 − a1 Pm(t ) − a3T (t )]FS (t ) −
a2
FS (t ) 2
2
(37)
in which Pm(t) is the U.S. corn spot price, FS(t) is the stockholding level and T(t) is a
deterministic time trend. Given the definition in (37), we exploit (18) and obtain the
following equation which gives us an estimable relationship for the convenience yield:
φ (t ) = a0 + a1 Pm(t ) + a 2 FS (t ) + a3T (t )
(38)
The direct simultaneity between spot price and marginal net convenience yield is
modelled. While the relationships among spot price, stockholding level and marginal
convenience yield have been discussed in previous sections, the introduction of a
deterministic time trend necessitates of an in-depth investigation. Indeed, time trend has
been modelled in order to capture the discrepancy between future and spot price in the
sub-period 1972-1983 derived from the macroeconomic shock in oil. Consequently, in
the adoption of future contracts, the market reacts to macroeconomic uncertainty by
19
requesting a higher risk premium and, in turn, increasing the futures price which
embodies the risk premium (Fama and French, 1987). According to the previous
framework, we can derive from the marginal storage value (λ(t)) the stock premium as a
wedge between the marginal storage value and the equilibrium price (Graph 2).
In order to close the model, we have to measure the storage depreciation rate (δ) which
is an essential variable for the empirical part of this work because it is involved in the
definition of the marginal storage value, in equation (24), from which the q(t) ratio is
derived in equation (30). The theoretical measurement relationship underlying the
storage depreciation rate has to satisfy particular features: i) this function takes values in
the interval [0, 1]; ii) it has to be related to the storage technology (STH(t)) and includes
institutional factors (IF(t)). Thus, in our work, storage depreciation is not assumed to be
constant as in Deaton and Laroque (1992), since we quantify and explain it over time by
a set of technological and institutional factors. Hence, we have to overcome the lack of
observations for δ. The estimation of δ is done by the following procedure10. First, we
set δ=0.05 : this value is considered to be plausible after looking at simulation exercises
performed in many works (Deaton and Laroque, 1992). Second, we build the series for
the wasted stock as follows:
Wasted Stock = δFS(t)
where
δ=0.05
(39)
Hence, we exploit the theoretical intuition introduced in ii): the storage depreciation rate
and, in turn, the wasted stocks depend on the storage technology and loan rate.
Assuming a linear relationship, we can estimate the next equation by OLS:
δFS (t ) = β1 STH (t ) + β 2 Loan(t ) + ω (t )
(40)
The goodness of such a regression is ensured by the high R2 value, 0.77, in which β 1 is
negative and β 2 is positive; this shows that a certain technology reduces the
depreciation rate, whereas a reduction in the loan rate, under the direction of U.S.
agricultural reform, increases the efficiency in the markets by the substitution of
obsolete storage technologies or by an increase in the public stocks. By equation (40),
we obtain a measure of the path of δ over time.
10
The estimation results from this procedure are available upon request to the Authors.
20
Once the estimation procedure is ran, we can carry out the fitted values for the wasted
stock and, consequently, filter the estimated series for δ as follows:
δˆ =
δFˆS (t )
(41)
FS (t )
As shown in Graph 3, the storage depreciation rate (δ) values are negligible in the whole
sample except for two particular years (i.e. 1982 and 1995); we can observe that the (1-
δ) is always above 0.90. This fact ensures that stockholding activity, perceived as
investment (Kahn, 1981), is not affected by irreversibility and the last statement is
strengthen by the high development of financial commodity markets since 1970s.
