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Copper and the Negative Price of Storage


Just as the price of a call option contains a premium based on price variability, so the shadow price of

Donald FrederickLarson

inventoriescontainsa dispersionpremium

Public Disclosure Authorized

associated with the unplannedcomponentof inventories.When inventory levelsare low, the valueof the premium increasesto the point where inventorieswill

be heldevenin the faceof a

Public Disclosure Authorized

fully anticipatedfall in price.

The World Bank InternationalEconomicsDeparmnent InternationalTrade Division April 1994



Summary findings Commodities are often stored during periods in which storage returns a negative price. Further, during periods of "backwardation," the expected revenue from holding inventories will be negative. Since the 1930s, the negative price of storage has been attributed to an offsetting "convenience yield." Kaldor, Working, and later Brennan argued that inventories are a nece-ssaryadjunct to business and that increasing inventories from some minimal level reduces overall costs. This theory has always been criticized by proponents of cost-of-carry models, who argue that a negative price for stor2 e creates arbitrage opportunities. Proponents of the cost-of-carry model have asserted that storage will occur only with positive returns. They offer a set of price-arbitrage conditions that associate negative returns with stockouts. Still, stockouts are rare in commodity markets, and storage appears to take place during periods of "backwardation" in apparent violation of the price-arbitrage conditions. For copper, inventories have always been available to the market regardless ot the price of storage. This is true whether the market is broadly defined at the U.S. or

world level, or more narrowly defined as the New York Commodities Exchange or the London Metal Exchange. Larson argues that although inventories may provide a Kaldor cost-reducing convenience yield, inventories also have value because of uncertainty. just as the price of a call option contains a premium based on price variability so the shadow price of inventories contains a dispersion premium associated with the unplanned component of inventories. Larson derives a generalized price-arbitrage condition in which either a Kaldor-convenience and/or a dispersion premium may justify inventory holding even during an expected price fall. He uses monthly observations of U.S producer inventories to estimate the parameters of the price-arbitrage condition. The estimates and simulations he presents are ambiguoLs with regard to the existence of a Kaldor-convenience but strongly support the notion of a dispersion premium for copper. And although the averagt value of such a premium is low, the value of the premium increases rapidly during periods when inventories are scarce.

This paper-a product of the International Trade Division, International Economics Department- is part of a larger effort in the departriient to understand international commodity markets. Copies of the paper are available free from the World Bank, 1818 H Street NW, Washington, DC 20433. Please contact Anna Kim, room S7-038, extension 33715 (78 pages) April 1994.

The Policy ResearchWorking Paper Seriesdisseminatesthe findings of work in progressto encouragethe exchangeof ideasabout developmentissues.An objectiveof theseriesis to getthe findi-gs out quickly,evenif the presentations are lessthan fully polished.The paperscarry the namesof theauthors and shouldbe usedand citedaccordingly.Thefindings,interpretations,and conclusionsare the authors'oum and should not beattributed to the WorldBank, its ExecutiveBoardof Directors,or any of its membercountries.

Produced by the Policy Research DisseminationCenter

Copper and the NegativePrice of Storage by Donald FrederickLarson

I wou!dlike to thankRamon Lopez and Ron DuncRnfor their guidanceand supportduring this project. I would also like to thank HowardLeathers, NancyBockstael,BruceGardner, and Karla Hoff for useful comments.

Table of Contents





Refied Copper Prices and the NegativePrice of Storage



The OptimizationModel and the Price-ArbitrageCondition



Empirical Results







Annex 1: The Derivation of the Price-ArbitrageCondition


Annex 2: Necessary Border Conditions


Annex 3: The Quadratic-PlusModel


List of Figures 2.1 2.2 2.3 2.4 3.1 3.2 4.1 4.2 4.3 4.4 4.5 4.6

Copper prices, August 1978 to December 1989 Future price profile for high-gradecopper Future prices over time with constant future price expectations The marginal cost-of-storagefunction The effect of an increase in price on inventoriesand production Convex shadowprice for inventories Monthlyconstant US producer prices and producer-heldinventories for October 1979 through December 1989 Discountedand deflated near-by spreads on the last day of the month mapped against closing producer inventory levels for October 1979 to December 1989 Discountedand deflated second-positionspreads on the last day of the month mapped against closing producer inventorylevels for October 1979 to December 1989 Simulateddispersionpremiums for March 1980 to December 1989 Simulatedmarginal productioncosts Simulatedmarginal storage costs

3 6 12 18 21 24 29 30 31 40 42 43

Lit of Tables 2.1 2.2 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 A3.1

Spreads,stocks,and producdonof refinedcopperin the UnitedStates on the first businessday of January, 1974-1983 Implicitinterestfrom lendingrefned copperstocksusingSeptember24, 1991 closingCOMEXprices for refinedcopper Distributionalcharacteristics of monthlyproducerpricesand COMEXend-of-monthcopperfuturesprices Instrumentsused for fittedvaluesof the log of inventories and the log of sales Estimatedparametersand t-scores Estimatedparametersof marginalcost functionsexpressedas elasticides, and estimatedaverageddispersionpremium Simulatedvaluesfor the marginalcost of production(C.), the marginalcost of storage(Cz),and the dispersionpremium Averagespreadsand simulateddispersionpremiumsduringperiodsof backward and forwardCOMEXmarkets Summarystatisticson the estimateddispersionpremiumunderalternative estimadonmethodsand alternativelag-lengthsfor the price-variance measurefor the log-linearmodel Tests on the effectsof excludingthe price of electricityfromthe marginal cost-of-production function Summarystatisticson the estimateddispersionpremiumunderalternative estimationnethods and alternativelag-lengthsfor the price-variance measure for the quadraticmodel Estimatedparametersand t-scores(in parentheses)for base modeland model treatedfor assumederrors-in-variables problem Hausmantest for misspecification basedon LIML,2SLSand 3SLSestimates for varyingmoving-variance calculations Tests relatedto the marginalcost of storage Tests on convexityof the shadowprice for refinedcopperinventoriesunder non-jointcostsand constantmarginalstoragecosts Summaryof the empiricalresults Parameterestimationsfor the quadratic-plus model


10 16 32 36 37 38 39 41 44 46 48 50 52 53 55 57 78

1.0 Introduction

The purposeof this paper is to explain producer-heldinventoriesof refined copper durirg an anticipatedfall in prices. Commoditiesare often stored during periods in which storage returns a negativeprice -- that is, when the expectedprice change(as indicatedby futures prices)does not coverthe time-valueof moneyplus storageexpenses. In fact, during periods of backwardation(whenthe settlementprice for a near-byfuture contractis greater than the price for contractswith moredistant settlementdates),the anticipatedrevenuefrom holding inventorieswill be negative. In the case of copper, inventorieshave always been available to the market regardlessof the price of storage. This is true whetherthe market is broadly defined at the US or world level, or more narrowly defined as the New York CommoditiesExchange,or the LondonMetal Exchange. Sincethe 1930s,the negativeprice of storagehas beenattributedto an off-setting"convenienceyield". Kaldor(1939),Working (1948)and, later, Brennan(1958)arguedthat inventoriesare a necessaryadjunctto carrying out businessand, at some minimallevel, increasinginventoriesreducesoverall costs. This theory has always been criticizedby proponentsof cost-of-carrymodels,who argue that a negative price for storage creates arbitrage opportunities. More recently, Williams and Wright (1991) have correctlyarguedthat "convenienceyields" have not been derived from a formal optimizationmodel. Proponents of the cost-of-carrymodel have asserted that storage will occur only with positivereturns and offer a set of price-arbitrageconditions. Williamsand Wright have augmentedthe cost-of-carryargumentby stressingthe effects of stock-outson price. Nonetheless,Williamsand Wright (1991, p.140) note there have been no stock-outs in the Chicago wheat markets during the last 120 years, and concede that storage appearsto take place duiringperiods of backwardation,in apparentviolationof the


price-arbitrageconditions. This paper argues that while inventories may provide a Kaldor cost-reducing convenienceyield, inventoriesalso have value becauseof uncer.ainty. Just as the price of a call optioncontainsa premium based on price variability,the shadowprice of inventories conte;ns a dispersionpremiumassociatedwith the unplannedcomponentof inventories. A goneralizedprice-arbitrageconditionis derivedin whicheither a Kaldor-convenienceand/or a dispersion premium may justify inventory-holdingeven during an expected price fall. Monthlyobservationsof US producerinventoriesare used to estimatethe parametersof the price-arbitragecondition. The estimationand simulationspresentedprovide little evidence for supportingor dismissingthe existenceof a Kaldor-convenience,but stronglysupport the notion of a dispersion premium for copper. Further, while the average value of such a premium is low, the value of the premium increases rapidly during periods in which inventoriesare scarce. Following this introduction,the remainderof the paper is organized as follows: Section 2 reviews the behaviorof prices for the spot and futures marketsin refined copper and discussesbackwardationand a negativeprice for storagein the context of the economic literature. In this section, the argumentsof Working,Brennan,Keynes,and Williams and Wright are more fullyexplained. Section3 developsthe formal modelof inventory-holding in terms of a stochastic-controlproblemfor the ending-valueof inventories. The generalized price-arbitrageconditionsare derivedfrom the optimizationproblem'sfirst-orderconditions. Section 4 contains a description of the data and an estimation of the parametersof the generalizedprice-arbitragecondition. Section 5 concludes.


2.0 Reflned Copper Prices and the Negative Price of Storage

Refining copper is a risky businessand refined copper prices are characterizedby volatile price movements. Price swings can be large and sudden while low price levels can linger persistently. The movementin refinedcoppekis especiallysignificantgiven the smallprofits obtained throughrefining-- typically6 to 10 per cent of the final cost of refinedelectrolytic copper(Brook Hunt & Associates,1986). Sincethe refinermust buy scrap or blister copper, the spreadbetweenthe two pricescontainsthe implicitfee for processingcopper(Figure2.1). Yet, because of the time it takes to process the copper, rapid price movements can dramaticallyinflateprofitsor generatelossesevenwhenthe spreadbetweeninputsand output

Pricesfor UScopper 180


Au-79 Jul-6

May-62 -



Rfmwe comr -+- Scrp oopper

Figure 2.1: Copper prices, August 1978to December 1989. 3








price remainsconstant. There are several options availableto the refiner. He can contract the processingservicesdirectly,charginga fee but returningthe refinedcopperto the original owner of the blister or scMpcopper (tolling);he can implicitlymake the same arrangement by buying spot in the blister or scrap marketand simultaneouslyselling (shorting)a futures contract in New York or London;he can contractwith a semifabricatorfor future delivery while buying in the spot market, etc. The value of finishedinventoriesheld by refiners is subject to the same volatility. Gains or losses genemtedby wide swings in the value of held inventoriesmay affect the firm's profitabilitymoregreatly thanreturnsfrom flows(productionand sales). Optionsopen to the refiner includercducinginventoriesto near-zerolevelsor contractingto sell inventories either explicitly, or implicitlythrough the futures market. In addition,even when forward contracts have been writLan,further revenue can be generated by implicitly "lending" inventories. The mechanicsof such an arrangementare discussedlater; howeverthe practice has obvious and certain positive returns during periods of market backwardation. Markets for refinedcopper and other metals are somewhatdistinctivein their proclivityfor backwardationin futures pricing. Backwardationoccurs when the price of futures contracts with a more distantdelivery date are discountedwith respect to near-bycontracts. A more commonand more general conditionis when inventoriesare held at less-than-fullcarrying charges,which Working(1949)termed a negative price for storage. A negativeprice for storage occurs when the difference between the expected rate of change in the price fcr immediate delivery and full carrying costs (interest, insurance, warehousing fees, and spoilage)is negative. As a practicalmatter,a situation in which the difference betweenthe price for near-byand moredistant futurespricesfalls below full carryingchargesis taken as


ClosugprioencNMCopprFuturs fortwodatesin 991











0. . ... . . . . 6 lbw

.s .;



. . . . .


Mm* IlIb1_31b

deliey dates Figure 2.2: Future price profile for high-gradecopper. a negativeprice for storage. Backwardmarketsnecessarilycarrya negativeprice fo: storage since revenuesfrom et rage are negative. Figure 2.2 gives the profile of closing future prices on high-gradecopper for two separatedates in 1991. From July I to September23 the entire price curve rose, while the futuresmarket remainedin backwardation.While the situationdepictedin Figure 2.2 cannot be characterizedas "normal",it is not uniquefor copper. From 1980to 1989the last-day-ofthe-monthspreads on COMEXcopper futures were backward29 times -- or roughly 24% of the time. A negative price for storagehas been even more common. Jeffrey Williams (1986) writes: In nine of the ten years from 1974 through 1983 at least some of the spreads' in copperfailed to displayfully carrying charges, sometimes, as in 1974 and 1980, by a considerable margin. That any spreads in copper were less than full

'A spread,in thiscontext,is thepriceof a futurescontractwitha moredistantdeliverydateminus the priceof a futurescontractwithan earlierdeliverydate. 5

carrying charges is surprising in itself Because copper is a natural resource, having little seasonality in production or demand, and large reserves of scrap, the presumption would be strong that i:s price should Increase to cover interest ana storage costs, as presumedin most models of natural resources.

