Internet Appendix for: Waves in Ship Prices and Investment

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1 Inene Appendx o: Waves n Shp Pces and Invesmen Robn Geenwood and Samuel G. Hanson Havad Busness School Augus 14 A: Addonal Empcal Analyss 1. Pesen value calculaon Shp eanngs ae pessen a sho hozons, bu he coelaon beween eanngs n yea and yea +k dsappeas o k geae han o equal o yeas. To consuc a smple esmae o he pesen value o eanngs, we assume ha he shp owne enes a 1-yea me chae. We oecas he goss eanngs om a me chae 1 yea lae by egessng uue goss eanngs on cuen g g goss eanngs:. 1 ab u 1 Sang n yeas, we assume ha goss eanngs equal he sample aveage o $5.4 mllon. Ae he shp s 15 h yea, we assume he owne eceves 85% o he sample aveage eanngs o he emande o he shp s le. Fnally, when he shp eaches 5 yeas o age, we assume s scapped and he owne eceves scap value. 1 Ths yelds he ollowng pesen value calculaon: PV g ˆ g g 8 g 1 ˆ 1 (11/ (1 ) ) (1 1/ (1 ) ) Scap.85, 1 1 (1 ) (1 ) (1 ) (1 ) (A1) whee denoes he dscoun ae.. Alenae measues o shp euns a. Compason o ou eun sees wh Sopod (9) Fgue A1 compaes an annual veson o ou eun sees (sampled n Decembe) wh he annual eun on shppng nvesmens sees compued by Sopod (9): he coelaon beween he wo s.9. 1 In ecen yeas, Clakson povdes an esmae o he scap value. Scap value s hghly coelaed wh nomnal seel pces, and hus when we ae mssng daa on scap value we esmae om a poecon o scap value on seel pces. A-1

2 8% 6% 4% % % -% -4% -6% -8% -1% -1% -14% Fgue A1: Compason o ou Reun Sees wh Sopod (9) Sopod Geenwood-Hanson b. Eanngs and pces on Handymax and Capesze shps Table A1 shows ha we oban smla esuls usng esmaes o eanngs, pces, and euns on 3, DWT Handymax shps, 15, DWT Capesze shps, o he ene bulke lee. Speccally, Table A1 shows ha he key vaables used n he pape (Panaxmax eanngs, Panamax pces, and lee-wde nvesmen) also oecas he euns on Handymax shps, Capesze shps, and he ene bulke lee. We also show ha eanngs and pce vaables based on Handymax shps, Capesze shps, and he oveall lee each oecas shppng euns. Fo nsance, boh Panamax pces (P Pana ) and Capesze pces (P Cape ) oecas Capesze euns. Ths s naual gven he homogeney o shppng capal: he aveage pawse coelaon beween he ou eanngs sees s.84, ha beween he ou pces sees s.96, and ha beween he ou eun sees s.87. Oveall, Table A1 suggess ha he eun pedcably we nd on Panamax shps s epesenave o he oveall bulk cae lee. Howeve, euns on lage Capesze shps ae somewha moe pedcable ha he es o he lee, whle hose on smalle Handmay shps ae somewha less pedcable. 3. Tme-sees obusness a. Sub-sample esuls Table A below shows subsample esuls o ou eun oecasng egessons usng X = P, and Inv. We show ull sample esuls, esuls doppng he 6-1 supe-cycle, and esuls dvdng he sample no wo equal halves. A-

3 Table A1: Alenae Measue o Bulk Cae Pces, Eanngs, and Reuns Ou consucon o 3, DWT Handymax and 15, DWT Capesze eanngs and pces ollows ou pocedue o compung he analogous Panaxmax sees. Howeve, we assume ha he daly opeang coss o Handymax shps ae $5, n consan 11 dollas and hose o Capesze shps ae $8,. To oban an esmae o lee-wde eanngs and pces, we aggegae he Handymax, Panamax, and Capesze sees usng daa on he composon o he lee shown n Fgue 1. We use eanngs daa o 3, DWT Handymax shps and pce daa o 3, Handymax shps. We use eanngs daa o 17,5 DWT Capesze shps and pce daa o 15, Capesze shps. Panel A: Foecasng Panamax euns usng Panamax eanngs and pces x + on Panamax X= Pana X=P Pana X=Inv B [] [-3.7] [-3.4] [-.4] T R Panel B: Foecasng alenae euns usng Panamax eanngs and pces x + on Handymax x + on Capesze x + on Flee X= Pana X=P Pana X=Inv X= Pana X=P Pana X=Inv X= Pana X=P Pana X=Inv B [] [-.9] [-1.75] [-.3] [-4.87] [-4.88] [-3.9] [-3.6] [-.7] [-.49] T R Panel C: Foecasng alenae euns usng alenae eanngs and pces x + on Handymax x + on Capesze x + on Flee X= Hand X=P Hand X=Inv X= Cape X=P Cape X=Inv X= Flee X=P Flee X=Inv B [] [-1.1] [-.1] [-.3] [-1.6] [-5.37] [-3.9] [-.36] [-.84] [-.49] T R Doppng he supe-cycle has only a modeae mpac on he esuls. (Ths has he lages mpac on he eanngs egessons whee he sascal sgncance s somewha weake even hough he coecen magnudes ae slghly lage). Tunng o he s- vesus second-hal spls, we see ha he esuls ae ypcally sgncan n boh he ealy and he lae sub-sample and ha he coecens ae aly smla n magnude. The only excepon hee s ne nvesmen whch was a weake sgnal o uue euns n he lae hal o he sample. To exploe he ssue o sub-sample sably uhe, we epo he -sascs o he null hypohess ha he coecens ae he same n he s and second hal o he sample, labeled as A-3

4 sable below. These ess sugges ha he magnude o eun pedcably has declned modesly snce he s hal o ou sample. Why ae he esuls o pces and eanngs a b songe n he s hal han n he second hal? Pces and eanngs wee aleady hgh compaed o he long-un aveage n 5 6, bu hen he ndusy eceved a second posve demand shock. Ths suggess ha even hough expeced euns may have been low n 6, ealzed euns n ha yea wee hgh. O couse, ealzed euns ulmaely moe han caugh up n 8 and ealy 9. In hs sense, he egesson was ulmaely coec, bu slghly o n he pecse mng o he collapse o he bubble. Relaedly, we suspec ha Inv may be weake pedco n he lae hal o ou sample, n pa, because he congeson-nduced ncease n me-o-buld delays dung he supe-cycle (Kaloupsd (14)) made a nose poxy o ms cuen nvesmen plans. Table A: Sub-sample Foecasng Resuls (1) () (3) (4) (5) (6) (7) (8) (9) (1) (11) (1) Full sample (1976-1) Dop 6-1 Fs Hal Second Hal P Inv P Inv P Inv P Inv Foecasng 1-monh euns b [] [-.37] [-.65] [-3.3] [-.56] [-.] [-4.4] [-.93] [-1.93] [-5.65] [-.6] [-1.91] [-.68] [ sable] [-.1] [.7] [3.51] T R Foecasng 4-monh euns b [] [-3.7] [-3.4] [-.4] [-1.91] [-.7] [-.9] [-3.1] [-3.46] [-.79] [-.83] [-.39] [-.33] [ sable] [1.9] [.11] [1.66] T R Nex Fgue A shows he dsbuon o eun oecasng coecens usng X = P, and Inv acoss all -yea peods n ou 35 yea sample. Speccally, we show he coecens o each -yea peod endng om 1995 o 1. The key oecasng coecens ae negave n almos all o hese -yea subsamples and ae oen sgncanly negave even n hese sho samples. Oveall, hese subsample esuls sugges ha ou ndngs ae no solely due o a handul o oulyng daa pons. Insead, eun pedcably appeas o be a aly obus eaue o he daa. Ths -sasc s obaned by esmaed a ull sample egesson wh a consan, a dummy o he second hal o he sample, he covaae o nees, and he covaae neaced wh he second hal dummy. We epo he -sasc on hs nal neacon em. A-4

5 Fgue A: Foecasng Coecens o -yea Samples Endng om Foecasng 1-monh euns X = Foecasng 4-monh euns X = X = P X = P X = Inv X = Inv A-5

