explaining the persistence of commodity prices

Computational Economics16: 149–171, 2000.© 2000Kluwer Academic Publishers. Printed in the Netherlands.

149

Explaining the Persistence of Commodity Prices

SERENA NG1 and FRANCISCO J. RUGE-MURCIA21Department of Economics, Boston College, Chestnut Hill, MA 02467, U.S.A.,E-mail: [email protected];2Department of Economics and C.R.D.E., University of Montreal,C.P. 6128, Montreal (Quebec) H3C 3J7, Canada, E-mail: [email protected]

Abstract. This paper extends the Competitive Storage Model by incorporating prominent featuresof the production process and financial markets. This extension seems necessary since the basicmodel does not successfully explain the degree of serial correlation observed in actual data. Togenerate a high degree of price persistence, the model must incorporate agents that are willing to holdstocks more often than predicted by the basic model, so we include characteristics of the productionand trading mechanisms to provide the required incentives. Specifically, we introduce (i) gestationlags in production with heteroskedastic supply shocks, (ii ) multiperiod forward contracts, and (iii )a convenience return to inventory holding. Rational expectations solutions for twelve commoditiesare solved numerically. Simulations are then used to assess the effects of these extensions on thetime-series properties of commodity prices. The results indicate that each feature accounts partly forthe persistence as well as the occasional spikes observed in actual data. Evidence is also presentedthat the precautionary demand for stocks might play a substantial role in the dynamics of commodityprices.

Key words: commodity prices, persistence, speculative storage

1. Introduction

This paper studies the determination of commodity prices when, in addition tothe producer and consumer, there is a rational, profit-maximizing agent that cancarry the good as inventory into future periods. This framework has been em-ployed, for example, by Newbery and Stiglitz (1982), Williams and Wright (1991),and Deaton and Laroque (1992) to examine the effect of speculative inventorieson commodity prices. The models developed here also give inventory holdingsprominence in determining the time-series properties of the price process but, inaddition, incorporate (i) gestation lags in production with heteroskedastic shocks,(ii ) multiperiod, overlapping forward contracts, and (iii ) a convenience motive forinventory holding. It is hoped that the addition of these realistic aspects of theproduction process and the trading mechanism will result in a model able to captureimportant characteristics of the data ignored by the basic model.

Earlier research documents two such features. First, prices are subject to oc-casional, dramatic increases or ‘spikes’. Deaton and Laroque (1992) model thesespikes as arising from stockouts. That is, in the absence of the smoothing effect of

150 SERENA NG AND FRANCISCO J. RUGE-MURCIA

inventories, the price is solely determined by the available harvest and consumptiondemand. Second, prices exhibit a high degree of serial correlation. Chambers andBailey (1996) and Deaton and Laroque (1996) explain this by assuming seriallycorrelated production shocks but still find substantial unaccountable persistence.

The specifications proposed here retain stockouts as a possible explanation forprice spikes but allow news about future production to affect current prices inways that could replicate pronounced price increases. In addition, we also exploitfeatures of the production process and financial markets to explain the persistenceof commodity prices. Loosely speaking, these features (production lags, contracts,and the convenience return) all increase the incentives for profit-maximizing agentsto hold stocks, thereby resulting in fewer periods when the intertemporal price rela-tionship would be severed by stockouts. Thus, these models should predict greaterpersistence than the basic storage model.1

The study of the factors that determine the price of primary commodities isimportant for several reasons. First, many less-developed countries depend on theexport of a small set (sometimes one) of agricultural products for most of theirforeign currency earnings. Second, several countries spend significant resourcesregulating agricultural sectors through price support mechanisms and regulationboards. A good understanding of the determinants of the price process helps assessthe effects of these government policies. Finally, the price behavior of industrialcommodities (e.g, copper, iron ore, oil) can have important implications for outputand business fluctuations.

As noted above, three extensions are examined. First, we consider ‘time tobuild’ as a source of persistence in commodity prices. On the one hand, gestationlags in production can increase the possibility of stockouts if there are periodsin which no new harvest is brought to the market. On the other hand, profit-maximizing speculators can view output seasonality as a reason to transfer someof the good to the period without production. Thus overall, production lags canreduce the probability of stockouts. Gestation lags provide a convenient and reas-onable way to introduce heteroskedasticity in harvest shocks that, under certainconditions, can also be associated with seasonality in consumption. Finally, inunderstanding the formation of price expectations, gestation lags are interestingbecause they allow news about the incoming crop to convey information about thevalue of next period’s output and thereby to influence the price of the commoditycurrently traded (see Lowryet al., 1987).

Second, we suggest the presence of forward, multiperiod contracts as a plausibleexplanation for the serial correlation observed in prices. The intuition for this de-rives from earlier work by Fischer (1977) and Taylor (1979, 1980), where staggeredlabor contracts yield a significant degree of serial correlation in nominal wages.Multiperiod contracts increase predicted price persistence by adding supply inevery period and unambiguously reduce the probability of stockout. Moreover, theoverlapping nature of the contracts means that, even if weather shocks are serially

EXPLAINING THE PERSISTENCE OF COMMODITY PRICES 151

uncorrelated, their effect on the commodity price lasts longer than the contractperiod.

Finally, we allow inventories to be demanded for other than speculative pur-poses. This is achieved by relaxing the assumption of zero convenience yield. Theconvenience yield (Kaldor, 1993) is a catch-all term for any return that accrues toconsumers and producers for being able to use the stored commodity wheneverdesired. Since the convenience yield could partially compensate inventory holdersfor their expected loss when the basis is below carrying charges, the model predictsa smaller number of stockouts and a larger degree of serial correlation in prices.

An important characteristic of basic storage models is that the demand for spec-ulative inventory will be positive if and only if the price expected by optimizingspeculators is high enough to cover carrying costs. This non-negativity constrainton inventories implies that the equilibrium price function is non-linear,2 takingthe form of a kink at the price at which inventory demand becomes positive. Theextensions proposed in this article preserve this non-linear structure of the priceprocess, and so we employ numerical methods to establish the rational expectationsolution. To asses the competing specifications in a unified framework, the mod-els are calibrated for a set of 12 commodities and their empirical properties arecompared with those of the actual data.

The plan of the article is as follows. Section 2 revises the basic storagemodel and introduces the main concept employed. This section also examines abenchmark case where serially correlated disturbances are incorporated into thecompetitive storage model. Sections 3, 4, and 5 introduce the specifications withgestation lags, forward contracts, and a convenience yield, respectively. Section 6presents the main empirical results and evaluates the different extensions. Finally,Section 7 concludes and suggests avenues for future research.

