Commodity storage and the autocorrelation of prices

Carlo Cafiero

Universita di Napoli Federico II, Italy

cafiero@unina.it

Eugenio S. A. Bobenrieth H.

Universidad de Concepcion, Chile

Juan R.A. Bobenrieth H.

Universidad del Bıo-bıo, Chile

Brian D. Wright

University of California at Berkeley, USA

draft: September 29, 2005

1. Introduction

For storable commodities with stochastic supply, intertemporal arbitrage can smooth the effectsof temporary gluts and, when stocks are available, temporary shortages. Qualitative features ofthe price behavior of some important commodities have long been seen to be consistent with sucharbitrage. In this paper we show that speculative arbitrage in simple models in the tradition ofGustafson (1958) can induce the degree of correlation and variation observed in widely-studiedprice histories of seven major commodities.

In establishing this claim, we must confront the weight of evidence that has accumulated againstit in recent years, nicely summarized by Deaton and Laroque (2003, p. 290): “[T]he speculativemodel, although capable of introducing some autocorrelation into an otherwise i.i.d. process, ap-pears to be incapable of generating the high degree of serial correlation of most commodity prices.”In doing so, we must address three related challenges, one numerical, the second theoretical, andthe third empirical. The first issue is raised by Deaton and Laroque (1995, p. S28): “In Deatonand Laroque (1992) our simulations failed to generate autocorrelations as high as those in thedata... [T]his result did not reflect our inability to choose the right parameters in the simulations,but is a general feature of the model.” Using a simple numerical model they simulated in Deatonand Laroque (1992), we show first that even their “high variance” numerical example (Deaton andLaroque 1992, p. 11) fails to generate (when storage is possible) as much price variation as observedin most of the commodities they consider. With a decrease in consumption demand elasticity from

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0.2 to 0.067 (at mean consumption), the model generates levels of correlation and variation of pricethat are much closer to those observed for most of the commodities they consider.

In these numerical examples, the harvest distribution is a discrete approximation to a normaldistribution. Maximum harvest is greater than the horizontal intercept of consumption demand.Price would be negative in the absence of storage, but with costless storage, arbitrage assuresthat prices are positive, given any positive interest rate. Market demand, p(0)(z) in Figure 1,converges on the horizontal axis, ensuring price is always positive. If storage cost consists solely ofproportional decay, as in Deaton and Laroque (1992, 1995, 1996), then the same argument ensuresprice is always positive.

−5 0 5 10 15 20 25 30−100

0

100

200

300

400

500

600

700

800

900

with free disposal

without free disposal

p = 100 − 150 c;k = 5

Fig. 1.— Market demands for fixed marginal and average storage costs

However models in the tradition of Gustafson (1958) have generally assumed that the costof storage includes a positive marginal cost. Assume this marginal cost is fixed at k > 0. Thenif the price at which consumption equals maximum harvest is less than k, in equilibrium market

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price will sometimes be negative, as shown in Figure 1. When available supply is sufficiently high,storers must either pay a subsidy for consumption, or store at a loss. In reality, we do not observenegative prices in commodity markets. Recognition of the possibility of free disposal will ensurethat prices cannot be negative in this model. In Figure 1, numerical solution of the model withk = k∗ and free disposal results in a market demand which closely follows the demand without freedisposal to the left of z∗, then follows the horizontal axis. For this modified model, there is no proofof existence and uniqueness of equilibrium. Since we do not want to assume that the price whenconsumption equals maximum harvest exceeds the marginal storage cost, we must prove existenceand uniqueness for this model.

Given the proof of existence and uniqueness, we are in a position to match the model withthe commodity price data, using a reasonable estimate of the short-run real risk-free interest rate,following closely the empirical approach described in Deaton and Laroque (1995, 1996). We wereable to get useable estimates for seven major commodities, coffee, copper, jute, maize, palm oil,sugar, and tin. In all cases, the results indicate that there is a positive marginal cost of storage,but decay is insignificant. Sample distributions of autocorrelations and covariances generated bysimulation indicate that the commodity price timer series in general do not reject the model, forthese commodities, at any reasonable level of confidence. Therefore, we have no reason to rule outan important role for speculative arbitrage in the dynamics of the prices of major commodities.

2. Can Storage Generate High Serial Correlation?

We begin by focusing on a preliminary question: can a simple storage model in the tradition ofGustafson (1958), with i.i.d. disturbances, generate price autocorrelations that are similar to thoseobserved in time series for major commodities?

To address this question, we use the type of simple storage model used by Deaton and Laroque.Production is given by an i.i.d. sequence ωt (t ≥ 1) with bounded support. Available supply attime t is zt ≡ ωt + xt−1, where xt−1 ≥ 0 are stocks carried from time t− 1 to time t. Consumptionct is the difference between available supply zt and stocks xt carried forward to the next period.The inverse consumption demand F (c) is strictly decreasing. There is no storage cost apart froman interest rate r > 0. Storage and price satisfy the arbitrage conditions:

xt = 0 if(

11 + r

)Etpt+1 < pt

xt ≥ 0 if(

11 + r

)Etpt+1 = pt,

where pt represents the price at time t, and Et is the expectation conditional on information attime t. The above complementary inequalities are consistent with profit-maximizing speculationby risk-neutral price-takers. Deaton and Laroque (1992) prove in this case that, given additionaltechnical conditions, there exists a unique stationary rational expectations equilibrium, that is,

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there is a price function p that describes the current price as a function of available supply zt, andsatisfies for all zt,

p(zt) = max[(

11 + r

)Etp{ωt+1 + (zt − F−1(p(zt)))}, F (zt)

].

