Seasonal Harvests and Commodity Prices: Some analytical results. Clare Kelly 1 Centre for Applied Microeconometrics, University of Copenhagen, and

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1 Seasoal Harvess ad Commody Prces: Some aalycal resuls Clare Kelly Cere for Appled Mcroecoomercs, Uversy of Copeage, ad Gauer Lao Scool of Maageme ad Ecoomcs, Quee's Uversy Belfas, ad CRESTENSAI, Rees, Frace Jauary 2004 Absrac: Ts paper caracerses e aalycal soluo o a o perod compeve sorage model of commody prces perodc arves dsrbuos, exedg e aalycal resuls Cambers ad Baley (996). e prese a suffce codo s for e geeral model suc a socks are ever depleed e arves perod. Te paral aalycal soluo for e prce fuco e arves perod s derved uder e assumpo of a lear demad fuco. Ts soluo provdes e codos uder c sockous e oarves perod ca become a absorbg sae. JEL Classfcao: D20 Key ords: Compeve sorage model; perodc arvess, eerogeeous dsrbuos. Correspodg auor: Cere for Appled Mcroecoomercs, Uversy of Copeage, Sudesræde 6, DK455 Copeage K, Demark. Tel: Fax: , Emal: clare.kelly@eco.ku.dk

2 . Iroduco Rece exesos o e compeve sorage model of commody prces of (Deao ad Laroque (DL), (992)) ave corporaed perodc arves dsrbuos (Cambers ad Baley (CB) 996, Osbore, 2003) o allo bo for dffere arves szes across me ad perods c a arves does o occur, bu e commody s raded ad cosumed. I s dffcul o caracerse e resulg equlbrum prce fucos sce aalycal soluos are o avalable ad so e leraure as reled o umercal soluos. Ts oe preses aalycal resuls c caracerse e equlbrum prce fucos a lo levels of sock a o perod model. Te perodcy s represeed as e arves occurrg every secod perod. Frsly, a suffce codo s derved so a sockous ever occur a perod e ere s a arves. Secodly, e paral aalycal soluo for e prce fuco e arves perod s derved e e demad fuco s assumed o be lear. Ts mples a a sockou ca reoccur every oarves perod eve a e maxmum possble arves. Hece, depedg o e parameers of e model, sockous e oarves perod ca become a absorbg sae. Ts propery of e model arses from e commo assumpo of a lear demad fuco. Seco 2 brefly preses e compeve sorage model of Cambers ad Baley (996), upo c e aalyss s based. Seco 3 preses e suffce codo suc a sorage s alays profable e arves perod. Seco 4 preses e aalycal soluo ad oer resuls for e lear demad case. Seco 5 cocludes. 2. Te Compeve Sorage Model Perodc Dsurbaces Follog CB (996) assume a me perods (deoed ) ca be grouped o epocs, dffere me perod ypes (deoed ) eac epoc. Te smples represeao s o assume o me perods eac epoc, oe e e arves occurs (=, e arves perod) c s folloed by a perod o arves, (=, e oarves perod). A equally vald represeao s a large arves folloed by a small arves. Uceray arses from e arves realsao (z) ad from demad socks (v). Tese o elemes cao be separaely defed ad are deoed z + v. Te follog assumpos are made regardg e relave produco eac seaso, cosumer beavour ad e capal marke: () () as compac suppor, expecao deoed ω Q d. as compac suppor, expecao deoed ω Q d. ad cumulave dsrbuo fuco ad cumulave dsrbuo fuco Q, Q, 2