2. The Basic Storage Model

The rational-expectations competitive-storage model describes the optimal inven-tory decisions by risk-neutral speculators. This model is now briefly summarized.Let harvest,zt , be given by

zt = z+ ut ,

where z is a constant, andut is a disturbance term assumed independent andidentically distributedi.i.d. (0, σ 2) with compact support.3 The disturbanceut ismost intuitively interpreted as a weather shock. Denote byIt−1,t inventories heldfrom periodt − 1 to periodt . Let r be the real interest rate andδ be a non-negativedepreciation cost, so that the storage technology yields (1 –δ) units of the goodin period t for each unit stored in periodt – 1. It is further assumed that (i) theconvenience yield is zero, so that the only demand for inventories is speculative,and (ii ) inventories are costly to hold,i.e. (1 − δ)/(1 + r) < 1. With the above


notation, the quantity of commodity available at timet (denoted byxt ) can bewritten

xt = zt + (1− δ)It−1,t . (1)

Letpt be the price of the commodity at timet andD(pt ) be a deterministic demandfunction with inverse demand functionP(xt ). Then, market clearing requires

xt = zt + (1− δ)It−1,t = D(pt)+ It,t+1 .

The demand for one-period-ahead inventoriesIt,t+1 is the result of intertemporalarbitrage by risk-neutral speculators. Formally, letEtpt+1 be the expectation ofpt+1 conditional on information available at timet. The information set is assumedto containpt , zt , and the distribution of supply shocks. Arbitrage implies (seeScheinkman and Schechtman, 1983)

It,t+1 ≥ 0 if (1 − δ)/(1 + r)Etpt+1 ≥ pt ,It,t+1 = 0 otherwise.

(2)

The interpretation of this first-order condition is straightforward. Profit maximizingstockholders demand positive inventories if and only if the expected future price ishigh enough to cover carrying costs. Otherwise, inventory demand is zero.

Deaton and Laroque (1992) demonstrate the existence of an equilibrium for thismodel and show that prices follow an ergodic process leading to a non-zero probab-ility of stockouts. Since inventories are an intertemporal link between two periods,prices follow a linear first-order autoregressive process when there is stockholding,with a conditional variance that increases as the level of stocks diminishes. How-ever, when stocks are not held, the market price is independent of history, beingcompletely determined by the contemporaneous harvest. Accordingly, when thereare stockouts, prices follow a white noise process. The overall price process is anon-linear first-order Markov process with a kink at the price

p∗ = (1− δ)/(1+ r)Etpt+1

that is constant under the assumption that shocks to harvests arei.i.d.Because of the non-negativity constraint on inventories, the price process is non-

linear, and it is typically not feasible to find a closed-form solution for the agents’price forecast. To address this problem, Deaton and Laroque (1992) obtain nu-merical solutions by iterating on the equilibrium price function until convergence.Other methods for solving non-linear rational expectations model are discussed inTaylor and Uhlig (1990). This article employs the procedure proposed by Williamsand Wright (1991) (see also Den Haan and Marcet, 1990) that parameterizes theconditional price expectation in terms of the model’s state variables. More pre-cisely, the agents’ price forecasts are expressed as a finite, low-order polynomial ofthe state variable. For the basic storage model, there is only one state variable: the


current level of inventories. Therefore, one could represent the agents’ forecasts ofthe future price as a polynominal in the level of stocks with an associated set ofcoefficients.4 Accordingly, we postulate the function

Etpt+1 = ψ1(It,t+1) , (3)

whereψ1 (·) denotes a finite polynomial. Presumably,ψ1 (·) is a decreasing andconvex function of the level of inventories, so that the larger the current level ofstocks, the larger the availability in the future period and the lower the expectedprice. Given a set of structural parameters and an initial grid of inventories andharvest shocks, it is possible to combine ordinary least squares estimates of thepolynomial coefficients with Newton’s method to adjust these coefficients until therational-expectations solution obtains.5

The basic speculative storage model can replicate some of the features of thedata. For example, the simulated model displays the periodic price spikes that areoccasionally observed in the actual time series (see Deaton and Laroque, 1992,p. 14). However, reduced-form estimates (Ng, 1996) and model evaluations basedon estimation of the structural parameters (Deaton and Laroque, 1992) find greaterpersistence in price data than that predicted by theory. This paper seeks to explainthis persistence by incorporating realistic features of the financial markets and theproductive process. Both theoretical and empirical results demonstrate that the ex-tensions of the basic storage model proposed below can, in part, explain the highserial correlation in prices.

2.1. ADDING SERIALLY CORRELATED SUPPLY/DEMAND SHOCKS

Before proceeding to the economic extensions of the competitive storage model,we consider a useful benchmark that entails the mechanical introduction of seriallycorrelated shocks in the supply or demand of the commodity. Deaton and Laroque(1996) and Chambers and Bailey (1996) have also examined this possibility whenweather shocks follow an AR(1) process and show limited success in accountingfor the serial correlation in prices. This section complements and extends their res-ults by considering a MA(1) specification for the model shocks. As shown below,using MA(1) disturbances is considerably more tractable than postulating AR(1)errors and, for invertible specifications, it captures the serial correlation of produc-tion/demand shocks in a more parsimonious way than a higher-order autoregressiveprocess.

Under this specification, the (inelastic) harvest in periodt is

zt = z+ ut ,where, as before,z is a constant, but now the weather disturbanceut is assumed tofollow the MA(1) process

ut = et + ρet−1 ,


whereet is i.i.d. (0,σ 2) with compact support. In this simple extension of the basicstorage model, the market-clearing and the (profit-maximizing) arbitrage condi-tions are still given by (2.1) and (2.2), respectively. Thus, prices still follow thenon-linear process described above for the competitive storage model. However,as in Chambers and Bailey (1996), the price kink (p∗) is no longer constant and isinstead a function of the current observed harvest (or weather shock). This resultarises because, with serially correlated disturbances, speculators’ form their priceforecasts using information now contained in the current production shock. Thisimplies that, for the model with MA(1) errors, the set of state variables should beenlarged to include the current observation ofet .

Thus, for the purpose of parameterizing the speculators’ conditional expecta-tions, we postulate the function

Etpt+1 = ψ2(It,t+1, et ) , (4)

where the coefficients forψ2 are to be determined numerically as described in theprevious section.

The assumption that shocks are serially correlated is a very simple and naturalway to introduce persistence into the model. For example, in the case of tree crops(like coffee), damage to the plants due to bad weather can reduce not only current,but also future production. In the case of industrial commodities, serial correlationin demand might arise as a result of business cycle fluctuations. Rather generally,it is difficult to identify the origin of the serial correlation of prices without fur-ther assumptions on the structure of the model. But we shall see in the next threesections that the persistence in commodity prices can be replicated even withi.i.d.shocks once the model is enriched with more realistic features of production andpreferences.

3. Gestation Lags and Heteroskedastic Supply Shocks

The production of commodities takes time. Cash crops such as cotton and teaare not harvested immediately after planting, and the eventual quantity of outputdepends on the sequence of (weather) shocks between planting and harvest. Like-wise, industrial commodities such as iron and copper require time to be mined andrefined. The model we present below is best suited to agricultural crops, which con-stitute most of our sample, but its implications are easily generalized to industrialcommodities.

