Internet Appendix for Informational Frictions and Commodity Markets

Transcription

1 Internet ppendix for Informational Friction and Commodity Market MICHEL SOCKIN and WEI XIONG In ti appendix, we preent in detail an extended model wit a future market, in upplement to te ummary of te model extenion in Section IV of te main paper.. Model Setting We introduce a new date t = 0 before te date t = and in te baeline model, and a centralized future market at t = 0 for delivery of te commodity at t = ll agent can take poition in te future market at t = 0, and can cooe to revie or unwind teir poition before delivery at t = Te ability to unwind poition before delivery i an advantage tat make future market trading appealing in practice. We keep all of te agent in te baeline model iland oueold, good producer, and commodity upplier and add a group of nancial trader. Tee trader invet in te commodity by taking a long poition in te future market at t = 0 and ten unwinding ti poition at t = witout taking delivery. To focu on information aggregation troug trading in te future market, we aume tat tere i no pot market trading at t = 0. t t = ; a pot market naturally emerge troug commodity delivery for te future market. Commodity upplier take a ort poition in te future market at t = 0 and ten make delivery at t = Supplier marginal cot of upplying te commodity determine te pot price. Wen a trader cooe to unwind a future poition at t = ; i gain/lo i determined by ti pot price. Citation format Sockin, Micael, and Wei Xiong, Internet ppendix for "Informational Friction and Commodity Market," Journal of Finance. Pleae note Wiley-Blackwell i not reponible for te content or functionality of any upporting information upplied by te autor. ny querie (oter tan miing material) ould be directed to te autor of te article.

2 Table I.I Timeline of te Extended Model t=0 t= t= Future Market Spot Market Good Market Houeold Trade/Conume Good Producer Supplier Oberve Signal Long Future Sort Future Take Delivery Produce Good Oberve Supply Sock Deliver Commodity Fin Trader Long/Sort Future Unwind Poition Table I.I peci e te timeline of te extended model. We keep te ame peci cation for te iland oueold, wo trade and conume bot ome and away good at t = a decribed in Section I. of te main paper. We modify ome of te peci cation for good producer and commodity upplier and decribe our peci cation for nancial trader below.. Good Producer in te main model, we allow good producer to ave te ame production tecnology and receive teir private ignal at t = 0. Eac producer take a long poition in te future market at t = 0 and ten commodity delivery at t =. Te timing of te producer information ow i key to our analyi. t t = 0; producer i information et Ii 0 = f i ; F g include it private ignal i and te traded future price F. t t =, it information et Ii = f i ; F; P X g include te updated pot price P X. We allow te producer to ue it updated information et at t = to revie it future poition for commodity delivery. Tat i, it production deciion i baed on not only it private ignal and te future price but alo te updated pot price. Tu, it i not obviou tat noie in te future market can a ect te producer production deciion and commodity demand. We examine ti key iue wit our extended model. t t = ; te producer optimize it production deciion X i (i.e., commodity demand)

3 baed on it updated information et I i max X i E P i Y i j I i P X X i + (P X F ) ~ X i Te rt two term above repreent te producer expected pro t from good production and te lat term i te gain/lo from it future poition. Te producer optimal production deciion i ten X i = n E X j Ii i. P X o =( ( )) (I.) Wen deciding it future poition at t = 0; te producer face a nuanced iue in tat, becaue it doe not need to commit it later production deciion to te initial future poition, it may engage in dynamic trading. In oter word, it could cooe a future poition to maximize it expected trading pro t at t = 0. Ti trading motive i not eential for our focu on analyzing te aggregation of te producer information but igni cantly complicate derivation of te future market equilibrium. To avoid ti complication, we make a implifying aumption tat te producer are myopic at t = 0 Tat i, at t = 0; eac producer cooe a future poition a if it commit to taking full delivery and uing te good for production, even toug te producer can revie it production deciion baed on te updated information at t =. Wile ti implifying aumption a ect eac producer trading pro t, it i innocuou for our analyi of ow te future price feed back to te producer later production deciion becaue eac producer till make good ue of it information and te future price i informative by aggregating eac producer information. Speci cally, at t = 0 te producer cooe a future poition ~ X i to maximize te following expected production pro t baed on it information et I 0 i max E P i Y i j Ii 0 ~X i F ~ X i ; were it treat ~ X i a it production input at t =. Trougout te ret of ti appendix, we ue a tilde to denote variable and coe cient aociated wit te future market at t = 0; we maintain te ame notation witout te tilde for variable related to te pot market at t = Te producer future poition i ten n ~X i = E X ~ j Ii 0 i. F o =( ( )) (I.). Financial Trader 3