6 b. Ovelappng euns and sandad eos Because we use monhly daa and oecas 1-, 4-, and 36-monh cumulave euns, s mpoan o accoun o he ovelappng naue o he eun obsevaons. The egessons ha oecas 1-monh euns use a Newey-Wes wndow lengh o 18 monhs; he 4-monh eun egessons use a wndow o 36 monhs; he 36-monh eun egessons use a wndow o 54 monhs. Ths s n keepng wh he ule o humb ha he wndow lengh should be 1.5 he oecasng hozon when wokng wh ovelappng monhly euns because he Newey-Wes esmao downweghs hghe ode auocovaances (e.g., Cochane and Pazzes (4)). Table A3: Alenae Sandad Eos and Annual Samplng X = Real Eanngs X = Used Shp Pce P X = Ne Invesmen k = 1-y -y 3-y 1-y -y 3-y 1-y -y 3-y Monhly daa [Baselne NW -sa] [-.37] [-3.7] [-.6] [-.65] [-3.4] [-.79] [-3.3] [-.4] [-1.9] [Moe NW lags] [-.7] [-3.38] [-.63] [-3.7] [-3.15] [-.88] [-.8] [-.35] [-1.11] Annual Samplng June [NW -sa] [-.3] [-.74] [-.7] [-1.66] [-.56] [-.6] [-.75] [-.1] [-.7] [HH -sa] [-.3] [-.4] [-.54] [-1.66] [-.9] [-.37] [-.75] [-1.97] [-.69] Annual Samplng Decembe [NW -sa] [-.7] [-.99] [-1.85] [-.68] [-3.] [-.33] [-.78] [-1.94] [-1.7] [HH -sa] [-.7] [-.67] [-1.77] [-.68] [-.69] [-.4] [-.78] [-1.84] [-.99] N (monhly) N (annual) Table A3 povdes como ha ou neences hee ae obus. The s wo ows show he esuls om he pape. The second se o ows shows wha happens we ncease he numbe o Newey-Wes lags n ou monhly oecasng egessons: hee we use 3 lags o he 1-monh egessons, 48 lags o he 4-monh egessons, and 66 lags o he 36-monh egessons. Addng moe lags geneally ases he -sascs and suggess ha we ae beng consevave. Ths s because addng moe lags han s necessay when compung Newey-Wes sandad eos oen leads o lage -sascs. 3 3 I he esduals ae seally coelaed, he pobably o makng a Type 1 eo s alls as one nceases he numbe o lags, beoe evenually sng as one adds uhe lags. Addng moe lags educes he bas o he vaance esmao, bu makes he esmao moe volale. Mnmzng he pobably o a Type I eo oughly coesponds wh mnmzng he mean squae eo o ou vaance esmao.e., o balancng hese wo oces. See, o nsance, Kee and Vogelsang (5) o Newey and Wes (1994). A-6

7 The emanng ows show he esuls we sample he daa annually. We show annual samplng n boh June and Decembe. Hee we use obus sandad eos when oecasng 1-yea euns (hee s no poblem o ovelappng euns hee), a Newey-Wes wndow lengh o -yeas when oecasng -yea euns, and a Newey-Wes wndow lengh o 3-yeas when oecasng 3- yea euns. We also show Hansen-Hodck sandad eos ha accoun o he 1 yea o ovelap when oecasng -yea euns ( yeas o ovelap when oecasng 3-yea euns). The oveall ake-away s ha boh ou esmaes and epoed -sascs appea o be obus. c. Sambaugh bas As shown by Sambaugh (1999), we esmae usng a me sees o T obsevaons whee, hen he expeced ne-sample bas s ˆ 13 E E ˆ O T (A4) uv uv [ ] [ ] (1/ ) v v T Thus, / > as one would expec when oecasng euns wh eanngs, pces, o nvesmen Sambaugh s eec would lead o a downwad bas n ne samples. We use Amhud and Huvch s (4) bas-adused esmao o see whehe ou esuls ae beng dven by Sambaugh bas. Table A4 shows aw (uncoeced) and bas-coeced esmaes usng annual daa sampled n Decembe. We pesen aw OLS esmaes along wh bas-coeced esmaed. Noe ha Bas[] = Bas[] /. Table A4: The Role o Sambaugh Bas X = Real Eanngs X = Used Shp Pce P X = Ne Invesmen 1-y -y 1-y -y 1-y -y Coeced Coeced [] [-1.76] [-.3] [-.1] [-.58] [-.] [-1.99] Raw Raw [] [-.7] [-.99] [-.68] [-3.] [-.78] [-1.94] Bas[] Bas[] uv / v Obsevaons Usng hs bas-adused esmao nsead o he sandad OLS esmao has almos no mpac on he magnude o sgncance o ou esmaed coecens. Sambaugh s mnmal A-7

8 because ou oecasng vaables ae ehe no vey pessen (n whch case Bas[] s small), o when hey ae moe pessen, nnovaons o ou oecasng vaables ae no sucenly coelaed wh euns (n whch case / s small). In ac, mos o he egessos have a hgh degee o mean eveson, especally when compaed wh he vaables eseaches have use o oecas aggegae sock euns. d. Conollng o a me ends Fnally, n Table A5 we nclude a me end as a conol n he egessons, esmang speccaons o he om x a bx c u. (A5) k k Thee s no good heoecal uscaon o ncludng a me end, bu we do o check ha ou esuls ae no dven by some omed slow-movng vaable. Includng a end has lle mpac on he esuls excep o he wo ode book vaables (NeConacng and Odes) whee he oecasng esuls ae songe when we conol o he secula gowh o he ode book ove me. A-8

9 Table A5: Conollng o a Tme Tend Table A5 epos me-sees oecasng egessons o he om xk a b X c u, k whee x +k denoes he k-yea log holdng peod excess eun on a Panamax dy bulk shp. In Panel A, X alenaely denoes eal eanngs, he cuen eal pce o a 5-yea used shp P, o he eanngs yeld /P. In Panel B, X alenaely denoes ne conacng acvy ove he pas 1 monhs, he sze o he ode book, delvees ove he ollowng 1 monhs, o demolons ove he pas 1 monhs, each scaled by he cuen lee sze. In he ghmos se o columns, we oecas euns usng ne nvesmen, Inv = Delvees +1 Demolons., -sascs ae based on Newey-Wes (1987) sandad eos allowng o seal coelaon a up o k monhly lags.e., we allow o seal coelaon a up 18, 36, and 54 monh lags, especvely, when oecasng 1-, -, and 3-yea euns. Panel A: Foecasng Shp Reuns Usng Shp Eanngs and Pces X = Real Eanngs X = Used Shp Pce P X = Eanngs Yeld /P k 1-y -y 3-y 1-y -y 3-y 1-y -y 3-y b [] [-.3] [-3.36] [-.8] [-.93] [-3.69] [-3.55] [-.8] [-1.3] [-1.34] c [] [.3] [.4] [.46] [.78] [.83] [.81] [.15] [.8] [.8] T R Panel B: Foecasng Shp Reuns Usng Indusy Invesmen Invesmen Measued Based on Ode Book (1996-1) Invesmen Measued Based on Changes n Flee Sze (1976-1) X = NeConacng X = Odes X = Delvees +1 X = Demolons X = Inv k 1-y -y 3-y 1-y -y 3-y 1-y -y 3-y 1-y -y 3-y 1-y -y 3-y b [] [-.56] [-.43] [-3.81] [-.9] [-4.85] [-4.91] [-.86] [-.39] [-.55] [1.78] [1.57] [1.68] [-3.35] [-.3] [-1.9] c [] [1.4] [1.5] [.64] [.77] [5.41] [4.] [.75] [.38] [.16] [.] [.1] [.1] [.7] [.35] [.13] T R A-9