Fama and French (1988),lookingat daily interest-compensatedspreadsbetweenspot and 3month,6-month,and 12-monthforwaidcoppercontractson the LMEbetween1972and 1983 reported negativeaverage spreadsfor copper. The general failure of the spreadsin futures prices to cover full carryingchargesis well known to the literature. Holbrook Working noted the phenomenon in 1934, and Nicholas Kaldor wrote about a "convenienceyield" to stocks in 1939. In 1948, Working provided a theory on "inverse carrying charges" which included a supply of storage at a negative price: Another important condition is thatfor most of the potential suppliers of storage, the costs are joint; the owners of large storage facilities are mostly engaged either in merchandising or in processing, and maintain storage faciliti2s largely as a necessary adjunct to their merchandising or processi;.g business. And not only are the facilities an adjunct; the exercise of the storing function itself is a necessary adjunct to the merchandising or processing business. Consequently, the direct costs of storing over some specified period as well as the indirect costs may be charged against the associated business which remains profitable, and so also may what appear as direct losses on the storage operation itself. For any such potential supplier of storage, stocks of a commodity below some fairly well recognized level carry what Kaldor has aptly called a convenience yield. This convenience yield may offjet what appears as a fairly large loss from exercise of the storage function itself

The concept of a convenienceyield has become a popularand enduringpart of the literatureon inventorybehavior. (See Howell,1956;Brennan,1958;Telser, 1958;Weymar, 1974; Gray and Peck 1981, Thompson,1986;Williams, 1986; Tilley and Campbell, 1988; Thurman,1988;Fama and French, 1988;and Gibsonand Schwarz,1990.) However,as with many economicterms, there is a conflict in the literature over its definition. Kaldor and Working, for example, apply convenienceyield to mean an implicit return to holders of 6

inventorieswhosemarket valueequalsthe differencebetweenfutures price spreau;sand full carryingcosts. The valueto theholderderivesfromjointnessbetweeninventoryholdingand related merchandisingor processing. More recently,"the" convenienceyield, has taken % moreoperationaldefinition,comingto meantheevaluationitself or, moreprecisely,the array of spreads between futures prices adjusted for carrying costs. Often warehousefees and transactioncosts are ignoredso that carrying chargesare exclusivelycomposedof interest charges. (See, for example,Famaand French, 1988). in such a case, the convenienceyiold is an empiricalentity,constantlychangingmuch like a tond yield curve. In this paper,I will retain the definition given by Kaldor and Working.

A similar idea, in keeping with

Working's definiti-:. of convenience,is that inventories are essential to production and therefore inventory-holdingcf inputs is joint with productionas well as with marketing. Ramey (1989) used such an argur..entto justify inclusionof inventorylevels in modeling productiontechnology. Jeffrey WVilliarnr and Brian Wright (Wright and Williams, 1989; Williams and Wright, 1991) have been especially critical of literature concerning a convenienceyield. Williams and Wright justifiably argue that convenienceyields are not derived from "first 2 They go on to argue that the "observationof principles" (i.e. optimizationconditions).

storage under backwardationis an aggregationphenomenon"and that "a spreadbelow full carrying charges can emerge only when there is no storage of that commodity....Profit3 (Wright maximizingstoragetakesplace only at full carryingcharges,properlycalculated."

2Rameyis an exceptionnot notedby Williamsand Wright.

'Williams(1986)creditsHiginbotham (1976)withsuggesting thattheparadoxesof spreadsresult from misleadingaveraging. 7

and Williams,pp. 3-4). By includingthe costs of transportinginventoriesfrom one location to another,or the costs of transformingone close substituteto the relevantcommodity(say dirty wheat to clean wheat),Wrightand Williamsargue,the convenienceyield disappears. Backwardationin futures markets arises when the probabilityof a stock-out for the near periodis greater than for a later period.4 In William. and Wright (1991,p. 140)they soften their stance, stating that "Empiricallyit does seem that storage takes place when the spot spread is in backwardation,if for no other evidencethan the disquietingfact that stock-outs are never observed." At the same time, in their simulationmodel, they maintaina set of price-arbitrageconditionsthat precludesstorageat less-than-fullcarrying charges. Since inventoriesare often reported aggregatedover location,if not over type and grade of commodityas well,it is difficultto test the Wright and Williamsassertionin many instances. However, the inventory-holdingbehaviorof certified warehouseinventories of refined copper provides a clear counter-exampleof the type which Williams and Wright recognizedin their 1991 statement. At numeroustimes throughouttheir history, both the LME and the COMEXhave reportedabundantsuppliesof certifiedwarehousestocks while simultaneouslyreportingbackwardationin futurespricing. The added value of certification is that certified inventories can be used to fulfill commitments resulting from futures contracts. Should a holder of an open short futures contract in copper decide to deliver copper rather than close his position,he need only transfera receipt for inventoriesheld in any certified warehouse. No transportationor transformationis involved. Table 2.1 reproducesdata reportedby Williams (1986) on copper spreadson a single day in January

4Bresnahan and Suslow(1986)makea similarargumentspecificaUy in the caseof copperand backwardation in the LordonMetalExchange.


cver a ten-yearperiodand supplementsit with data on inventorylevels. For the observations reported,it was more commonthan not for refinedcopperinventoryto be held at a negative price, that is, below full carrying costs. At no time did a "stock-out" occur, although inventorieswere low in January 1974. At the sametime, however,it is difficultto arguethat effective stock-outsoccurred in 1979 and 1980 but did not in 1975 and 1976. And copper is not the only commodityto retain certified stocksin backwardmarkets. Williams(1986) notes that, despite the fact that there is no reason why elevators cannot be completely emptied,certifiedwheat and soybeanstocks,preciselythose eligible for delivery on futures contracts,have neverfallen to zero. For wheatin Chicago,this includess periodof over 120 years.5

While Williamsand Wright recognizethe tension betweenthe cost-of-carrymodel and observedperiods of negativestoragewith positiveinventorylevels, a negativeprice for storage is routinely excluded from the possible range of solutions in the literature on commoditypolicyissues. The generalinter-temporalarbitrageconditionsare oftengiven as:

VWith regardto the 1973soybeancrop,Williamswrites(pp.*6-37): In 1973, the cost of keepinathe more than 3 millionbushels in store in Chicagowas more than $2.1 million. This expensewas even larger nationwide. The total stock of old-cropsoybeansas of I September1973was 60 millionbushels,the smallestcarryoverof the decade(althoughit was still some 4% of the crop being thenharvested.) As of I August 1973,the spreadbetweenthe spot price and the Augustfuturescontract,on whichdeliverieswere eligibleuntilthe end of the month, was -$1.30per bushel. Thissuggeststhat the holdersof those60 millionbushelspaid on the order of $78 million,quite apartfrom physical storagecosts,for the privilegeof keeping them in store that one extra month. 9

Table 2.1: Spreads.stocks, and producion of refined copperin the UnitedStateson the first businessday of January. 1974-1983. Prisd of spread











iaauaay-March march-may May-July July,Sepem*ber Seplomber-Decembaer

4.60 -1.40 -1.00 -.60 -.60

.45 .55 .60 .60 .65

.40 .50 .45 .45 .35

.40 .45 .45 .45 .45

.40 .50 .50 .45 .45

.75 .75 .70 .55 .45

1.45 .15 .10 .15 .15

1.30 1.30 .95 .85 .80

.80 .85 .85 .85 .80

.60 .55 .55 .60 .70

Price of December conract











.00 .00 -.20 -.30 -.40

-.10 -.10 -.10 -.10 -.15

.00 -.05 -.0S -.05 .00

Spreadsafkerremovalof carryingcharges janry-Maub March-May May-July July-Sepsember Sephmbe-December

-5.20 -2.10 -1.65 -1.20 -1.15

.00 .00 .00 .00 .00

.00 .00 .00 .00 -.05

.00 .00 .00 -.05 -.05

-.05 -.05 -.05 -.05 -.05

.00 .00 -.05 -.25 -.35

.00 -1.30 -1.30 -1.25 -1.20

Stcks and mventoryof refined copperexpresaedm short tous ComexWarehousestocks 5,873 TotalUS stos 49,098

43,214 194,851

100,102 360,700

200,953 473,800

184,390 471,100

179,572 367,900

98,856 186,300

179,770 253,000

186,920 338,600

272,999 484,500

US productionin Januay 157,700










Note: Spreds in copperae basedon the closingpricesas of the first businessday in Januaryand are expressedm cenls per poundper month,taken from Wllias (1986).Inventoryvahlescorresponig to Januaryare December3 t ;evels. Inventoryand producton da were compiledfrom variousissuesof Meal Slatdcs.


p, s p,',/(l r) - k tpt > pe,'11(l +r) - k-



zt - °-

where z is inventorylevel, p is price, r is a discountrate, k is a constant positiveper unit storagecost, and a superscriptc denotesexpectations.RecentexamplesincludeMirandaand Helmberger(1988) and Glauber, Helmberger,and Miranda(1989). Another argument, related to a negative price for storage, is Keynes's (1931) notion of normal Keynes

backwardation. argued markets








backward in equilibrium (at

constantprice expectations),

_ Corm




implying negative reveizue for storage, because of the Figure 2.3: Futures prices over time with constant future price expectations. way in which risk is transferred from commodity producers, who are naturally long in the commodity, to speculators. Hedgingproducerstake a short positionin the market (to offset their naurally long positions)whilespeculatorstake the oppositeposition. Keynesarguedthat speculators would decline taking an opposing long positionunless the expected rate-of-retumon long positions exceededthe riskless rate-of-return. In order for this to be the case, the expectd rate-of-returnon the futures positionwouldhave to be less than the expectedspot price and 11

rise as the contractmatured;that is, futures priceswould have to be discountedto expected prices. Keynescalled this price-timerelationshipnormalbackwardation. (See Figure 2.3.) Conversely,if hedgers are, on net, long in futures contracts,then speculatorswould require a higher-than-risklessrate of return on short positions. In this instance, the futures price would have to contain a premium over the expectedprice, which declines as the contract matures. Keynes referredto this relationshipas normalcontango. Despite its implications of a negative price for storage, Keynes'stheory of normal backwardationhas little to say aboutinventoriesand arbitragepossibilities. This omissionhas formedthe basis of criticism by cost-of-carryproponents(see, for example,Figlewski, 1986). Regardlessof the validity of Keynes's arguments,the theory set off a number of empirical studies aimed at verifyingthe presenceof a downwardbias in futures (that is, a tendercy for the price of a futures contract to "rise up" to expected price levels as the contractapproachesmaturity). Houthakker(1957),usingtrade statisticson corn and wheat, concluded that naive small traders could benefit by blindly following a long-side trade position. Tessler(1958, 1960)found no evidenceof normalbackwardation.Cootner(1960, 1967)then presentedseveralcases in supportof normal backwardation,while Dusak(1973), using a portfolio approach,noted that even gains from arbitrating backwardmarkets were smallwhen comparedto investmentsin the stock marketwhich carried similarlevelsof risk. Two more recent studies, however, have emergedwhich provide support for Houthakker's original findings. In the first, Carter,Rausser,and Schmitz(1983)modifyDusak'sportfolio approachto allow for systematicrisk and found non-zeroestimatesof systematicrisk for most of the speculative return series examined. More recently, using non-parametric techniques,Eric Chang (1990) foundstatisticalsupportfor normal backwardationin a study


based on wheat, com, and soybean futures. In studies relating directly to the market tor metals, Hsieh and Kulatilaka(1982), using monthlydata on LME forward contracts from January 1970 to September1980,reportedaverage risk premia of 2.8% for copper, 17%for tin, 12.7% for zinc, and 16% for lead. In a later study, MacDonald and Taylor (1989) reported evidencefor a "time-varyingpremium"in the forward prices for tin and zinc. While cditicisms of Keynes's theory of normal backwardation and Kaldores convenienceyield remain valid, the empiricalstudies offered in their support are often at odds with the simple cost-of-carryprice-arbitrageconditionsgiven in 2.1. When inventoriesare being held at less than full carryingchargesor when the price on futures contracts is below expected prices, incentives are created to inter-temporally arbitragethe marketunder thecost-of-carrymodel. Themost obviouscourse of acdonwould be for holdersof inventoriesto reducetheir costsby sellingimmediatelyintothe spot market, bringing near-prices down relative to more distant delivery dates. Such inter-temporal arbitratingshouldcontinueuntil the expectedreturns to arbitrageequal the expectedreturns to inventoryholding,or a "stock-out"occurs,wheninventoriesare reducedto some near-zero minimumlevel. Similar arbitrage opportunities are available even when inventories have been contractedfor futuredelivery. For example,considera holderof inventories(saya producer) who has sold copperforwardfor deliveryin 60 days. Regardlessof the terms of the forward contract,the seller can generaterevenuein a backwardmarket,or reduceholdingcosts when storagereturns a negativeprice, by "lending"inventoriesto the market. Althoughdirect loan marketsare currentlyrelatively rare for commodities,they do exist for uranium, and a brokeragefirm, the Nuclear Exchange Corporation(NUEXCO),


exists to facilitatesuch ioans. In addition,there are active explicit loan marketsfor equity 6 Williams(1986)documentsan active loan marketin grain shareson manystock exchanges.