3 () < < ad < <. (v) Q exbs Frs Order Socasc Domace over requre a e suppors of e o dsrbuos do o overlap). Q, (aloug s does o (v) Cosumer demad s represeed by e fuco D( p ) e verse deoed P q D q. (v) Te rage of e demad fuco D( p ) s bouded above, suc a ( 0) defed, ad a e rage of D s P q s, +. (v) Te capal marke eres rae s r, ad e rae of asage caused by sorage s 0< / + <. δ ; e dscoued cos of sorage θ s suc a θ ( δ) ( r) Deoe e equlbrum spo prce as me perod of ype, for =,, as f x, ere p ad e equlbrum prce fuco ay x s curre sock of e commody gve by e arves ad ay veory carred o e perod. Assume eac equlbrum prce fuco s oegave, ocreasg ad couous. Cosumers ad rsk eural speculaors joly deerme demad for a commody, aloug cosumers beave passvely e marke. Speculaors form a (raoal) expecao of prce e follog perod of ype j, for j =,. If s expeced prce s lo, e all sock s sold o cosumers, ad speculaors ave zero demad for veory. A sockou occurs e perod, ad e spo prce s gve by e verse demad fuco, p f ( x ) P( x ) = =. If e expeced prce s g.e. greaer a e prce from sellg all curre sock o cosumers, e speculaors ( ) demad e commody for sorage. I s case, a amou ( ) e (posve) veory level s e x D f ( x ) ( ) j s x ( δ ) x D f ( x ) + + D f x s sold o cosumers, ad e avalable sock e ex perod +. Te level of veory s cose o equae e spo prce ad e dscoued expecao of prce e ex me perod gve by ( ) θ ( δ) j j j p = f x = f j + + x D f x Q d. CB prove e exsece of uque saoary prce fucos f, c are couous, oegave, ad ocreasg a sasfy ( ) { j } θ ( δ) max j j p, = f x = f + x D f x Q d P x. () 3

4 Tus, equlbrum, e spo prce s eer e expeced fuure prce or e prce gve by e demad curve, cever s greaer 2. Le j p deoe e dscoued expeced prce f o veores are carred over from perod, c s gve by θ p = θef f Q d. (2) j j j j j Follog DL (992), j p s e curre prce a c, o veory demad, a u eld o e ex perod ould make zero expeced prof, ad s e prce a c e soluo sces beee e o possble regmes gve (). Te equlbrum prce fuco s above e demad fuco e e prce from sellg all curre sock s less a s crcal prce.e. j = > e P( x ) p p f x P x <. I s case, carryg veores s profable, p j < p, e equlbrum prce fuco s j gve by e frs erm (), ad p = f ( x ). Aleravely, p f ( x ) P( x ) + + = =, e j j > p, c case o veores are eld, ere s a sockou ad p f ( ) P x =. + + I summary, as e amou of veory carred beee e perods vares, e equlbrum prce fuco f x sces beee e dscoued expeced prce (posve veores) ad e demad fuco (zero veores), a e crcal prce j p. Ts crcal prce also defes e maxmum value of e sock for c e equlbrum prce fuco s e demad fuco. For values of e sock belo x, ere j x P ( p ), e perod prce fuco s gve by e demad curve. Fgure porrays a possble soluo o e model (smlar o Fgure 2 CB). Above e crcal prces p ad p, e fucos f x ad f x, are e demad fuco, respecvely. Belo ese prces e fucos le above e demad fuco. I e example so, e fucos do o ersec ad f ( x) f ( x). CB (996) caracerse e relaosp beee e crcal prces eac perod ype. By sregeg e assumpo of FOSD o oe of ooverlappg suppors of e dsrbuos (.e. < < < < < ), ey so a p > p.e. e mmum prce a c s o profable o carry veory o e oarves perod ( p ) s greaer a e mmum prce a c s o profable o carry veory o e arves perod, ( Hoever ey asser a eve s resul s p ). 2 I e ermology of CB: arves correspods o odd ad oarves correspods o eve. Te resuls of CB apply for ay umber of perod ypes eac epoc. 4

5 Fgure Possble soluo o o perod model perodc dsurbaces p p p f f f ( x) ( x ) ( x ) P ( x) x x x suffce o defy e relaosp beee e prce fucos emselves, ad a ay furer aalycal resuls ould requre addoal assumpos o e dsurbaces ad/or e demad fuco. Te remader of s oe addresses s ssue. 3. A Suffce Codo for Posve Sorage e Harves Perod. 3 I s seco, Proposo preses a suffce codo for e arves perod prce fuco o le above e demad curve for all values of e sock. Te, sockous ever occur e arves perod ad erefore veory s alays carred o e oarves perod. 4 Proposo. P < θ E P Proof. See Appedx. s a suffce codo for f ( x) P( x) >, over all x, +. Ts codo requres a e prce gve by e verse demad curve a e ors arves mus be less a e dscoued expecao of e verse demad curve over e oarves perod dsrbuo. If sasfed, f ( x) P( x) = s ever a opmal soluo e arves 3 Te saoary of e fucos allos us o om e me subscrp e remader of e paper. 4 Osbore (2003) sos a a sockou s mpossble a perod pror o oe ere ere s o arves or demad uceray, c rvally mples a f ( x ) s ever e demad fuco. Tus s aalyss s mos applcable e e dsrbuo of e arvess s eerogeeous across perods.e. small ad large arvess alerae. Te e codos uder c a sockou occurs e arves perod s orval. 5