To complete the information background in order to estimate the model, we need to
compute the marginal net convenience yield (φ’(t)). Following Pindyck (2002), we build
the convenience yield series as follows:
φ (t ) − k = φ (t )' = [1 + IR (t )]Pm (t ) − Pf (t )
(42)
where Pf(t) is the future price with one year delivery. To carry out the series of the gross
marginal rate, we have to estimate the per unit storage cost which is assumed to be
constant. Therefore, we set:
k = min φ (t ) '
(43)
and, once the value for k is quantified, we immediately obtain the series for the gross
marginal convenience yield:
φ (t ) = φ (t ) ' + k
(44)
The dynamic framework derived in Section 3 and, in particular, the definition of q(t) in
(30) can be employed to forecast the stockholding level. The algorithm we propose is
based on the observation that the ending stocks in t-1 coincide with the starting stock in
period t. Therefore, we can forecast the ending stocks in period t by applying the
function q(t) which describes the evolution of stockholding level in t to the initial stock
level. However, because the variables involved in q(t) are not determined at the
beginning of the forecast period, we replace them with the expected value according to
equations (27) and (38). Hence, we can write the following recursive algorithm:
Fˆ S (t ) = FS (t − 1) qˆ (t )
(45)
in which:
21
Pˆ m(t ) − φˆ(t ) + k
1 − δˆ
qˆ (t ) =
Pˆ m
(46)
The initial condition, FS(t-1), is modulated by the expected q(t) and gives the value of
the ending stocks in t. Therefore, given the initial condition, the ending stocks are
affected by the variables of spot and storage markets. Moreover, replacing the variables
with the expected value obtained from the equations resulting from an optimization
program can be explained by agent rationality.
Other than carrying out the stock computation as shown in (45), we can also give an
algorithm by which a fitted series can be obtained for the U.S. corn futures price. Using
the empirical estimates for the spot price and gross convenience yield and considering
the value assigned to the marginal storage cost, the computation for the futures is as
follows:
Pˆ f (t ) = [1 + IR(t )]Pˆ m(t ) − φˆ(t ) + kˆ
(47)
Exploiting the estimates obtained from (27) and (38), we can implement (47) and find
the estimates for the futures price. In the next Section, we present and comment on the
dynamic properties of the series and the estimation results.
5. Empirical Illustration: Data and Results
The variable descriptions and sources of the U.S. corn market used in the estimation are
reported in the Appendix. The sample period frequency is represented by the annual
time series (1973-2000). Graph 4, describes the evolution of spot and futures corn price,
from which we extract information about the expected behaviour of the agents by the
spread corn series; the evidence shows a wide spread in the 70s, whereas the value
decreased starting in the middle 80s. Applying the usual interpretation seems to be
relevant the role of the macroeconomic shocks, that strongly affect to the differences
between expected and current corn prices. Gilbert and Perlman (1987) suggest that the
extraordinary events surrounding the first major oil price rise in 1973-4 may have
generated a psychological impact on other primary commodity markets. Therefore, the
22
risk premium included in the futures price and associated with these shocks drives the
futures price up by the increasing uncertainty.
We attempted to account for the exogenous shocks of U.S. corn market by defining the
deterministic trend (T) in the storage function. If we have successfully accounted for the
whole impact of uncertainty on the corn storage market (and consequently in the spot
market) through this deterministic variable, we should be able to not reject the statistical
significance of the time trend coefficient in (27). Moreover, the spot price path shows
that two other transitory peaks were met in the sample: the first occurs in the 1985
agricultural reform (Miranda and Helmberger, 1988), whereas the second anomaly is
due to the world production crisis in 1995 and the consequent increase in the corn spot
price.
Graph 5 proposes the production and stock dynamics. It is relevant to notice that the
stock path (FS(t)) indicates a stationary behaviour. A formal result will be presented
below through the implementation of the unit root ADF and KPSS test. Moreover, the
U.S. agricultural reform in 1985 ignited a change in the stock behaviour, confirming the
previous results that the shocks does not affect the long run dynamics. The market
agents revised down their estimate after few periods. In this direction, the shock
induces a modification in the market stabilization, increases the role played by the
public stocks and, temporarily, the price volatility (Miranda and Helmberger, 1988).
Finally the second peak in the world corn production is accounted explicitly in the price
and convenience yield equations by two dummy variables (D95 for production and D96
for convenience yield). As we have extensively discussed in this section, the reason for
these dummies is explained by considering that the impact of this shock affects the spot
price at the time of the production is realized and, with a temporal lead (i.e. one period),
the rational storage behaviour is conditioned by the production plans in both the
speculators and processors decisions.