warehouse receipts in the UnitedStates in the 1860s. While no explicit loan market exists for refined copper, loan transactionscan be accomplishedin an equivalentmannerusingfutures and spot markets,obviatingthe need for an explicit market. Once the decisionhas beenmade to hold inventoriesfor a spancovering the deliverydate of at least one futures contract,lendinginventoriescan be accomplishedin a straightforwardmanner. Holders of inventories can sell into the spot market while simultaneouslycontractingto repurchasethe copper (by purchasinga futures contract)at a future date for a fixed price. In a backwardmarket,this sum will be positiveand constitutes a positive interest paymenton an implicitcopper loan. The effect on the copper market is to increase supplies available for immediatedelivery and increase the demand for future deliveries,arbitratingthe backwardation.It should be notedthat not only does the supplier of inventoriesreceive a payment,but he also eliminatesthe need to store. As a result,even when the futures market does not exhibit backwardationbut does return a negativeprice for storage, a holder of inventories can reduce his costs by lending supplies to the market. Therefore, in cost-of-carry models, a negative price for storage generates arbitrage opportunitiesas well,encouragingsuppliesto be freed from inventories.' As with the direct 6The LondonExchangeprovidesthe moststraight-forward example.In Londontheexchange

settleseveryfortnightratherthaneveryday as in NewYork. Buyersof stockshavecontractedto receivethestocksona particularday,but,if compensated, mayagreeto delaytakingdeliveryof the stock. If there is sufficientpressurefor immediatedelivery,the personagreeingto postponea contracteddeliverymayreceiveeithera concessionary rateon marginloansor a feefromtheseller. and is equivalentto intereston a loanof deliverablestock. Thefee is calleda "backwardation" 7Theflip-sideof lendingis,of course,borrowing.Becauseof the !wo-stepnatureof the implicit

loanproceduregivenabove,first sellingnearwhilebuyinglong,each"lender"neednot correspond witha singleborrower.Forexamplea fabricatormaybe purchasingthecopperfromthespotmarket 14

inter-temporal arbitrage, holders of inventories can be expected to lend into markets exhibiting backwardationor negative storage prices until the expected return on lending equals the expectedreturn on holdinginventories. The implicitreturnsto loans from inventoriesof refinedcopperwere calculatedfrom futures price spreads for September24, 1991 and are given in Table 2.2. The returns are probablyunderestimated,as warehousefees, whichwouldincreasethe returnsto lending,and transactioncosts, which would decreasethe returns slightly,have been omitted. Recalling that total profits from refining may equal less than 10% of the price of refined copper, the incentivesto lend are significant. Put another way, the price of borrowing inventoriesis high. In the following section, it is argued that a convex shadowprice for inventories, combinedwith uncertaindemand,gives rise to a dispersionpremiumfor inventories. When inventoriesare low,the effect of a higher-than-expectedsales level willhave a greatereffect on price than when inventoriesare more plentiful. In addition,when inventoriesare low, the range of possible price outcomesbecome more skewedtoward higher prices. By carrying inventoriesinto the period,the producerhas the optionof taking advantageof higherprices, shouldthey materialize,withoutincreasingproductionand incurringincreasinglyexpensive marginalcosts. Becauseof the asymmetry,the drop in the value of the carried inventories, when sales arecorrespondinglylower-than-expected,is not as severe. This relationshipgives rise to a generalizationof the price-arbitrageconditions in which it is rational to hold

whilea tollermaybe lockingin a futuresales pricefor refinedcopper. However,if desired,the mechanismis readilyavailablefor a refinerto borrowinventorieswhicharereplenished fromfuture production.This mechanismallowsindividualfirmsto holda negativeinventoryof copper,even thoughstocksmustbe non-negative in theaggregate. 15

inventories in the face of expected price declines. In its emphasison nonlinearities,the modelis similar to Gardner's(1979) modelof optimal grain storage. Gardner used recursiveprogrammingtechniquesto demonstratethat non-linearites in the sh&tdow price for inventoriescan result, in part, from nonlinearitiesin Table 2.2: Implicitinterestfrom lendingrefinedcopperstock using September24, 1991cloing COMEXpos for refinedcopper. Delivery Date

Sottlement Price



------------------- cents per pound (% annualizedreturn) September'91



















100.85 100.40






3.03 (33.0) 1.92 (21.4) 1.77 (19.9) 1.76 (20.1) 1.06 (12.3) 1.30 (15.1) 1.10 (12.9) 1.10 (13.0) 0.94

4.94 (26.9) 3.68 (20.5) 3.52 (19.9) 2.81 (16.0) 2.35 (13.6) 2.40 (13.9) 2.19 (12.8) 2.04 (12.0) 1.88

6.69 (23.9) 5.42 (20.1) 4.56 (17.2) 4.10 (15.6) 3.44 (13.2) 3.48 (13.5) 3.13 (12.2) 2.97 (11.7) 2.81




0.94 (11.2)



0.94 (11.2) 0.49

1.88 (11.2)

1.43 (8.5)


Note: Implicitinterest calculatedon an annualized5.98%discountrate where: interest= pf(t)- p'(t+n)em,n = 1,2,3 and where p' representsettlementprices for futurs contractsdeliverableat t and t+n.


2.36 (9-4!

the underlyingobjectivefunction. SinceGardner'stopic was the optimalstorage of grains, the primary source of asymmetrywas the fact that supplies of grairncan be withheld for futureconsumptionwhile a symmetrictransferfrom the future to tlie present is impossible. Nonetheless,Gardneralso showedthat nonlinearitiesin demand,for example,would affect the valuationof the shadowprices as well. As will be seen later, the convex relationship between price and inventories is suggested by simply plotting the two series and is a standardfeature throughouta long history of economicliterature. It is


often statedas a stylized fact that the marginalvalueof inventoryincreases with scarcity,decliningin a nonlinear manner to zero as inventory levels

! Nz

0 o z

of inventories at low levels of

of inventories le~vol >~~~~


increase. The marginallyhigh value

of nenoe


physicalstocksis usuallyascribedto convenience and is often the Figure 2.4: The marginalcost-of-storagefunction. departurepointin textbooksand appliedstudies(for example,Stein, 1987;Famaand French, 1988;Gibsonand Schwartz,1990). Conversely,whenholdersof inventoriesappearwilling to retain inventoriesdespitearbitrageopportunities,as in backwardmarketsor other markets with a negativeprice for storage,the holders are presumedto receivea higher "convenience yield" as compensation. Working(1948) and Brennan(1958)providea number of reasons why the value of inventoriesabovecarryingchargesmight rise to very high valuesat scarce levels and fall to zero at sufficientlyhigh stock levels. Workingarguesprimarilythat stocks


are often an adjunct of business and provide convenience and cost savings through a reductionin restockingcosts and an abilityto quicklymeet orders. As stock levels increase, the marginalcontributionof additionalstocks goes to zero. As inventoriesbuild, storage facilities reach capacity levels and marginalstoragebecomesincreasinglyexpensive. Generally,the marginalcost-of-storagefunctionis depictedas shown in Figure2.4, where z' marks the inflection point. However, in addition to the notion of convenience, Working also offered some argumentsbased on expectationsand probability,noting that "Merchants who deal in goods that are subject to whims of fashion, or to sudden obsolescencefor other reasons,must lay in stocks and carry them in expectationthat some part of the stocks will have to be sold at a heavy loss." Additionally,he points out that as stocks become low the probabilityof a "squeeze"in the futures market increases. Brennan also arguesthat the "convenienceyield"derivesprimarilyfrom fewer delaysand lower costs (becauseof less frequentorderingand stocking)in deliveringgoodsto consumers. However, Brennanalso discusseswhat he calls a "risk-aversionfactor",notingthat the larger the level of inventories,the greater the effect of a price change and the revaluationof held stocks. In the next section a formal model is developed to show that convex inventoryshadowprices for coppercan arise regardlessof whetherinventoriesare cost-reducing. The model does allow for a cost-reducingKaldor-convenienceyield as well. A generalizedset of price-arbitrageconditionsis then derivedfrom the first-orderconditionsof the model. The modelis then used to test r ipiricallyfor convexityin the shadowprice of inventorieswhich gives rise to an estimatabledispersionpremium. The existenceof a cost-reducingeffect for inventoriesis also examined.


3.0 The Optmizaton ModWand the Price-ArbitrageCondition In this section, the formal model is summarized from the detailed derivation containedin Annex 1. A generalizedprice-arbitrageconditionis derivedfrom the first-order conditionsof the optimizationproblemwhich is consistentwith inventory-holdingduring an anticipatedprice fall. The copper-refiningproblem is characterizedas a continuoustwocycle problem with uncertainfuture demand. In the current period, the producer knowsthe current sales price. By deciding how much to produce and sell, he determineshow much inventoryhe will bring into the next period. The expectedmarginal value, or the shadow price, of the inventory in the next period is not known, but contains a stochastic element since demand is uncertain. The effects of randomdemand shockson the shadowprice of inventoriesmay be asymmetric-- that is, a positive random shock may increaseprices by more than an equally sized negativerandom shock. In such a case, the shadow price of inventorywill carry a dispersionpremium so th;t the shadowprice of inventoriesincreases with the varianceof the stochasticcomponentof sales. Such a premiumis analogousto the volatilitypremium in an optionsprice and can result in positive inventorylevels even when price declinesare expected. The solution to the copper refiners profit maximizationproblem can be found by solving the Hamiltonian: Max,,,H a [ps - C(y,z)]e




where p is the sales price; s is the sales level; C is a joint cost functionfor storage and productionwhere z is level of inventoriesand y is the productionlevel; r is the discountrate. A is the change in profit due to a marginal change in the inventory level, or the


shadowpriceof inventories.The producermaximizesprofitsby settingthe marginalcost of productionand the narginalrevenueof salesequalto the valueof a marginalchangein the levelof inventoriesat the end of theperiod. Pricesare knownin the currentperiod,andinventorylevelsare deterninedonce sales andproductionlevelsare decided,but the endingvalueof inventoriesis not knowr. exactly. Rather,the valueof stocks,andthe shadowpriceof inventories,X, are basedon expectations aboutfutureprices,production,andsales. Whensales containa randomelement,inventorylevelswill also be stochasticand changesin inventorieswill containa plannedand unplanoedcomponent.The difference betweenplannedandactualinventorieswill be the differencebetweenexpectedand actual productionminussales. For the momnent, assu,nethat the changein inventoriescan be expressedas the followingprocess: ;z;_z

Et ,s [y,-s, l ]



where E has an expectedvalueof zero and a variancea2. Rewritingthe constrainton inventoriesin continuous-time notation,the valueof endinginventoriesat time t, is the solutionto the followinginfinite-horizon problem: e"'V(z,l) * Max, EF{[ps - C(yz)]t.

d = E(y-)d



The termdvm u(t)dt112 is a Wienerprocess,whereu(t) - N(O,1). Thesolutionto theproblemmakesuse of stochasticcalculus,themechanics of which are somewhattediusandhavebeenrelegatedto Annex1. Howeverthe relevantfirst-order conditionscan be summarizedgraphically.The producersolvesfor plannedproduction, 20

sales and inventoriesby settingerpected marginalcosts equal to expectedprice equalto the shadowprice of inventories, V; - which is itself an expectedvalue. Becausethe producer has the optionof either producingmore oLstoring less, he must solve on two margins. This is shown graphically in Figure 3.1 where marginal costs we drawn convex in y and the shadowprice of inventoriesis drawn convexin z. Whir. tSere is an increasein the expected price, inventories are reduced until the shadow price is equal to the new expectedprice

E[p]. V

Elpj, V




-mz al cost








Figure3.1: The effectof an increasein priceon inxentoriesandproduction. for currentconsumption.At thesametime,plannedpioductionincreases freeinginventories and expecte,d marginalcostsincreaseuntilmarginalcostequalsexpectedprice. conditons The first-orderconditionscanbe manipulatedto expressthe optimization in termsof expectedprice:


E[dp/dt]= rE(p] + E[Ct] - 2V0



for z,s,y > 0


Equation3.11 is a generalizationof price-arbitrageconditionsgiven in cost-of-carrymodels such as Williamsand Wright(1991, p. 27). The arbitrageconditionstatesthat the expected changein price will be equal to rE[p] -- intereston investingthe moneyelsewhere-- plusC, - the costs of physical storage and any amenityfrom storage-- minus lV_2

If Va is

positive, then this last term constitutes a dispersion premium that increases with the variability of the stochasticcomponentof inventorieso2. The last two componentsof 3.4 have importart implicationsfor holding inventories in the face of less-than-fullcarrying charges. Accordingto the condition, it still may be optimalto hold inventorieswhen the

market is in backwardation-- E[dp/dt] < 0 - if inventories provide a cost-reducing Kaldor-convenience(that is, if Cz is sufficientlynegative)and/or the dispersion premium, 5 V=C2, is sufficientlypositive.

The two componentsare not mutuallydependent. Kaldor-

conveniencealone can potentiallyexplain inventory-holdingin backwardmarkets, as can a dispersion premium. When V. =0 and C. is positive, 3.4 reduces to the cost-of-carry price-arbitragecondition. For the cost-of-carrymodel, no inventoriesare held when the sum of the current price plus a constant physical storage cost is greater than the expected discountedfuture price. Puttingthis constraintinto a continuous-timecounter-part: E [/dpt


rp +k, fir z >0.


The two arbitrage conditionsdiffer in two respects. In 3.12, storage costs are treated as a constantpositivemarginalcost, and separate from other activitiessuch as sales or production. Cost-of-carrymodels are usually based on the activities of professional 22

speculatorswho presumablyreceive no conveniencefrom holding inventoriesand do not participatein production. Generally,however, there is nothingfundamentalto the derivation of cost-of-carry models which requires fixed marginal storage costs and the marginal storage-costfunctioncould be written in a more flexiblemanner. The second difference between the two arbitrage conditions is the presence of a dispersion premium in the generalizedprice-arbitrageconditions. This comesfrom treatingthe valueof the inventories as stochastic. Cost-of-carrymodels use expected prices (or futures prices representing expectedprices)but do not treat the price changesthemselvesas a stochasticprocess. This differs from option-pricingmodelswhere the varianceof the underlyingcommodityprice enters explicitlyinto the evaluationof the option. The dispersion premium, 2 V o2, can be interpreted as the expected difference between the stochasticand deterministicvalue of inventories. To see this, start with the marginal-valuefunction of inventories defined in terms of a deterministic component (plannedinventories)plus a randomelement, and calculatea Taylor-seriesapproximation: V(Zd+E)




V4E + V6e zz 2


+ I V,e 6 = E


When e has an expected value of zero, and is symmetrically distributed, where

E[e] = E[e3] = 0 and E[e2] = o,2, the expecteddifferencebetween the stochasticand deterministiccomponentof the shadowprice for inventoriesis approximatelythe dispersion premium: EEVC(zd+e) - V(zd)]

_ IV,n2


Even when demandand inventoriesare treatedas stochastic,there is nothingin the


first or second-ordermnaximization conditionsthat would require V= to be positive. In fact, V could certainlybe quadraticin . so that V. need not exist. However, in much of the literatureon inventoriesthe shadowprice of inventoriesis often describedas convex in z - at least imnplicitly.'Usually it is stated that there is some type of "pipeline"mninimum stock level (Figure 3.2). As stock-levelsare drawn down and approach pipeline levels, larger and larger price increasesare requiredto depletediminishedinventories. Usuallythe convexity is

attributed to


convenienceyield. Accordingto this argument, pipeline levels



required to carry out business in an orderly fashion, while additional stocks, up to a point, can still j facilitate transactions or minimize costs suchas re-orders, deliveries,or restocking.


pipel ine


As it turns out, V= will exists if either of the marginal cost Figure3.2: Convex shadowprice for inventories. functions, C,, the cost of physical storage, or C , the cost of production, are nonlinear. This is true whetheror not the cost function is joint or whetherstocks are cost-reducingat any level. A fundamentalquestionis why the marginalcost function should be convex in either storageor productionlevels. Copper storage is simplicityitself, and requires only a secure and dry area. However, if storage and productionare joint and storage, at some