10 B: Model Soluon and Omed Poos 1. The opmzaon poblem o he epesenave m We begn by devng equaon (13) n he man ex. Each m chooses cuen ne nvesmen o maxmze he expeced ne pesen value o eanngs. Each m s Bellman equaon s E J q, A 1, Q 1 A, Q Jq, A, Qmax V( q,, A, Q) 1 E V( q,, A, Q) A, Q (1 ) whee > s he consan dscoun ae o equed eun used by ms. The s ode condon, (B1) o m ne nvesmen s 1 J q, A 1, Q 1 P k E A, Q 1 q 1 The Envelope Theoem mples ha 4,, 1 J q, A 1, Q 1 J q A Q E A, Q. q 1 q 1 (B) (B3) Assumng he sandad no bubbles condon holds, 5 we can eae (B3) owad o oban 1 J q, A 1, Q 1 E A, Q E A, Q. (B4) 1 q 1 1 (1 ) Ths shows ha m ne nvesmen s gven by he amla q-heoy ype nvesmen equaon P( AQ, ) P, (B5) k 4 To see hs, suppose ha s he opmal polcy acon so ha 1 J q, A, Q V( q,, ) (1 ),, 1 1,, A Q E q A Q A Q J We hen have 1 (1 ) E J q, A, Q / q A, Q q J q, A, Q / by s ode condon 1 P k E J q A Q q A Q q (1 ),, /, ( / ). 5 The no bubbles o ansvesaly condon s lm Π,. A-1

11 whee P s he eplacemen cos o a shp and 1 1 E P( A, Q ) A, Q PAQ (, ) 1 E A, Q (1 ) E A BQ C P A, Q, (1 ) (B6) s he make pce o a shp. The Bellman opeao o hs poblem sases Blackwell s Sucen Condons and s a Conacon Mappng. Theeoe, he Conacon Mappng Theoem mples ha hee s a unque soluon o he Bellman Equaon. Thus, we can guess and vey a soluon o he Bellman equaon, hen hs mus be he unque soluon. Speccally, usng equaons (B5) and (B6) s easy o check ha he ollowng uncon solves he Bellman equaon n (B1):,, (, ) J q A Q q P A Q 1 E P( A, Q ) P A, Q. k (1 ) (B7) As n sandad n q-heoy, he value o a m s he sock o cuen shps valued a cuen pces (assumng he m does no adus s capal sock) plus a em ha elecs he value deved om dynamcally adusng s capal sock.. Model equlbum Poposon 1 (Equlbum nvesmen and pces): Thee exss a unque equlbum such ha he ne nvesmen o he epesenave m s x y A z Q and equlbum shp pces ae P P kx ky A kz Q. The wo slope coecens (.e., y and z ) ae a uncon o ve exogenous paamees: k,,,, and B. In addon o hese ve paamees, he necep em (.e., x ) also depends on A, C,, and P. Invesmen and shp pces ae deceasng n he cuen lee sze ( z ). Fuhemoe, () nvesmen and pces eac moe aggessvely o he cuen lee sze when ms undeesmae he compeon (.e., z / ); () ms esponse o he cuen lee sze ndependen o he A-11

12 peceved pessence o demand (.e., z / ); () nvesmen and pces eac moe aggessvely o he cuen lee sze when he demand cuve s moe nelasc (.e., z / B ); (v) nvesmen and pces eac less aggessvely o cuen lee sze when equed euns ae hghe (.e., z / ); and (v) when adusmen coss ae hghe, nvesmen eacs less aggessvely o cuen lee sze, bu pces eac moe aggessvely (.e., z / k and ). ( zk )/ k z k ( z / k) Invesmen and shp pces ae nceasng n cuen demand ( y ). Fuhemoe, () nvesmen and pces eac moe aggessvely o cuen demand when ms undeesmae he compeon (.e., y / ); () nvesmen and pces eac moe aggessvely o cuen demand when demand s moe peceved o be pessen (.e., y / ); () nvesmen and pces eac less aggessvely o cuen demand when he demand cuve s moe nelasc (.e., y / B ); (v) nvesmen and pces eac less aggessvely o cuen demand when equed euns ae hghe (.e., y / ); and (v) when adusmen coss ae hghe, nvesmen eacs less aggessvely o cuen demand, bu pces eac moe aggessvely ( y / k and ). ( yk )/ k y k ( y / k) Poo: We s solve o he equlbum coecens. We hen pove he compaave sacs dscussed above and hen chaaceze he sysem dynamcs. Solve o he equlbum coecens: We conecue ha x y A zq P( A, Q ) P kx ky A kz Q (B8) The equlbum mus sasy P kx ky A kzq E A 1 BQ 1 C P P kx kya 1 kzq 1 A, Q 1 (1 ) A A B Q ( x y A zq) CP P kx ky((1 ) A A) kz Q ( x ya zq ) 1 (B9) A-1

13 Boh he le-hand and gh-hand sdes ae lnea n A, and Q. Thus, by machng coecens n (B9), we can hen solve o he xed-pon values o x, y, and z. Speccally, machng coecens on Q shows ha he equlbum value o z sases ( ) z whee () z k z² ( kb ) z B. (B1) We wan he negave oo o he quadac n (B1), whch s z k B k B B, k k k (B11) when and z B/( k) when. Gven hs soluon o z, machng coecens on A and he consan shows ha he equlbum values o y and x ae gven by y k(1 ) k( Bkz ) k(1 ) B/ z, (B1) and A(1 ) A( y B/ z ) C( ) P x ya zq. B/ z (B13) Unqueness o he equlbum: We now show ha he lnea equlbum descbed n Poposon 1 s he unque saonay equlbum o he model. To show hs s he case, we conecue an alenae saonay equlbum o he om x y A zq g( A, Q ) P( A, Q ) P kx ky A kz Q kg( A, Q ), (B14) whee gaq (, ) s an abay uncon o he sae vaables. Poceedng as above, we agan oban condons (B11), (B1), and (B13) as well as a unconal equaon chaacezng gaq (, ): 1 g( A, Q) E g( A, Q ) A, Q B/( kz ) (B15) Ieang on (B15) and makng use o he law o eaed expecaons, we have k 1 g( A, Q) lmk lm k Eg( A k, Q k) A, Q. 1 B/( kz ) (B16) A-13

14 Snce he conecued equlbum s saonay, we have lm (, ), E g A Q A Q c o k k k some consan c especve o he nal values o A and Q. Thus, snce 1 /( ) 1 B kz equaon (B16) shows ha we mus have gaq (, ) n any saonay equlbum. Compaave sacs o z z : Beoe poceedng we s show ha 1. (B17) Usng equaon (B11), one can show ha equaon (B17) s equvalen o k(1 ) B 1 k(1 ) B 1 B. k k k Ths holds snce 4 ( ) XY Z X Y X Y Z and usng X k(1 ) B / k, Y 1/( ), and Z B/( k ), we have 1 B 4 B XY Z. k k Wh (B17) n hand, we now poceed o he compaave sacs o z. Snce az bz c whee ak, b kb, and c B. Thus, by he Implc Funcon heoem, we have a b c z z z a b c z, z az b o any pmve model paamee. Thus, we have: z z k z B / ; (B18) z / ; z B z / 1 ; z / kz ; z k z z / ; z ( B kz ) ( kz)/ k z k ( z / k) kz kb ; A-14

15 Compaave sacs o y : Recall ha y k(1 ) k( Bkz ) k(1 ) B/ z. Thus, we have y / : Obvous snce y / ; y / B : Obvous snce z / ; z / B ; y / : Obvous snce y / k : Obvous snce z / ; ; ( kz )/ k : Noe ha ( ky )/ k y k ( y / k) ky (1 ) B/ ( kz ) Snce ( kz )/ k., he denomnao s deceasng n k, whch shows ha ( ky )/ k. 3. Sysem dynamcs and seady-sae dsbuon The ue sysem dynamcs peceved by he economecan can be summazed usng a veco auo-egesson: A 1 (1 ) A A 1. Q 1 x y 1 z Q Theeoe, he ue seady sae s gven by 1 A A A 1 (1 ) A, x y A 1 y 1 A C ( ) P Q z x z B assumng ha 1 so he max s nveble. (B19) (B) The sysem dynamcs ae govened by sgn and magnude o he second egenvalue o he max n (B19). Speccally, he wo egenvalues ae 1 and z. Speccally, A-15 1 he dynamcs ae oscllaoy and non-oscllaoy ; and he sysem has convegen dynamcs, whch eun o he seady sae 1 and dvegen dynamcs 1. Above we showed ha 1 z 1. Thus, n he aonal expecaon case ( 1) we always have non-oscllaoy and convegen dynamcs. When 1, we can have ehe non-