Gestation lags have several important implications. First, there can be periodswith no production. For agricultural commodities this could occur if harvestingtakes place in a particular season(s).Ceteris paribus, this limits supply to themarket and increases the probability of stock-outs. However, profit-maximizingspeculators have an incentive to transfer some of the good (via inventory) to peri-ods without production, thereby decreasing the probability of stockouts. Thus, it ispossible for the net effect of gestations lags to reduce the number of periods where


the intertemporal price link is severed and, consequently, to increase the estimatedserial correlation.

Second, the periodicity in production permits the straightforward introductionof heteroskedasticity in supply shocks. Intuitively, the variance of weather shocksmight differ across months or seasons. Under certain conditions, this heteroske-dasticity can be also interpreted as modeling seasonal effects inconsumption.Specifically, in the case when the inverse demand function is of the linear formP(x) = a + bx, the parameterb cannot be separately identified from the variance ofthe harvest (see Deaton and Laroque, 1996, pp. 905–906). Since a researcher usingprice data can only identify the ratiob/σ , changes inσ across seasons would beobservationally equivalent to changes in the slope of the demand functionb acrossseasons.

Third, while suppliers cannot revise their production schedules once the crophas been planted, speculators can observe shocks in between harvests. This inform-ation allows agents to update their forecasts about the size of the incoming harvestand demand inventories consistent with their revised expectations of the one-period-ahead price. Thus, the trading of commodities takes place on a continuousbasis in spite of the periodicity in the production process.

Finally, to the extent that news is allowed to have an effect on prices, a modelwith gestation lags might generate large price increases without stockouts inthe contemporaneous period. Consider, for example, the situation when a ‘bad’weather shock has affected the current crop. The anticipation of a smaller harvestand a higher price in the incoming period prompts rational agents to increase theircurrent demand for inventories to take advantage of theex-anteprofit opportunity.Thus, the current price rises. For a large negative shock, the price increase couldbe enough to replicate the large price spikes observed in commodity data.6

To formalize this idea, assume that a production cycle takes two periods. Thecrop grows during the odd periods while the harvest and subsequent planting takeplace in the even periods.7 Accordingly, harvestzt is affected by shocks duringthe growing and harvesting seasons. Letz be mean output anduj denote thei.i.d.supply shock in periodj. Assume that the distribution generating the observationson u has a variance that could differ if the period is odd (σ1) or even (σ2).8 Then,the harvest in the periodt is

zt = z+ utand the availability is

xt = zt + (1− δ)It−1,t .

Comparable expressions hold for the other even periods, namelyt + 2, t + 4, etc.In the odd periods, when there is no harvest, the only source of availability ispreviously accumulated stocks. Specifically in periodt + 1,

xt+1 = (1− δ)It,t+1 ,


with similar expressions holding for the other odd periods,t + 3, t + 5, etc. Marketclearing in even and odd periods requires that

t : xt = zt + (1− δ)It−1,t = D(pt)+ It,t+1

t + 1 : xt+1 = (1− δ)It,t+1 = D(pt+1)+ It+1,t+2 ,

whereD(pt) is a deterministic demand function. Speculative stockholding is nowgoverned by a pair of arbitrage conditions:

It,t+1 ≥ 0 if (1− δ)/(1− r)Etpt+1 = pt ,It+1,t+2 ≥ 0 if (1− δ)/(1+ r)Et+1pt+2 = pt+1 .

(5)

Notice that the arbitrage conditions (5) imply (as in the basic model) that pricesfollow a linear first-order autoregressive process with constant coefficient (1+ r)/(1−δ) when there is stockholding and a white noise process where there are stock-outs. However, once heteroskedastic production shocks are allowed, the thresholdpricep∗ is no longer unique. Instead, there are two distinctp∗s associated with theodd and even periods (see Chamber and Bailey, 1996, p. 938).

While gestation lags do not fundamentally alter the non-linear structure of theprice process, they can still increase serial correlation. This can happen becauseboth the periodic absence of new production and the information revealed byweather shocks have the effect of inducing speculators to hold inventories, therebyreducing the probability of stockouts. Since the intertemporal price link would thenbe severed less frequently, the predicted price persistence would be larger thanimplied by the basic storage model.9

Although the interpretation of the above arbitrage conditions is similar to thatin the basic storage model, there is an important difference here in the informationavailable to agents when constructing their price forecasts. In the basic storagemodel, speculators always observe current period output. In the model with gesta-tion lags, speculators observe the current supply shock only in odd periods and thelevel of output in even periods. Moreover, the shocks (or news) in the odd periodsreveal information about the size of the harvest and the price in the upcomingperiod. Since agents are rational and make use of all information available, theinformation revealed by harvest shocks in the odd periods is exploited by agentswhen making their inventory-holding decisions for next period. More formally,there are two state variables in the odd periods without harvest (the current levelof inventories and the current supply shock) and only one state variable in the evenperiods with harvest (the level of inventories). Thus,

t : Etpt+1 = ψ3(It,t+1) ,

t + 1 : Et+1pt+2 = ψ4(It+1,t+2, ut+1) .(6)

The coefficients forψ3 andψ4 are to be determined numerically as describedabove.


4. Multiperiod Forward Contracts

One institutional feature of trading in commodity markets is that contracts withdifferent holding periods are available. For example, the Chicago Board of Tradelists forward contracts for coffee for delivery as short as three months and as long asone year. This section examines the empirical implications of multiperiod contractson commodity prices. Earlier research in macroeconomics (Fischer, 1977; Taylor,1979, 1980) shows that in labor markets, overlapping wage contracts increase thedegree of rigidity in the endogenous variable. Thus, one should expect that overlap-ping forward contracts in commodity markets could increase the degree of serialcorrelation in pricesvis á visthat predicted by the basic storage model (where thisfeature is absent).

To model multiperiod contracts, letIt+i,t+j denote inventories carried in periodt + i for delivery in periodt + j. The basic storage model is the special case wherej –i = 1. For tractability, we concentrate on contracts with a maximum holding periodof two. Thus,j – i = 1 for one period contracts, andj – i = 2 for two period contracts.Here, the availability in each period consists of contemporaneous production plusthe inventories contracted in periodst – 1 andt – 2 to be delivered in the currentperiod, that is,

xt = zt + (1− δ)It−1,t + (1− δ)2It−2,t .

As in the simple storage model, the harvest is assumed to be described byxt =z + ut , whereut is ani.i.d. disturbance with mean zero and constant variance. LetD(pt ) be a deterministic demand fuction and note that market clearing requires

zt + (1− δ)It−1,t + (1− δ)2It−2,t = D(pt)+ It,t+1+ It,t+2 .