In order to specify the robust empirical estimates, we investigate the dynamic properties
of the series. The Augmented Dickey Fuller test (ADF) is implemented for current price
(Pm(t)), convenience yield ( φ (t ) ), discrete change in stocks (∆FS) and stockholding
level (FS) with optimal lags and presence/absence of deterministic components. We
23
confirm the ADF results in our sample computing - under the null hypothesis of
stationary variables - the KPSS test.
Table 1 shows the results. The estimation of the unit root is implemented with a
constant for price (Pm), discrete change in stock (∆FS) and stockholding level (FS),
whereas the convenience yield ( φ (t ) ) is corrected for the significant relevance of the
time trend that justifies the previous specification (37). Two remarks have to be
discussed. Statistically, the optimal number of lags selected by the usual non-parametric
criteria show the consistency in annual data of a simple Dickey-Fuller test. Second, the
current price and discrete changes in stock are stationary at significance level of a one
percent, whereas, for the other variables, the performed tests are not fully informative.
Indeed, in the stockholding level the ADF test rejects the non-stationarity hypothesis
only at the ten percent level, whereas KPSS test does not reject stationarity. Conversely,
the convenience yield shows an ambiguous result in the KPSS test, whereas it is
consistent with the expectations in the ADF test. However, from the empirical results,
we prefer to reject the unit root of (Pm) and ( φ (t ) ) and specify the simultaneous model
(27) and (38) in levels.
In table 2 the first column includes in preliminary OLS estimations of price and
marginal convenience yield equations. The estimated equations broadly conform with
the pattern anticipated in Graphs (4) and (5). Both exchange rate and unit production
costs are seen as having a direct impact on commodity prices, as far as the effects of the
negative stock changes are consistent in the stockholding models. In the convenience
yield equation, the expected signs in current price, stockholding level and deterministic
trend are coherent with the theory, showing a high statistical significance. The positive
current price effect is well determined, which suggests simultaneous relationships with
the convenience yield.
To estimate simultaneously the state variables in the cash and storage markets, by (27)
and (38) respectively, we use the Full Information Maximum Likelihood (FIML); the
procedure applied to the model for corn uses the price equation as an instrument for
estimating the parameters in the convenience yield equation. The estimated parameters
are reported in Table 3.a and 3.b. The first column shows that the equation price, written
in implicit form, is robust to the sign changes and statistically significant. The (∆FS)
24
parameter also has an important impact at the ten percent level of significance.
Disregarding what causes the stock reduction, (increased consumption, less production
in the spot market or lower futures price expectations) the result is an increase in the
spot market price. As in the single spot price estimations, the exchange rate and unit
production cost have signs which are consistent with the theoretical expectations and are
statistically significant at the usual levels.
The convenience yield equation is characterized by the price, trend and stock variables,
which appear in equation (38). The coefficient estimates are all positive and consistent
with a well-behaved marginal value of storage function. Even if the specification is
different, we find analogous statistical confirmations of estimation in Pindyck (2002)
for crude oil and heating oil. Only the stock coefficient is not completely significant,
showing a loss of efficiency in the (FS) parameter when the simultaneous estimation is
obtained.
Table 3 also contains a battery of diagnostics on these estimated equations. The LM test
for residual serial correlation (Godfrey, 1978) is indicated in both to the Durbin-Watson
statistic (DW). The tests are consistent with the a priori hypothesis of not rejecting
autocorrelation; finally, a few of misspecification is held in the current price equation.
The next set of statistics test for structural stability considers the post-sample residuals
for these periods. The natural statistical index for evaluating the fits is the R2 ; it is high
enough in the convenience yield equation (0.64), but is low in the price equation (0.33).
In the price equation this poor in-sample performance is due to a large negative
(positive) residual in the 1995. The influence of weather in 1995 in the corn export
countries, as Argentina and Brazil, to a brief period of very high world corn price at the
end of 1995. Even if these events are considered by the agents as the transitory shock,
they make an adaptive revision of the behaviour. As noted in the previous theoretical
section, the expected price has an impact on the convenience yield because current and
futures price values affect current and expected futures demand of the commodity (and
its substitute) and the stockholding equation. Coherent with a high residual level in the
price equation, we introduce the relative dummy into the price and convenience yield
equations. Hence, this specification provides a more powerful test of the temporary
shocks in the corn market, given the accessibility of the information. The table 3.b
25
reports the F-test and the results of introducing the dummy variables into the system.