'For example,Working(1948),p. 19 and Brennan(1958),p. 54. 24

level, is cost-reducing, then there is probably some range over which the convenience diminishesrapidly as inventoriesgrow. In addition, marginal production costs are also likely to be increasing at an increasingrate over some production range, leading to nonlinearitiesin the cost function. The refining industry is constantly changingand there is a wide diversity to the scale on which refiners operate. For example, one of the newest refiningplants in the US opened in 1982 in SouthCarolina (AT&TNasaue RecycleCorp) with a relativelysmall capacityof 70,000 refined tons per year. Since then capacity has grown to 87,500 tons. For these small plants, which attempt to control costs by runmning near peak capacity, there remains little room for productionincreases. Alternatively,the country's largest plant, ASARCO's Texas plant, expandedits huge 420,000 ton/year capacityto 456,000 in 1987. At the same time, parts of the facility are quite old. Certainly under normal circumstancesASARCO couldprobably increaseproductionby 20,000 tons more readily than AT&T Nasaue. Still, for the industry as a whole, increasing production means bringing on line older and increasingly less efficient facilities, operating with double shifts or otherwise using increasinglymore expensive inputs, both of which are likely to lead to convex marginal costs. Earlier it was stated that stochasticdemandwill give rise to stochastic inventories; however, little was said about the relationshipbetweenthe meansand variancesof the two distributions. As it turns out, the problemcan be cast in terms of either stochasticdemand or stochasticprice. This result comes from the finance literature (for example, Cootner, 1964, and especiallyMerton, 1992). There is long history of decomposinginventoriesinto expected and random components (planned and unplanned inventories), particularly in


Keynesianmacroeconomnics.Stochasticprices are often employed in finance and capital literature (for example, Blackand Scholes, 1973, or Abel, 1983). For the copper-refiningproblemat hand, either interpretationis appropriate. Since there are no real restrictionson trade or the recovery of scrap copper, US producers are subjectto eventsand decisionsmade aroundthe world. On the demand side, copper is used in products such as plumbing components,wiring, and brass. Not only does the derived demand for these goodsfluctuatewith fluctuationsin those associatedindustriesworld-wide (for example, the constructionof new homes), but demand is also subject to cross-price effects from competing materials (for example, plastic plumbing components, or fiberoptics). In addition, US refiners are not the only agentswho hold inventories. Speculators hold inventories at both major exchanges(COMEX and LME), fabricators (for example, brass-mills)often hold inventories at their plants, and non-US refiners hold inventories overseas. Domesticmarket conditionsare also influencedby the flow of new supplies, either in the form of imports or in the supply of recycledcopper. Refinerswould preferto knowexactlyabouteventsin other sectors which influence price and sales in their own market. Ultimately, this is impossibleand some components of final demandand price will not be knownwith certainty. In an abstract sense, the refiner can be thoughtof as knowingthe deterministiccomponentof his demandscheduleand the general properties, but not the value of the stochasticcomponent. The productionfunction presents less of a problem for the producersince, unlikeagriculturewhere weather,disease, and pests createuncertainty, "ie yield of refinedcopperfrom a given set of inputs is known. Still, the final level of production- after adjustmentsto unanticipatedchangesin demand and price - is not known aheadof time, but is conditionalon final demandand price. The


initial stochastic term therefore originates in the demand schedule but will ultimately influencesales as well. Normalizingthe demandschedulewith respectto quantity, sales are stochastic;normalizingwith respect to price, price is stochastic. In the next section, estimationresults are presentedbased on monthlydata for US copper



m(t) * p(t) -p(t-l)-r(t-l)p(t-l),







the followingtwo equationsare estimatedjointly:

m(t) - Cp(t)-*




3(t) = C,(t) + up(t)

The first equationin 3.8 is a discrete version of the price-arbitrageconditiongiven in 3.11 plus an error term u,. The secondequationis taken from the first-orderconditions given in Annex 1, and simply states that the producer will produce to the point where marginalcost equalspiice.. Merton's result is especiallyhandy in this application,sincethe second of the two equationsprovides a formal model for up, the stochasticcomponentof price. By using instruments,consistent(in the econometricsense) estimatedvalues ofup can then be used to form a logically self-consistentestimate of


As noted earlier,

expressingthe constraint in terms of the unanticipatedcomponentof price rather than sales or inventories is essentially a simple re-scaling of the constraint. The symbol P= is therefore used to reflectthe re-scaledvalue of V=. The term Pm is treated as a constant and estimateddirectly as a parameter. This treatment is equivalentto assumingthat the underlyingmarginalcost functioncan be reasonablyapproximatedby a quadratic.2 These results are employedin Section4 of this paper to test for convexityof the



shadowprice for copper and provide an estimateof the dispersionpremiumassociatedwith copper inventories. In addition, the model can also provide a test of jointness between production and inventory-holdingwhich forns the economic incentive for a Kaldorconveniencebenefit.


4.0 EmpiricalResults

In this section, results from estimating the price-arbitrage and marginal cost functions, using monthly data on sales, production, and inventory levels for US copper refiners, are presented. Beforethe estimatingprocedureand resultsare discussedhowever, the underlyingdata is presentedin terms of the price-arbitrageequationdevelopedearlier in Section 3. Figure 4.1 plots the US producer price for refined copper againstinventoriesheld by producers. The data readily suggestsa convex relationship. This relationshipis not

Copperprices and inventorylevels held by producers 1.6 -

1.5 01.3 2

0i. -2



~0.6 c150

2 .


1 0

idO 2600 producerinventories (thousand tons)



Figur 4.1: MonthlyconstantUSproducerpricesandproducer-held inventoriesfor October 1979throughDecemnber, 1989.


uniquefor copper, and the convex relationshipbetweenprice and inventories is discussed in muchof the literatureon inventories. For authors such as Workingand Brennan,or more recentlyStein and Famma, this relationshipis based on a convenienceyield. In most costof-carry modelsthe relationshipis ignored, althoughan exceptionis Williamsand Wright, who argue that high prices result from the increasingpotentialfor a stock-out. Figure 4.2 mapsthe spread' betweenthe closingprices of the two nearest COMEX copperfutures contractson the last day of the month againstmonthlyclosingproducer-held inventories. Figure 4.3 maps the spread between the two futures prices with second and third closestsettlementdates. The spreadsbecome negativewhen inventorylevels are low, rising to near-zero positive levels with larger inventorylevels. Again, this relationshipis not uniqueto copper. Working(1948)describedthe same relationshipusing wheatdata for 1926. Brennan's 1958 article provides a number of similar graphs for eggs, cheese, butter,

wheat, and oats.


Brennan and others attributed such negative spreads convenience




b__w_n_two_c__ st_C_Xcontracts 0

a O


.9 a



- ,s5

-20 -20

subtracted estimates of the



100 0


260 250

physical storage costs from the I anddeflatednear-byspreadson thelast Figure4.2: Discounted spreads to quantify the day of the monthmappedagainstclosingproducerinventory levelsforOctober1979to December 1989. 'The spread is expressedin constant1985cents per poundand was calculatedas spread a [F(t+1)/(1+r) - F(t)i/PPJ.,wherethefuturewiththe closestsettlementdateis subtractedfromthe moredistantfuture'sprice. Thedifferenceis thendeflatedby themonthlyUSproducerpriceindex. 30

convenienceyield. Williamsand Producr hsnvutorri end the sprd

Wright argue that backwardation

betwe e 2nd I

in futures prices comes from increasedprobabilityof a stockout.




d3rddrce con.t.cte










such 1-20.

circumstances is not fully



0 5

explained; however, Wright and






WAilliaistargue that observedI Wlawre spreadson 4.3: Discountedand deflatedsecond-position storage is, in some case,


the last day of the month mappedagainst closing producer inventorylevelsfor October1979to December1989.

aggregationphenomena. They

arguethat while the market in New York may promisenegative returns to storage, prices are forwardin the geographicallydiversemarketswhere the inventoriesare actuallystored. The generalized price-arbitrage condition developed in Section 3 suggests two possible reasons for storage in backward markets. First, the condition formalizes the argumentsput forward by Kaldor and Working. As inventorylevels are drawn down, the remaininginventoriescouldpotentiallyconfergreater and greater convenienceyields-- that is, the marginal cost-of-storagefunction may become increasinglynegative. The second reason is that the convex shape of the shadow price function, which arises from nonlinearities in the cost function, conveys a dispersion premium when demand is uncertain. For example,when inventoriesare low and demandis unexpectedlyhigh, marginalincreases in production become increasinglyexpensiveand prices rise increasinglyquickly. At the same time, stocks will be drawn down. Followingthe reductionin stocks, prices will rise even more dramatically should a consecutive period of high demand materialize.


Alternatively,whilea drop in demandmay be just as likely,the effectson price are not symmnetric.Fromanypointon the convexmarginalcostcurve, the absolutevalueof the pricedroprequiredto lowerproductionby a unit is lessthan the priceincreaserequiredto increaseproductionby a unit. Likewise,a greaterpricechangeis requiredto drawdown inventoriesby a singleunit thanto increaseinventoriesby a singleunit. This asymmetry skewsthe possibleoutcomeof prices(andthereforethe shadowpriceof inventories)toward higherpricesandgeneratesa dispersionpremium. As a result,not onlyshouldpricesbe higherwhen inventorylevelsare low, but the distributionof prices in generalshouldbe skewedtowardhighervalues. The distributionalcharacteristics of the real monthlyproducerpricesas well as the real futuresprices(asobservedon thelastdayof themonth)for January1980to December 1989are givenin Table4.1.2 The spotpricesare USproducerprices(whichare usedin the empiricalestimation reported later in this section),


Table 4.1: Distributional characteristics of monthly producer prices and COMEXend-of-monthcopper futuresprices.


futures :,rices are from

Standard Skewness Mean Deviation

the COMEX in New York. Becauseof the

Producerprice Nearbyfuture One-aheadfuture

0.91 0.83 0.83

0.24 0.25 0.23

0.97 1.06 1.01

difference in market locations,the meansare slightly

Note: Futures prices are for COMEX high-gradecopper while spot prices are US producerprices. All prices are deflatedby the producer price index, 1985-100.


2 Thecoefficient of skewnessgivenin Table4.1 is E[(p

symmetricdistr.ibtion is 0.


- pT)]lo, so theexpectedvaluefor a

However, the spot and the futures series exhibit similar standarddeviations, and all three series are skewedwith the long tail of the distributionextendingtoward higher prices. In order to obtain the parametersof the price-arbitrageequationderived in Section 3, as well as the parametersof the marginal cost functionsand the dispersion premium, variations of the followingtwo generalizedequationswere estimated: M(t)




¢,,(t) -


(t) +V'M


+ 5(t) is the

where a. and u. are random errors, where m(t) *p(t)-p(t-1)-r(t-l)p(t-1)

money-returnto storage,p is the producerprice, C, and C, are marginalcost functionsfor inventories,z, and production,y. Severalfunctionalforms were consideredfor C, andC,, and these will be discussedin greater detail later. As discussedearlier, the variance in the stochasticdifferentialequation associated with the change in copper inventoriescan be expressed in terms of the variance in the unanticipatedcomponentof the price for copper. In turn, the unanticipatedcomponentof price is given by u, in 4.1, which can be used to construct an estimate of



estination process involvesseveral stages. First, to avoid simultaneitybiases, instruments were used to provide fitted values for endogenoussales and production. Next, substituting fitted values for endogenousright-hand-sidevariables, the second equation of 4.1 was estimatedusing least-squares. One resultis an unbiasedestimateof the residualvector, 3 An Il-month moving-varianceestimate,




of the variance of the unanticipated

the2SLSestimates mayvarywithtime, u, willbe heteroskedastic.Nonetheless, inefficient,willbe unbiased.(SeeKennedy,1992,p. 114.) for il, alithough 33

component of producer price O2(t)

was then constructed from 4,


6,(t), m


p(n)] /(l-l) and i,(t)O

Following the construction of




,(n) is the moving-mean.

2p(t), both equations of 4.1 were estimated together

simultaneouslyas the final stage of a three-stage least-squaresprocedure. As discussed earlier, P.

is treated as a constant and estimated directly as a parameter. The entit-

an estimateof the dispersion-premnium. Qdconstitutes 62 The choice of a functional-formfor the underlying cost-function, C(z,y), had to meet several criteria. First, the form had to be flexible enough to allow third-order derivativesand jointness between inventoriesand production, that is CI,, Co, C, I Cm, etc. had to be potentiallynon-zero. The followinglog-linearequationswere used initially to approximate the marginal cost functions C, and C, from 4.1, since they meet the criterionwith a paucityof estimatedparameters: C, = bo +blnz +b2 ny +bx

+b4lnw +b5 nf


C = Co+ clnz + c2 lny + c 3lnx + c4 inw+ c lnf 5

where the b1 and cl for i =0,,..._ are fixedparameters. The choiceof functionalfonn has implications -- especially for the underlying marginal cost-of-storagefunction. These implications,along with more general and more restricted functionalforms, are discussed later in this section. The process of refiningcopper, explainedin Section2, is relatively simple, which helps limit the numberof parametersin 4.2. In additionto inventorylevelsand production levels, the price of electricity,x, the price of raw input copper, w, and the capacityof the refiningplants,f, are givenas argumentsof the cost function. All prices are deflatedby the US producerprice indexwhichserves as a proxy for omittedprices (primarilywages). The 34

cost-functionC(y,z) is initiallymodeledas joint. Thesecond-derivativefunctions,C,, and CX are not constrainedto be symmetric,althoughsymmetryis later tested. Monthlydata coveringa period from September1978through December 1989was used in the estimation. Production,sales, and producer-heldinventorieswere taken from various issues of World Metal Statisdcs. The price of refinedcopper was taken as the US producerprice for refinedcopperfrom various issuesof Metl Statisdcs. The price of No. 2 Scrap (New York) served as the price of raw input copper and also came from Metal Statstis.