In this case speculative stockholding must satisfy a pair of arbitrage conditions thathold simultaneously in every period to determine the demand for one-period andtwo-period ahead inventories. Specifically,

It,t+1 ≥ 0 if (1− δ)/(1+ r)Etpt+1 = pt , (7)

It,t+2 ≥ 0 if [(1− δ)/(1+ r)]2Etpt+2 = pt .As before, speculators carry inventories if and only if the expected capital gainsuffices to cover storage costs. It is assumed that speculators holding two-periodinventories are contractually prevented from rolling them over. That is, once a stockholder agrees to deliverIt,t,+2 units of the good in periodt + 2, she must physicallyhold the stocks during the contract period and cannot modify her stockholding inlight of new information available after her decision is made.10 Thus, the pres-ence of overlapping contracts means that, even if the weather shocks are seriallyuncorrelated, their effect on the commodity price lasts longer than the contractperiod.


The presistent effects of weather shocks can also be seen by noting that inthis set up there are two different threshold valuesp∗. More precisely, spec-ulators hold one-period contracts whenever the current price is belowp∗ =(1− δ)/(1+ r)Etpt+1 and two-period contracts whenever the price is lower thanp∗ = [(1 − δ)/(1+ r)]2Etpt+2. As seen in Section 6, the fact that contracts ofdifferent maturity are subject to different arbitrage conditions makes it possible todecompose the level of stocks into those held for one and two periods.

In this model with multiperiod contracts, the number of state variables is sub-stantially larger than in the models considered above. The agents’ forecasts of thefuture price depends on all types of inventories currently held by speculators, andin theory, the one- and two-period ahead expectations should be parameterizedasEtpt+1 = ψ5(It−1,t+1, It,t+1, It,t+2), andEtpt+2 = ψ6(It−1,t+1, It,t+1, It,t+2),respectively. Notice however, that, from the agents’ perspective,It,t+1 andIt,t+2

contain exactly the same information. Thus, the inclusion of both the OLS es-timation of the expectations polynomial produces colinearity between these tworegressors. Therefore, in calculating the expectations coefficients, the followingparameterization is employed

Etpt+1 = ψ ′5(It−1,t+1, It,t+1) , (8)

and

Etpt+2 = ψ ′6(It−1,t+1, It,t+2) , (9)

where the coefficientsψ ′5 andψ ′6 satisfy the assumption of rational expactationsand are numerically estimated for each commodity in the sample.

Multi-period contracts can increase the predicted persistence in prices becausethey provide additional sources of supply in every period and unambiguously re-duce the probability of stockout. The effect of overlapping contracts on the timeseries properties of prices will depend on the agent’s willingness to carry multi-period inventories. We show in Section 6 that, in contrast to earlier literature onstorage, where contracts are absent and stockholders are free to roll-over theirinventories, a model with overlapping contracts can partially explain the high serialcorrelation in prices.

5. The Convenience Yield

The arbitage conditions implied by the various models above all indicate that thedemand for inventories is positive only when the basis (i.e., the difference betweenthe future expected price and the current spot price) is enough to cover storagecosts. As a direct consequence of this, the models admit zero as the lower boundfor inventories. This differs from earlier literature on storage that indicates that incertain instances individuals and firms could be willing to carry stocks even whenthe difference between the future expected price and the current price is less than


the cost of storage.11 Three possible explanations have been advanced to accountfor this apparent negative return to storage. First, Vance (1946) suggests that agentsmight unduly discount the future relative to the present. To support this view, Vancecites empirical evidence collected by Vaile (1944) that indicates that future pricesare consistently lower at the beginning than at the close of the trading period foreach contract considered.

A second explanation (that also accounts for Vaile’s observation) is that futuresprices might be downwardly biased as a result of a risk premium. This explana-tion is related to the theory of normal backwardation proposed by Keynes (1930)and postulates that, since production must be planned well ahead of consumption,hedgers as a whole have a tendency to go short in futures and, consequently, apositive risk premium is required to persuade speculators to take the correspondinglong position. Subsequent empirical research has sought to detect the presence andevaluate the magnitude of the expected risk premium with inconclusive results (seeTelser, 1958; Cootner, 1960, 1967; Dusak, 1973; Breeden, 1980; and Fama andFrench, 1987).

A third explanation is proposed in Kaldor (1939), which suggests that the returnof storage might include a component of ‘convenience’; that is, the possibility ofusing the stored materials whenever desired. For a producer, this return could arisebecause inventories (i) reduce the need to revise production schedules, (ii ) allow thesupplier to take advantage of unexpectedly higher prices to meet demand quicklywithout increasing production, (iii ) diminish the probability of having to turn downbuyers, or (iv) reduce replenishment costs and time delays in delivery. In the caseof a consumer, a convenience yield could arise if either the timing or the level ofconsumption is stochastic. Thus, regardless of whether the stockholders are playersin the supply or the demand side of the market, there is likely a convenience returnthat could partially offset the physical and financial costs of carrying stocks.

The proposition that inventory holders might derive a convenience yield fromholding inventories is embedded in the classical storage supply function examinedempirically in Working (1934, 1949), Brennan (1958), Fama and French (1987,1988), and Miranda and Rui (1996). Pindyck (1993) examines the extent to whichthe current price of a commodity is explained by the present discounted sum ofexpected convenience returns. As a whole, this body of research appears to in-dicate there can be both a speculative and a convenience motive for inventoryholding. However, the effect of introducing a convenience yield on the time seriesproperties of commodity prices remains to be investigated. In particular, it wouldbe interesting to examine how the introduction of a convenience yield alters thedegree of serial correlation in prices. Recall that the basic storage model predicts alarge number of stockouts, and the weak intertemporal link served by inventories isresponsible for the low degree of serial correlation in prices. Since the convenienceyield could partially compensate inventory holders for the expected loss when thebasis is below carrying charges, a model that allows for a convenience yield shouldpredict a smaller number of stockouts. One would therefore expect the demand


for inventories for convenience purposes to strengthen the intertemporal link andhence result in a higher persistence in prices.

Since the sole determinant of the convenience yield is the level of inventoriesheld, one could mathematically express the convenience return as a function only ofthe level of stocks. Letφ(It,t+1) be the convenience yield from holding inventoriesbetween periodt andt + 1. Economic reasoning suggests that the marginal value ofholding inventories should increase with scarcity, and diminishing marginal returnsuggests a declining and concave relationship between stocks and the convenienceyield. The latter arises from the fact that, as inventories accumulate, marginal stor-age costs increase. Henceφ′(It,t+1) > 0 andφ′′(It,t+1) < 0. For example, onecould parameterize the (gross) marginal convenience yield as

φ′(It,t+1) = θ + (1− θ)g(It,t+1) , (10)

where

θ = (1+ r)(1− ε)(1− δ) ,

for an arbitrarily smallε and a normalizing constantc≥0, and

g(It,t+1) = It,t+1

It,t+1+ c .