16 oscllaoy dynamcs abou he seady-sae 1 o oscllaoy dynamcs 1. When z 1, we can have 1 1, coespondng o dvegen, oscllaoy dynamcs () s z sucen close o and () B s sucenly lage o k s sucenly small. Obvously, n ou model smulaons and esmaon we ocus on he empcally elevan case wh convegen dynamcs. We can ewe equaon (B19) as A 1 A A A 1. Q y 1 1 Q z (B1) Q Q Takng vaances o boh sdes o (B1), he vaance o he sysem abou he seady-sae sases z A, AQ, A, AQ, y. AQ, Q, y 1 z AQ, Q, 1 z Solvng (B) o he 3-unknown paamees, we oban, A, A, A, 1 (B) (B3) and snce y ( (1 ) ), 1 (1 ) A, AQ, y A, z AQ, AQ, z ( y ) (1 z ) y (1 z ) Q, A, Q, AQ, (1 z ) 1 ( y ) 1 (1 ), 1 (1 ) 1 (1 ) A, z Q, z z n any saonay dsbuon nduced by he model. Saghowad algeba shows ha he vaance o eanngs s (B4) (B5) B B, A, Q, AQ, A, 1 (1 z ) (1 (1 z By )) B ( y ), 1 (1 z ) 1 (1 z ) and he vaance o pces s (B6) ( ky ) 1 ( ) ( ) ( )( ) 1 (1 )1 (1 ) A, P, ky A, kz Q, ky kz AQ, z z (B7) We can also use (B1) o chaaceze he pah and auo-covaance o eanngs n he model. Speccally, we have A-16

17 A A [, ] 1 1 E A Q B y z Q Q ( ) (1 ) ( ) (1 ) ( ) (B8) 1 1 l l A A By l z A A B z Q Q (1 z ) By (1 z ) ( A ) (1 ) A B z ( Q Q ). Thus, he auo-covaance o shp eanngs s (1 z ) Cov[, ] Cov By ( A ) (1 ) ( ),( ) ( ) A B z Q Q A A B Q Q (1 z ) (1 z ) By A, B AQ, B z AQ, B Q, (1 z ) (1 ) (1 z ) By A, B(1 z ) Q, (A9) (1 z ) And he auo-coelaons o eanngs ae gven by (1 z ) A, Q, Co[, ] By B(1 z ). (1 z ),, 4. Sysem dynamcs and seady-sae dsbuon peceved by ms The sysem dynamcs peceved by ms ae A 1 (1 ) A A 1, Q 1 x y 1 z Q so he peceved seady sae s gven by 1 A A 1 (1 ) A x ya Q 1 y 1 z x z. (B3) (B31) Thus, ms peceve he same seady-sae as he economecan. Howeve, snce 1 z 1, ms expec he dynamcs o be convegen and non-oscllaoy. We can ewe equaon (B3) as A A A A Q y 1 Q z Q Q Takng vaances o (B3), he peceved vaance o he sysem abou he seady-sae s (B3) A-17

18 A, AQ, A, AQ,. AQ, Q, y 1 z AQ, Q, y 1 z Thus, we oban A,, 1 y A, AQ,, 1 (1 z ) (B33) (B34) (B35) and ( y ) 1 (1 z ). 1 (1 ) 1 (1 ) A, Q, z z (B36) The peceved vaance o eanngs s B B, A, Q, AQ, A, 1 (1 z ) (1 (1 )) ( ), z By B y 1 (1 z ) 1 (1 z ) and he peceved vaance o pces s ( ky ) 1 ( ) ( ) ( )( ) 1 (1 ) 1 (1 ) A, P, k y A, k z Q, k y k z AQ, z z (B37) (B38) We can use (B3) o chaaceze he pah o eanngs expeced peceved by ms. (1 z ) E [ A, Q] B y ( A ) (1 ) ( ). A B z Q Q (1 z ) Thus, he peceved auo-covaance o shp eanngs s (B39) Cov B y (1 z ) B B z B [, ] (1 ) A AQ AQ Q (1 z ),,,, (1 z ) By A, B(1 z ) Q,. (1 z ) And he auo-coelaons o eanngs peceved by ms ae gven by (1 z ) A, Q, Co[, ] By B(1 z ). (1 z ),, 5. Obsevables ae nvaan n B/k Noe ha (B4) A-18

19 1 1 B 1 1 B 1 B z, k k k only depends on B/k. Fuhemoe, kx and ky only depend on B/k. Thus, we change B and k popoonaely holdng B/k consan, s easy o see ha hs has a popoonal eec on I and Q, bu has no eec on I /Q,, P, o R +1, and hus no eec on any o he daa sascs used n ou ndec neence esmaon pocedue. I s also saghowad o see ha a change n A o C has an addve eec on he lee sze Q, bu has no eec on I,, P, o R +1. As a esul, a change n hese paamees has a small mpac on I /Q. 6. A useul lmng case I s easy o show ha he cobweb model s a lmng case o ou model when ms compleely neglec he compeon ( = ) and adcally ove-exapolae demand ( = 1). To clealy solae he ole o compeon neglec, hee we consde he moe geneal case whee capal s nnely lved and shs n demand ae pemanen and deemnsc,.e., whee = =1 and C. Snce = =1, hee s no scope o ove-exapolang demand. In hs case, z s gven n equaon (B11) and s also easy o see ha y z / B and x z ( p )/ B, whch mples ha pces ae P P kz B P and nvesmen s ( / )( ) I z B P. Gven an nal level o demand A, he seady-sae lee sze s ( / )( ) Q ( A) ( A P )/ B, and seady-sae eanngs ae. In he lmng case whee =, P we have z B/( k) whch mples P / and I ( / P)/ k. A Now suppose hee s an unexpeced shock a = ha pemanenly ases demand o. The new seady-sae lee sze s Q ( A ) ( A P )/ B, and he seady-sae enal ae and shp pce ae unchanged. Followng he nal shock, he aggegae lee sze (Q ) and eanngs ( ) evolve accodng o I Q Q ( z / B)( P) and A BQ (B41a) A-19

20 Snce z, nvesmen s posve when eanngs ae above he seady-sae. Ieang on (B41a), we can show ha (B41b) Q Q 1 1 ( Q ) ( z / B ) (1 z ) and P 1 z (1 z ). Thus, 1 z 1, he sysem conveges o s new seady-sae ollowng he shock.e., lm Q Q ( Q ) and lm P. Fgue B1 uses hs smpled veson o he model o conas ndusy dynamcs n hee cases: he cobweb model ( ), aonal expecaons ( 1), and paal compeon neglec ( 1). In Fgue B1, he sequence o equlbum (Q, ) pas ae maked wh dos. Vecal movemens n he gue show he deemnaon o spo eanngs gven he cuen lee sze, Q. These movemens ae dcaed by he demand cuve (eanngs and lease aes ae he same n hs case). Laeal movemens n he gue depc ms nvesmen esponse o cuen eanngs. The laeal movemens show he eanngs ha each m expecs o peval nex peod and, gven hose expecaons, he quany ha each m chooses o supply. When ms sue om compeon neglec, acual eanngs de om expeced eanngs because acual ndusy nvesmen des om he ndusy nvesmen ha ms had expeced. Panel A llusaes Kaldo s (1938) cobweb model n whch ms choose he quany o supply n peod +1 unde he naïve assumpon ha hee wll be zeo compeve supply esponse (.e., ), so eanngs wll always be he same as hey wee n peod. Speccally, when compeon neglec s complee, I k 1 ( / P), so he laeal movemens n Panel A ae peecly hozonal, and he adusmen pocess aces ou he cobweb-lke paen n pce vesus quany space. These cobweb dynamcs can be conased wh he aonal expecaons equlbaon pocess ha obans when 1. Wh posve adusmen coss, eanngs and shp pces mus eman above he seady sae levels o seveal peods o nduce ms o nves. Howeve, snce expeced eanngs ae he same as acual eanngs unde aonal expecaons, he adusmen pocess aces ou a sagh lne. Panel B o Fgue B1 llusaes he case whee compeon neglec s sevee bu no complee (.e., s close o zeo). Whle ms coecly ancpae he deconal change n eanngs, hey undeesmae he magnude o ha change ollowng a demand shock. Ths A-