The price for industrialelectricitywas taken from various issuesof the United

StatesDepartmentof EnergypublicationMonthly Energy Review. The interest rate used was the 30-day US Treasury Bill rate. All prices and the interest rate were convertedto constant 1985-dollarvalues, using the US Producer Price Index. The Treasury Bill and Producer Price data was taken from the InternationalMonetaryFund's FinancialStatistics data base. The data on copper refinery capacity is collectedby the AmericanBureau of Metal Statistics, Inc and reported in various issues of Non-Ferrous Metal Data. The capacityis reportedat the plant level, based on surveysof plantmanagers. For most periods the combinedcapacityof all plantsgreatlyexceededproductionlevels. Plant managersmay have an incentiveto exaggeratecapacitylevels (to forestall new entrants)and have no real incentiveto be accurate. In addition,the sur"ey is only reported once a year. As a result, the data was treated as suspicious. However,using instrumentsand a smoothingtechnique designed to detect errors-in-variables problems did not affect the estimation results materially. Resultsof this procedure are reported later in this section. During the period a majorstrikeby labor unionsdramaticallyreducedrefineryoutput from July 1980through variable, k, was used to designate the observations October 1980. An intercept-dunmmy


associatedwith the strike. These atypical observationswere, however, illustrative of the relationshipbetweeninventoriesand uncertainty,as will be seen later during the discussion of the simulationresults. The initial modcl was estimatedfrom: m(t) - bo +b9,.t)


p(t) - co+ 1,ln(t) + c2ln(t)

+b,lx(t) +b.lnw(t) +bslqt) + c.lnx(t) +c4lnw(t)+CsW(t)

+bsk(t) - , V02(t) +ut) (43) +clk(k) + .(g)

In the first-stageof the estimationprocess,fitted valuesfor Int(t) and In9(t) were obtained by regressing the log of inventoryand production levels on the instrumentsgiven in Table 4.2 using OLS. The regressions resulted in R2 s of .93 and .69, respectively. In the second-stageof the estimationprocess, the fittedvaluesfor the log of productionand the log of inventories were substituted into the second equation of 4.3, and

Table 4.2: and Instruments for fittee valuesof the log of invenwories the logofused sales.

estimates of the random-component of price, fi(t)

were estimated by

interestrate priceof electricity scrap-copper price fixedcapacitylevel log of the priceof electricity log of the scrap-copper price log of fixedcapacitylevel monthof the observation year of the observation laggedproduction level laued inventorylevel lag of productionlevelsquared lag of inventorylevelsquared

least-squares. A six-month moving variance, d:, was constructedfrom the resulting residuals and included on the RHS of the first equation in 4.3.4


estimating both equations as the


the six-monthmoving varianceare emphasizedhere becausethe model providcd the maximumlikelihoodestimateamongthe lag-lengthchoices. At a practicallevel,however, models

usingfour, five,seven,or eight-month moving-variances werenotstatistically differentfromthesixmonthmodelwhensubjectedto a likelihoodratiotest. 36

third-stage in a threestage



yielded the

results given in Table

Tabb 4.3: Estinated paruneters andt-scores.








Marginal cost of storge (C.) coefficients

4.3, which gives the estimated


and Table 4.4, which expresses the coefficients

as elasticities.


results, for the most

bo. intercept



bl, inventories



b,, production

b3, electricityprice

0.06 0.43

1.25 3.12

bs. capacity b5, strike

-0.11 0.06

-2.22 1.23

b,, scrapcopperprice



Murtinal cost of production{C> coeffickjis




c,, inventories



part, were insensitive to






alternative choices for

ccs, copperprice 4, sCMp capacity c,, strikce

0.66 40.23

10.82 -3.66

the length of the lag




used to construct 6, as well as the estimationmethod. Alternativeestimatesare discussedlater. The estimateof the dispersionpremium is positiveand significantat a greater-than 99% level of confidence. All of the arguments to the marginal cost functions C, are significantat 97%+ confidencelevel, with the exceptionsof the interceptand the coefficient on the strike dummy. At the mean, marginal productioncosts are rising with additional production and are reduced by additionsto capacity. Evaluated at the mean, additional inventoryslightlyincreasesthe marginalcost of production. A 1% increase in the price of scrap copper increases marginal production costs by 739%. The elasticity of marginal production costs with respect to the price of electricity is negative, implying the unlikely


Table 4.4: Estimatedparametersof marginal cost hinctions expressed dispersionpremium.

elasticities,and estimatedaveraged

Estimated hfctions evabuatedat seris' mea: 'Me elsticity of with respect to: inventories production electricityprice scrap copper price refinery capacity strike dummy Average dispersion premium In cents per pound:



1.93 3.32 22.75 11.79 -5.82 3.21

0.09 0.18 0.51 0.73 -0.25 0.16


Note: Elasticitiesand dispersionpremiumcalculatedat muns. a, is constructedby a six-monthmoving varianceof second-stageprice-forecastresidualsfor refinedcopper.

conclusionthat marginalproductioncosts fall as utilitiesraise their rates. While a negative elasticity is unlikely, it may be that, in the short-run for a given fixed set of plant and equipment,energy costs are relatively fixed so that the elasticity is properly zero. As it turns out, constrainingthe elasticity on the price of electricityto zero has little effect on subsequenthypothesistesting or simulationresults. This topic is discussedmore fully later in this section. The estimatesassociatedwith the marginal cost of storage are more mixed. The elasticity with respect to inventory levels has the expectedsign, and is significantwith a 93%+ level of confidence. The elasticity with respect to production levels and the 5 . The electricity, scrap copper, and coefficienton the strike dummy are not significant

refinery capacityelasticitiesare significant. The sign on the refinery capacityis negative,

5 Although whensubjectedt? a Wald-testthey individually, bothstrikedummiesarenotsignificant

arejointlysignificantwitha 99%confidencelevel. 38

which is reasonablesincelarger refinerycapacityis probablyassociatedwith a larger facility with more storage space. The electricityelasticity is positive, which is also reasonableif the storagefacilitiesare either lightedor heated. Mysteriously,the elasticityassociatedwith the price of scrap copper is also positive and significant. Later, results are presented from constrainedversions of the marginal cost-of-storagefunction includingconstant marginal storage costs, as asserted in cost-of-carrymodels, and non-jointcosts. As it turns out, the significanceand the size of the dispersionpremium varies little under alternativeversions of the model. The estimatedmodelwas simulatedin order to calculateaveragemarginalproduction costs, average marginal storage costs, and the average dispersion premium. The average levels, along with the minimum and maximumvalues simulated for these functions, are reported in Table 4.5. The marginalstoragecosts are on-averagelow but reasonableat 1.9 cents per pound. The range, from around 5

Table 4.5: Simulated values for the marginal cost of production (Cr), the marginalcost of storage(C), and the dispersionpremium.

cents to 11 cents is


reasonable andl suggests some






Cy Premium



cents per pound ------------




87.7 1.4

56.5 0.0

126.0 9.4

convenience-yield. The average



ranges of the marginalcost of productionand the dispersionpremium appeared reasonable as well. Figure 4.4 mapsthe simulateddispersionpremiumagainstactualproducer inventory levels. With the exceptionof about 15 observations,the simulateddispersion premiums





i0.080.07 -

, 0.06 0.05-0.04

0.03 -i








producer Inventorles(thousomdtons)

Figure 4.4: Simulateddispersionpremiumsfor March 1980 to December 1989. behave as expected, climbing when inventories are low and dropping off quickly when inventoriesgrow. The outliers were generatedby either a strike, the threat of a strike, or war, all of which resultedin a higher dispersionpremium. In the figure, the outliersmarked with a 1 represent the three months leading up to the anticipated1980 copper strike, and observationsmarked with an s are monthsduring the strike. The observationsmarkedwith a 2 representa period in 1983when another labor strike seemedlikely, but which endedas negotiationswere successfullyconcluded, starting with Kennecott's provisionalagreement with its unions in April (Crowserand Thompson, 1984). Finally, the observationsmarked with an f represent the Falkland crisis (April to June, 1982). For those months, price variabilityand the dispersionpremiumgrewdespitemoderateinventoriesof refinedcopper. Although estimatedfrom data on US producerproduction, price, and inventories, the model behavesas expectedvis-a-vis the futures market in New York. The simulation results indicatethat, in the case of copper, informationaboutprice spreads in NewYork can be used to predict dispersionpremiumsfor US producers scatteredacross the country. As


shown in Section3, inventoriesare rationallyheld when a price fall is anticipatedprovided the dispersion prenmumis sufficiently large. Further, in the absence of a dispersion premium, backwardationin the futuresmarket generatespowerfulincentivesto arbitragethe market intertemporally.As a result,the dispersionpremiumfor inventory-holders,including US producers, should be higherduring periods of extendedbackwardationon the COMEX. Otherwise,inventorieswouldflowto NewYorkto takeadvantageof arbitrageopportunities. To test this aspect of the model's performance,the sample periodwas divided into two subsamples.

Observations were categorized as "backward market" observations if the

discounted spread between the second and third nearest COMEX contracts (the spreads shown graphically in Figure 4.3) was negative; the observation was categorized as a "forwardmarket" observationif the spread was positive. The spread between the second and third closest contracts was chosen to categorize the observationssince it would allow time for the copper to be physically shipped. Of the observations, about 48% were "backward-market"observations.Averagespreadsand dispersionpremiumswere calculated from the two samples and are reported on the first line of Table 4.6. observations associated with wars, strikes, or anticipated strikes were removed from both samples, and

means were

Table 4.6: Average spreads and simulated dispersion premiums during periods of backward and forward COMEX markets.

backward markets average average Sample spread premium

calculated for the "purged" samples as well.

In the


of the

observations were

In addition,

Full Purged

-3.22 -3.38

1.85 1.80

forward markets average average spread premium per pound 0.44 0.40

0.98 0.48

Note: In the purged sample, observations associated with anticipatedstrikes, strikes, or wars were removed. Spreads are discountedand reported in constantcents per pound.


categorizedas "backwardmarket" observations. The means from this sampleare reported in the second line of Table 4.6. For the "backwardmarket" sample, the spread averaged 3.22 cents and the simulatedvalue of the dispersionpremium averaged 1.85 cents. For the "forward market" sample, the spread averaged 0.44 cents and the average dispersion premium dropped in half to 0.98 cents. Purging the samplesof war and labor unrest, the difference was more dramatic, with the average dispersiondropping from 1.80 cents in the "backwardmarket" sampleto 0.48 cents in the "forward market" sample. Figure 4.5 mapsthe simulatedmarginalcosts againstproductionlevels. Theoutliers again are the strike months, when US productionwas constrained. Interestingly,some of the highest marginal costs occurred in the months prior to the strike as inventorieswere accumulatedto substitutefor anticipateddrops in production. Figure 4.6 maps the simulated cost of storage against inventory levels. The estimatedelasticityof marginalstoragecosts with respect to inventorylevels is positivebut insignificant,and it is difficultto discernany clear pattern from the plotted data. If refined

Simulated marginal production costs 1.30




1.00 _1.1







_ I

' ,,'



I,L1 I


U0 .7 0 If



80 60



100 140 1S0 productionleveis(thousandtons)

Figure 4.5: Simulatedmarginalproductioncosts. 42

200 220

Simulatedmarginolstoragecosts 0.120.10 0.08

-, 0.06 0.04












150 tons) Inventorylevels(thousand



4.6: Simulatedmarginalstorage costs.

copper inventories do indeed generate a Kaldor-convenience, then negative marginal storage costs should be associated with low levels of inventories. Further, after inventory levels reach some rninimal levels, the marginal costs should become positive as the marginal

"convenience"disappears. In simulation, however,iany

of the most negativevalues for

the marginal cost of storage are associated with relatively high inventory levels. Later in this section, the issues of jointness and Kaldor-convenience are examined under alternative

specifications. The estirnation and simulation results presented to this point are not sensitive to the lag-length chosen to generate the movir,g-variance for the random component of producer

prices. Nor are the results sensitiveto the estimationtechniquechosen. In additionto the using two three-stageleast-squaresestimaes presentedearlier, the modelwas also estimnated 43

Tabl 4.7: Summarystatisticson the estimateddispersionpremiumunder alternativeestimationmethodsand alternativelag-lengthsfor the price-variancemeasurefor the log-linearmodel. maximum ..... 2SLS----lag on t-score premiumpremium variance on V, mean std 4 5 6 7 8 9

7.01 5.09 5.21 4.81 3.78 3.53

1.21 1.29 1.46 1.44 1.33 1.38

2.13 1.99 2.09 1.98 1.77 1.77

----LIML----t-score premiumpremium on V. mean std 4.96 5.93 8.03 8.03 7.60 7.63

1.21 1.29 1.46 1.44 1.33 1.38

2.13 1.99 2.09 1.98 1.77 1.77

----3SLS----t-scorepremiumpremium on V,, mean std 5.94 4.65 4.97 4.67 3.65 3.37

1.11 1.22 1.41 1.39 1.28 1.30

1.96 1.88 2.00 1.90 1.70 1.68

Note: the hypothesisthat the shadow-priceof copper inventoriesis not strictlyconvexcould be rejected with a 99% + level of confidenceunder all estimationmethodsand all moving-variancechoices. The mean and standarddeviationsreportedin the table are from simulationsof the estimatedmodels.

alternative instrumental variable techniques


two-stage least-squares and limnited-

information-maximnum-likelihood. Becauseendogenousvariablesappearas regressorsin4. 1, instrumentsor full-informationlikelihoodtechniquesare requiredto avoid biasedestimates. However, all instrumentalvariabletechniquesshouldbe consistent. If the modelis correctly specified, three-stage least-squaresshould produce the most efficient unbiased estimates, sincetheprocedureincludesinformationon the contemporaneousvariance-covariancematrix of the structural equations' dIsturbances. In practice, with limited sample sizes, the superiorityof one method over another becomes less clear. Kennedy (1992, pp. 166-167) reports the results from several Monte Carlo studies that examine the sensitivity of estimatorsto changesin sample size, specificationerrors, multicollinearity,etc. Kennedy concludes that Monte Carlo studies consistently rank two- and three-stage least-squares estimatorsquite high in terms of robustness. At a practical level, the estimatesshould be similar. In fact, Hausman(1978)has argued that large differencesbetweenestimatorsthat are consistentis, in itself, a sign of misspecification. 44