This specification forφ′ (It,t+1) contains no convenience yield as a special caseby settingc = 0, satisfies the requirement that the marginal convenience yield bea decreasing, convex function of the level of inventories, and insures that it willbe appropriately bounded between(1+ r)/(1− δ), whenIt,t+1→0, and 1, whenIt,t+1→∞. The property thatφ′(It,t+1) is bounded between (1+r)/(1−δ) and 1 isimportant because it guarantees the existence of the equilibrium price function (seeDeaton and Laroque, 1992) and preserves the non-linearity of the price process.That is, there might be (negative) values of the intertemporal price spread that can-not be compensated by the convenience return, in which case, the non-negativityconstraint on inventories binds. Miranda and Rui (1996) postulate a cost functionthat admits an unbounded marginal convenience yield. In this case, inventories arealways strictly positive. Since the intertemporal price link created by stockholdersis never severed by stockouts, this model predicts a high serial correlation in prices.However, it is unclear how a model without stockouts explains another fundamentalfeatures of commodity prices, namely the presence of price spikes.

Following Gibson and Schwartz (1990), Schwartz (1997), and Routledge,Seppim and Spatt (1997), we introduce the convenience yield multiplicatively.Stockholding is now governed by the arbitrage condition.

It,t+1≥0 if φ′(It,t+1)(1− δ)/(1+ r)Etpt+1 = pt . (11)

As before, the stockholder carries inventories if and only if the expected return cov-ers the cost of storagenetof the convenience yield, and prices follow a white noise


process when there are stockouts. However, a unique implication of the conveni-ence yield is that prices will follow an AR(1) process with time-varying coefficient(1+r)/[(1−δ)φ′(It,t+1)]while there is stockholding. Furthermore, the price abovewhich inventory demand would be zero isp∗ = φ′(It,t+1)(1− δ)/(1+ r)Etpt+1.To the extent that the level of stocks is time-varying,p∗ is no longer constant asin the basic model. Notice that, for a current price abovep∗, even the conveniencereturn is insufficient to induce a demand for stocks.

In principle, the introduction of a convenience yield allows us to decomposethe demand for inventories in two components: speculative and precautionarydemand. For speculative demand,φ′(It,t+1) = 1, and the (gross) return is(1 −δ)/(1+ r)Etpt+1. For precautionary demand,φ′(It,t+1)≥1, and the (gross) returnis φ′(It,t+1)(1 − δ)/(1 + r)Etpt+1. However, the above parameterization of themarginal convenience yield implies thatφ′(It,t+1) > 1 for any non-negative levelof inventories and, consequently, implies that all stocks will be held for precau-tionary reasons. Therefore, we also consider an alternative, linear specification forφ′(It,t+1) that satisfies the boundary conditions and yet permits the identificationof speculative and precautionary inventory demand. Let

φ′(It,t+1)+ θ + (1− θ)g(It,t+1) , (12)

where

θ = (1+ r)(1− δ) ,

g(It,t+1) = min

{γ

θ − 1It,t+1,1

},

and γ > 0. In this case, the demand for inventories will have two kinks. Thefirst is the constantp∗∗ = (1 − δ)/(1 + r)Etpt+1, and speculative demand iszero whenp > p∗∗ (but precautionary demand is still positive). The second is thetime-varyingp∗ = φ′(It,t+1)(1− δ)/(1+ r)Etpt+1, and whenp > p∗ even theconvenience return is insufficient to induce a positive demand for inventories. Sinceφ′(It,t+1)≥1,p≥p∗∗ for all t. Thus, whenever speculators are in the market (in thesense that they demand positive levels of stocks), precautionary holders can alsobe in the market, but their gross convenience return would be 1. For values of thecurrent price strictly larger thanp∗∗ but smaller thanp∗, only precautionary holdersdemand stocks. Finally, if the current price exceedsp∗, neither speculators norprecautionary holders demand inventories, and a stockout occurs. While it is clearthat the existence of a second price kink is an artifact of the parameterization ofthe marginal convenience yield, this specification proves useful below in assessingthe relevance of the convenience yield for explaining the persistence of commodityprices.

The state variables for this model are the same as in the basic storagemodel, namely, the current level of inventories. Thus,Etpt+1 is parameterized


by ψ ′7(It,t+1), where the coefficients of the polynomialψ7· satisfy the rationalexpectations solution and are numerically estimated as in the model above.

6. Empirical Assessment

The specifications proposed above introduce features of the production process andfinancial markets that provide a theoretical explanation of the observed persistencein commodity prices. In order to assess the qualitative and quantitative implica-tions of these extensions in a unified framework, we calibrate the models usingparameter values estimated by Deaton and Laroque (1996) for a set of agriculturaland industrial commodities. The calibrated model is used to simulate a time seriesof commodity prices, subject to the restrictions of the theoretical model, whoseproperties may be compared by a set of statistics with those of the actual data. Astatistic of primary interest is the first-order autocorrelation of prices.

Before proceeding, we first examine whether the method of parameterized ex-pectations we use to obtain the rational expectations solution can replicate theresults in Deaton and Laroque (1992, 1996). To this end, the basic storage model isfirst calibrated for the parameters values used in Deaton and Laroque (1992, p. 11).There, the authors assume a linear inverse demand functionP(x) = a + bx andexamine the degree of serial correlation in prices for an arbitrary set of structuralparameters (a, b, δ, r). As noted in Deaton and Laroque (1996), the parameteracannot be identified separately from the mean of harvest, nor the parameterb fromthe variance of harvest. Thus, variations ina across commodities can arise becauseof variations in autonomous demand, the mean of the shock, or both. With this inmind, the method of parameterized expectation yields an estimated first-order auto-gressive coefficient of 0.087 for the values (a, b, δ, r) = (200, –1.0, 0.05, 0.056).For the set (600, –5.0, 0.0, 0.05), the estimated value is 0.441. These statistics arevery close to the values of 0.08 and 0.48 reported in Deaton and Laroque (1992).

Since we desire to compare the properties of the extended models with those ofthe basic storage model, we also calibrate the latter using the estimates (a, b, andδ) reported in Deaton and Laroque (1996, p. 911) for a set of 12 commodities.To be consistent with Deaton and Laroque, the rate of interest is fixed tor =0.05. For each calibrated model, 5000 observations are simulated and the first-order autoregressive coefficient calculated. For this sample size, the autoregressivecoefficient has a standard error of 0.014. These results serve as our base case andare presented in Table I. Results reported in Deaton and Laroque (1996) are alsoreproduced in Table I for convenience. While we use the method of parameterizedexpectation to find the rational price forecast whereas Deaton and Laroque iterateon the arbitrage condition, the reader can verify that both methods yield quantit-atively similar estimates. The largest absolute difference is 0.057 for the case ofcopper and maize, and the average difference is only 0.013. Thus, the low degreeof serial correlation predicted by the basic model appears robust to the choice ofsolution method. It seems reasonable to conclude that, to the extent that our models


Table I. Comparison of estimates of serial correlation

Demand =P(x) = a + bxz∼N(0, σ = 1).