21 geneaes he dampened cobweb-lke paen depced n Panel B o Fgue B1. Speccally, he laeal movemens ae dagonal, elecng he ac ha ms paally ancpae he compeve esponse. As ses, ndvdual ms nceasngly ecognze how compeos ae lkely o espond, so ndusy nvesmen becomes less sensve o devaons o eanngs om he seady sae. Fo small enough values o, he dynamcs can be oscllaoy as n Panel B. As appoaches 1, he dynamcs become non-oscllaoy.e., ndusy lee sze seadly ses o he new seady sae, and eanngs seadly all back o P. Ths s shown n Panel C, whch compaes he dynamcs unde modeae compeon neglec wh hose unde aonal expecaons. Whle he dynamcs ae no oscllaoy n hs case, hee s sll excess volaly n shp pces and nvesmen. In each o he scenaos descbed above, we can compue he eun om ownng and opeang a shp. The eun beween me and +1 along he equlbum pah ollowng he nal shock s gven by 1P1 ( Bkz )( z / B)( P) 1 P P k( z / B)( P) 1 R (1 ) (1 ). (B41c) Snce hs example s deemnsc, ealzed euns and he euns expeced by he economecan ae he same. In he aonal expecaons case ( 1 ), expeced euns R +1 equal equed euns, especve o. In conas, when ms sue om compeon neglec ( 1 ), expeced euns ae less han equed euns when cuen eanngs ae above he seady sae.e., P, hen R 1 and vce vesa. And snce eanngs, pces, and nvesmen all conan he same nomaon n hs smple case, analogous saemens hold o used pces and nvesmen.e., P P P I R 1. The nuon o hese esuls s naual. In he aonal expecaons equlbum, ms expec any eanngs n excess o he seady-sae level o be pecsely ose by capal losses om holdng shps. Howeve, when ms sue om compeon neglec, nvesmen oveeacs o devaons o eanngs om he seady sae. Because ms ae supsed by he ndusy supply esponse, uue ealzed euns ae below equed euns. A-1

22 Fgue B1: Cobweb Dynamcs Ths gue llusaes model dynamcs n he case whee = C = and = = 1 ollowng a shock o demand. Panel A: Complee compeon neglec (= ) vesus aonal expecaons (= 1) Eanngs () Flee Sze (Q) Complee Compeon Neglec Raonal Expecaons Eanngs () Eanngs () Panel B: Sevee compeon neglec vesus aonal expecaons Flee Sze (Q) Sevee Compeon Neglec Raonal Expecaons Panel C:Modeae compeon neglec vesus aonal expecaons Flee Sze (Q) Modeae Compeon Neglec Raonal Expecaons A-

23 7. Equlbum expeced euns Expeced euns wll geneally no equal ms equed euns. Speccally, equaon (B9) nsues ha E 1 R A, Q 1 E A 1BQ1C P P k( x y A 1zQ1) A, Q P kx ky A kz Q P(1 ) kx C(1 ky)((1 ) A A) ( Bkz) Q ( x y A zq) 1, P kx ky A kz Q (B4) by consucon. In wods, he epesenave m expecs ha holdng peod euns wll equal he equed eun on capal,. Howeve, he ue expeced eun peceved by he economecan who does no sue om compeon neglec and does no oveesmae he pessence o demand shocks wll geneally de om 1 +. Speccally, we have E 1 R A, Q 1 E A 1BQ1C P P kx ky A kz Q P k( x y A 1zQ1) A, Q P(1 ) kx C(1 ky)((1 ) AA) ( Bkz) Q ( x y A zq) P kx ky A kz Q (B43) Subacng (B4) om (B43) shows ha (1 )( B kz )( x y A z Q ) ( )(1 ky )( A A) 1, P kx ky A kzq E R A Q (B44) Equaon (B44) gves he geneal expesson o expeced euns when 1 and. Alhough (B44) shows ha expeced euns can be decomposed no a em ha vanshes when hee s ull compeon awaeness ( 1) and a em ha vanshes when hee s no demand oveexapolaon ( ), hese wo bases do neac n ou model. Speccally, snce y / A-3

24 and y /, demand ove-exapolaon naually amples he eun pedcably due o compeon neglec and vce vesa. Snce he laen demand pocess, A, s no eadly obsevable, s useul o ecas equaon (B44) n ems o obsevables, namely, ndusy ne nvesmen ( I ) and ne eanngs ( ), whch conan he same nomaon as A and Q. Noe ha ( C P) 1 BA I x y z Q A 1 BI ( x ) z z( C P). Q z By ( I x ) y y( C P) (B1) mples z By (1 ky ) z (1 ). Thus, usng ou expesson o A and (B13), yelds 1 ky (1 ky )( A A) B( I x ) z z ( C P ) A( z By ) Theeoe, we oban z By 1 (1 ) 1 (1 ) 1 (1 ) ( B / z ) I ( B / z ) x A(1 )(1 ky ) ( C P) ( B / z ) I ( P) ( B / z ) I ( ). ( B kz ) I ( ) ( ) ( B / z ) I E[ R 1 N, ] (1 ) I. P ki 1 P ki (B45) Poposon (Foecasng egessons): Consde a oecasng egesson o euns n a neghbohood o he seady-sae. (a) Consde a mulvaae egesson o euns on demand ( A ) and lee sze ( Q ). I 1 o, hen E[ R 1 A, Q]/ A. I 1, hen E[ R 1 A, Q]/ Q. (b) I 1 o, hen nvesmen ( I ), pces ( oecas euns n a unvaae egesson. P), and pos ( ) wll each negavely (c) Consde a mulvaae egesson o euns on nvesmen ( I ) and pos ( ). A-4

25 () I hee s compeon neglec bu no demand ove-exapolaon ( 1 and ), hen E[ R 1, I ]/ I and E[ R 1, I ]/ ; () I hee s demand ove-exapolaon bu no compeon neglec ( 1 and ), hen E[ R 1, I ]/ and E[ R 1, I ]/ I ; () I hee s boh compeon neglec and ove-exapolaon, (.e., 1 and ), hen we always have E[ R 1, I ]/. Fuhemoe, compeon neglec s elavely mpoan n he sense ha B z B kz ( )/(1 ) /( ( )) (1 ), hen E[ R 1, I ]/ I. Ohewse, compeon neglec s elavely unmpoan, hen E[ R 1, I ]/ I. Poo: We have, ( ) E R 1 A Q 1 P (1 ) P y( Bkz) ( ( ) ( )) A P ky A A kz Q Q 1 PP ( kz( Q Q)) ( ) P (1 ky). ( P ky( A A) kz( Q Q )) and, E R 1 A Q 1 P P (1 ) z( Bkz) ( ( ) ( )) Q P ky A A kz Q Q 1 P ( A A) +( ) P (1 ky) kz ( P ky( A A) kz( Q Q )) Thus, n a neghbohood o he seady sae, ( AQ, ) ( AQ, ), we have E R 1 A, Q E[ R 1 AQ, ] 1 1 (1 ) P y( Bkz) ( ) P (1 ky), A A (B46) and E R 1 A, Q E[ R 1 A, Q ] Q Q 1 (1 ) P z( B kz). (B47) Pa (a) ollows om nspecng (B46) and (B47). A-5