There is a marginal rejection of the statistical test in the price equation, but in the
convenience yield equation the influence of international sectoral shocks in the corn
production is robust. The remarkable difference, when we insert the dummy variables to
control for the shocks, is the statistical significant value in the convenience yield
equation of the FS parameter. There is a strong evidence of a stock level induction in
the agent decisions of accumulation inventories by the dummy. The increase in the spot
price in 1995 has probably generated a positive expectation in the futures price in the
short term, so that the speculative agents have programmed more stocks. Speculative
motives have to be considered along with the precautionary explanation. and are
important in explaining in part the accumulation process. In fact, the uncertainty may
also have induced the processors to accumulate stocks when, using the opportunity
costs, the marketing costs were greater than the storage costs.
In Graph 6, the comparison between the observed and fitted of spot price equation is
illustrated. As we can see, the difference in the early 70s is followed by a period in
which the spot price is well-fitted by our model (middle 80s), thus we can conclude that
the shock caused by the agricultural reform is effectively accounted. Moreover, when
the world production shock in 1995 is incorporated, the model provides to reduce the
estimated errors.
The derivation of q(t) shows us the movements of optimal stockholding determinants
and the behaviour of agents (Graph 7). High macroeconomic uncertainty in the first part
of the sample generates q(t) values that are much greater than one. This evidence is
always explained by the macroeconomic uncertainty in the 70s, that leads to an
increasing risk premium in the futures markets.
By the algorithm described in Section 4, we obtain Graph 8, in which the fitted stock
values (with or without the 1995 Dummy) are compared with the observed stock series.
Until 1981, the fitted values with and without the dummy are systematically greater
than the observed data values. Nevertheless the wedge represented by the stock
premium, the stockholding dynamics is fully captured by our model, as for as the
government intervention in which is evident that the agents have needed of time in
order to adjust their stockholding choices, even if the agricultural reform does not
26
change the structural behaviour. In the 1986-1988 period, the predicted stocks are
higher than the observed ones. Once completed the learning process, the predicted
stocks revert to the market data series.
In Graph 9 a powerful result is achieved by using equation () to carry out a fitted values
series which matches the six-month delivery futures price. It is worth noticing that our
estimation price captures the dynamic in the futures price, verifying the statistical
robustness of the model and the economic coherence.
6. Concluding remarks
The models for spot price dynamics of agricultural commodities should be able to trace
changes in current or expected futures values caused by macroeconomic variables or
institutional factors. The speculative stockholding models, which are currently used to
dynamically solve price equations in agricultural primary commodity markets, yield
unsatisfactory results and present some difficulties in endogenously accounting for
agent reactions due to exogenous shocks. This is the case if we consider a market that
is based solely on speculators.
The aim of this paper has been to extend the interactions between agents in the spot
market to the stockholding decisions made by the storage market. We have investigated
the short-run determinants in the U.S. corn market and have proposed a continuous
optimisation of the total discounted profit function, in which the production and
stockholding levels are considered the control variables which leads to evaluating
simultaneously the optimality in both spot and storage markets. Moreover, as in
Pindyck (2002), a marginal convenience yield was also taken in account in order to
investigate the microeconomic foundation of the storage market. We have extended the
function in three directions: (a) the storage market is considered to be populated by both
speculators, which make arbitrage by speculating on the difference between current and
expected prices, and processors, whose aim is to transform the corn. Dynamic
simulations of the model show that a general shock has an instantaneous effect on
speculator behaviour, whereas, for precautionary motives, the processor behaviour is
sticky. (b) The linkage between the cash and storage markets is given by the net change
27
in stocks. This variable is relevant in the structure of this paper because it included the
storage depreciation rate which is modelled as a time-varying variable and is
endogenous to the model. The assumption of a constant storage depreciation rate is
relaxed: indeed, the storage depreciation rate is not only influenced by technological
arguments, but also from the institutional factors. The explicit formalization of
institutional factors in the depreciation rate leads to endogenous checks for changes in
the government intervention in the markets. (c) A criterium based on the Tobin idea has
been proposed to dynamically evaluate the stock accumulation by the agents. The first
order conditions give the marginal storage value, from which the q(t) is derived. Finally,
the recursive computations for stock and futures price are used to predict series for the
U.S. corn futures price.