Commodity a b δ D&L Parameterized

procedure expectations

Cocoa 0.162 –0.221 0.116 0.298 0.352

Coffee 0.263 –0.158 0.139 0.242 0.219

Copper 0.545 –0.326 0.069 0.392 0.335

Cotton 0.642 –0.312 0.169 0.192 0.173

Jute 0.572 –0.356 0.096 0.302 0.289

Maize 0.635 –0.636 0.059 0.356 0.413

Palm oil 0.461 –0.429 0.058 0.416 0.397

Rice 0.598 –0.336 0.147 0.224 0.237

Sugar 0.643 –0.626 0.177 0.264 0.266

Tea 0.479 –0.211 0.123 0.230 0.213

Tin 0.256 –0.170 0.148 0.256 0.238

Wheat 0.723 –0.394 0.130 0.259 0.250

Notes: All simulations are based on a sample size of 5000 observations.The conditional price expectation was parameterized using a polynomialof order 3 in the current level of stocks.

produce estimates of the autoregressive coefficients that differ from those of thebasic storage model, this difference is not attributable to the use of an alternativeprocedure for modelling the agent’s forecast.

The results for the extended models are reported in Tables II, III and IV. First,we see from Table II that, a comparison of the serial correlation for the actualdata (column 1) and for the base case (column 2) indicates that the commodityprices simulated from the basic model display only about one-third of the serialcorrelation of the actual time series. It is course, the objective of our analysis toaccount for this discrepancy.

In Table II, we see from the last three columns that the naive introduction of seri-ally correlated shocks does indeed increase the persistence predicted by the model,as expected, and while for plausible values of the moving average coefficient (ρ =0.2 andρ = 0.5), this increase is substantial, the serial correlation is still below thatobserved in actual data. Only when the coefficient acquires the more extreme valueof ρ = 0.8 does the predicted serial correlation rise enough to be comparable to theempirical one for a small set of commodities.

Of course, introducing serially correlated shocks is but a crude way of handlingthe possible misspecification of the basic model, so we now examine the economicmodels of inventory holding that are the main focus of this paper. It is hoped thatthese specifications can eliminate some (or all) of the sources of misspecificationin the competitive storage model.


Table II. Estimates of serial correlation with MA(1) shocks

Demand =P(x) = a + bxz∼N(0, σ ).

Commodity Actual Basic MA(1) shocks

model ρ = 0.2 ρ = 0.5 ρ = 0.8

Cocoa 0.834 0.352 0.447 0.547 0.609

Coffee 0.804 0.219 0.367 0.489 0.576

Copper 0.838 0.335 0.418 0.529 0.619

Cotton 0.884 0.173 0.328 0.468 0.564

Jute 0.713 0.289 0.391 0.522 0.589

Maize 0.756 0.413 0.481 0.584 0.644

Palm oil 0.730 0.397 0.479 0.587 0.637

Rice 0.829 0.237 0.327 0.494 0.579

Sugar 0.621 0.266 0.364 0.504 0.583

Tea 0.778 0.213 0.335 0.478 0.571

Tin 0.895 0.238 0.377 0.521 0.567

Wheat 0.863 0.250 0.352 0.504 0.602

Notes: All simulations are based on a sample size of 5000 observa-tions. The conditional price expectation was parameterized using apolynomial of order 2.

Consider first, the model with gestation lags. It is apparent from Table III thatproduction lagsalonecannot account for the persistence observed in the time seriesof commodity prices. In the case of maize, sugar, tin and wheat, the degree ofpersistence decreases rather than increases. Interestingly, these tend to be com-modities with high estimates forδ. Less of what is stored is made available to themarket when the depreciation rate is high, and this weakens the intertemporal linkin prices.

However, a model with gestation lagsand heteroskedastic supply shocks doespredict a higher price persistence than the basic model. Increasingσ2/σ1 from 1to 1.5 raises the calculated first-order serial correlation by over 30 percent in thecase of cocoa and by as much as 70 percent in the case of wheat. An additional7 to 30 percent increase is observed asσ2/σ1 is increased to 1.8.12 Note also thatthe improvement tends to be larger whenb is relatively small in absolute value. Inspite of this, this model does not generate adequate price persistence to match thevalues observed in the actual series. The gain from this extension is particularlysmall with commodities such as cotton, rice and wheat, whose depreciation ratesare estimated to be high.

To examine further the role of heteroskedasticity in producing price persistence,we solve the model for cocoa with increasingly large value ofσ2/σ1. In addition tothe values already considered (1, 1.5, and 1.8), we also examine 2, 2.5, 3, and 5.


Table III. Estimates of serial correlation with gestation lags


Commodity Actual Basic Gestation lags

model σ2/σ1 = 1 σ2/σ1 = 1.5 σ2/σ1 = 1.8

Cocoa 0.834 0.352 0.353 0.457 0.511

Coffee 0.804 0.219 0.256 0.385 0.433

Copper 0.838 0.335 0.353 0.471 0.526

Cotton 0.884 0.173 0.196 0.278 0.365

Jute 0.713 0.289 0.298 0.417 0.486

Maize 0.756 0.413 0.385 0.604 0.620

Palm oil 0.730 0.397 0.416 0.593 0.640

Rice 0.829 0.237 0.219 0.343 0.398

Sugar 0.621 0.266 0.239 0.355 0.427

Tea 0.778 0.213 0.218 0.361 0.428

Tin 0.895 0.238 0.226 0.380 0.428

Wheat 0.863 0.250 0.222 0.384 0.411

Notes: The standard deviationσ1 was fixed to 1 while the standard deviationσ2 was allowed to take the values 1, 1.5 and 1.8. All simulations are basedon a sample size of 5000 observations. The conditional price expectation wasparameterized using a polynomials of order 2. (except for maize and palm oilwhere a polynominal of order 1 was employed for the casesσ2/σ1 = 1.5 andσ2/σ1 = 1.8).

Figure 1. Serial correlation and heteroskedastic supply shocks.


Table IV. Estimates of serial correlation


Commodity Actual Basic Overlapping Convenience yield

model contracts c = 50 γ = 0.01 γ = 0.1 γ = 1

Cocoa 0.834 0.352 0.462 0.522 0.537 0.416 0.357

Coffee 0.804 0.219 0.385 0.530 0.477 0.365 0.231

Copper 0.838 0.335 0.394 0.608 0.497 0.375 0.333

Cotton 0.884 0.173 0.337 0.473 0.458 0.341 0.212

Jute 0.713 0.289 0.365 0.545 0.530 0.393 0.333

Maize 0.756 0.413 0.418 0.623 0.557 0.482 0.380

Palm oil 0.730 0.397 0.438 0.625 0.589 0.432 0.399

Rice 0.829 0.237 0.334 0.475 0.441 0.344 0.254

Sugar 0.621 0.266 0.370 0.424 0.456 0.379 0.279

Tea 0.778 0.213 0.302 0.509 0.458 0.352 0.246

Tin 0.895 0.238 0.355 0.472 0.486 0.391 0.281

Wheat 0.863 0.250 0.368 0.505 0.508 0.357 0.270

Precautionary stocks 0% 0% 100% 100% 84.2% 22.6%

Notes: The last row denotes the average of the percentage of precautionary stocks for all12 commodities. All simulations are based on a sample size of 5000 observations. The con-ditional price expectation was parameterized using a polynomials of order 2 (overlappingcontracts) and 3 (convenience yield).