4 We introduce a group of nancial trader, wo trade in te future market at t = 0 and unwind teir poition at t = before delivery. For implicity, we aume tat te aggregate poition of nancial trader and good producer i given by te aggregate poition of producer multiplied by a factor e log + e log + Z ~X i ( i ; F ) d (" i ) ; were te factor e log + repreent te contribution of nancial trader. Ti multiplicative peci cation i ueful for enuring te tractable log-linear equilibrium of our model. We allow te contribution of nancial trader e log + to contain a component log ; were > 0; to capture te poibility tat te trading of nancial trader i partially driven by teir knowledge of te global fundamental log. Te trading of nancial trader alo contain a random component, wic i unobervable by oter market participant. Ti aumption i realitic in two repect. Firt, in practice, te trading of nancial trader i often driven by portfolio diveri cation and rik-control purpoe unrelated to fundamental of commodity market. participant cannot directly oberve oter poition. Second, market Speci cally, we aume tat a a normal ditribution independent of oter ource of uncertainty in te model, wit mean and variance v N ; ; Te preence of nancial trader introduce an additional ource of uncertainty to te future market, a bot good producer and commodity upplier cannot oberve at t = 0 t t = ; nancial trader unwind teir poition, and commodity upplier make delivery only to good producer. From an economic perpective, ti peci cation implie tat te poition of nancial trader tend to expand and contract wit producer future poition, wic i broadly conitent wit te expanion and contraction of te aggregate commodity future poition of portfolio invetor and edge fund in te recent log + commodity price boom-and-but cycle (e.g., Ceng, Kirilenko, and Xiong (0)). lo note tat e can be le tan one. Ti implie tat nancial trader may take a net ort poition at ome point, wic i conitent wit ort poition taken by edge fund in practice. Depite te fact tat large trader need to report teir future poition to te Commoditie Future Trading Commiion (CFTC) on a daily bai, ambiguity in trader clai cation and netting of poition taken by trader wo are involved in di erent line of buine nevertele make te aggregate poition provided by te CFTC weekly Commitment of Trader Report to te public imprecie. See Ceng, Kirilenko, and Xiong (0) for a more detailed dicuion of te trader clai cation and netting problem in te CFTC Large Trader Reporting Sytem and a ummary of poition taken by commodity index trader and edge fund. 4

5 .3 Commodity Supplier Commodity upplier take a ort poition of ~ XS in te future market at t = 0 and ten make delivery of X S unit of te commodity at t = We maintain te ame convex cot function for te upplier ditribution N ; k e =k (X +k S ) +k k ; were te upply ock a a Gauian We aume tat te upplier oberve teir upply ock only at t = ; wic implie tat te upply ock doe not a ect te future price at t = 0 and intead it te pot market at t =. Due to ti timing, te upply ock provide a camou age for te unwinding of nancial trader aggregate future poition at t = Tat i, even after nancial trader unwind teir poition, te commodity pot price doe not reveal teir poition. 3 In ummary, te upplier information et at t = 0 i I 0 S = ff g, and at t = i I S = ff; P X; g t t = ; te upplier face te following optimization problem k max P X X S X S + k e =k X +k k S + (F P X ) X ~ S ; were tey cooe X S te quantity of commodity delivery to maximize te pro t from delivery in te rt two term. Te lat term i te gain/lo from teir initial future poition. Te upplier optimal upply curve i ten given by X S = e PX k ; wic i identical to teir upply curve in te baeline model. t t = 0; like te good producer, te upplier alo face a nuanced iue related to dynamic trading. teir initial future poition doe not necearily equal teir later commodity delivery, tey may alo cooe to maximize te trading pro t from t = 0 to t = To be conitent wit our earlier aumption about te myopic beavior of good producer, we aume tat at t = 0 te upplier believe tat good producer will take full delivery of teir future poition and tat te upplier cooe teir initial ort poition to myopically maximize te pro t from making delivery of e ( log +) ~ XS unit of te commodity to good producer i max E F e ( log +) XS ~ I 0S ~X S E k + k e =k +k e ( log +) k XS ~ 3 Ti timing may appear pecial in our tatic etting wit only one round of future market trading followed by pyical commodity delivery, a tere i no particular reaon to argue weter letting te upplier oberve te upply ock at t = 0 or t = i more natural. However, if we view ti etting a one module of a more realitic etting wit many recurrent period and a upply ock arriving in eac period, ten tere i alway a upply ock itting te market wen nancial trader unwind teir future poition. I0 S 5