Before estimating their model on the commodity price data, Deaton and Laroque (1992) askwhether there exist, within the parameter space of the model, specifications of the above simplemodel that yield price behavior characteristic of observed commodity markets. This question canbe investigated by solving the model by numerical approximation of the equilibrium price function,and then deriving the implications for time series of price behavior. Deaton and Laroque (1992)present the results of this exercise for five sets of parameterizations with interest rate r = 0.05,positive decay rate d ≥ 0, no other storage cost (i.e., k = 0), linear or isoelastic demand, andnormal or lognormal harvest distributions. They found that the highest autocorrelation of pricewas produced by their “high variance” case taken from an example in Williams and Wright (1991), with no physical storage costs, linear consumer demand, P (x) = 600 − 5x, and harvests witha discrete approximation to the normal distribution (with mean 100 and standard deviation 10).They found this case to imply a price autocorrelation of 0.48, far below the sample correlationscalculated from the 88-year time series of prices of thirteen commodities, which are all in excess of0.6. They concluded that perhaps the autocorrelation observed in commodity prices needs to beexplained by phenomena other than storage (Deaton and Laroque 1992, page 19).

We solved the storage model for the same specification, and simulated a time series of 100, 000periods. The first and second order autocorrelations over this long sample are 0.46 and 0.30, closeto the values obtained by Deaton and Laroque (1992) for the invariant distribution (0.48 and 0.31,respectively).

In order to assess the implications of the model for samples of the same length as those of theobserved commodity price series used for this paper, we took successive samples of size 88 from thesimulated series, starting from period t = 1, the second with period t = 2, and so on. The resultsare presented in Figure 2.

The median of the first order autocorrelations is 0.45. The 90-th percentile is 0.61, a littlebelow the lowest value in the commodities price series, which is 0.62, for sugar, which lies at the92.55 percentile of the simulated samples. All twelve others are above 0.7, the 98.63 percentile; itis clear that the example does not match the data for these others at all well. The same criticismapplies to many of the other examples in Wright and Williams (1982), and Williams and Wright(1991), with similar specifications.

However this “high variance” case has another problem. It does not generate sufficient pricevariation to match the values for most of the commodities in the 88-year samples. The “longrun” estimate of the coefficient of variation of price is 0.25. This value, however, is not directlycomparable with the value of 0.50 that Deaton and Laroque (1992, Table 2) report, for the latteris the value for the case with no storage. Storage reduces the price variance by about one half. We

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present the histogram of simulated sample coefficients of variation for the model in figure 2. Thecoefficients of variation measured for the time series of the commodity prices lie above the 98thpercentile of the sample distribution implied by their “high variance” case for samples of this size,for all commodities they considered but bananas and tea. It is clear that this “high variance” case ofDeaton and Laroque, with zero storage costs, in fact has unrealistically low variation. Insufficientlysteep demand, or, equivalently, insufficiently variable harvest, means too little storage, too manystockouts, too little variance, and too little smoothing to match the data.

To increase the price variation in the model, we changed the slope of consumption demandfrom −5 to −15 (a change in elasticity, at mean consumption, from −0.2 to −0.067). Once again wesolved the model and generated a simulated sample of 100, 000 periods. The results are presentedin Figure 3.

The median of the sample coefficients of variation is in this case 0.46, quite close to theobserved values of many of the commodities. Only tin and bananas have values less than the tenthpercentile of the generated sample distribution. Median of the first-order sample correlations is0.60. The values for 5 commodities, jute, maize, palm oil, sugar, and tea lie between 10th and the90th percentiles.

Figures 2 and 3 together show that tripling the price variation that would occur withoutstorage leads to sufficiently greater arbitrage that the actual price variation only doubles. Thegreater arbitrage is also reflected in much higher serial correlation. Clearly there is no reason tobelieve, on the basis of numerical result, that storage is incapable of generating the degree of serialcorrelation observed in most major commodities.

The simulations we have discussed favor storage by assuming no storage cost other than interestcharges. But, as noted before, physical storage costs are not in general zero. Before moving to adiscussion of estimation of the model, we discuss our choices of storage cost specification and realinterest rate for the estimated model.

3. The cost of storage

Deaton and Laroque (1992) proved the existence of a stationary rational expectations equi-librium in a model in which, as in Samuelson (1971), storage cost consists only of interest and“shrinkage” or deterioration of the stock, proportional to the value of the amount stored. Sincestorage costs go to zero as consumption price goes to zero, arbitrage assures there is never excesssupply, or negative price, in their model. Econometric implementations of their model (Deatonand Laroque 1992, 1995, 1996; Ng 1996; Michaelides and Ng 2000) maintain the assumption thatstorage cost consists solely of proportional deterioration.

In the preliminary simulations, we followed Deaton and Laroquein assuming a real interest rateof five percent. However, given this assumption, estimates of deterioration tend to be negative, as

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99.840

1000

2000

3000

4000

5000

6000

7000prices

0.310.450.590

5000

10000

150001st order autocorrelation

0.19 0.240.280

5000

10000

15000coeff. of variation

Fig. 2.— Price characteristics implied by the “high variance” case of Deaton and Laroque (1992)

99.430

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000prices

0.42 0.6 0.760

2000

4000

6000

8000

10000

120001st order autocorrelation

0.33 0.45 0.570

2000

4000

6000

8000

10000

12000

14000coeff. of variation

Fig. 3.— Price characteristics implied by a model with steeper demand

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noted below. In our estimates, we chose a lower lower fixed real interest rate of one and one halfpercent, at the upper end of estimates of the short run risk-free cost of capital, based on surveysof the evidence in the United States and the United Kingdom in the last century presented inGoetzmann and Ibbotson (2005); Campbell (1999); Shiller (2005).