In the empirical part, we use the data for the 1973-2000 period to validate the general
model and introduce the previous features as the sensitivity analysis. Two possible
limitations of our analysis may account for this finding linked with the econometric
specification; the macroeconomic oil shocks are treated in a deterministic view and a
sectoral production shock is treated as a punctual phenomenon. With regard to these
empirical facts, different specifications are estimated; the results are statistically robust
and coherent with the economic theory with respect to the macroeconomic and sectoral
shocks. The empirical results show that the stockholding behaviour is fully driven by
the q-ratio variable: as shown, even if a shock causes heterogeneous behaviour of the
speculators who instantaneously adjust their financial positions, while the processors,
who demonstrate a precautionary behaviour due to the productive process, do not react
instantaneously and, therefore, the temporary lack of observed stock series away from
the maximizing path is justified.
28
Appendix A: Data and Statistical Sources
Data Table – Variables and Statistical Sources
U.S. Corn Market Data (Sample Period: 1973-2000)
Variable
Production Cost
Spot Price
Production
Imports
Exports
Consumption
Stocks
Storage Technology
Loan Rate
Interest Rate
Exchange Rate
Future Price (6 months)
Symbol
Ctd
Pm
Q
M
X
Con
FS
STH
Loan
IR
EXC
Pf
Statistical Source
U.S. National Service for Economic Research
U.S. National Service for Economic Research
U.S. National Statistical Service for Agriculture
U.S. National Statistical Service for Agriculture
U.S. National Statistical Service for Agriculture
U.S. National Statistical Service for Agriculture
U.S. National Statistical Service for Agriculture
U.S. National Statistical Service for Agriculture
U.S. National Statistical Service for Agriculture
U.S. Federal Reserve Bank
U.S. Federal Reserve Bank
U.S. Federal Reserve Bank
29
Tables
Table 1 - Unit root tests
ADF
Value of test statistic
Pm
FS
∆FS
φ
-3.5139
-2.5947
-5.1576
-4.101
Optimal number of lags
Akaike Info Criterion
1
0
0
Final Prediction Error
0
0
0
Hannan-Quinn Criterion
0
0
0
Schwarz Criterion
0
0
0
Asymptotic critical values: -3,43 (1%); -2,86(5%); -2,57(10%); Davidson et al. (1993)
0
0
0
0
KPSS
Value of test statistic
Pm
FS
∆FS
φ
-0.17
0.2155
0.064
0.4322
Asymptotic critical values: 0,739 (1%); 0,463(5%); 0,347(10%); Kwiatkowski et al.(1992)
Table 2- OLS Preliminarly Estimation
Dependent Variable: Convenience Yield
Variable
Coefficient t-statistic
p-values
Intercept
-6.1519
-5.5468
0.0000
Price
2.2963
5.9081
0.0000
Trend
0.1452
11.5342
0.0000
Stock
0.0003
3.5067
0.0017
R-squared
0.8491
Dependent Variable: Price
Variable
Coefficient t-statistic
Change in
-0.0001
-1.5728
Stocks
Exchange
0.0131
2.4586
Rate
Unit Prod. 73.3938
2.4545
R-squared
p-values
0.1283
0.0212
0.0214
0.2449
30
Table 3.a: FIML Estimation (without Dummy)
Dependent Variable: Convenience Yield
Coefficient p-value
Variable
-3.6682
0.1040
Intercept
1.4149
0.0770
Price
0.1367