6 Since i independent of and log ; it i eay to derive ~X S = e =k n E e ( log +) I 0 S =E e +k k ( log +) I 0 S io k F k ; (I.3) wic i a function of te future price F.4 Joint Equilibrium of Di erent Market We analyze te joint equilibrium of a number of market te good market between eac pair of matced iland at t =, te pot market for te commodity at t =, and te future market at t = 0. Equilibrium require clearing of eac of tee market t t = ; for eac pair of randomly matced iland fi; jg, te oueold of tee iland trade teir produced good and clear te market of eac good C i + Cj = X i ; Ci + C j = X j t t = ; te commodity upply equal te good producer aggregate demand Z t t = 0; te future market clear X ( i ; F; P X ) d (" i ) = X S (P X ; ) e log + Z ~X i ( i ; F ) d (" i ) = ~ X S (F ) B. Te Equilibrium Te good market equilibrium at t = remain identical to tat derived in Propoition for te main model. Te future market equilibrium at t = 0 and te pot market equilibrium at t = alo remain log-linear and can be derived following a imilar procedure a te derivation of Propoition. Te following propoition ummarize te key feature of te equilibrium wit explicit expreion for all coe cient given in Section D. Propoition t t = 0; te future market a a unique log-linear equilibrium te future price i a log-linear function of log and, log F = ~ log + ~ + ~ 0 ; (I.4) 6

7 wit te coe cient ~ > 0 and ~ > 0, wile te long poition taken by good producer i i a log-linear function of it private ignal i and log F, log ~ X i = ~ l i + ~ l F log F + ~ l 0 ; (I.5) wit te coe cient ~ l > 0 t t = ; te pot market alo a a unique log-linear equilibrium te pot price of te commodity i a log-linear function of log ; log F; and, log P X = log + F log F ; (I.6) wit te coe cient > 0, F > 0, and < 0, wile te commodity conumed by producer i i a log-linear function of i ; log F; and log P X, log X i = l i + l F log F + l P log P X + l 0 ; (I.7) wit te coe cient l > 0 and l F > 0; and te ign of l P undetermined. Tere are two round of information aggregation in te equilibrium. During te rt round of trading in te future market at t = 0, good producer take long poition baed on teir private ignal. Te future price log F aggregate producer information, and re ect a linear combination of log and, a given in (I.4). Te future price doe not fully reveal log due to te noie originated from te trading of nancial trader. Te pot price tat emerge from te commodity delivery at t = repreent anoter round of information aggregation by pooling togeter te good producer demand for delivery. a reult of te arrival of te upply ock ; te pot price log P X doe not fully reveal eiter log or, and intead re ect a linear combination of log and, a derived in (I.6). Depite te updated information from te pot price at t =, te informational content of log F i not ubumed by te pot price, and till a an in uence on good producer expectation of log. a reult of ti informational role, equation (I.7) con rm tat eac good producer commodity demand at t = i increaing wit log F; a l F > 0, and equation (I.6) ow tat te pot price i alo increaing wit log F, a F > 0 Ti i te key feedback cannel troug wic future market trading a ect commodity demand and te pot price depite te availability of information from te pot price. Te implifying aumption we make regarding te myopic trading of good producer and commodity upplier at t = 0 are innocuou to te informational role of te future price 7