Next we considered the specification of the physical cost of storage. When a commodity suchas corn, coffee or tin is stored in a commercial warehouse, the warehouse receipt specifies the gradeand quantity of the commodity. Shrinkage of quantity, or other deterioration, is typically coveredin the cost of storage. We were unable to locate a comprehensive storage cost series for any of thecommodities we consider. However there is evidence for some commodities that, within the sampleinterval, the cost of commercial storage per unit has been constant and independent of commodityprice movements for significant periods of time.1

Consistent with the above observations, specifications of the storage model with constantmarginal and average storage cost are found in the pioneering model of Gustafson (1958), and inNewbery and Stiglitz (1979, Chapter 29), Gardner (1979); Wright and Williams (1982); Mirandaand Helmberger (1988) and Frechette and Fackler (1999). Here we likewise adopt a constantmarginal and average cost of storage. Having done so, we must confront the problem that, whenthe available supply exceeds the sum of consumption at a price of zero plus storage demand ata discounted price equal to storage cost, the market does not clear at a non-negative price. (SeeFigure 1.) Indeed existence has not been established for the model with demand with unrestrictedprice domain and constant, positive marginal storage costs.

Thus, before estimating the model, we must first prove existence and uniqueness of equilibriumin this class of models, assuming free disposal.

4. A model with k > 0 and free disposal

We model a competitive commodity market with constant marginal and average storage cost(aside from any depreciation) and free disposal. All agents have rational expectations.

Supply shocks ωt are i.i.d., with support in R that has lower bound ω ∈ R. Storers are risk

1For example Holbrook Working reports that daily charges for wheat storage in public elevators in Chicago were

constant from December 1910 through December 1916 (Working 1929, p. 22). This implies that deterioration costs,

which would be proportional to price, were not significant enough to justify price-dependence in the storage charges.

A detailed analysis of the cost of storing a number of major commodities aound the decade of the 1970s is found in

UNCTAD (1975). For cocoa, which spoils more easily than major grains, warehouse storage charges in New York

stayed around $5 per ton per month from 1975 through 1984 while the cocoa price fluctuated wildly, between $1063

and $4222 per ton. (Williams 1986, pp. 213-214) For metals, shrinkage and storage costs are negligible. In Oklahoma,

Texas, and Arkansas and Kansas where warm weather and associated pest problems encourage storers to use public

elevators, their storage cost per bushel stayed constant at 2.5 cents per bushel per month from 1985-2000. (Anderson

2005).

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neutral and have a constant discount rate r > 0. Stocks physically deteriorate at rate d, 0 ≤ d < 1,and the cost of storing xt ≥ 0 units from time t to time t+ 1, paid at time t, is given by kxt, withk > 0. The state variable zt is total available supply at time t, zt ≡ ωt+(1−d)xt−1, zt ∈ Z ≡ [ω,∞[.Inverse consumer demand, F : R → R. F , is continuous, strictly decreasing, {z : F (z) = 0} �=∅, limz→−∞ F (z) = ∞, and

(1 − d

1 + r

)EF (ωt) − k > 0, where E denotes the expectation taken

with respect to the random variable ωt.

A stationary rational expectations equilibrium (SREE) is a price function p : Z → R whichdescribes the current price pt as a function of the state zt, and satisfies for all zt,

pt = p(zt) = max{(

1 − d

1 + r

)Etp{ωt+1 + (1 − d)xt} − k, F (zt)

}(1)

where:

xt =

{zt − F−1(p(zt)), if zt < z∗ ≡ inf{z : p(z) = 0}z∗ − F−1(0), if zt ≥ z∗.

(2)

Existence and uniqueness of the SREE, as well as some properties are given by the followingTheorem:

Theorem. There is a unique stationary rational expectations equilibrium p in the class of contin-

uous non-increasing functions. Furthermore, if p∗ ≡(

1 − d

1 + r

)Ep(ω) − k, then:

p(z) = F (z), for z ≤ F−1(p∗)

p(z) > max{F (z), 0}, for F−1(p∗) < z < z∗

p(z) = 0, for z ≥ z∗.

p is strictly decreasing whenever it is strictly positive. The equilibrium level of inventories, x(z), isstrictly increasing for z in [F−1(p∗), z∗].

The proof of the Theorem (in Appendix 1) follows the structure of the proof of Theorem 1 inDeaton and Laroque (1992).

The following Proposition parallels Proposition 1 in Deaton and Laroque (1996, p.906), andallows identification of the model when only prices are observed, by arbitrarily setting the meanand the standard deviation of the supply shocks ωt to be zero and one, respectively.

Proposition. Consider a model with discount rate r, stocks deterioration parameter d, constantmarginal and average storage cost k, supply shocks ωt, and inverse demand function F. Any othermodel with discount rate r, stocks deterioration parameter d, constant marginal and average storagecost k, supply shocks ωt ≡ σωt + µ, and inverse demand function F satisfying F (σz + µ) = F (z),has the same rational expectations price process as the base model.

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The proof of the Proposition (in Appendix 2) follows the structure of the proof of Proposition1 in Deaton and Laroque (1996).

5. Estimation

We estimate the model described in section 4 using the pseudo-likelihood maximization pro-cedure of Deaton and Laroque (1995, 1996).

First, the SREE price function (1), (2) is solved numerically on an equally spaced grid of pointsover a suitable range of values of z, and interpolated using cubic splines as in Deaton and Laroque(1995, 1996). The lowest value of the range of z is lower than the lowest possible production. Theupper bound of the range for approximation should be large enough to ensure that the approximatedfunction would cover even the lowest data point. Finding this required some experimentation forthe various commodities, with results as reported in Table 1. The use of cubic splines to interpolatemight induce non-negligible errors due to the fact that p is kinked (See Cafiero 2002; Michaelidesand Ng 2000, p.243). To reduce approximation errors around the kink point, we use a fine grid of200 points.