0.0000
Trend
0.0002
0.1830
Stocks
R-squared
0.6432
Dependent Variable: Price
Coefficient
Variable
Change in Stocks
-0.0001
Exchange Rate
0.0136
Unit Production Cost
70.9675
R-squared
0.3303
p-value
0.0980
0.0140
0.0190
Table 3.b: FIML Estimation (with Dummy)
Dependent Variable: Convenience Yield
Variable
Coefficient
p-value
Intercept
-3.2479
0.1090
Price
1.2631
0.0960
Trend
0.1458
0.0000
Stocks
0.0002
0.0880
Dummy 1997
-1.8038
0.0210
R-squared
0.8067
Dependent Variable: Price
Variable
Coefficient
Change in Stocks
-0.0001
Exchange Rate
0.0132
Unit Production Cost
69.9813
Dummy 1996
0.7125
R-squared
0.5574
p-value
0.0770
0.0020
0.0040
0.1140
Graphs
Graph 1- The market adjustment process after shocking an exogenous variable
Panel a)
FS
Panel d)
∆FS=0
q(t)
A
∆2FS=0
B
C
∆2FS’=0
φ( t )− k
∆FS
Panel b)
Panel c)
Pm
Pm
Pm(t) *
E
Pm 2
Pm(t)
λ& ′ =
D
0
λ =0
Pm 0
φ( t) - k
∆FS
Graph 2- The role of δ in the marginal storage value
λ
λ∗
Pm
Equilibrium Stock Premium
for a given value of δ
δ∗
δ
1
Graph 3- The (1-δ) dynamic path
1.00
0.95
0.90
1-d
0.85
0.80
Year
19
99
19
97
19
98
19
95
19
96
19
93
19
94
19
92
19
90
19
91
19
88
19
89
19
86
19
87
19
85
19
83
19
84
19
81
19
82
19
79
19
80
19
78
19
76
19
77
19
74
19
75
t
19
73
0.75
Graph 4- Future and Spot Price: A Comparison
7.00
6.00
U.S. Dollars
5.00
4.00
U.S. Corn Price
U.S. Corn Future Price (6 Months Delivery)
3.00
2.00
1.00
19
99
19
97
19
95
19
93
19
91
19
89
19
87
19
85
19
83
19
81
19
79
19
77
19
75
19
73
0.00
Year
Graph 5- Production and Stock behaviours in U.S. Corn Market
12000.00
10000.00
8000.00
Corn Production
6000.00
Stored Corn
4000.00
2000.00
19
73
19
74
19
75
19
76
19
77
19
78
19
79
19
80
19
81
19
82
19
83
19
84
19
85
19
86
19
87
19
88
19
89
19
90
19
91
19
92
19
93
19
94
19
95
19
96
19
97
19
98
19
99
20
00
0.00
Year
34
Graph 6- A Comparison of the Estimates for U.S. Corn Price
4
3.5
3
U.S. Dollars
2.5
2
U.S. Corn Fitted (with Dummy) Price
1.5
U.S. Corn Fitted (without Dummy) Price
U.S. Corn Price
1
0.5
19
99
19
97
19
95
19
93
19
91
19
89
19
87
19
85
19
83
19
81
19
79
19
77
19
75
19
73
0
Years
Graph 7- Different Computations for the q(t) Ratio
2.5
2
1.5
Computed (with Dummy) q
Computed (without Dummy) q
1
0.5
19
73
19
74
19
75
19
76
19
77
19
78
19
79
19
80
19
81
19
82
19
83
19
84
19
85
19
86
19
87
19
88
19
89
19
90
19
91
19
92
19
93
19
94
19
95
19
96
19
97
19
98
19
99
20
00
0
Years
35
Graph 8- Stock Computations
8000.00
7000.00
6000.00
Thousand of Bushel
5000.00
U.S. Corn Stocks
4000.00
U.S. Corn Fitted (with Dummy) Stocks
U.S. Corn Fitted (without Dummy) Stocks
3000.00
2000.00
1000.00
19
99
19
97
19
95
19
93
19
91
19
89
19
87
19
85
19
83
19
81
19
79
19
77
19
75
19
73
0.00
Years
Graph 9- A Comparison of the Estimates for U.S. Corn Future Price (Six Months Delivery)
7
6
4
U.S. Corn Future Price
U.S. Corn Fitted (with Dummy) Future Price
U.S. Corn Fitted (without Dummy) Future Price
3
2
1
Years
36
19
99
19
97
19
95
19
93
19
91
19
89
19
87
19
85
19
83
19
81
19
79
19
77
19
75
0
19
73
U.S. Dollars
5
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