8 at t = long a good producer trade on teir private ignal, te future price would aggregate te information, wic in turn etablie te future price a a ueful price ignal for te later round at t =. Our implifying aumption ave quantitative conequence for good producer trading pro t and te e ciency of te future price ignal, but ould not critically a ect te qualitative feedback cannel of te future price, wic we caracterize in te next ubection. 4 Interetingly, Propoition alo reveal tat l P can be eiter poitive or negative, due to te o etting cot e ect and informational e ect of te pot price, imilar to our caracterization of te main model. C. Implication C. Feedback on Commodity Demand nancial trader do not take or make any pyical delivery, teir trading in te future market doe not ave direct e ect on commodity upply or demand. However, teir trading a ect te future price, troug wic it can furter impact commodity demand and pot price. By ubtituting equation (I.4) into (I.6), we expre te pot price log P X a a linear combination of primitive variable log, ; and log P X = + F ~ log + F ~ + + F 0 ~ + 0 (I.8) Te term arie troug te future price. F > 0 and ~ > 0; te noie from nancial trader trading in te future market, ; a a poitive e ect on te pot price. Furtermore, by ubtituting te equation above and (I.4) into (I.7), we obtain an individual producer commodity demand a log X i = l i + l F ~ + l P + F ~ log + (l F + l P F ) ~ + l P + (l F + l P F ) ~ 0 + l P 0 + l 0 ; and te producer aggregate demand a Z i log X ( i ; F; P X ) d (" i ) = l + l P + l F ~ + l P F ~ log + (l F + l P F ) ~ + l P + (l F + l P F ) ~ 0 + l P 0 + l 0 + l (I.9) 4 Note tat depite te di erent information content of te future price and te pot price, tere i no arbitrage between te two price becaue te two price are traded at di erent point in time and te pot price i expoed to te upply ock realized later. 8

9 By uing equation (I.8) in te proof of Propoition, te coe cient on in te aggregate commodity demand i l F + l P F = k F > 0 Tu, alo a a poitive e ect on aggregate commodity demand. Te e ect of on commodity demand and te pot price clarify te imple yet important conceptual point tat trader in commodity future market, wo never take or make pyical commodity delivery, can nevertele impact commodity market troug te informational feedback cannel of commodity future price. C. Market Tranparency Information friction in te future market, originating from te unobervability of te poition of di erent participant, are eential in order for te trading of nancial trader to impact te demand for te commodity and pot price. Te following propoition con rm tat a (i.e., te poition of nancial trader become publicly obervable), te pot market equilibrium converge to te perfect-information bencmark. Propoition ; te pot price and aggregate demand converge to te perfectinformation bencmark. Propoition ow tat by improving tranparency of te future market, one can acieve te perfect-information bencmark becaue by making te poition of nancial trader publicly obervable, te noie no longer interfere wit te information aggregation in te future market. a reult, te future price fully reveal te global fundamental, wic allow good producer to acieve te ame e ciency allowed by te perfectinformation bencmark. Ti nice convergence reult relie on te aumption tat te upply noie doe not a ect te future market trading at t = 0 and it te pot market only at t =. Nevertele, ti reult igligt te importance of improving market tranparency. 5 5 Wile our analyi focue on te noie e ect of teir trading, nancial trader can alo contribute to information aggregation. increae, te future poition of nancial trader build more on te global economic fundamental log, in wic cae te future price log F become more informative of log. Ti i becaue one can prove baed on Propoition tat ~ = ~, te ratio of te loading of log F on log and, increae wit. 9

10 Impoing poition limit on peculator in commodity future market a occupied muc of te pot-008 policy debate, wile improving market tranparency a received muc le attention. By igligting te feedback e ect originating from information friction a a key cannel for noie in future market trading to a ect commodity price and demand, our model ugget tat impoing poition limit may not addre te central information friction tat confront participant in commodity market and tu may not be e ective in reducing potential ditortion caued by peculative trading. Intead, increaing te tranparency of trading poition migt be more e ective. D. Tecnical Proof D. Proof of Propoition We follow te ame procedure a in te proof of Propoition in te main paper to derive te future market equilibrium at t = 0. We rt conjecture te log-linear form for te future price and eac iland producer long poition in (I.4) and (I.5) wit te coe cient ~ 0 ; ~ ; ~ ; ~ l 0 ; ~ l ; and ~ l F to be determined by equilibrium condition. Let z be a u cient tatitic of te information contained in F z log F ~ 0 ~ ~ = log + ~ ~ Ten, conditional on oberving i and F; producer i expectation of log i E [log j i ; log F ] = E [log j i ; z] = + + ~ a + i + ~ ~ ~ z = c 0 + c i + c F log F 0 ~ ~ ; (I.0) were c 0 = + + ~ ~ a ~ ~ 0 ~ + ~ ; ~ c = + + ~ ~ ; c F = + + ~ ~ ~ ~ 0