6 perod ad veory, x D f ( x) e dscoued expecao of ex perod prce = θ + ( δ), s alays posve. Te, e prce fuco s gve by ( ) f x f x D f x Q d. (3) Ts resul allos us o order e prce fucos over a resrced rage of e sock. For values of e sock x, ere e oarves perod s e demad fuco, f ( x) P( x) x [, x ] (recall ( x P p )), e prce fuco =. Hece, f e codo Proposo s sasfed, for ay gve value of x a rage, f ( x) f ( x) > ; e arves perod fuco s alays greaer a e oarves perod fuco. I Fgure, f x f x > P x s a possble arves perod prce fuco f Proposo s rue.e. for all x, ad ece f ( x) > f ( x) e f ( x) P( x) =. Hoever, s orderg of e prce fucos levels of e sock belo x, does o mply a spo prces are ger eer e arves or oarves perod, sce e level of sock ll o be cosa over me. Neer does mply a s orderg ll be maaed a values of e sock greaer a x, ere s possble a e fucos ll cross. 5 Te caracersao of e equlbrum prce fucos mpled by Proposo, exeds e resuls CB. Hoever er assumpo of ooverlappg suppors of e dsrbuos mples P( ) P( ) <, c s clearly more srge a P < E P. Noe oever a sce θ <, for a suffcely small value of θ e codo Proposo, P( ) < θ E P( ), s more demadg a P( ) P( ) <. Coversely, for θ close o s less demadg. Terefore, aloug s o possble o compare e resrcveess of e codos geeral, ere are values of θ for c Proposo s sasfed uder eaker codos a e assumpo CB. I e case ere a arves occurs every perod, DL prove Teorem 2 a e lm dsrbuo of veores as a compac suppor ad prce follos a reeal process e e codo P( ) θ EP( ) < s sasfed. Te proof requres a veores are posve some me perods bu become zero fe me.e. a sockou occurs probably equal o oe ad e veory does o become fe. I e perodc case, e aalogous codo for suc a proof s P( ) θ E P( ) <. Ts esures a a e maxmum arves, veores are ozero beee arves ad oarves perods, bu are 5 Numercal aalyss (o so ere) reveals a eve f Proposo s rue, e prce fucos ll mee for some values of e parameers a sock levels greaer a x. 6

7 depleed probably equal o oe e oarves perod. Gve P( ) P( ) <, because P (). s a decreasg fuco, e codo s sasfed rvally e e codo Proposo olds. Tus e soluo o e eerogeeous arves case sares mpora properes of e smpler operodc case. 4. Lear Demad Case Te resul Proposo provdes a caracersao of e prce fucos e geeral case. Hoever, as suggesed CB, s ecessary o make addoal assumpos abou fucoal form o ga furer koledge of e prce fucos. I s seco e paral aalycal soluo o () for e arves perod s preseed, based o e assumpo a e demad fuco s lear s argume. Proposo 2. Assume a e verse demad fuco s gve by P( x) = ax+ b, ere 0 a <, b > 0, ad < b / a. From (), e oarves perod equlbrum prce fuco s e demad fuco, for all () x x [, ],.e. p f ( x) P( x) = =. Cosder a level of sock e arves perod ( ) > 0, ad ( δ ) x D f x ( ) x [ x, x ] suc a + x D f x x, for all. Te e equlbrum prce fuco e arves perod s lear ad s gve by e expecao of e demad fuco over e oarves perod dsrbuo, dscoued by θ,.e. ( ) = θ + ( δ) f x P x D f x Q d, c ca be re as ere () f x = α x+ β, (4) ( δ) ( δ) θ θ θ + θ δ α = a> a; β = aω + b; + θ + θ δ + θ δ Te maxmum level of e sock e arves perod, x, suc a a sockou alays ( ) occurs e oarves perod.e. ( δ ) defed as () + x D f x x, for all, s ( ) ( ) x = θ + p P b θp ω a δ Te maxmum arves leads o a sockou e oarves perod (.e.. (5) < x ), f 7