26 These appoxmaons ae no subsanve n naue. Consde equaon (B46) above. Usng ou paamee esmaes, Fgue B shows he acual expeced euns mpled by he model vesus he lnea appoxmaon as a uncon o he demand sae vaable A a Q = Q. We show hs o hose values o demand ha would ase 95% o he me condonal on Q = Q. As shown below, he lnea appoxmaon s hghly accuae o values o demand ha ae lkely o ase n equlbum. Noneheless, we do no use hs appoxmaon when we use he model o smulae he daa o ou ndec neence esmao. 6 Fgue B: Accuacy o Reun Appoxmaons To pove pa (b), we appoxmae expeced euns as 1 E 1 R 1 A, Q P (1 ) y( Bkz) ( )(1 ky) ( A A) 1 P (1 ) z( Bkz) ( Q Q ). Theeoe, Cov[ R, I ] Cov[ E R A, Q, I ] 1 1 P (1 )( Bkz )[( y ) ( z ) y z ] 1 A Q AQ P ( )(1 ky )[ y z ] 1 A AQ (B48) (B49) (B49) s negave snce [( y) A ( z) Q yzaq] Va[ I ] and y z y [(1 )/(1 (1 z ))]. Smlaly, we have A AQ A 6 Howeve, we have checked and we oban nealy dencal paamee esmaes, nsead o usng smulaon o compue he means, vaances, and oecasng elaonshps mpled by he model, we used lnea appoxmaons o hs so o compue he quanes analycally usng appoxmae closed-om expessons. A-6

27 Cov[ R, ] Cov[ E R A, Q, ] 1 1 P (1 )( Bkz )[ y Bz ( By z ) ] 1 A Q AQ P ( )(1 ky )[ B ]. 1 A AQ (B5) (B5) s negave snce algeba shows [ y Bz ( By z ) ] and [ B ]. A Q AQ To pove pa (c), noe ha E[ R 1, ] ( ) I P ( B/ z) P k( ) (1 )( B kz ), I ( P ki ) 1 ( P ki ) and A AQ ( 1 ) I E [ R, ] 1 1 P ki. Thus, n a neghbohood o he seady sae, (, ) (, ) I, we have [ 1, ] [ 1, ] 1 P 1 Bkz P B z I I 1 E R I E R (1 ) ( ) ( / ). (B51) and N [ 1, ] [ 1, ] 1 P 1 E R I E R. (B5) Pa (c) ollows om nspecng (B51) and (B5). Poposon B1 chaacezes how hese pedcably esuls vay wh he undelyng model paamees. Speccally, we ake compaave sacs on he oecasng esuls nea he model s seady-sae. Howeve, when compung hese compaave sacs, we allow he seady-sae o he model o change whee elevan. These esuls ae dscussed n he ex, bu ae saed and poven omally hee o he sake o compleeness. Poposon B1 (The ole o compeon neglec, demand ove-exapolaon, nelasc demand, and elasc supply): (a) Reun pedcably becomes songe when compeon neglec s moe sevee (.e.,, / (1 ) E R 1 A Q A and E 1 R A, Q / Q (1 ) ) o 1 demand ove-exapolaon s moe sevee (.e., E [ R A, Q ]/ A, bu E[ R1 A, Q]/ Q ). 1 A-7

28 (b) The pedcably due o compeon neglec becomes songe when demand s moe nelasc and weake when supply s moe nelasc. Fomally, when 1 and =, E[ R 1 A, Q]/ AB, E[ R 1 A, Q]/ Qk. E[ R 1 A, Q]/ QB, E[ R 1 A, Q]/ Ak, and (c) The pedcably due o demand ove-exapolaon becomes weake when demand s moe nelasc and songe when supply s moe nelasc. Fomally, when 1 and >, E[ R 1 A, Q]/ AB and 1 E [ R A, Q ]/ Ak. (d) In a mulvaae egesson o euns on eanngs and nvesmen, he coecen on eanngs becomes moe negave when demand exapolaon s moe sevee (.e., E[ R 1, I ]/ ); he coecen on nvesmen alls when compeon neglec s moe sevee and ses when demand exapolaon s moe sevee (.e., E[ R 1, I ]/ I(1 ) and 1 E [ R, I ]/ I ). Fnally, when compeon neglec s elavely mpoan, he coecen on nvesmen becomes moe negave when ehe demand o supply s moe nelasc. Poo: Pa (a) ollows om deenang (B46) and (B47). Speccally, we have E R A, Q y P y( Bkz) (1 ) P ( Bkz) A 1 z 1 y (1 ) P ky ( ) P k, E R 1 A, Q 1 y 1 1 y (1 ) P ( Bkz) P (1 ky) ( ) P k, A E, R 1 A Q z ( ) (1 ) ( ), 1 1 P z B kz P B kz Q and E, R 1 A Q. Q Pa (b) ollows om deenang (B46) and (B47) ae seng. Speccally, A-8

29 and and E R 1 A, Q 1 z y (1 ) P y 1 k Bkz A B B B 1 z (1 ) P 1 k ( y) k(1 ) /, B E R, 1 A Q 1 z z (1 ) P ( Bkz) z 1 k, QB B B E R 1 A, Q 1 ( kz) y (1 ) P y ( Bkz) A k k k 1 (1 )( Bkz) k ( B kz ) (1 ) P ( y), k ( B kz) E R 1 A, Q 1 z ( kz) (1 ) P ( Bkz) z Qk k k 1 z( B kz) (1 ) P. k ( B kz ) Pa (c) ollows om deenaon (B46) ae seng 1. Speccally, we have E R A, Q ( ky ) ( ), 1 1 P A k k E R A, Q y ( ). 1 1 P k A B B Fnally, pa (d) ollows om deenang (B51) and (B5). Speccally, we have E [ R I, ] z B z ( ) (1 ), P B kz P k P I 1 ( z) E[ R 1 I, ] 1 I (1 ) P ( B/ z ), 1 and E[ R 1 I, ] 1 (1 ) P 1. A-9

30 Fnally, deenang (B51) ae seng, we have E[ R 1 I, ] 1 z (1 ) P 1k, I B B and E [ R I, ] ( kz ) (1 ). 1 1 P I k k 8. Model exensons We have made a numbe o assumpons o keep he model acable so as o anspaenly model he logc o compeon neglec and demand-exapolaon. Hee we dscuss a numbe o possble exensons ha we have exploed. a. Tme-vayng eplacemen coss o shps We have also assumed ha shps have a consan eplacemen cos o P. I s saghowad o exend he model so shp eplacemen coss ollow an AR(1) pocess. In ha case, m nvesmen and shp pces would be lnea uncons o he hee sae vaables: he me-vayng eplacemen cos, P,, as well as A and Q. Invesmen would be deceasng n P, and shp pces would be nceasng n P,. (Boh pces and nvesmen would be nceasng n A and deceasng n Q as n ou baselne model). 7 Relaedly, we have assumed ha demand ollows a saonay pocess. We could easly add a deemnsc me end o demand and nsead assume ha devaons o demand om end ae saonay. b. Addng ms wh unbased beles We could add sophscaed ms who hold unbased expecaons o ou model. Ths would have he eec o dampenng he boom-bus cycles and aenuang he pedcably o shppng euns. Speccally, consde an exenson n whch unbased ms ae eaed smlaly o based 7 Thus, we had me-sees daa on he eal pce o aw maeals used o consuc a shp (hs s no he same as he pce o a new shp whch would elec a poenally me-vayng mak-up ove cos) we could exend he model and consde hs exenson n ou esmaon. A-3

31 ms: hey ae allowed o opeae shps, nves by puchasng shps n he new buld make, and o dsnves by scappng shps. These unbased ms wll endogenously choose o lean agan he wnd, nvesng less (moe) when based ms ae ove-nvesng (unde-nvesng). Ths conaan nvesmen esponse wll educe he magnude o eun pedcably because mpacs omoow s lee sze and hence omoow s eanngs. The degee o dampenng would be a uncon o he numbe o sophscaed ms and he nvesmen adusmen coss (.e., adusmen coss would play an analogous ole o abageu sk aveson n behavoal nance models). c. Ievesble nvesmen and asymmec adusmen Fnally, we can noduce a wedge beween he scap value ealzed when and he eplacemen cos o a new shp ncued when, so ha ms opmally choose o a non-degeneae se o values o A and Q. We have also assumed adusmen coss ae symmec. Obvously, any deences beween he coss o nceasng o deceasng he capal sock would geneae asymmees n he speed o adusmen o posve o negave demand shocks. Howeve, he key poposons ha we emphasze n he pape connue o hold n a moe geneal veson o he model ha ncopoaes nvesmen asymmees. The cos o hs genealy s ha we ae no longe able o solve he model n closed om. To noduce evesbly and asymmey, suppose ha pe-peod m pos ae V q A Q q A Q P P k k (,,, ) (, ) ( 1 [ ] 1 [ ]) ( 1 [ ] 1 [ ]) /, whee ( A, Q) A BQ C P s cuen shp eanngs, 1 [ ] s an ndcao o, and 1[ ] s an ndcao o. We allow he new buld pce o exceed he scap pce ( P P ). A smple way o capue he dea ha he sock o shps wll be scke down han up s o assume ha he adusmen coss assocaed wh dsnvesmen o exceed hose assocaed wh nvesmen,.e., k k. Relaedly, a ough way o capue Kaloupsd s (14) dea ha me-o-buld delays ncease dung nvesmen booms s o make he oppose assumpon,.e., k k. Unde hese moe geneal assumpons, he opmal nvesmen o he epesenave m s whee PAQ (, ) P PAQ (, ) P 1[ P( A, Q) P ] 1[ P( A, Q) P ] k k A-31