11 Producer i conditional variance of log i ~ ;i = V ar [log j i ; log F ] = + + ~ ~ (I.) By ubtituting equation (I.5) into producer i optimal production deciion in equation (I.), we obtain log ~ X i = ( ) log + ( ) ~ l ~ l ( ) c 0 + c i + c F log F ~ + ( ) + ~ l ~ l F log F ( ( )) ~ ;i + ~ l ( ( )) For te above equation to matc te conjectured equilibrium poition in equation (I.5), te contant term and te coe cient on i and log F ave to be identical ~ l0 = ~ l = ~ lf = ( ) ~ l 0 + ~ l + ( ( )) + ~ l c ; ( ) + ~ l ( ) + + ~ l c 0 + ( ( )) ~ ;i log ; (I.) ( ) ( ) ~ l F ( ) + + ~ l ( ) c F (I.3) (I.4) By ubtituting equation (I.3) into (I.4), we ave ~ l = + ( ) ~ l F ( ) By manipulating equation (I.3), we alo ave tat ~ ~ (I.5) ~ l = + ( ) + ~ ~ ( ) (I.6) We now ue market clearing of te future market to determine tree oter equation for te coe cient. ggregating equation (I.5) give te producer aggregate poition, Z ~l ~X i ( i ; F ) d (" i ) = exp + ~ l F ~ log + ~ l F ~ + ~ l 0 + ~ l F 0 ~ + ~ l (I.7)

12 Equation (I.3) give ~ X S De ne z log F ~ 0 ~ a ~ = ~ ~ (log a) + Te upplier conditional expection of i ten E [ j log F ] = E [ j z ] = + ~ " ~ + ~ # log F 0 ~ ~ ~ a ; ~ ~ wit conditional variance V ar [ j log F ] = + ~ ~ Teir conditional expectation of log i E [log j log F ] = E [log j z ] = + ~ " ~ a + ~ # log F 0 ~ ~ ~ ; ~ wit conditional variance V ar [log j log F ] = + ~ ~ Tu, we obtain an expreion for log X ~ S tat i linear in log and Next, te market-clearing condition log e R log + X ~ i i ( i ; F ) d (" i ) = log X ~ S require tat te coe cient on log and and te contant term be identical on bot ide + ~ l + ~ l F ~ = k ~ + + ~ ~ ~ + ; (I.8) ~ + ~ l F ~ = k ~ + + ~ ~ ~ ~ + ; (I.9) ~ ~ ~ l0 + ~ l F 0 ~ + ~ l = k ~ ~ ~ + ~ ~ + ~ ~ ~ + ~ ~ ~ a + =k (I.0) ( + k) k + ~ ~ ( + k) k + ~ ~ Equation (I.9) directly implie tat ~ lf = k + + ~ ~ ~ ~ ~ (I.) Equation (I.8) and (I.9) togeter imply tat ~ l = ~ ~

13 By combining ti equation wit (I.6), we arrive at ~ l 3 + ~ l + + ( ) + ~ l ( ) = 0 (I.) By furter making te convenient ubtitution L = ~ l + ; called te Tcirnau tran- 3 formation, we obtain te depreed cubic polynomial were L 3 + pl + q = 0; p = + q = 3 ( ) 3 ; 3 ( ) 7 3 ( ) It i eay to verify tat q + p3 > 0 and terefore L 4 7 i a real root of ti depreed cubic polynomial, wic a one real and two complex root. Following Cardano metod, te one real root of equation (I.) i given by r ~ l = 3 q q p q r q 4 + p3 7 Since te coe cient of equation (I.) cange ign only once, by Decarte Rule of Sign te real root mut be poitive. Since ~ l = ~ ~ ; we ave ~ = ~l + ~ ; 3 wic, togeter wit our expreion for ~ l and equation (I.5) and (I.), implie tat ~ = 0 ( )) + ~l + + C + k ( ) ~ l ~l + (I.3) and terefore ~ = 0 ( )) + ~l + + C + k ( ) ~ l ~l + (I.4) 3