8 θ ( ω ) > θ + ( ) P P p P δ Proof. See Appedx. (6) Te aalycal soluo for e arves perod prce fuco gve (4) s defed over levels of e arves perod sock suc a a sockou does o occur e arves perod. All of e veory carred o e oarves perod s depleed (a sockou alays occurs e oarves), rrespecve of e realsao of e demad sock. Te maxmum level of e arves perod sock for c (4) apples, x, s defed (5), ad s deermed by e parameers of e demad fuco, e asage rae δ, ad e dsrbuo of oarves perod dsurbaces. Te erm p P( ) s e dfferece beee e prce a c a u of veory ould make zero expeced prof ad e maxmum possble prce e oarves perod. Te secod erm, b θp( ω ), s e dfferece beee e ercep of e lear verse demad fuco ad e verse demad curve a e mea of e oarves perod dsurbaces, dscoued by e sorage cos 6. Furermore, f e codo (6) s sasfed, e oce a sockou occurs e oarves perod, ad eve f e maxmum arves occurs e ex arves perod, ere ll be a sockou e oarves perod. Tus sockous become permae e oarves perod. I s case, prce follos a reeal process eac arves cycle because socks are depleed every oarves perod. I addo sce x s a creasg fuco of δ, a ger asage rae creases e rage of arves over c s absorbg sae caracersc domaes. To be more precse, eever.e f < x, ad for e loes values of e arves perod sock,,, e soluo s e demad curve, f ( x) P( x) x x =. For x x, x, e x s e dscoued expecao of e demad curve gve (4), ad ere s a sockou e oarves perod. Fally, e x> x, e arves perod soluo s e dscoued expecao of e oarves perod fuco gve by (3), e fuco e oarves perod s o e demad curve, ad e do o ko o o caracerse e aalycal soluo for f x. Fgure 2a preses e umercal soluo o e model ere e parameer values are cose o so ese ree regmes of e equlbrum prce fuco (δ 6 I e lear case P( ) Q ( d ) = P( Q ( d ν )) = P Proposo 3., ad s smplfcao s used 8

9 a =.5, 5 Fgure 2a Numercal soluo lear demad fuco b =, ~ U [ 0.4,0.85], ~ [ 0.8,.25] U, δ = 5%, r = 3%, θ = p p x x x STOCK a =.5, 5 Fgure 2b b =, ~ U [ 0.4,0.85], ~ [ 0.8,.25] U, δ = %, r = 3%, θ = s exaggeraed ad se equal o 5%). Noe a STOCK Q exbs FOSD over Q, e suppors of e dsrbuos overlap ad ω > 0. For s combao of e parameers, x = 2.23, c s greaer a =.25. Tus (6) s sasfed mplyg a sockous are a absorbg sae. Noe also a e prce fucos cross s case. I Proposo 3 e specalse Proposo o e lear demad case. 9

10 Proposo 3. If e ( θ ) > θω, (7) a b >, for all x, +. f x P x Proof. See Appedx Te codo e akes a smpler form, (7). If olds e > x ad e soluo gve (4) apples for e rage of arves perod sock x, x, x. Furermore, sorage s alays posve a e ed of e arves perod ad e arves perod fuco les srcly above e oarves perod fuco e equals e demad fuco. Te codo requres a e mmum of e arves dsrbuo, s greaer a a eged average of e mea of e oarves perod dsrbuo, by e demad curve s zero, b/ a. ω, ad e sock value a c e prce gve Ts loer boud s creasg e mea of e oarves dsrbuo ω, e slope of e verse demad fuco a, ad e asage rae δ. Tus a ger ω or δ, or a more elasc demad fuco all mply a (7) s arder o sasfy ad sockous are more probable e arves perod, ce. par. Te effec of e elascy of demad s cosse e umercal aalyss DL. Te codo Proposo 3 s less resrcve a e ooverlappg suppor assumpo of CB, ( > ), for values of θ suc a θ e umercal example s 0.92, mplyg a (7) s eaker a + b/ a θ < θ <. Te value of ω + b / a > for asage raes up o 5% e e eres rae s 3%. Te umercal soluo o e model s preseed Fgure 2b usg e prevous parameer values bu for a loer asage rae, (δ = %), c esures Proposo 3 s sasfed. Hece e arves perod prce fuco s above e demad curve for all values of e sock, sog a eve a e loes possble arves, speculaors alays demad e commody for veory. e loer asage rae, (6) s o sasfed ad sockous e oarves perod are o a absorbg sae, ( =.25 ad x = 0.96 ). 5. Cocluso 0