Table 4.7 reports the t-score6associated with estimates of P estimatorsunder alternative specificationsfor d:(t).

from the three

The table also reports the average

simulatedpremium calculatedfrom the estimate, as well as the standard deviationof the simulatedpremium. The estimatedcoefficienton the moving-variance, J=, was always positi - and significant- that is, the hypothesisthat the shadowprice of inventoriesis not strictly positive could be rejected regardless of the lag-length chosen in the variance calculation and regardless of the estimation technique. Further, the distribution of the simulatedpremiumswasquite stableacrossvariance-choicesand estimationtechniques. This was less true for the extreme values of the various simulations. Minimumvalues for all simulationswere slightlygreater than zero, whilemaximumvaluesranged betweenslightly above 7 cents to slightly under 14 cents. Extremevalues consistentlydeclinedas the laglength increased, regardlessof estimationtechnique. As discussedearlier, the unconstrainedestimationof the model retumed the wrong signfor the price elasticityfor electricity. The modelwas re-estimatedunder the assumption that energycostsare detem-inedprimarilyby plantand equipmentand that the correct shortrun coefficient value should be zero. Since the t-score on the original estimate was significant, the constraint is a binding one; however, imposingthe constraint had little practical consequence.Regardlessof the estimationprocedureor lag-lengthchosen for the moving variance, P.

remained positive and significant. The three-stage least-squares

estimatesfor a range of lag-lengthchoicesare reported in Table 4.8. Further, Wald tests

6 Thet-scoresreportedin Table4.6 (andlater in Table4.7) for the two- and three-stage least-

squaresresultsare computedby TSPVersion6 to be robustin the presenceof heteroskedasticity using White's method. This proceduretendedto lower the t-score comparedto alternative calculations. 45

were constructed to see Tabb 4.8: Tests on the effectsof excludingthe price of electricityfrom the marginal cost-of-productionfmction.

if the values of the remaining








electricity was excluded

maxium lag In moving variance

t-score on V.

4J test score

4 5 6 7

6.51 4.56 4.91 4.62

1.04 0.48 0.55 0.64







from the marginal costof-production un tion. The test was cfinstructed by pairing each of the remaining

Note: Under the maintainedhypothesisthat electricity use is a fixed cost, the hypothesisthat shadow prices are not convex can be rejected at a 99%+ level of confidencefor all moving-variancechoices. The hypothesisthat excludingthe prke of electricityfrom the marginalcostof-production function does not change the remaining coefficient estimatescannotbe rejectedat any reasonablelevel of confidence. The Wald-testis distributedas X2with I degree-of-freedom.


parameter estimatesfrom the three-stageleast-squaresestimatesof the two models - one model includingthe price of electricity, and one model excludingthe price of electricity. Whenthe exclusiondoes not affect the estimatesfor the fourteenparameters,the sum of the differences between the parameters will be jointly zero. As can be seen in the second columnof Table 4.8, the hypothesisthat the estimatedparametersare not differentcouldnot be rejectedwith any reasonablelevel of confidence. As discussedearlier, the existence of the dispersionpremium depends in part on non-linearitiesin the marginalcost-of-storageand productionfunction. The log-linearform chosen for the marginal cost functions(4.2) has the advantageof being nonlinear while requiring few estimatedparameters. However, there is a tradeoff between the practical advantageof havingfewer parametersto estimateand fiexibilityin the functionalform. This is particularlytrue given that the marginalcost-of-storagefunctionis usually depictedas s-


shaped (recall Figure 3.4). For this particular functionalform, a positive coefficienton inventoriesin the cost-of-storageequationimpliesthat the marginalcost-of-storagefunction is increasing as inventoriesaccumulate,but at a decreasing rate -- that is, C. = b2/z and C=

-b2 /z2 . Often inventory-storagecosts are expectedto rise first at a declining rate

(C= is at first negative), but then to rise increasinglyrapidly (so that C. is positive)as storage facilities reach full capacity. While storage capacity for some grains or frozen commoditiesis strictly limited, refined copper is easily stored and not subject to the same type of storage-capacitylimits. Nonetheless,in order to test whether the results reported above are dependent on the functionalform chosen, a more general form, qtiadraticwith third-order terms for z and y, was estimated as well. More specifically, the following marginalcost functionswere substitutedfor 4.2: 5


Cs=bo+ b1Xi+


bXIX,j + b7t 3 (4.2b)

, 1 1

C, - Co+

cxi +

+ C793 C jVZIX

where Xi,for i 1,2,...,5 representsz, y, x, w, and f from 4.2. The estimationprocedure was repeatedon: 55


m(t) =bo+

+b6k(t) +b723EbuX,Xj biX,+ 111~~~~~~~~~

2 d2 + CI(t)

(4.3b) p(t) - co+

cX +

rcuXX +¢ 6 k(t) + C7 9



using the three estimationtechniquesand the six specificationof 6(t) The more generalquadratic-plusform generatedan additionalthirty-twoparameters to estimate. For the base model(three-stageleast-squaresusing a six-monthlag to generate 47

6;(t), only seventeenof the forty-sevenestimatedparameterswere individuallysignificant at a 90% confidencelevel. These parameters and their t-scores are given in Annex II. Nonetheless, P. proved relentlesslysignificantregardless of the estimationtechniqueor lag-lengthchoice. In simulation,the quadraticversion producedlower average premiums, primarily due to lower maximum values. The maximum simulated values across the estimatedmodelsdid not declineas the lag-lengthon the variance was increased. Rather, the range was more limited, from 4.92 cents to 7.16 cents, and showed no clear pattern. Table 4.9 provides the summaryresults for the quadratic-plusestimations. Evaluatedat the meanof the series, the linear functionCm turnedout to be negative at a 97%+ level of confidence,although, in simulation, C= did take on small positive values when inventories were extremely high. At the same time, C., which should be positive, turned out to be negativeon average, althoughnot significantlyso. In summary, despitethe fact that manyof its parametersproved insignificant,the estimatedquadratic-plus

Table 4.9: Summarystatisticson the estimateddispersionpremiumunder alternativeestimationmethodsand

alternativelag-lengthsfor the price-variancemeasure for the quadraticmodel.

m.aximum lag on variance 4 5 6 7 8 9

---- 2SLS -t-score premiumpremium on V. mean std

2.72 1.65' 2.43 2.69 2.71 2.85

0.42 0.42 0.54 0.52 0.51 0.55

1.12 1.01 1.26 1.19 1.14 1.18

---- LIML t-score premiumpremium on V. mean std 2.47 2.62 3.28 2.61 2.42 2.42

0.42 0.42 0.54 0.52 0.51 0.55

1.12 1.01 1.24 1.19 1.14 1.18

-- 3SLS t-scorepremiumpremium on V. mean std

2.39 1.63' 2.43 2.68 2.71 2.83

0.39 0.42 0.54 0.52 0.51 0.55

1.02 1.01 1.24 1.19 1.14 1.18

Note: the hypothesisthat the shadow-priceof copper inventoriesis not strictlyconvexcould be rejected witha 99%+ level of confidenceundermost estimationmethodsand moving-variancechoices. The two scores marked with an * were significantat a 94%+ level of confidence. The mean and standard deviationsreported in the table are from simulationsof the estimatedmodels.


model supportsthe notion of a dispersionpremium, while suggestingthat the extreme-case premiumsare quantitativelysmaller than those simulatedby the log-linearmodel. At the same time, while the marginalcost-of-storagefunctionwas given greater flexibility, it did not trace out the classic path of rising slowly to an inflectionpoint, and then rising more rapidly. Despitethe consistencyof the estimatesand simulations, one potential problem is shared by all of the specifications. Since industry-widerefinery capacityis only published once a year, the annual figure was repeated for all twelve months of the year when estimating the model. On the one hand, shadow prices for inventories are based on expectations, and it could well be that the market expectations must be based on the publishedrefinery data sinceno better source of informationexists. Alternatively,sincethe number of refiners is small, industry participantsmay form better subjectiveestimatesof capacity. Regardless,there is certainlythe potentialfor measurementerror in the variable. The practical consequencesof an errors-in-variablesproblem cannot be readily known. If the measurementerror is distributed independently of the disturbance terms, then t"1e estimates remain consistent. However, if not, the estimates may be biased even asymptotically. Since the exact nature of an errors-in-variablesproblem can never be known, the correct course of action cannotbe known. In "correcting"the problem, the cuie may be worse than the disease-- if indeeda diseaseexists. The general prescriptionfor an errors-in-variablesproblem is to use instrumentsfor the problem regressoror to somehow average out the effects of the measurementerrors. (Kennedy, 1991 pp. 137-40.) The rationale is that by averagingout the data, the measurementerrors are averaged as well, reducing their impact. Table 4.10 reports the parameter estimates after addressing the


Table 4.10: Estirnatedparametersand t-scores (in parentheses)for base modeland modeltreated for assumed errors-in-variablesproblem.

base model

Durbin rank method

in-year smoothing

1.81 (4.09) 0.04 (1.89) 0.06 (1.25) 0.43 (3.12) 0.22 (4.91) -0.11 (-2.22) 8.94 (4.97) 0.06 (1.23)

1.28 (2.76) 0.02 (1.22) 0.04 (0.80) 0.48 (3.58) 0.22 (4.68) 0.01 (1.47) 8.10 (4.96) 0.04 (0.72)

1.84 (4.04) 0.03 (1.25) 0.06 (1.06) 0.42 (2.62) 0.21 (4.52) -0.11 (-1.49) 8.91 (4.50) 0.07 (1.12)

0.39 (0.68) 0.09 (4.11) 0.16 (2.21) -0.46 (-2.70) 0.66 (10.82) -0.23 (-3.66) 0.14 (1.86)

-0.87 (-1.44) 0.05 (3.07) 0.14 (2.03) -0.36 (-2.24) 0.66 (10.62) 0.04 (4.1) 0.12 (1.82)

1.07 (1.86) 0.13 (5.00) 0.14 2.06 -0.65 (-3.51) 0.64 (10.61) -0.39 (-4.86) 0.14 (2.04)

In equation m(t) coeffcient on: intercept: inventories: production: electricity: scrap copper: capacity: price variance: strike dummy: In equation p(t) coefricientcn: intercept: inventories: production: electricity: scrap copper: capacity: strike dummy:

errors-in-variablesproblemtwo ways. The first columnrepresentsthe coefficientsfrom the base model, repeated from Table 4.3.

The second column lists parameter estimates

followinga treatmentfor the errors-in-variablesproblemsuggestedby Durbin (see Kennedy, 50

p. 140, or Johnston, pp. 430-2). Followingthis procedure, the observationson refining capacitywere rankedfrom highestto lowest. Observationson the rank were then substituted as an instrumentfor the observationson capacity. Since the capacity was ranked from highestto lowest, the sign on the capacitycoefficientsswitchedand, of course, the scale of the capacity coefficientschanged. Otherwise, the effects of treating the estimation for measurementerrors in the capacityvariablewere negligible. Columnthree of Table 4.10 reports estimationresults in whichthe annual observationswere smoothed. The smoothing procedure was to measure the differencebetween refining capacityfor any two years and, from the January observationfor the first year, add one-twelfthof the difference to each monthlyobservation. Again, the effectsof the smoothingprocedureon the estimationresults are quite marginal. As mentionedearlier, Hausmanhas argued that estimatesobtainedfrom alternative but consistentestimatorsshouldthemselvesbe consistent. Basedon this principal,Hausnian proposed the followingtest for misspecification. Two sets of consistent estimates are differencedand then standalrdizedby the differencein the covariancematricesof the two sets of estimates. The resultingquadraticform is asymptoticallychi-squared,with the degrees of freedomequal to the numberof linearlyindependentrows in the differenced covariance matrix. The model fails the test, signaling misspecificationof the model, when the hypothesisthat the two estimatesets are the same can be rejected. The estimationpackage TSP Version6.0 provides a matrix procedure for calculatingthe Hausmantest. Table 4.11 reports the results of the Hausman test based on the difference between the three-stage (consistentand efficient)parameter estimatesand both two-stageand limited-infonration (consistent)parameterestimates. Sincedifferentalgorithmsare used to numericallycompute


Tabk 4.11: Hausmantest for misspecificationbasedon LIML, 2SLS and 3SLSestimatesfor varying movingvariancecalculations. - three-stage least-squares compared to alternative estimators maximumlag in moving variance

twotage kast-squares degrees of freedom test-score

4 5 6 7 8 9

2 2 1 1 1 1

Ihmited-lnformatlonmaxinum-likelihood degrees of freedom test-score

0.34 0.21 0.06 0.05 0.01 0.00

6 5 7 8 8 8

0.02 0.01 0.01 0.02 0.02 0.00

Note: Test-scoresae distributedas X2. The hypothesisthat the two sets of estimatesare equal could not be rejected with any reasonablelevel of confidence.

the three estimates, they will in practice yield different estimates. The Hausman test

measureswhetherthe estimatesare criticallydifferent. The test was repeatedusing various lag-lengths in the moving-variancecalculation. Under all versions of the model the hypothesisthat efficientand consistentestimateswere equal couldnot be rejected with any reasonablelevel of confidence. Earlier simulation and estimation results cast some doubt on the existence of a Kaldor-conveniencefor refinedcopperinventories. Kaldorand, later, Workingand Brennan argued that a convenienceyield exists when storage is a necessary adjunct to business, which, in the case of copper refining, implies a jointness in the cost of production and storage. Further, if this convenienceyieldeffectivelyexplainsperiods of backwardationand a negativeprice for storage, then the marginalcost of storage must be negativeover some range of low inventorylevels. Contrastedagainstthis is the assertionin many cost-of-carry modelsthat marginalstorage costs are fixed and positive.


Table 4.12 reports the test results on three hypothesesabout inventorycosts. All three tests are constructedas Wald tests so the test scores have a chi-squaredistribution. The symmetrytest tests the hypothesisthat C.


C, I evaluatedat the mean for the series.