14 Since by equation (I.), ~ l > 0; ~ and ~ mut ave te ame ign. Wit ~ and ~ determined, ~ l F i ten given by equation (I.), ~ 0 by equation (I.), ~ 0 = k ~ lf + 0 ~ lf = k + + ~ ~ ( ) ~ l ~ log + =k + ( + k) k + 0 ~ l + ( ) + ~ l + ( ) ~ ~ l + + ~ and ~ l 0 by equation (I.0), ~ l0 = ~ ~ + ~ ~ ~ l ( ) k ~ lf ~0 + =k + + ~ ~ ~ + ~ + ~ ~ ~ + ~ a ~ ~ ( + k) k + ~ ~ + ~ ~ ~ ; + ~ ~ ~ l a ~ l + ~ ~ ~l + ; We now derive te pot market equilibrium at t = We again rt conjecture tat te pot price P X and a good producer updated commodity demand take te log-linear form given in equation (I.6) and (I.7) wit te coe cient 0 ; ; F ; ; l 0 ; l ; l F ; and l P determined by equilibrium condition. to be Te mean and variance of producer i prior belief over log carried from t = 0 i derived in (I.0) and (I.). De ne z p = log P X 0 F log F = log + Ten, after oberving te pot price P X at t = ; te producer expectation of log i E [log j i ; log F; log P X ] = E [log j i ; log F; z p ] = 4 ~ ;i c i + log PX 0 F log F ; ~ ;i +

15 wit conditional variance ;i = V ar [log j i ; log F; log P X ] = ~ ;i + We ue (I.) to compute log X i ; and obtain a linear expreion of i, log F, and P X. By matcing te coe cient of ti expreion wit te conjectured form in (I.7), we obtain l 0 = l = l F = l P = log + ( + l ) ( ) ;i + + ( + l ) ;i ~ ;i c 0 ~ ;i c ( ( )) ;i ~ ;i c ; ( ) l c F ~0 + ~ ; + l ;i 0 + ( + l ) ;i ~ ;i c F F ; (I.5) ( + l ) ;i (I.6) Market clearing of te pot market require R X id (" i ) = X S ; wic implie (k l P ) log P X = l 0 + l + l log + l F log F By matcing coe cient on bot ide, we ave (k l P ) 0 = l 0 + l ; (k l P ) = l ; (I.7) (k l P ) F = l F ; (I.8) (k l P ) = (I.9) From equation (I.7) and (I.9), we ave tat l = and l F above, we alo ee tat ; and given our expreion for l 0 0 = k l P + + l ;i + ( + l ) ;i ~ ;i c 0 c F ~0 + ~ + l log + ( + l ) ( ) ;i + ( ) l ;i + l ; F = + l ;i (k l P ) + ~ ;ic F (I.30) 5

16 From our expreion for l above and l = = ; we ave l 3 + ~ ;i c ~ ;i c ~ ;i l ( ) ( ) = 0 (I.3) Ti i a depreed cubic polynomial woe unique real and poitive root i given by v v u l = t 3 ~ ;i a u ( ) + t ~ ;i a ( ) 7 3 ~ ;i a ~ ;i ( ) v v u + t u 3 ~ ;i a ut ~ ;i a + 3 ( ) 4 ( ) 7 3 ~ ;i a ~ ;i ( ) It follow tat l > 0 and from equation (I.9) tat = ( and, ince l = = > 0; ) l + ( + l ) (~ ;i + l ) l + ( ) k > 0; = + ( + l ) (~ ;i + l ) l + ( ) k < 0 We now prove tat l F > 0 Given te expreion for l F in (I.5) and given l > 0; it i u cient for l F > 0 if ~ ;i c F > F Given te expreion for F in (I.30), and recognizing tat ~ ;i > 0 and c F > 0; te above condition can be rewritten a > + l ;i (k l P ) + Furtermore, from te expreion for ;i and l P ; ti condition can be furter expreed a ( + k ( )) ~ ;i + + l > Since l =, given our expreion for < 0, te condition reduce to + l ~ ;i + l > 0; 6