Table 1: Parameters used in the estimation

Commodity minimum z maximum z pointscoffee -5 30 200copper -5 40 200jute -5 30 200maize -5 40 200palm oil -5 30 200sugar -5 20 200tin -5 45 200

Then, using the approximate SREE price function we calculate the first two moments of pt+1

conditional on pt :

m(pt) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

N∑n=1

p(ωnt+1 + (1 − d)[p−1(pt) − F−1(pt)]) Pr(ωn

t+1), if pt > 0

N∑n=1

p(ωnt+1 + (1 − d)[z∗ − z]) Pr(ωn

t+1), if pt = 0,

– 10 –

s(pt) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

N∑n=1

p(ωnt+1 + (1 − d)[p−1(pt) − F−1(pt)])

2 Pr(ωnt+1) −m2(pt), if pt > 0

N∑n=1

p(ωit+1 + (1 − d)[z∗ − z])2 Pr(ωn

t+1) −m2(pt), if pt = 0,

where ωnt+1 and Pr(ωn

t+1) are discrete values chosen to approximate the standard normal distribu-tion. We use the same approximation to a standard normal used in Deaton and Laroque (1995,1996): ωn

t+1 is restricted to take one of the conditional means of 10 equiprobable divisions of thestandard normal distribution, ±1.755, ±1.045, ±0.677, ±0.386, ±0.126. The restrictions of zeromean and unit variance for the distribution of the supply shocks are imposed to identify the model(see the Proposition in section 4).

To match the prediction of the model with the actual price data, we form the logarithm of thepseudo-likelihood function as:

lnL =T−1∑t=1

ln lt = 0.5

[−(T − 1) ln(2π) −

T−1∑1

ln s(pt) −T−1∑t=1

[pt+1 −m(pt)]2

s(pt)

](3)

Following Deaton and Laroque (1995, 1996), the interest rate is fixed and initially set atr = 0.05 (although see below) and the pseudo-likelihood function (3) is maximized with respect tothe vector of parameters θ ≡ {a, b, d, k} where b = −eb, k = ek, and d = ed. The trasformationis used to impose the restrictions that b < 0, k > 0, and d > 0. Even though (3) is not the truelog-likelihood (in presence of storage, prices will not be distributed normally), the estimates areconsistent (Gourieraux, Monfort, and Trognon 1984).

To estimate the variance-covariance matrix of the vector of parameters θ ≡ {a, b, d, k}, we firstobtain a consistent estimate of the variance-covariance matrix of the parameters θ by forming thefollowing expression

V = J−1G′GJ−1, (4)

where J is a consistent estimate of the expected Hessian of the log-pseudo likelihood and G is aconsistent estimate of the expected matrix of score contributions. In practice, we form the matricesJ and G with typical elements:

Ji,j =∂2 lnL∂θi∂θj

; Gt,i =∂ ln lt∂θi

,

by taking numerical derivatives of the log-pseudo likelihood, lnL, and its components, ln lt, allevaluated at the point estimates of the parameters θ (see Deaton and Laroque 1996, equation 18).2

2The numerical derivatives of the log-pseudo likelihood and of its components are calculated by implementing a

version of the routine described by Miranda and Fackler (2002, pp. 97-104).

– 11 –

A consistent estimate of the variance covariance matrix of the original parameters θ is obtainedusing the delta method, as:

V = DVD′

where D is a matrix made of the derivatives of the transformation functions:

D =

⎧⎪⎪⎪⎨⎪⎪⎪⎩1 0 0 00 −eb 0 00 0 ed 00 0 0 ek

⎫⎪⎪⎪⎬⎪⎪⎪⎭ .

5.1. Results

5.1.1. Parameter estimates

Our first estimation was an attempt to extend the estimation of Deaton and Laroque to thecase of positive constant marginal cost of storage in addition to physical deterioration. By imposingthe restriction that d > −r, the results of the estimates on the twelve series of commodity pricespreviously analyzed in the literature are reported in Appendix table 5.

These estimates are unsatisfactory. First, it must be noted the disturbing result that d isestimated at values below zero for each commodities, which poses a problem if they are to beinterpreted as decay rates,3 and second, for seven of the twelve commodities decay is estimated atthe lowest possible value of −0.05.

We then estimated the model by constraining d ≥ 0, while keeping r fixed at 5 percent,obtaining results for seven of the twelve commodities, while for the remaining our routine couldnot locate a maximum of the pseudo likelihood function. As expected, for all of the commoditiesfor which we got an estimate, the constraint on d turned out to be binding. For that reason, andto considerably ease the estimation procedure, we dropped the d parameter from the estimation,setting it equal to zero, and generate the results reported in Table 2.

We also estimated the model for lower interest rate, fixing it at one and one half per cent,obtaining the result reported in Table 3.

3Negative values for the decay rate are also implied by the GMM estimates of Deaton and Laroque (1992, Table

4) when assuming an interest rate of 0.05.

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Table 2: Estimation with k > 0, r = 0.05 and no decayCommodity a b k PL p∗

coffee 0.5560 -3.0313 0.0014 131.8950 2.1181(0.0257) (0.1903) (0.0004)

copper 0.9981 -2.4972 0.0008 96.8280 2.1863(0.0242) (0.1337) (0.0005)

jute 1.2697 -4.3459 0.0053 53.5867 3.4135(0.0905) (0.2520) (0.0016)

maize 1.2027 -3.0952 0.0068 41.8382 2.6677(0.0343) (0.0804) (0.0014)

palm oil 1.2575 -4.1394 0.0053 66.0259 3.2905(0.0435) (0.2234) (0.0007)

sugar 0.6036 -0.8865 0.0325 -2.4458 0.9347(0.0643) (0.0289) (0.0039)

tin 4.7264 -19.1825 0.0024 152.4536 17.1283(0.0194) (0.9930) (0.0003)

Table 3: Estimation with k > 0 and r = 0.015Commodity a b k PL p∗ no stockouts (%)coffee 0.2904 -1.5179 0.0045 132.7 1.1209 58.7