If the true underlying cost function is joint, then Young's theorem states that the partial second derivativesshouldbe equal. The hypothesisof symmetryis rejectedwhen the testscore exceedsa critical value. The results of the symmetrytest are mixedand are sensitive to the moving variance chosen in the estimationmodel. For lag-lengths4 and 9, the hypothesisof symmetrycould be rejectedat a 90% confidencelevel. The second column of Table 4.12 tests the hypothesis that the cost function is actually non-joint. This test is constructed to be quite weak in the sense that only the coefficientson productionin the marginalcost of storageand on inventoriesin the marginal cost of production (b2 and cl in equation 4.2) were constrainedto zero. This left, for

Table 4.12: Tests related to the marginalcost of storage.

maimum lag hi moving variance

tests on hypotheses of ---------------------------constant marginalcosts non-jointcosts


Wald test scores ----------------




2.04 1.37 1.32 2.30 279'

6 7 8 9

2.19 4.420 7.87" 10.86" 10.50" 8.77"


17.93" 24.79" 27.73" 26.02" 26.41"

2 Test scores Note: All tests wer constructedas Wald testsdistributedas X withone degree of freedom. 0 denotedwithan * are significantat a 90% + confidencelevel. Testscores markedwithan are significant at a 95%+ level of confidenceand test scores marked with - are significantat a greater-than99% confidencelevel.


example, the price of scrap copper as an cxplanatoryvariable in the marginal cost of storage. Despitethe weakdefinition,the hypothesisof a non-jointcost-functionwas rejected at a 95% + level of confidencefor all but the four-monthmoving-variancemodels. Using a stronger definitionof non-jointness,which wouldexclude the price of scrap copper from the marginalcost-of-storagefunctioncl =b2 = b4= 0, non-jointnesswasrejectedfor all models with a 99%+ level of confidence. The third column reports the results on the test that marginal storage costs are constant, as proponentsof cost-of-carrymodels often assert. Constantstorage costs imply that bi - 0, for i = 1,2,3,4,5 (from equation4.2) in the estimatedmodel. This hypothesis was rejectedat greater-than99% confidencelevels for all moving-variancechoices. Kaldor argued that the adjunct nature of selling and storing provides the economic rationale for convenience. For copper producers this means a jointness between the activities of producing, selling, and storing copper. Others have argued that inventory activity is less complex, and, in the case of most cost-of-carrymodels, storage costs can simply be best described by a constant marginal cos;. The test results provide strong evidencethat marginal storage costs are not constant, as asserted in cost-of-carrymodels. Further, the combinedresults on the tests of symmetryand jointnessprovidesomewhatless strong support to the notionthat costs are joint. Still, a Kaldor-convenienceyield relies on more thanjointness. Inventoriesare expectedto yield their highest marginalconvenience as inventoriesare initiallyaccumulated- that is, the marginalcost of inventoriesis expected to be negativeat low levels, but grow positive as additionalinventoriesbuild. None of the estimated models presented here displayedthat characteristic. Rather, the evidence here suggests that, although inventories are a necessary adjunct to business, they generate no


savings. As statedearlier, the existenceof a dispersionpremium is sufficientto justify storage during arnanticipatedprice decline and strong empiricalevidencewas presentedto support the existenceof such a premium. The estimationresults reported thus far have come from joint-cost models. Table 4.13 reports tests on the hypothesis that the shadow price for copperinventoriesis convexunder two maintainedhypotheses. The first columnreports test scores on the hypothesis that





Table 4.13: Tests on convexity of the shadow price for refined copper inventoriesundernon-jointcosts and constantmarginal storagecosts.



than or equal to zero

ing ~


maintained hypotheses --constant non-joint marginalcost cost function -

lag In te tmaximum moving variance


























--------- t-scores-------

hypothesisthat the cost-




4.69 3.88

function is not joint.










The results here include the



marginal storage costs

Note: The hypothesisthat shadowprices are not convexcan be rejected at a 99%+ level of confidenceunderboth maintainedhypothesesfor all

choices. moving-variance

have nothing to do with the cost of scrap copper, although similar results were obtained from weaker definitions. The hypothesis that the shadow price of copper inventories is not strictly convex was strongly rejected at a greater-than99% level of confidence. The second column tests the same hypothesisunder the assertionthat marginalcosts are constant. Again the alternative hypothesiswas strongly rejected. Generally,the estimationresults providestrong supportfor the notionthat the value


of inventoriescontains a dispersionpremium. This finding is empiricallyrobust under a variety of maintainedhypothesesand under a variety of estimationtechniques. Further, the model behavesin simulationas wouldbe expectedeven when simulatedthrough periods of rare events, includinga war and labor disputes. The existenceof a dispersionpremium is not guaranteed by the theory developed in Section 3, so empirical tests are crucial in explaining inventorieswhen a fall in price is anticipated. The model does not strongly support or contradictthe possibilityof Kaldor-convenience,which couldalso explainthe socalled negative price for storage. On the one hand, the hypothesis of jointness between production and inventory-holdingfor US copper refiners, a pre-condition for Kaldorconvenience,is generallysupportedby the modelresults. However, the explanationsof the

convenience- savings in deliverycosts, re-orderingcosts, etc. -- implythat inventoriesare likely to be marginallyless cost-savingas they accumulate. In simulation,the model did generate some negativemarginalvalues for the cost of inventory-holding;however, those values were not consistently associated with lower levels of inventories. Table 4.14 summarizesthe empirical results.


Table 4.14: Summaryof the empiricalresults. Results relating to Kaldor-convenlenceyields: -Jointnessbetweenproductionand storage,which providesthe economicrationalefor a Kaldorconvenienceyield is generallysupported. The result is not completelyrobustunder alternative models. -Symmetry between marginal production and inventory-holdingcosts, an implication of jointness, is generallysupported,althoughthe result is not completelyrobust under alternative models. -The hypothesisof constantmarginal storage costs, a featureof cost-of-carrymodels which excludesthe possibilityof jointness, is robustlyrejected. -In simulation,negativemarginalstoragecosts are generated. While negativemarginal storage costs implya Kaldor-convenience,the observationsdo not appear exclusivelywhen inventories are low, as Kaldor's theory wouldpredict. Results relating to the dispersion premium -A dispersionpremiumwill exist if the shadow-pricefunctionfor inventoriesis strictly convex in inventories. The alternativehypothesisof concavityis stronglyrejected with a high degree of confidence. -In simulation,the dispersionpremiumfor copper from 1980to 1989averagedaround 1.4 cents per pound. However,when inventorieswere low, when strikesappearedimminent,and, during the Falklandcrisis, the premium rose. In simulation,the premium ranged from near zero to about 9 cents per pound. -The estimatedvalue of the premiumas well as tests on the convexityof the shadowprice for copper inventorieswere extremelyrobust under alternativeestimationtechniques. -Althoughthe model was estimatedexclusively from data on US producers, in simulation, larger-than-averagedispersion premiums were associated with periods of backwardationin COMEXcopper futures. -Hausmanargues that since two-stageleast-squares,limited-information-maximum-likelihood, and three-stageleast-squaresestimatorsare all consistent, estimationresults from the various estimatorsshould be similar. The model passes Hausman's specificationtest even under a varietyof specificationsfor the price-error-varianceterm. -The test on the convexityof the shadowprice and the existenceof a dispersionpremiumwas robustundera varietyof specificationsfor the underlyingmarginalcost functions. Generalizing the functionalform greatlyincreasedthe number of parametersto be estimatedand generated a large number of individuallyinsignificantparameterestimates. Nonetheless,the dispersion premiumproved relentlesslysignifiant. -The simulateddispersionpremiums, as well as the tests regardingits existence,were robust even when non-jointnessor fixed marginal storagecosts were imposed.


5.0 Conclusion As with many primary commodities, the price of storage for refined copper is sometimesnegativeand copperfuturesmarketshaveremainedin backwardationfor extended periodsof time. These marketcharacteristicsare in direct violationof frequentlyused pricearbitrage conditionswhich maintain that storage only occurs when the price of storage is positive. In Section 3, a generalized price-arbitrage condition is developed which is consistent with observed inventory-holdingin the face of an anticipated fall in price. Potentially,there are two reasons why inventoriesmight be held under such circumstances. First, the marginalcost of storagemay indeedbe negativeat certainlow levelsof inventories - that is, inventoriesmayproducea Kaldor-convenienceyield. In the case of copper, while there was some evidencethat inventory-holdingare joint activitiesresulting in a joint cost function, there is little evidencein the simulationsof a Kaldor-convenienceyield. The second justification for inventory-holdingin backward markets stems from uncertaintyand a convexrelationshipbetweenthe shadow-priceof inventoriesand inventory levels. When stock prices are low, an unanticipatedpositive shift in the demand schedule will result in rapid price gains. While a negativeshift may be just as likely, the effect on price is not symmetric. This skews the distributionof potential price outcomes toward higher prices and generatesa dispersionpremium. Estimatesand simulationsin Section4 providestrongevidencethat the shadowprice functionof refined copper inventoriesis indeed convex in inventories. Under a variety of assumptions,the key parameterin testingfor convexityproved statisticallyrobust. Further, the estimated value of the dispersion premium proved robust as well. The premium remainedlow on average (around 1.4 cents per pound). Simulationsrevealedthat the low average premium masked the potential importance of the premium. Producer-held 58

inventories for the simulationperiod ranged from around 25,000 to 230,000 tons. For inventorylevels above 80,000 tons, the dispersionpremium fell to positive but near-zero levels in the absence of labor strife or war. However, as inventories fell further the premiumshot up rapidlyto a maximumvalue of about ninecents per pound. In simulation, the dispersionpremiumalso jumpedduring the monthsleadingup to the 1980refiner's labor strike and immediatelyfell at the conclusion of the strike. Similar movementsoccurred during a 1983 labor dispute and the 1982 Falkland War. Further, the model simulated larger dispersionsduring periods of backwardationin the copper futures market in New York (COMEX) even though U.S. producerprices, production,and inventorieswere used in estimatingthe model. Just as the price of a call option contains a premium based on the underlying variabilityof prices, the shadowprice of copperinventoriescontaina premiumbased on the variabilityof the unplannedcomponentof inventories. When inventorylevelsare low, the value of the premium increasesto the point where certain levels of inventorieswill be held even in the face of a fully anticipatedfall in price.


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Annex 1: The Derivation of the Price-Arbitrage Condition In this annex,the formalmodelis presented. A generalizedprice-arbitragecondition is derived from the first-order conditionsof the optimizationproblem which is consistent with inventory-holdingduring an anticipatedprice fall. The copper-refiningproblem is characterized as a continuoustwo-cycle problem with uncertain future demand. In the current period, the producer knows the current sales price. By deciding how much to produceand sell, he determineshow much inventoryhe willbring into the next period. The expectedmarginal value, or the shadowprice, of the inventory in the next period is not known,but containsa stochasticelementsincedemandis uncertain. The effects of random demand shockson the shadowprice of inventoriesmay be asymmetric- that is, a positive random shock may increase prices by more than an equally sized negative randomshock. In such a case, the shadow price of inventory will carry a dispersionpremiurn so that the shadowprice of inventoriesincreaseswith the varianceof the stochasticcomponentof sales. Such a premium is analogousto the volatilitypremniumin an optionsprice and can result in positive inventorylevels even when price declines are expected. The first-periodproducerproblem can be written as:


f[ps-C(y,z)]e-'dt + J(zl,t 1 )

st. z




where p is the sales price; s is the sales level; C is a joint cost function for storage and productionwhere z is level of inventories3nd y is the productionlevel; r is the discountrate and J(zl,tl) is the value of ending stocks. The cost functionis treatedjointly to allow for a potentialcost-reducingKaldor-conveniencefor inventories.


The solutioncan be found by solving the Hamiltonian: Max,,H a [ps - C(y,z)]el

+ A(y-s)


where the first-orderconditionsare given by: i) 8H1.3s- 0 - p - Ael, for to ststl,

U) a811o8y - 0 - Cy - Ael', for tostst 1 , M) a8lH& -

fv) A(t)


A - C,


A-e, for toits: 1 ,



v) z(to) = Zo

The producer maximizesprofits by settingthe marginalcost of productionand the marginal revenueof sales epal to the value of a marginalchangein the level of inventoriesat the end of the first period - that is, the shadowprice of inventoriesat time tS (from conditionsi, ii and iv in Al.3, evaluatedat t,). Evaluatingthe solutionat t, - O, p - C, X A * ai/8. Prices are known in the current period, and inventory levels are determinedonce sales and production levels are decided, but the endingvalue of inventoriesis not known exactly. Rather, the value of stocks J, and the shadow price of inventories, X, are based on expectationsabout future prices, production,and sales. When sales containa random element, inventorylevels will also be stochasticand changes in inventorieswill contain a planned and unplannedcomponent. The difference betweenplannedand actual inventorieswill be the difference betweenexpectedand actual productionminus sales. For the moment, assume that the change in inventoriescan be expressedas the followingprocess:



- E,-1[zt;-Z-] - et = ;-;-I

- Et d[yt-sg]


where e has an expected value of zero and a variance o2. Rewriting the constraint on inventories in continuous-timenotation, the value of ending inventoriesat time t, is the solutionto the followinginfinite-horizonproblem:

e Max, Ef{tps

- C(y,z)]e"K}dt.


A - E(y-s)t* + adv.

The term dv u(t)dt 11 2 is a Wiener process, where u(t) -N(O,1).


Because inventory

changesincludea randomcomponent, dz/dt does not exist in the usual sense and the


of stochasticcalculus apply. Evaluated at t,, the solution to A1.5 gives the value of ending stocks, that is, J(.zt 1 ) - V(zl)e '.

In the language of optimalcontrol theory, the firm's problem is a

stochastic infinite-horizonproblem, stretching from t, onwards. As a result, the shadow price for the end-of-periodinventoriesin the first stage of the producerproblemnis based on expectationsof an on-goingprocess of productionamid uncertaindemand. Generally, the solution for J(zl,tl) can be found by solving Bellman's equation, a partial differential equation: -aJ(zj,tr)1&8 = rV(z1 )e"'.