32 E ( A, Q ) A, Q 1 (1 ) PAQ (, ) s he pesen value o uue pos ha ms expec om a un o nsalled capal. Ths concluson s a specal case o he geneal q-heoy esuls n Abel and Ebeley (1994). Naually, any deence beween new-buld and scappng pces ( P P ) geneaes an nacon egon whee ms nehe acvely nvesmen no acvely dsnves. And he adusmen coss assocaed wh dsnvesmen exceed hose assocaed wh nvesmen ( k k ), nvesmen esponds moe aggessvely o hgh shp pces han o low shp pces. By conas, k because me-o-buld delays lenghen when odeng hgh, hen m nvesmen wll be concave n used shp pces. The elaonshp beween nvesmen and pces o he case whee k k k and P P s show n Fgue B below. Fgue B: Ne Invesmen Vesus Pce when k k and P P. Gven hese nvesmen polces, we can solve o equlbum pces. The law o moon o Q peceved by ms who sue om compeon neglec s P( A, Q) P P( A, Q) P Q 1 Q 1[ P( A, Q) P ] 1[ P( A, Q) P ]. k k By conas, he ue law o moon ses = 1 n he above equaon. As n ou baselne model, ms may also beleve ha he exogenous demand pocess (A ) s moe pessen han uly s. Thus, he equlbum pce sases he ollowng unconal equaon A-3

33 PAQ (, ) E ( A, Q ) P( A, Q ) A, Q PAQ (, ) P (, ) PAQ P (1 ) A AB 1 [ (, ) ] 1 [ (, ) ] Q PAQ P PAQ P C P k k 1 PAQ (, ) P PAQ (, ) P EP(1 ) A A, Q 1[ P( A, Q ) P ] 1[ P( A, Q ) P ] 1 k k 1 whee he expecaon s ove he dsbuon o +1. Usng sandad dscee gd appoxmaon echnques (Tauchen (1986)), we can numecally solve o he equlbum pce uncon by eang on he above mappng. Gven he non-lnea esponse o nvesmen o pce when ehe k k o P P, we nd ha he equlbum pce uncon s a non-lnea uncon o he sae vaables (A,Q ). Fo nsance, k k and P P, peods o below nomal pos ae expeced o las longe han peods o above nomal pos due o he asymmey n m s endogenous supply esponse. Ths mples ha shp pces wll be a concave uncon o (A,Q ) and ha pces wll be unomly lowe han hey ae wh symmec adusmen coss. Howeve, equlbum shps pces ae sll scly nceasng n cuen demand (A ) and scly deceasng n cuen lee sze (Q ). The case whee k k and P P s llusaed below. Fgue B3: Pces vesus Demand and Supply when k k and P P. By consucon, ms beleve ha he expeced eun o nvesng n shps s consan and equal o. By conas, an unbased economecan who knows he ue laws o moon o boh demand and he lee sze, expecs euns o de om. The expeced uue payo om puchasng a shp s A-33

34 ( 1, 1), ( 1, 1), E ( A, Q ) P( A, Q ) A, Q (1 ) P( A, Q ) E A Q A Q E A Q A Q E P( A, Q ) A, Q E P( A, Q ) A, Q I = 1 and =, hs expeced payo s always zeo. By conas, hs expeced payo s weakly deceasng n A ehe < 1 o < and weakly nceasng n Q < 1. Fuhemoe, he expeced payo wll be scly deceasng n A < o < 1 and and scly nceasng n Q < 1 and. 8 Thus, snce E ( A 1, Q 1) P( A 1, Q 1) A, Q (1 ) P( A, Q) ER 1 A, Q PAQ (, ) expeced euns wll nhe hese same popees n a egon o he seady sae. In hs way, ou man eun pedcably esuls, namely, ha E[ R 1 A, Q]/ A and 1 E [ R A, Q]/ Q, ae unchanged n hs moe geneal veson o he model. Addng nvesmen asymmees o ou model can also geneae skewed pce changes. Speccally, nvesmen s a convex uncon o pces (.e., k k and P P ), equlbum pces wll be a concave uncon o (A,Q ) as shown above. As a esul, pce changes wll be negavely skewed even he pmve demand shocks ( +1 ) ae dawn om a symmec dsbuon. Inuvely, because nvesmen asymmees mean ha supply glus ae expeced o pess o longe han supply shoages, a negave demand shock leads o a lage dop n pces han an posve demand shock o he same magnude. Howeve, hs negave eun skewness would ase whehe o no ms have based expecaons. 8 To see hs, ecall ha = I(A,Q ) s weakly nceasng n A and weakly deceasng n Q ; smlaly, P(A,Q ) s scly nceasng n A and scly deceasng n Q. Now we E ( A, Q ) P( A, Q ) A, Q (1 ) P( A, Q ) ( )( A A) B(1 ) I( A, Q ) E P( A ( A A), Q I( A, Q )) P( A ( A A), Q I( A, Q )) 1 1 E P( A ( A A), Q I( A, Q )) P( A ( A A) 1 A-34, Q I ( A, Q )). 1 Each o hese hee ems s weakly deceasng n A ehe < 1 o < and weakly nceasng n Q < 1. Fuhemoe, he s and second ems ae scly nceasng n A <. Smlaly, he s and hd ems ae scly nceasng n A and scly nceasng n Q ousde o he nvesmen nacon egon < 1.

35 C: Indec Ineence Pocedue We use ndec neence o esmae he paamees o ou model o ndusy cycles. We ae neesed n a L 1 veco o model paamees θ. Assume ha we have M L daa sascs o he om β ˆ T, (C1) whee he β ˆ T ae uncons o ou sample o T me-sees obsevaons (e.g., me-sees means, me-sees vaances, me-sees egesson coecens, ec.) and he βθ () ae he coespondng uncons o ou smulaed me-sees daa. By smulang a sucenly long me sees we can elmnae smulaon nose. We can hus egad he smulaed sascs as deemnsc and connuously deenable uncon o he unknown paamees βθ (). I we assume Now dene he esmao θˆ agmn ( βˆ β( θ)) W( βˆ β( θ )) (C) T θ T T Compacness: θ n( Θ ) whee Θ s a compac subse o Idencaon: βθ ( ) lm E[ β ˆ ] mples θ θ T T L R. Lmng Behavo: A Cenal Lm Theoem mples ha T( βˆ T β( θ)) N( S, ) Full Rank: Γθ () D βθ () exss, has ull ank L, and s a connuous uncon o. Then we wll have θ θ p ˆ θ, (C3) T.e., ou esmao wll be conssen o he ue populaon paamee o nees. d We now pove asympoc nomaly. The s ode condon o θ ˆ T s ( Γθ ( ˆ )) W( βˆ β( θ ˆ )). (C4) T T T By he nemedae value heoem, hs mples ( Γθ ( ˆ )) W( βˆ β( θ ) Γ( θ )( θˆ θ )) T T T T T( θˆ θ ) [( Γ( θˆ )) WΓ( θ)] ( Γθ ( ˆ )) W T( βˆ β( θ ), 1 T T T T T (C5) A-35