(0.0218) (0.0807) (0.0006)copper 0.6342 -1.8392 0.0062 100.6 1.6501 45.16

(0.0174) (0.0805) (0.0007)jute 0.8129 -3.0873 0.0126 55.5 2.5270 46.69

(0.0470) (0.1550) (0.0018)maize 0.8732 -2.8586 0.0135 44.0 2.4453 42.54

(0.0447) (0.0899) (0.0018)palm oil 0.7847 -3.1500 0.0104 69.3 2.5472 52.54

(0.0288) (0.1216) (0.0009)sugar 0.5888 -0.8682 0.0520 -2.9 0.9697 6.0

(0.0271) (0.0288) (0.0055)tin 0.4964 -2.4012 0.0040 156.0 1.8682 64.23

(0.0013) (0.0028) (0.0000)

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5.1.2. Assessing the time series properties of the estimated models

For each commodity, and using the estimated parameters, we have simulated very long seriesof prices of 100,000 periods and calculated the values of first order autocorrelation, second orderautocorrelation and coefficient of variation on all possible series of 88 consecutive periods. Toassess the question of whether or not the estimated models are able to replicate the observedcharacteristics of the commodity price series, we compare the values as measured on the 1900-1987price series with the corresponding empirical distributions generated by the estimated models.

The data in Table 4 include the values of first order autocorrelation, second order autocorrela-tion and coefficient of variation as measured on the actual time series of prices, the correspondingmedian values of the simulated empirical distributions, and the values of the percentiles of theempirical distributions corresponding to the observed values.

The histograms of prices, first and second order autocorrelation and coefficient of variation arereported for all commodities in the appendix.

Note that the observed correlations fit well within reasonable confidence bounds derived fromthe simulated samples. The same is true for the coefficients of variation, noting the significantexception of jute.

However one striking feature of the results is that, according to the estimated models, stockoutsoccurred over the sample interval only in sugar. For all other commodities the cutoff price forstorage, P*, is higher than the highest observed price. For these commodities, consumption is veryinsensitive to price. The frequencies of 0 stockouts in the simulated 88-period samples are over45% for all commodities other than sugar (see Table 3. Comparison with the log likelihood valuesfor AR(1) estimates in Deaton and Laroque (1995, Table III, p. S37) indicates that the pseudo loglikelihood values are higher for the model estimated here.

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Table 4: Characteristics of price series.Commodity ρ1 ρ2 CVcoffee

Observed values 0.80 0.61 0.45Median sample values 0.81 0.69 0.71Percentile 45 35 9

copperObserved values 0.85 0.66 0.38Median sample values 0.78 0.64 0.48Percentile 72 55 25

juteObserved values 0.72 0.45 0.32Median sample values 0.79 0.65 0.48Percentile 35 18 2

maizeObserved values 0.76 0.54 0.38Median sample values 0.77 0.63 0.49Percentile 48 34 10

palm oilObserved values 0.73 0.48 0.47Median sample values 0.80 0.67 0.61Percentile 33 19 23

sugarObserved values 0.63 0.39 0.60Median sample values 0.51 0.34 0.61Percentile 83 65 45

tinObserved values 0.89 0.75 0.41Median sample values 0.83 0.72 0.57Percentile 70 59 19

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6. Conclusion

It would, of course, be too much to expect to explain the evolution of prices of a diverse set ofmajor commodities over almost a century in a model with stationary linear consumption demand, afixed distribution of market disturbances, a constant interest rate, and no supply response. Indeed,it would be very surprising if long run market influences had no effect on the evolution of prices.

Here we restrict our attention to two preliminary questions that have been the focus of theempirical work thus far: First, can the simple standard storage model with i.i.d. disturbances, underreasonable specifications drawn from a parameter space including values considered in previousliterature related to this topic, generate sample distributions of autocorrelation that are, withreasonable probability, consistent with estimates of autocorrelation from the observed commodityprice time series? Second, can estimates of the standard storage model, using the econometricapproach of Deaton and Laroque (1995, 1996) yield sample distributions of autocorrelation thatare consistent with values calculated directly from the time series?

Our answers are in sharp contrast with the conclusions of the pathbreaking numerical andeconometric investigations that have focused on these questions. First, for the simple storagemodel with linear consumption demand and no storage cost, we are able to identify reasonablesets of parameter values that are capable of producing sample distributions of autocorrelationconsistent with values observed in the time series of thirteen commodities that they examined.Second, if we adopt an assumption of constant marginal and average storage cost, we are able toobtain econometric estimates for seven of the major commodities. Our estimates imply empiricaldistributions for the sample autocorrelations that in general cover, with reasonable degrees ofconfidence, the values calculated from the time series of observed prices.

We thus conclude that speculative arbitrage as represented in simple models in the traditionof Gustafson (1958) can indeed induce the degree of correlation and variation observed in widely-studied price histories of a number of major commodities.

REFERENCES

Anderson, K. (2005): “Personal communication,” Oklahoma State University.

Cafiero, C. (2002): “Matching theory with data: the case of serial correlation in commodity priceseries induced by storage,” Working Paper - Dipartimento di Economia e Politica Agraria.Universita degli Studi di Napoli Federico II.

Campbell, J. (1999): “Asset Prices, Consumption and the Business Cycle,” in Handbook ofMacroeconomics, ed. by J. Taylor, and M. Woodford, chap. 19. North-Holland, Amsterdam.

Deaton, A., and G. Laroque (1992): “On the Behaviour of Commodity Prices,” Review ofEconomic Studies, 59(1), 1–23.

– 16 –

(1995): “Estimating a nonlinear rational expectations commodity price model with unob-servable state variables,” Journal of Applied Econometrics, 10, S9–S40.

(1996): “Competitive Storage and Commodity Price Dynamics,” Journal of PoliticalEconomy, 104(5), 896–923.

(2003): “A model of commodity prices after Sir Arthur Lewis,” Journal of DevelopmentEconomics, 71, 289–310.