11 Ts oe as caracersed e equlbrum prce fucos a o perod compeve sorage model of commody prces, e arvess ave perodc dsrbuos. e prese a suffce codo for sockous o o occur e arves perod. Assumg a lear demad fuco allos us o derve e aalycal soluo for e arves perod prce fuco over a erval of e rage of e sock. I e lear demad case, e caracerse e codo ((6) above) suc a e sock reacg e arves perod s permaely zero. Te codos uderlyg e caracersao of e equlbrum prce fucos above are o drecly comparable o e assumpo made CB of ooverlappg suppor of e perodc dsrbuos. Hoever, f e asage ad eres raes are lo eoug, our resuls old uder eaker codos. 6. Refereces. Cambers M.J. ad Baley R.E. (996). A Teory of Commody Prce Flucuaos, Joural of Polcal Ecoomy, 04, Deao, A. ad Laroque G., (992). O e Beavour of Commody Prces, Reve of Ecoomc Sudes, 59, 23. Osbore, T., (2003) Marke Nes Commody Prce Teory: Applcao o e Eopa Gra Marke Reve of Ecoomc Sudes, forcomg. 7. Appedx Proof of Proposo All sock s cosumed e arves perod ad o veory s carred o e oarves perod e p p, c case = =, p f ( ) p f x P x =, + + from (), ere p = θ f ( ) Q ( d). Ts mples p P( x).e. e dscoued expeced prce e oarves perod s less a e prce aaable from sellg all curre sock. Coversely, sock s alays carred o e oarves perod ad P( x ) s ever e soluo e arves perod f max ( P( x) ) p < (A).e. e ges possble prce e curre perod s less a e dscoued expeced prce e oarves perod. By defo P( x) f ( x) θ e oarves perod, erefore P ( ) Q ( d) θ f ( ) Q d p. Sce arg max P ( ) codo for (A) s, + =, a suffce

12 < E P θ P Q d θ P. (A2) If (A2) s sasfed, oldg socks ul e oarves perod s alays profable. Proof of Proposo 2 () Recall (): e soluo e arves perod s gve by { ( ) } = θ + ( δ) max f x f x D f x Q d, P x If f ( x) = P( x), e e aalycal resul s drecly avalable. If f ( x) P( x) ( ) = θ + ( δ) f x f x D f x Q d. For all values of e sock x e oarves perod, suc a =, ( δ ) f x P x ( ) + x D f x x ad = θ + ( δ) ( ( )) >, e, x x [, ] f P x D f x Q d (A3) Te for ay dra of e arves e oarves perod, + ( δ )( ( )) = δ P x D f x, e soluo s gve as f x b a + x + b a. Smplfyg ad egrag over e oarves perod dsrbuo gves P ( δ )( x D ( f ( x ))) + Q ( d ) = a Q d + b+ ( δ )( ax f ( x) + b) Deoe Deog. (A4) ω =. Subsug (A4) o (A3) ad solvg for f Q d θ ( ) θ( δ) ω f x = a + b + ax+ b. + θ δ + θ δ ( δ) ( δ) θ θ θ + θ δ α = a, ad β = aω + b, + θ + θ δ + θ δ x gves gves Proposo 2(). f x = α x+ β. (A5) θ δ Clearly + θ δ < gvg α > a, because a < 0., e, () Le x deoe e maxmum value of e sock e arves perod, suc a = + β.e. f x α x 2

13 ( ) ( δ ) + x D f x x. (A6) Te for all,, ( + ( δ) ( ( ))) = + ( δ) ( ( )). f x D f x P x D f x Assume (A6) olds exacly a e maxmum arves e oarves perod ( ) ( δ ) + x D f x = x.,.e. Subsug e verse demad fuco, e equlbrum prce e arves perod from (4), usg x P ( p ) =, ad smplfyg gves p b a + ( δ) ( ax (( αx + β) b) ) = a, a ad solvg for x gves Nog ( a )( ) ( β ) ( a ) p a + b b x =. (A7) α δ α a a α = + θ ; b θ aθω b β =, + θ δ s smplfes furer o ( δ) ( δ) ( ) ( θ( ω )) + θ x = p a + b b a + b. (A8) a δ Nog P( x) = ax+ b, gves e resul Proposo 4. () To so < x, subsue (A8) ad P( x) = ax+ b, c gves ( ) a + b > θ + p P + θp ω δ ad some smple mapulao s becomes, P( ) θp( ω ) > θ + ( p P( )) δ. Proof of Proposo 3 by subsuo. 3