36 o some θ T ha s a convex combnaon o θ ˆ T and θ. Thus, snce θ ˆ T θ and snce Γθ () D βθ () s connuous, Slusky s Theoem mples ha θ T whee Γ D θ βθ ( ). d ˆ 1 1 ( T ) N(,( ) ( ) ), p θ θ Γ WΓ Γ WSWΓ Γ WΓ (C6) I s easy o show ha asympocally ecen esmaes can be obaned by usng W=S p whee S ˆ S. In ha case, equaon (C6) mples ha T d ˆ 1 1 ( T ) N(,( ) ), θ θ Γ S Γ (C7) whch has he smalles asympoc vaance n he class o esmaos dened by (C). Howeve, we do no use W=Sˆ 1 snce we woy ha, whle mpovng asympoc ecency n heoy, hs s lkely o lead o unneesng and poenally sange esuls n pacce. As n many applcaons, ou esmaed S max s nealy sngula because some o ou me-sees sascs ae hghly coelaed wh each ohe. As a esul, usng ˆ 1 ˆ 1 W=S n ou ndec neence esmao would place a huge amoun o wegh on mnmzng lnea combnaons o he momens ha ae esmaed wh suspcously hgh pecson n he daa. Howeve, hese pecsely esmaed lnea combnaons ae no necessaly he mos economcally neesng. See Cochane (5) o a uhe dscusson o he benes o usng smple pe-speced weghng maces nsead. Insead, we use ˆ 1 W= [ dag( S )] so ou esmao s θˆ ag mn [( ˆ ( θ )) / ˆ ] M T θ m 1 m m s m, whee sˆ [ S ˆ ].e., we wegh sascs nvesely by he esmaed vaances. Ths choce m mm o W pus all he momens n common (un ee) ems and accomplshes he basc ecency goal o assgnng geae wegh o sascs ha ae esmaed moe pecsely whle avodng he palls o ˆ 1 W=S ha ase when S s nealy sngula. Ou esmao o he vaance o he ndec neence esmao akes he om T Vˆ( θˆ ) T ( ΓWΓ ˆ ˆ) ΓWSWΓ ˆ ˆ ˆ( ΓWΓ ˆ ˆ), (C9) T whee Γˆ D βθ ( ˆ ) s a conssen esmao o Γ, and Ŝ s a conssen esmao o S. θ T A-36

37 Inuvely, an auxlay paamee ( ) m θ s nomave abou a paamee l he paal devave o he momen wh espec o he paamee s lage n absolue magnude.e., ( θ )/ s lage n magnude. I we have los o unnomave momens, hen ΓW Γ ml m l wll be an ll-condoned max wh los o nea-zeo values. Consequenly, ( ΓWΓ ) 1 wll be vey lage n magnude, mplyng ha he sandad eos on ou SMM esmaes wll be vey lage.e., hey wll be esmaed que mpecsely. Moe omally, we ollow Genzkow and Shapo (13) and examne he elemens o he nluence max, Λ, and he scaled-nluence max, Λ. The elemens o he nluence max SCALE ae smply he paal devaves o ou esmao wh espec o he sample daa sascs, evaluaed a he ue paamee veco. The elemens o he scaled nluence max ae a naual un-ee measue o nluence: [ Λ ] s he sandad devaon esponse o model paamee l o SCALE lm a one sandad devaon ncease n daa sasc m. Thus, Genzkow and Shapo (13) ague ha examnng SCALE s a naual way o assess souces o dencaon n non-lnea models such as ous. Snce ndec neence s a mnmum dsance esmao, we have Λ ( ΓWΓ ) 1 ΓW and 1/ 1 1/ Λ V ΓWΓ ΓW S whee V SCALE [ dag( )] ( ) [ dag( )] ΛSΛ s he asympoc vaance o ou esmao. Thus, ncopoaes nomaon boh abou how he smulaed daa sascs depend on he model paamees () as well as nomaon abou how he sample daa sascs ae weghed by he esmao (W). To oban a conssen esmao o S, we use a sysem OLS appoach (o seemnglyunelaed egesson amewok) usng Newey-Wes (1987) sandad eos ha accoun o he seal coelaon o esduals o esmae he on vaance o he sample daa sascs. Speccally, we can nepe ou daa sascs as conssng o a sysem o lnea equaons whee y Xβε (C1) y1, x 1, β1 1, y, 1,,, x, β y X β, ε. y x β M, M, M M, (C11) The sysem OLS esmao o s A-37

38 T 1 T T 1 1 β ˆ, XX Xy (C1) and, leng e y Xb OLS, he Newey-Wes (1987) syle vaance esmao o β ˆ T s 1 T T J T T 1ˆ 1 T T NW 1 S T L XX XeeX Xee X X e ex J 1 XX (C13) 1 1 We use (C13) allowng o seal coelaon o esduals a up o J = 36 monhs. D: Addonal Sucual Esmaon Resuls D.1. Robusness wh espec o he ou assumpon abou A Table D1 povdes obusness on ou assumpon on he A paamee whch deemnes he scale o demand and he equlbum lee sze. Fo bevy, we show only he unesced esmaon, alhough he message s he same n he esced speccaons. Table D1: Robusness on assumpon Ths able povdes obusness on he paamee whch deemnes he scale o demand and he equlbum lee sze. Speccally, we show he esuls o he unesced esmaes coespondng o deen levels o. Baselne: Smalle: Lage: = 5 = 5 = 75 b [] b [] b [].454 [3.16].66 [3.1].395 [3.4].98 [.7].111 [.6].86 [.79].6 [1.69].55 [13.7].638 [1.] [5.5] 4.4 [5.5] 4.44 [5.54] k 1.47 [8.14] [7.88].9 [8.9].11 [8.81].19 [8.8].111 [8.79] P 35.6 [7.31] 35.5 [7.5] [7.34] As shown above, he choce o A eally only aecs ou esmae o k whch conols he elascy o m s nvesmen esponse. The eason s ha ou esmae o k s hghly sensve o (I /Q P ), he slope om a egesson o I /Q on P. Snce 1 1 I / Q P kq k AC( ) P / B, A-38

39 a hghe assumed level o A anslaes no a lowe esmae o k. Founaely, he key behavoal paamees we emphasze, namely () and ( ), ae no vey sensve o modeae changes n k nduced by choosng deen values o A. D.. Model-mpled mpulse esponse uncons Fgues D1, D, and D3 show he model-mpled mpulse esponse uncons allowng o compeon neglec only, allowng o demand ove-exapolaon only, and allowng o boh bases, especvely. A-39

40 Fgue D1 Model-Impled Impulse Response Funcons: Compeon Neglec Only Ths gue shows he model-mpled mpulse esponse uncons ollowng a one-me shock o demand. The gue coesponds o he esmaes n column () o Table VI, whch allows o compeon neglec ( < 1), bu does no allow o demand ove exapolaon (we mpose = ). Followng an egh un demand shock a = 1, he gues conas he mpulse esponse unde aonal expecaons (mposng = 1) wh he mpulse esponse ancpaed by ms who sue om compeon neglec and he acual mpulse espond unde compeon neglec. A-4

41 Fgue D Model-Impled Impulse Response Funcons: Demand Ove-Exapolaon Only Ths gue shows he model-mpled mpulse esponse uncons ollowng a one-me shock o demand. The gue coesponds o he esmaes n column (3) o Table VI whch allows o demand ove exapolaon ( > ), bu does no allow o compeon neglec (we mpose = 1). Followng an egh un demand shock a = 1, he gues conas he mpulse esponse unde aonal expecaons ( = ) wh he mpulse esponse ancpaed by ms who sue oveexapolaon demand and he acual mpulse esponse unde demand ove-exapolaon. A-41

42 Fgue D3 Model-Impled Impulse Response Funcons: Boh Bases Ths gue shows he model-mpled mpulse esponse uncons ollowng a one-me shock o demand. The gues coesponds o he esmaes n column (4) o Table VI whch allows o boh compeon neglec ( < 1) and demand ove exapolaon ( > ). Followng an egh un demand shock a = 1, he gues conas he mpulse esponse unde aonal expecaons ( = and = 1) wh he mpulse esponse ancpaed by ms who sue om boh bases and he acual mpulse esponse when ms sue om boh bases. A-4