Frechette, D. L., and P. Fackler (1999): “What Causes Commodity Price Backwardation?,”American Journal of Agricultural Economics, 81, 761–71.

Gardner, B. L. (1979): Optimal Stockpiling of Grain. Lexington Books, Lexington, Mass.

Goetzmann, W. N., and R. Ibbotson (2005): “History and the Equity Risk Premium,” YaleICF working paper No. 05.04.

Gourieraux, C., A. Monfort, and A. Trognon (1984): “Pseudo-Maximum Likelihood Meth-ods: Theory,” Econometrica, 52, 681–700.

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Williams, J. C. (1986): The economic function of futures markets. Cambridge University Press,Cambridge, UK.

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A. Proof of the Theorem

To prove the Theorem, we first prove a series of preliminary results. Consider Y ≡ {(q, z) :z ∈ Z, q ≥ max{F (z), 0}}. Let g : Z → [0,∞[ be a continuous, non-increasing function, such thatg(z) ≥ F (z) ∀ z ∈ Z. Define G : Y → R by:

G(q, z) ≡(

1 − d

1 + r

)Eg{ω + (1 − d)x(q, z)} − k, (A1)

where

x(q, z) ≡{z − F−1(q), if z < z∗gz∗g − F−1(q), if z ≥ z∗g

(A2)

and

z∗g ≡ inf{z ≥ F−1(0) :

(1 − d

1 + r

)Eg{ω + (1 − d)(z − F−1(0))} − k = 0

}. (A3)

We denote by T the operator that assigns to the function g the function Tg which satisfies thefollowing functional equation:

Tg(z) = max {G(Tg(z), z), F (z)} . (A4)

A SREE is a function g such that Tg = g.

This preprint was prepared with the AAS LATEX macros v5.0.

– 18 –

Lemma 1 Assume that g : Z → [0,∞[ is a continuous, non-increasing function, such that g(z) ≥F (z), ∀ z ∈ Z. Then G : Y → R is continuous and non-increasing in both its arguments. Further-more, if z < z∗g ,

G(F (z), z) =(

1 − d

1 + r

)Eg{ω} − k.

Proof of Lemma 1. Trivial. Note that x is continuous and g is uniformly continuous.

Lemma 2 Assume that g satisfies the hypotheses of Lemma 1. Then:(i) There exists a unique function Tg which is the solution of (9). T g : Z → [0,∞[ is continuous,non-increasing and:

Tg(z) = F (z), for F (z) ≥(

1 − d

1 + r

)Eg{ω} − k

Tg(z) = G(Tg(z), z), for F (z) <(

1 − d

1 + r

)Eg{ω} − k

(ii) Furthermore, g1 ≥ g2 ⇒ Tg1 ≥ Tg2.

Proof of Lemma 2. (i) For a given z ∈ Z, Tg(z) is equal to the solution in unknown q, q ≥max{F (z), 0}, of:

ψz(q) ≡ max{G(q, z) − q, F (z) − q} = 0. (A5)

ψz(q) is strictly decreasing and continuous in q, and

ψz( [ max{F (z), 0},∞ [ ) = ] −∞, ψz(max{F (z), 0}) ] .

To evaluate ψz(max{F (z), 0}) we consider three cases:Case 1: For ω ≤ z ≤ F−1(0),

ψz(max{F (z), 0}) = ψz(F (z)) = max{(

1 − d

1 + r

)Eg{ω} − k − F (z), 0

}.

If F (z) ≥(

1 − d

1 + r

)Eg{ω} − k, then ψz(F (z)) = 0, and Tg(z) = F (z).

If F (z) <(

1 − d

1 + r

)Eg{ω}−k, then ψz(F (z)) > 0, T g(z) exists and satisfies Tg(z) = G(Tg(z), z).

– 19 –

Case 2: For F−1(0) < z < z∗g ,

ψz(max{F (z), 0}) = ψz(0) = max {G(0, z), F (z)} = G(0, z) > 0.

Then Tg(z) exists and satisfies Tg(z) = G(Tg(z), z).

Case 3: For z ≥ z∗g ,

ψz(max{F (z), 0}) = ψz(0) = max {G(0, z), F (z)}

= G(0, z) =(

1 − d

1 + r

)Eg{ω + (1 − d)(z∗g − F−1(0))} − k = 0.

Therefore Tg(z) = 0 and satisfies Tg(z) = G(Tg(z), z).

This proves the existence and uniqueness of the solution Tg(z). Continuity and monotonicityfollow from the continuity and monotonicity of

max{G(q, z) − q, F (z) − q}.

(ii) Let g1, g2 be two functions that satisfy the hypotheses of Lemma 1, and such that g1 ≥ g2. Then,z∗g1

≥ z∗g2. For z < z∗g2

, G1(q, z) ≥ G2(q, z) ∀ q ≥ max{F (z), 0}, and therefore Tg1(z) ≥ Tg2(z).For z ≥ z∗g2

, T g2(z) = 0, and therefore Tg1(z) ≥ Tg2(z).

Lemma 3 (i)If p is a SREE, and p is non-increasing in z, then p(ω) = F (ω).(ii) If g satisfies the assumptions of Lemma 1 and g(ω) = F (ω), then Tg(ω) = F (ω).

Proof of Lemma 3. Deaton and Laroque (1992, p.21).

– 20 –

Proof of the Theorem:(i)Consider two functions g1, g2 satisfying the hypotheses of Lemma 1, and such that there exists anon-negative constant a with g2 ≤ g1 + a. By Lemma 2 (ii),

Tg2 ≤ T (g1 + a)

For z < z∗g1,

T (g1 + a)(z) ≤ max{(

1 − d

1 + r

)Eg1{ω + (1 − d)(z − F−1(Tg1(z)))} − k, F (z)

}+

(1 − d

1 + r

)a

= Tg1(z) +(

1 − d

1 + r

)a.