17 wic i alway ati ed. Terefore, l F > 0 In addition, ince (k k > l P ; we ee from (k l P ) F = l F tat F > 0 We now examine te ign of l P By ubtituting l = into (I.6), we ave l P ) = l implie tat and te expreion of ;i and l P = ( + ( ) k) ~ ;i + l kl ( k) l ~ ;i Conequently, l P can be poitive or negative depending on te ign of kl ( k) l ~ ;i D. Proof of Propoition In (I.8), log P X i a linear expreion of log ; ; and. We need to ow tat a, te coe cient on log and converge to teir correponding value in te perfect-information bencmark (Propoition 3 of te main paper), and te variance of We rewrite equation (I.) a ~l + ~l + V = F ~ 0 + ( ) ~ l = ( ) become u ciently large, te rigt-and ide converge to zero and terefore, ince te cubic polynomial a a unique real olution, ~ l 0 By ubtituting equation (I.) into our expreion for ~ ; one can expre ~ a ( ) ~ = + k ( ) + ( ) ~ l ( ) ~ l ( ) ; ~ l 0; and tu ~ +k( rewrite (I.3) a l 3 + ~ ;i l = ( + l ) + ( ) ~ l ) In addition, by ubtituting for c ; we can ~ ;i ( ) Since (l + ) grow a increae, ~ ;i = + + (l + ) 0 a It ten follow tat l 0. By ubtituting (I.3) and our expreion for c into our expreion for ; we ave = + ( ) k ( ) + ( ) k (~ ;i l ) 7

18 ~ ;i +k( ; ~ ;i l = ( + l ) ( ) l 3 0; and terefore Given ) tat l = and given our expreion for l P ; we ave tat a ; te coe cient of in (I.8) equal l P = ( + l ) ;i l wic i it value in te perfect-information bencmark. + k ( ) ; Since l = ; and given tat a ; l 0 and ; we ave +k( ) 0 By ubtituting for ;i ; c F ; l P ; and ~ = ~ ; we can rewrite F ~ a F ~ = = ( ) + k ( ) (l + ) l ( ) + k ( ) ( + l ) ( ) ~ ;i l3= were we ue ubtitution wit equation (I.3). ; l 0; and ; wic i it value in te perfect- ) te coe cient on log in (I.8) + F ~ +k( information bencmark. By uing te expreion of F ; l P ; l ; c F ; ;i ; ~ l ; and ~ manipulating term, we ave F ~ = ( ) + k ( ) l ~l + Conequently, we can write V a ( ) V = + k ( ) l ~l + We can rewrite equation (I.) a ~l + = ( ) ~ l ( + ) l ; ~ ;i l3= 0, + ( ) By applying te Implicit Function Teorem to equation (I.), in Propoition and ~ = ~l + ~l ~l + ~l + ( ) < 0 Conequently, ~l + i growing in Now we can rewrite equation (I.3) by ubtituting for ~ ;i and c a v u t l ~l ~l + l r = + l 8 ( )

19 ; l 0 Tu, for ti equation to old, we mut ave (l + ). Te left-and ide (LHS) of te above equation can ten be expreed a v u t l ~l + l + + ~l + + l r t ~l + l 3= + o ~l + + l Ti ugget tat l 3= mut be rinking at approximately te ame rate a ~l + i growing for te LHS to remain nite. Terefore, l mut be rinking at a fater rate and V 0 a. In ummary, we ave own tat a, log P X converge wit it counterpart in te perfect-information bencmark. We can imilarly prove tat te producer aggregate demand alo converge. Reference Ceng, Ing-aw, ndrei Kirilenko, and Wei Xiong, 0, Convective rik ow in commodity future market, Review of Finance, fortcoming. 9