The first-order serial correlations are estimated using simulated samples of 5000observations. The calculated values are presented in Figure 1. These results clearlysuggest that introducing heteroskedasticity in supply shocks has a substantial effecton the serial correlation predicted by the model, but the increase is marginal to nilonceσ2/σ1 exceeds 2.5. Thus, even increasingσ2/σ1 to unrealistic values cannotreplicate the persistence observed in the actual data.

The results for the model incorporating overlapping contracts are reported inTable IV. Compared to the base case, the degree of serial correlation is increasedfrom a modest 1 percent in the case of maize to an impressive 94 percent in thecase of cotton. While these results are encouraging, the degree of persistence isstill substantially smaller than that estimated using actual the data.

As noted in Section 4, the introduction of contracts is conjectured to reducethe probability of stockouts by providing additional sources of availability. Forexample, even if speculators are unwilling to carry one-period inventories in thecurrent period, the decision taken in the preceding period (t – 1) to carry two-period inventories can help avoid a stockout in the next period (t + 1) should abad harvest occur. To examine whether this hypothesis is supported by the data,Figure 2 presents a subsample of 100 observations simulated using the model for


Figure 2. Comparison of inventory holdings.

cocoa. The figure suggests that, while two-period contracts are demanded by specu-lators, they complement rather than replace one-period ahead contracts. Moreover,this type of inventory is held less frequently than shorter term maturities. One-period contracts are held 96.3 percent of the time, while two-period contracts areheld only 29.3 percent of the periods and usually simultaneously with the shortermaturity contracts. This underscores the role of inventories as a mechanism fortransferring goods from abundant to scare periods and provides some insight intothe role contracts play in determining commodity prices. However, it also indicatesthat contracts alone might not fully explain the high degree of serial correlationobserved in the actual data.

The results of the simulations for the model with a convenience yield arepresented in the last four columns of Table IV. Consistent with the previous two ex-tensions, serial correlation is higher for commodities with a low depreciation rate,and although it is still lower than that of the observed data, it appears that the stor-age model with convenience yield produces the largest increase in serial correlationamong the models considered. Indeed, with a suitable choice of parameters (suchasc andγ ), both specifications ofφ′(It,t+1) are capable of increasing the serial cor-relation substantially. Recall that in the gestation-lag model, even an unreasonabledegree of heteroskedasticity does not increase the serial correlation to levels closeto those observed in the data. With the introduction of a convenience yield usingthe smooth parameterization ofφ′ in (10), maize and palm oil now have a serialcorrelation coefficient of over 0.6, which is much closer to the observed value of0.7 than the estimates obtained from the previous extensions. A further analysis ofthe simulations reveals that introducing a convenience yield significantly reducesthe probability of stockouts and the number of spikes observed in a finite sample.With this additional reason for holding inventories, agents might carry stocks evenif the expected return is lower than the carrying cost. Consequently, in a given


sample, the number of observed stockouts diminishes and the number of periodsfor which inventory holding creates an intertemporal link in prices increases. Thus,the estimated serial correlation in prices rises.

With φ′ defined as (12), the ability of the convenience yield to increase serialcorrelation in prices depends on the value ofγ . For the case whenγ = 1, themarginal convenience yield rapidly decreases with inventories and hence does notsignificantly strengthen the intertemporal link in prices. Hence the estimates ofserial correlation are numerically close to the vales predicted by the basic model,where the precautionary motive is absent. For the case whenγ = 0.01, the serialcorrelations predicted by the model are comparable to ones obtained with thesmooth parameterization ofφ′ defined (10). But the autoregressive coefficient im-plied by the two parameterizations ofφ′ are (different) time-varying functions ofthe level of stocks, accounting for the differences in the time series properties ofprices implied by the two parameterizations.

To disentangle the demand for precautionary and speculative inventories and tohelp examine the effect of buffer-stockholding behavior on commodity prices, thefraction of inventories held for convenience purpose is calculated using the modelwith the piece-wise linear marginal convenience yield function described in (12)for various values of the coefficientγ . This parameter measures the reduction inthe marginal convenience yield as a result of an unitary increase in the level ofstocks and is smaller the stronger is the desire for holding buffer stocks. We seethat, for all commodities, the price persistence is higher the lower the value ofγ .Further, the last row of Table IV reveals that the largerγ , the lower the proportionof stocks held by speculators. The intuition behind these results is straightforward:for lower values ofγ the marginal convenience yield decreases more slowly withthe level of stocks. Thus, there is a larger range of values of stocks for which theprecautionary motive dominates the speculative motive for inventory holding. Tothe extent that agents are willing to hold stocks for both the expected capital gainand the convenience return, the probability of observing stockouts decreases andthe price persistence rises. The average proportion of precautionary stocks for all12 commodities is smaller (22.6%) than in the two other cases considered (100%whenγ = 0.01 and 84.2% whenγ = 0.1).

The role of precautionary stockholding in the persistence of commodity pricesis further documented in Figure 3. Here the model for cocoa is solved and sim-ulated for a large range of values ofγ . The serial correlations are based onsamples of 5000 observations. This graph clearly suggests that price persistenceis monotonically increasing in the proportion of stocks held for precautionarymotives.

Finally, it is important to note that the model with a convenience yield givespredictions for price serial correlation that are (i) numerically similar to those ob-tained by mechanically introducing MA(1) shocks (withρ = 0.2, 0.5) but (ii ) basedon a more solid economic rationale: the precautionary nature of inventory demand.


Figure 3. Precautionary storage and serial correlation.

7. Conclusions

The goal of this analysis is to generalize the basic storage model to one better ableto reproduce the degree of serial correlation observed in actual data. The proposedspecifications preserve important features of the basic model, such as the non-negativity constraint on inventories and the possibility of stockouts that accountfor occasional price spikes. However, they explicitly incorporate realistic aspectsof the production process and financial markets that could explain the persistenceof commodity prices. The results are only partially successful. The first two exten-sions are capable of reducing the discrepancy between the serial correlation in thedata and the one predicted by the model from a factor of 3 to 2. The introductionof gestation lags with heteroskedastic shocks is somewhat more successful, butsubstantial persistence remains unaccounted for.