For z∗g1≤ z < z∗g1+a,

T (g1 + a)(z) ≤ max{(

1 − d

1 + r

)Eg1{ω + (1 − d)(z∗g1

− F−1(0))} − k, F (z)}

+(

1 − d

1 + r

)a

= Tg1(z) +(

1 − d

1 + r

)a.

For z ≥ z∗g1+a,

T (g1 + a)(z) = 0 ≤ Tg1(z) +(

1 − d

1 + r

)a.

Therefore, T (g1 + a) ≤ Tg1 +(

1 − d

1 + r

)a. We conclude that :

Tg2 ≤ Tg1 +(

1 − d

1 + r

)a. (A6)

Let G ≡ {g : Z → [0,∞[ / g is continuous, non-increasing, g ≥ F, g(ω) = F (ω)}. Lemma 2and Lemma 3 imply that T (G) ⊆ G.

d(g1, g2) ≡ supz∈Z

|g1(z) − g2(z)|, g1, g2 ∈ G

is a metric on G. For any g1, g2 ∈ G, taking a = d(g1, g2) in (11), we conclude that:

d(Tg1, T g2) ≤(

1 − d

1 + r

)d(g1, g2)

Thus the operator T is a contraction in the complete metric space (G, d) , and therefore it hasa unique fixed point p ∈ G.

– 21 –

(ii) p is strictly decreasing whenever it is strictly positive:If not, since p is non-increasing, there is an interval where p is a positive constant. We have twocases:First case: Suppose that there exists a first interval I ≡ [z′, z′′] where p is constant. Let B ≡p(z′). ∀ z ∈ I,

B = p(z) =(

1 − d

1 + r

)Ep{ω + (1 − d)(z − F−1(B))} − k

Since p is non-increasing, p{ω + (1 − d)(z − F−1(B))} is constant (≤ B), for z ∈ I. Therefore,

B ≤(

1 − d

1 + r

)B − k, a contradiction.

Second case: Suppose that there is no first interval where p is constant. Let

I ≡ {I : I is an interval where p is constant}

and let p ≡ sup{p(z) : z ∈ I and I ∈ I}. Since there is no first interval where p is constant, p isaccumulated by a sequence of values of p in I, I ∈ I.

Take any ε > 0 and consider an interval I such that the value of p in I is ≥ p − ε. LetB ≡ value of p in I. ∀ z ∈ I,

B = p(z) =(

1 − d

1 + r

)Ep{ω + (1 − d)(z − F−1(B))} − k.

Since p is non-increasing, p{ω + (1 − d)(z − F−1(B))} is constant for z ∈ I and p{ω + (1 − d)(z −F−1(B))} ≤ p. Therefore,

B ≤(

1 − d

1 + r

)p − k, and then,

B ≤(

1 − d

1 + r

)(B + ε) − k.

Since ε > 0 is arbitrary, we obtain a contradiction.

(iii) The equilibrium level of inventories, x(z), is strictly increasing for z in [F−1(p∗), z∗] :Let z1 < z2 in [F−1(p∗), z∗]. Since p is strictly decreasing in this interval, p(z1) > p(z2). Therefore,(

1 − d

1 + r

)Ep{ω + (1 − d)x(z1)} − k >

(1 − d

1 + r

)Ep{ω + (1 − d)x(z2)} − k,

which implies that x(z1) < x(z2).

– 22 –

B. Proof of the Proposition

Consider the base model with discount rate r, stocks deterioration parameter d, constantmarginal and average storage cost k, supply shocks ωt, and inverse consumption demand F. By theTheorem, there exists a unique stationary rational expectations equilibrium p(z) characterized by:

p(z) = max{(

1 − d

1 + r

)Ep{ω + (1 − d)x(z)} − k, F (z)

}(B1)

where

x(z) =

{z − F−1(p(z)), if z < z∗ ≡ inf{z : p(z) = 0}z∗ − F−1(0), if z ≥ z∗

(B2)

Consider the alternative model with discount rate r, stocks deterioration parameter d, constantmarginal and average storage cost k, supply shocks ωt ≡ σωt + µ, inverse consumption demand Fsatisfying F (σz + µ) = F (z), and unique stationary rational expectations equilibrium p(z).

Let p1(z) ≡ p(σz + µ). It suffices to prove that p1 satisfies the functional equation (12)− (13).

If z < z∗1 ≡ inf{z : p1(z) = 0}, thenEp1{ω + (1 − d)(z − F−1(p1(z)))} = Ep{σ[ω + (1 − d)(z − F−1(p(σz + µ))] + µ} == Ep{(σω + µ) + (1 − d)σ(z − F−1(p(z)))} = Ep{ω + (1 − d)(z − (F )−1(p(z))},where z ≡ σz + µ. Therefore,

max{(

1 − d

1 + r

)Ep1{ω + (1 − d)(z − F−1(p1(z)))} − k, F (z)

}=

max{(

1 − d

1 + r

)Ep{ω + (1 − d)(z − (F )−1(p(z))) − k, F (z)

}= p(z) = p(σz + µ) = p1(z).

If z ≥ z∗1 , thenEp1{ω + (1 − d)(z∗1 − F−1(0))} = Ep{σ[ω + (1 − d)(z∗1 − F−1(0))] + µ} =Ep{ω + (1 − d)(z∗ − (F )−1(0))}. Therefore,

max{(

1 − d

1 + r

)Ep1{ω + (1 − d)(z∗1 − F−1(0))} − k, F (z)

}=

max{(

1 − d

1 + r

)Ep{ω + (1 − d)(z∗ − (F )−1(0))} − k, F (z)

}, where z∗ ≡ inf{z : p(z) = 0}

= p(z) = p(σz + µ) = p1(z).