Arbitrarily setting tO-0 simplifies the equation

somewhatso that the solutionto the inventoryproblem from t1 onward can be represented by the Hamilton-Jacobiequation of dynamicprogramming: rV(z 1 ) = Max,,E [Ps - C(y,z) + VZ(y-s) + V(A1.6

The first-order conditionsfor the maximizationof A1.6 are:


i) E(p - VS) . 0 a) E(-Cy + V.) - 0


L) E(A) - E(y-S)dt tv) z(t1 0) - Zi

The producer solves for plannedproduction,sale and inventoriesby setting expectedmarginalcostsequalto expectedpnce equalto the dsdow priceof inventories,V, -

which is itself an expectedvalue.

It is worth noting that the distributionof the error term, especiallya2, is of the decisionvariables. The varianceof the error is regardedas a stateof independent natureandis not subjectto choiceon the part of the producer. Thisassumptionis implicit thoughout the paper and works well empirically. Further,the derivationof stochastic differenialequationsare premisedon the fact that the distributionis dependentsolelyon rraet-periodinformation(see, for example,Malliaris nd Brock,1982,p. 67, or Merton, 1992,p. 67). Thedistinctionis a subtleone, sinceit is reasonableto thinkof the variance as somefunctionof demandor supply. However,in termsof the optimizationproblem,it is the expectedvalueof the function,not the functionitself,that is relevant. Several additionalasuumptionsmust be made to guaranteethat the first-order conditionsdo indoedprovidea maxinmu. V mustbe concavein z; the solutionvaluesof z, y, ad s must be positivel(otherwisebordersolutionsmust be considered);and the

' For theempiricalpblem at had, refner ientories of refinedcopper re allpositiveu are

at both the COMEXad LME haveremainedpostive quaties sold d produced. Inventories

thzoughoutthebsty of thou insditutios well. However,to be complete,stock-outsneedto be consorintsintoducedto the moximizaon onsideredin the thoretical modelad non-negativity

problem.Thee aregivenin Anex 2. 68

transversality-at-infinityconditionmust hold.2 An expression for the marginal value of inventories is found by applying the envelopetheory (Lopez, 1989)to the Hamilton-Jacobiequationgiven in Al.6:

rVg - E-C, + V,(y -S) + .!V



At the solution, Etp] - Vz, (as part of the first-orderconditionsin A1.7) so that: (Al.9)

EBdpetkJ - V1[*k/d*j

Rewritingpart iii of A1.7 provides E[dz/dkJ - Ely-asl. These results can be combinedto form a price-arbitragecondition. First, from rearrangingA1.8 and using iii from Al.7: V,xELy-sl = VYE[dzdd - rV, + E[C3J -



CombiningAl. 10 with A1.9 provides: - 1

Edpl/d] - rE[p] + E[C,]

2 V02,

for z,a,y > 0


Equation Al.11 is a generalizationof price-arbitrage conditions given in cost-of-carry modelssuch as Williamsand Wright (1991, p. 27). The arbitrge conditionstates that the

2 Forthe infinite-horizon autonomous problemgivenabove,the tverslity

lrm V,(t)Z(t)#


liin JS(t)Z(t)rf




Benvenisteand Schnkmn (1982) owed that the conditionis necoy midsffiient for the solutionof A1.6 to be opdmal. he logicis thatany posidvesock level g hve no valuea the problemapoaes infinity. Otherwe, the firmcouldfurhe Ices proft by eitherproducing lessor sellingmorein the lastperiod. Invene havevaluebeuse, uldtmay, twy canbe sold. If somepriceexistsat whichnocoppercanbe sold,thm meupperboundmu eist fortheuhdow price of inventories.If so and if sock levels, z, re limitedby pysic sore or naural endowmu, thendioun wil assurethatthe laaverality conditionholds. SeeBrock(1987) for ftrther detils on the geal condition. 69

expectedchangein price willbe equal to rE[pl -- intereston investingthe moneyelsewhere plus C, - the costs of physicalstorageand any amenityfrom storage-- minus 2 V= 02. If V= is positive, then this last term constitutesa dispersion premium that increases with the variability of the stochasticcomponentof inventories02. The last two componentsof Al.11 have important implicationsfor holding inventories in the face of less-than-full carrying charges. Accordingto the condition, it still may be optimalto hold inventories when the market is in backwardation- E[dplb] < 0 -- if inventoriesprovide a costreducing Kaldor-convenience(that is, if C, is sufficientlynegative)and/or the dispersion premium, 2 Vo


, is sufficientlypositive. The two componentsare not mutuallydependent.

Kaldor-conveniencealone can potentiallyexplain inventory-holdingin backwardmarkets, as can a dispersionpremium. When V, = 0 and C. is positive, Al. 11 reducesto the costof-carry price-arbitragecondition. For the cost-of-carrymodel, no inventories are held when the sum of the current price plus a constant physical storage cost is greater than the expected discountedfuture price. Puttingthis constraintinto a continuous-tirnecounter-part:

E[d,pldt]= rpk jbfr z




The two arbitrage conditionsdiffer in two respects. In Al.12, storage costs are treated as a constant positivemarginalcost, and separatefrom other activitiessuch as sales or production. Cost-of-carrymodels are usually based on the activities of professional speculatorswho presumablyreceive no conveniencefrom holding inventoriesand do not participatein production. Generally,however,there is nothingfundamental to the derivation of cost-of-carry models which requires fixed marginal storage costs and the marginal storage-costfunction could be written in a more flexiblemanner. The second difference 70

between the two arbitrage conditions is the presence of a dispersion premium in the generalizedprice-arbitrageconditions. This comesfrom treatingthe value of the inventories as stochastic. Cost-of-carrymodels use expected prices (or futures prices representing expectedprices) but do not treat the price changesthemselvesas a stochasticprocess. This differs from option-pricingmodels where the variance of the underlyingcommodityprice enters explicitly into the evaluationof the option. The dispersion premium, 2 Va


can be interpreted as the expecteddifference

between the stochasticand deterministicvalue of inventories. To see this, start with the marginal-valuefunction of inventories defined in terms of a detemiinistic component (plannedinventories)plus a randomelement, and calculatea Taylor-seriesapproximation: VZ(zd+e) s

VZ(Zd) + Vege +


+ 6


When e has an expected value of zero, and is symmetrically distributed, where E[e] = E[e 3 j = 0 and E[e2 1 = o2, the expected difference between the stochastic and

determinikticcomponentof the shadowprice for inventoriesis approximatelythe dispersion premium: E|Vz+[

V(zd)] 0 I V



Even when demandand inventoriesare treatedas stochastic,there is nothingin the first or second-ordermaximiza.on conditionsthat would require V= to be positive. In fact, V couldcertainly be quadraticin z so that V= need not exist. However, in much of the literature on inventoriesthe shadowprice of inventoriesis often describedas convex in


3 Usually it is stated that there is some type of "pipeline" minimum z - at least implicitly.

stock level (Figure A1.2). As stock-levelsare drawn down and approachpipeline levels, larger and larger price increasesare required to depletediminishedinventories. Usuallythe convexityis attributedto a convenienceyield. Accordingto this argument, pipeline levels are required to carry out business in an orderly fashion, while additionalstocks, up to a point, can still facilitate transactions or minimize costs such as re-orders, deliveries, or restocking. As it turns out, V= will exists if either of the marginalcost functions,Cs, the cost of physical storage, or C,,, the cost of production, are nonlinear. This is true whetheror not the cost functionis joint or whetherstocks are cost-reducingat any level. To see this, recall that E[p] = E[C,J


V. from the first-orderconditions(Al.7) and that: d2 p = V

Z2 = d2C,


From equation A1.15 it is easy to see that if the joint cost function is linear or quadraticin z and y so that d2 C. is z. ro, then V= = E[d2p/dz2l will always equal zero. Earlier it was stated that stochasticdemandwill give rise to stochasticinventories; however, little was said aboutthe relationshipbetweenthe means and variancesof the two distributions. As it turns out, the problem can be cast in terms of either stochasticdemand or stochasticprice. This result comes from the finance literature (for example, Cootner, 1964, and especiallyMerton, 1992). There is long history of decomposinginventoriesinto expected and random components (planned and unplanned inventories), particularly in Keynesianmacroeconomics. Stochasticprices are often employed in finance and capital


19and Brennan(1958),p. 54. 72

literature(for example, Blackand Scholes, 1973, or Abel, 1983). For the formal model, the only distinctionbetween the two concepts involves a normalizationof the stochastic difference equation, which constrains the optimization problem. This result is due to Merton (1992, pp. 57-75). Consider first the process: v(t)





where k denotesobserved values, so that the partial sums Sn1 ?jv(k) form a martingale. Now let F(t) =f(X,t), where f is some non-linear function. F(k)-F(k-1) X(k)


Merton shows that

will give rise to essentially the same stochastic difference equation as


This result is used frequently in finance literature -- for example, when

valuinginstrumentssuch as warrants in terms of the varianceof the underlyingstock price. For the problemat hand, the result is useful sinrceit allows the differential equation associatedwith the optimizationproblem to be expressedeither in terms of stochasticsales or stochasticprice. Empirically,it turns out to be convenientto treat price as stochastic sincethere is an explicitfirst-orderconditionfrom which the price-expectationerror can be estimated. Returningto the copper-inventoryproblem,when sales are treated as stochasticthen z= z,_l y[p(s)]


s. Utilizing Merton's result, the stochastic differential equation can be

written as: dz = E(y-s)dt


(yPpP-l)QS(t)u8(t)dt11 2


When the demand equation is inverted and price is treated as stochastic, = Zt

*y(p) +s(p) and the associatedstochasticdifferentialequation can be written as:

Either equationA1.17 or AI. 18 can be normalizedto yield a version similarto the constaint






+ (yp _SP)aO(,)uP(,)d,112

in A1.5: dz = E(y-s)dt

+ oa(t)u(t)dt112 .


In the next section, estimationresults are presentedbased on monthlydata for US copper








the followingtwo equationsare estimatedjointly:

m(t) a p(t)-p(t-l)-r(t-l)p(t-l), I"(t)



C,(t) -. P 2


+ U,(t)


p(t) = C(t) + u(t) The first equation in A1.20 is a discrete version of the price-arbitrage condition given in Al.11 plus an error term u.

The second equatiou is a combinationof i and ii

from the first-order conditionsgiven in Al.7 plus an error term up. Merton's result is especiallyhandy in this application,since the secondof the two equationsprovidesa formal model for u, the stochasticcomponentof price. By using instruments,consistent(in the econometric sense) estimated values of up can then be used to form a logically selfconsistent estimate of a2. As noted earlier, expressing the constraint in terms of the unanticipatedcomponentof price rather than sales or inventoriesis essentiallya simple rescalingof the constraint. The symbol P. is thereforeused to reflect the re-scaled valueof V..

The term P. is treated as a constantand estimateddirectly as a parameter. This

treatment is equivalent to assuning that the underlying marginal cost function can be reasonablyapproximatedby a quadratic.'

'Recallequation3.15. 74

These results are employed in Section 4 of this paper to test for convexity in the shadowprice for copperand providean estimateof the dispersionpremiumassociatedwith copper inventories. In addition, the model can also provide a test of jointness between production and inventory holding, which forms the ecomonic rationale for a Kaldorconveniencebenefit.


















Annex 3: Tho QuadratI-PIn Modd In order to test whether the results rqorted above are dependenton the functional form chosen, a more general form, quadratic with third-order terms for z and y, was estimated. Specifically,the followingmarginalcost functionswere substitutedfor 4.2:

XiS +S b X,

C,bo+ I

C,. co+ E 1



+b, (A3.1)


Ec £Xx 1


+ 79

where Xi, for i =1,2,...,5 representsinventories(z), production(y), the price of electricity (x), the price of scrap copper(w), and refiningcapacity(f). The estimationprocedure was repeatedon: 5 5



mbo+E biyxl+E E b"XX,+bk(t) +b7 * ClX + 1

+d (t) (A3.2)



P(t) "Co +


Z CUX,X +C.k(t) +C 7 4 dt(t) 110t

using the three estimationtechniquesand the six specificationsof 0;(t).

Paameters from

one of the eighteenestimations,where a six-monthmoving average was used to calculate 6 (t), are given in Table A3.1. The parameters were estimatedusing three-stageleastsquares.


TabbA3.1:Paameterestimdonsfor tbe quadratic-plus model. Parameter CZ O CZ Z CZ Y CZ X CZ W CZF





0.72 -I.OOE-02 *0.03 51.69 -7.05 0.24E-02 4.39E-04

0.16 -1.45 -2.06' 0.33 -2.44' 3.10' -1.92*




1.001-03 O.JOE-05 0.57E-05 0.48 0.02 -0.48E46 4120.33 93.13 40.05 1.12 -0.50E43 0.52E07 0.84-07 0 10



2.50* 0.47 1.44 0.48 2.03* 2.83* -0.20 -0.69 1.96* -3.36* 1.62 -1.68* 0.51 1.77* 3.00*



-0.82 0.93E-02 -0.02 13.09 7.89 -0.34E-03 0.401-05

-0.17 1.36 -1.05 0.08 2.97' -0.40 0.82 40.67 -2.24* *0.43 1.47 0.34 0.54 -0.85 1.72* 0.06 -2.32' 0.86 -0.33 -1.37 -2.12* -0.41 0.31

-L.OOE-05 *0.22

-O.IIE-02 0.12E-05 0.48E-04 0.17 -0.54E402 0.48145 66.20 -98.85 0.01 -0.21 -0.43E-03 -0.16E-06 -0.16E-06 0.01


Note: T-scormsmarkodwith an * ar sinifiant at a 90%+ levelof confidence.The parametersare namedusing the folowing convention:CZ_W is the parameterin the marginalcost of storale function (C.) on w, CY_WFis the parmmeterin the marginal cost of productionfunction (C,) on the product of w andf, CZ ZZZ is the coefficie.ntin C, on z', where z is inventories,y is producdon, x is the price of elctricity, w is the price of scrap copper, f is refining capacity,and k is the strike dummy.V ZZZ is the coefficient on the six-month movingvarince, - 2(t).


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