The third specification considers the more general situation when agents candemand inventories for both precautionary and speculative reasons. This modelpredicts values of serial correlation that are closer to the actual ones, explainsalmost 65% of the observed price persistence, and unambiguously highlights therole of a convenience yield in the dynamics of commodity prices. At the theoreticallevel, these results suggests that explicitly modeling the agents’ risk aversion (thatunderlies the convenience yield) could produce a model of storage that capturesthe features of the data better. At an empirical level, the evidence indicates that asuccessful explanation for the time series properties of commodity prices wouldrequire the introduction of a non-speculative demand for stocks.

Additional modifications to the basic model might also be necessary. The in-clusion of all the above features within a single model could be numericallychallenging but could also provide more accurate predictions than each separatespecification on its own. Further, given the substantial differences in institutional


and physical features across commodities and the idiosyncracies of their produc-tion and consumption processes, it might be necessary to construct commodity-specific models to provide more accurate empirical predictions. The specificationsproposed above could be useful here. They allow analysis with a variety of tradingand production mechanisms that suitably model the length and type of contracts,seasonality in production, and heterogeneity of depreciation rates, price elasticityof demand, and spoilage costs.

Acknowledgements

We have benefited from helpful comments by seminar participants at Johns Hop-kins University. Jean-Philippe Laforte and David Servettaz provided excellentresearch assistance. Financial support by the Social Science and HumanitiesResearch Council of Canada (SSHRC) and the Fonds pour la Formation de Cher-cheurs et l’Aide à la Recherche du Québec (FCAR) is gratefully acknowledged.

Notes1 The model developed by Deaton and Laroque (1992) yields a high probability of stockouts and,

consequently, predicts a far larger number of spikes than observed in actual data.2 The idea that inventories are bounded below by zero was originally discussed by Samuelson

(1957) and Gustafson (1958). Muth (1961) obtains a linear rational expectations equation only underthe assumption that speculative inventories can be either positive or negative.

3 This specification implicitly assumes that the commodity supply is inelastic with respect to theexpected price. This postulate is solely made for tractability and does not affect the basic implicationsof the model.

4 Judd (1990) rigorously defends the use of polynomials to approximate non-linear functions.5 Williams and Wright (1991, pp. 81–90) describe the algorithm in detail for the cases with and

without supply elasticity.6 A recent example of this alternative explanation of price ‘spikes’ was the large increase in current

and future prices of coffee in June of 1994 following unusually low temperatures in Southern Brazil(seeThe Economist, 2 July 1994, p. 96).

7 Lowry et al. (1987) develop a quarterly model for soybeans with storage within and across cropyears. In their model, the grain is harvested only in one of the quarters.

8 Chambers and Bailey (1996, p. 940) show the existence and uniqueness of the equilibrium pricefunction with periodic disturbances.

9 Notice that the periodic arrival of the harvest might reduce the agents incentive to carry invent-ories from the odd to the even periods. Thus, gestation lags might also reduce the serial correlationin prices. Which one of the two effects dominates is an empirical matter to be examined in Section 6.

10 The assumption is equivalent to the one in Fischer (1977), where two-period labor contractscannot be renegotiated at the end of the first term.

11 This observation is not universally shared. Williams and Wright (1991) dismiss the empiricalobservation of stockholding below full carrying charges as an aggregation phenomenon (see also,Williams, 1986, and Wright and Williams, 1989).

12 For the cases of maize and palm oil, the numerical procedure fails to find solutions that involvepolynomials of order higher than 1 for the model with heteroskedastic disturbances. Since the linear


approximation might not be as accurate as the paramerizations that include higher order terms, theresults for these two commodities must be interpreted with caution.

References

Breeden, D.T. (1980). Consumption risks in futures markets.Journal of Finance, 35, 503–520.Brennan, M.J. (1958). The supply of storage.American Economic Review, 47, 50–72.Chambers, M.J. and Bailey, R. (1996). A theory of commodity price fluctuations.Journal of Political

Economy, 104, 924–957.Cootner, P.H. (1967). Returns to speculators: Telser vs. Keynes.Journal of Political Economy, 68,

396–404.Deaton, A.S. and Laroque, G. (1992). On the behavior of commodity prices.Review of Economic

Studies, 89, 1–23.Deaton, A.S. and Laroque, G. (1996). Competitive-storage and commodity price dynamics.Journal

of Political Economy, 104, 896–923.Den Haan, W.J. and Marcet, A. (1990). Solving the stochastic growth model by parameterizing

expectations.Journal of Business and Economic Statistics, 8, 31–34.Dusak, K. (1973). Futures trading and investor returns: An investigation of commodity market risk

preminums.Journal of Political Economy, 81, 1387–1406.Fama, E.F. and French, K.R. (1987). Commodity futures prices: Some evidence on forecast power,

premiums, and the theory of storage.Journal of Business, 60, 55–73.Fama, E.F. and French, K.R. (1988). Business cycles and the behavior of metal prices.Journal of

Finance, 43, 1075–1093.Fischer, S. (1977). Long-term contracts, rational expectations, and the optimal money supply rule.

Journal of Political Economy, 85, 191–205.Gibson, R. and Schwatz, E.S. (1990). Stochastic convenience yield and the pricing of oil contingent

claims.Journal of Finance, 45, 959–976.Gustafson, R.L. (1958). Carryover levels for grains: A method for determining amounts that are

optimal under specified conditions.USDA Technical Bulletin, 1178.Judd, K.L. (1990). Minimum weighted residuals methods for solving dynamic economic models.

Hoover Institution,Mimeo.Kaldor, N. (1939). Speculation and economic stability.Review of Economic Studies, 7, 1–27.Keynes, J.M. (1930).A Treatise on Money, Volume II: The Applied Theory of Money. Macmillan,

London.Lowry, M., Glauber, J., Miranda, M. and Helmberger, P. (1987). Pricing and storage of field crops: A

quarterly model applied to soybeans.American Journal of Agricultural Economics, November,740–749.

Miranda, M.J. and Rui, X. (1996). An empirical reassesment of the commodity storage model. OhioState University,Mimeo.

Muth, J.F. (1961). Rational expectations and the theory of price movements.Econometrica, 29, 3–22.Newbery, D. and Stiglitz, J. (1982). Optimal commodity stockpiling riles.Oxford Economic Papers,

34, 403–427.Ng, S. (1996). Looking for evidence of speculative storage in commodity markets.Journal of

Economic Dynamics and Control, 20, 123–143.Pindyck, R.S. (1993). The present value model of rational commodity pricing.Economic Journal,

103, 511–530.Routledge, B.R., Seppi, D.J. and Spatt, C.S. (1997). Equilibrium forward curves for commodities.

Graduate School of Industrial Administration, Carnegie Mellon University,Mimeo.Samuelson, P.A. (1957). Intertemporal price equilibrium: A prologue to the theory of speculation.

Weltwirtschafliches Archiv, 79, 181–219.

explaining the persistence of commodity prices

Documents