C. Tables

– 23 –

Table 5: Estimation with k, and d ≥ −rCommodity a b d k PL p∗

cocoa 0.3719 -1.4213 -0.0231 0.0072 132.8176 1.1238(0.2628) (0.7434) (0.0366) (0.0150)

coffee 0.4482 -1.8347 -0.0139 0.0044 132.3314 1.4069(0.1885) (1.6978) (0.0332) (0.0089)

copper 0.7246 -1.2415 -0.0399 0.0159 100.5674 1.3717(0.1992) (0.8160) (0.0257) (0.0237)

cotton 1.2764 -1.9880 -0.0500 0.0364 78.9417 2.3402(0.3728) (0.7692) (0.0000) (0.0202)

jute 0.9079 -1.3467 -0.0500 0.0408 55.8583 1.5838(0.2039) (0.2550) (0.0000) (0.0239)

maize 1.0006 -1.4626 -0.0500 0.0413 44.9756 1.7424(0.0615) (0.0899) (0.0000) (0.0123)

palm oil 0.8427 -1.3398 -0.0500 0.0327 71.9831 1.5358(0.0849) (0.4280) (0.0000) (0.0124)

rice 1.3701 -2.5041 -0.0500 0.0418 63.3237 2.7220(0.2231) (1.3452) (0.0000) (0.0000)

sugar 0.6041 -0.8803 -0.0060 0.0357 -2.4348 0.9381(0.1677) (0.3554) (0.1546) (0.1049)

tea 5.2569 -3.2473 -0.0497 0.1224 95.5926 6.8302(0.8345) (0.3786) (0.0067) (0.0470)

tin 0.4520 -0.7181 -0.0470 0.0147 156.4531 0.8239(0.1190) (0.4268) (0.0446) (0.0203)

wheat 1.3362 -2.3417 -0.0500 0.0431 56.0497 2.5871(0.3138) (0.6058) (0.0000) (0.0303)

Table 6: Comparison of maximized Pseudo Likelihood valuesCommodity Deaton and Laroque i.i.d. storage Deaton and Laroque full model our modelcoffee 111.0 131.2 132.7copper 73.9 103.1 100.6jute 44.8 53.3 55.5maize 32.1 46.2 44.0palm oil 22.2 58.9 69.3sugar -10.7 -3.9 -2.9tin 108.9 152.3 156.0

– 24 –

D. Graphs

– 25 –

0.2940

2000

4000

6000

8000

10000

12000

14000Histogram of prices

0.5703 0.776 0.91640

0.5

1

1.5

2x 10

4 1st order autocorrelation

0.4149 0.6559 0.84470

2000

4000

6000

8000

10000

12000

140002nd order autocorrelation

0.46190.75761.11810

0.5

1

1.5

2

2.5x 10

4 Coeff. of variation

Fig. 4.— Empirical distributions of prices, first and second order autocorrelation and coefficient ofvariation for coffee

– 26 –

0.62850

2000

4000

6000

8000

10000

12000

14000Histogram of prices

0.5339 0.745 0.90710

2000

4000

6000

8000

10000

12000

140001st order autocorrelation

0.3788 0.619 0.82980

2000

4000

6000

8000

10000

120002nd order autocorrelation

0.3080.51370.76330

2000

4000

6000

8000

10000

12000

14000

16000

18000Coeff. of variation

Fig. 5.— Empirical distributions of prices, first and second order autocorrelation and coefficient ofvariation for copper

– 27 –

0.80820

2000

4000

6000

8000

10000

12000

14000Histogram of prices

0.5385 0.7497 0.90790

5000

10000

150001st order autocorrelation

0.3851 0.6245 0.83050

2000

4000

6000

8000

10000

120002nd order autocorrelation

0.40670.66290.96590

0.5

1

1.5

2x 10

4 Coeff. of variation

Fig. 6.— Empirical distributions of prices, first and second order autocorrelation and coefficient ofvariation for jute

– 28 –

0.8660

2000

4000

6000

8000

10000

12000Histogram of prices

0.5259 0.7359 0.90140

2000

4000

6000

8000

10000

12000

140001st order autocorrelation

0.3691 0.6077 0.81950

2000

4000

6000

8000

10000

120002nd order autocorrelation

0.37950.6151 0.88920

0.5

1

1.5

2x 10

4 Coeff. of variation

Fig. 7.— Empirical distributions of prices, first and second order autocorrelation and coefficient ofvariation for maize

– 29 –

0.77940

2000

4000

6000

8000

10000

12000

14000Histogram of prices

0.5509 0.763 0.91580

2000

4000

6000

8000

10000

12000

14000

160001st order autocorrelation

0.3986 0.6409 0.84350

2000

4000

6000

8000

10000

120002nd order autocorrelation

0.39280.65050.96170

0.5

1

1.5

2x 10

4 Coeff. of variation

Fig. 8.— Empirical distributions of prices, first and second order autocorrelation and coefficient ofvariation for palm oil

– 30 –

0.58670

1000

2000

3000

4000

5000

6000

7000Histogram of prices

0.3560.51380.6690

5000

10000

150001st order autocorrelation

0.16420.33810.51380

2000

4000

6000

8000

10000

12000

140002nd order autocorrelation

0.49790.6237 0.76090

2000

4000

6000

8000

10000

12000

14000

16000

18000Coeff. of variation

Fig. 9.— Empirical distributions of prices, first and second order autocorrelation and coefficient ofvariation for sugar

– 31 –

0.49180

5000

10000

15000Histogram of prices

0.5959 0.7968 0.93270

0.5

1

1.5

2x 10

4 1st order autocorrelation

0.436 0.6831 0.87310

2000

4000

6000

8000

10000

12000

140002nd order autocorrelation

0.36140.62250.95970

0.5

1

1.5

2

2.5

3x 10

4 Coeff. of variation

Fig. 10.— Empirical distributions of prices, first and second order autocorrelation and coefficientof variation for tin