This work is distributed as a Discussion Paper by the STANFORD INSTITUTE FOR ECONOMIC POLICY RESEARCH. SIEPR Discussion Paper No.

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1 This work is disribued as a Discussion Paper by he STANFORD INSTITUTE FOR ECONOMIC POLICY RESEARCH SIEPR Discussion Paper No Producion Targes By Guillermo Caruana CEMFI And Liran Einav Sanford Universiy June, 2005 Sanford Insiue for Economic Policy Research Sanford Universiy Sanford, CA (650) The Sanford Insiue for Economic Policy Research a Sanford Universiy suppors research bearing on economic and public policy issues. The SIEPR Discussion Paper Series repors on research and policy analysis conduced by researchers affiliaed wih he Insiue. Working papers in his series reflec he views of he auhors and no necessarily hose of he Sanford Insiue for Economic Policy Research or Sanford Universiy.

2 Producion Targes Guillermo Caruana CEMFI Liran Einav Sanford Universiy and NBER June 20, 2005 Absrac We presen a dynamic quaniy seing game, where players may coninuously adjus heir quaniy arges, bu incur convex adjusmen coss when hey do so. These coss allow players o use quaniy arges as a parial commimen device. We show ha he equilibrium pah of such a game is hump-shaped and ha he final equilibrium oucome is more compeiive han is saic analog. We hen es he heory using monhly producion arges of he Big Three U.S. auo manufacurers during and show ha he hump-shaped dynamic paern is presen in he daa. Iniially, producion arges seadily increase unil hey peak abou 2-3 monhs before producion. Then, hey gradually decline o evenual producion levels. This qualiaive paern is fairly robus across a range of similar exercises. We conclude ha sraegic consideraions play a role in he planning phase in he auo indusry, and ha saic models may herefore under-esimae he indusry s compeiiveness. Keywords: Differenial games, adjusmen coss, Courno, quaniy compeiion, dynamic oligopoly games. JEL classificaion: C72, C73, D43, L13, L62. An earlier version of his paper had he ile Quaniy Compeiion wih Producion Commimen: Theory and Evidence from he Auo Indusry. We hank Susan Ahey, Tim Bresnahan, Uli Doraszelski, Ken Judd, Ignacio Palacios-Huera, Chris Snyder, and especially Eseban Rossi-Hansberg for helpful conversaions, and seminar paricipans a IO Fes 2004, Bilbao, Carlos III, CEMFI, ECARES, and Olin School of Business for commens. We are exremely hankful o Maura Doyle and Chris Snyder for so kindly sharing heir daa wih us. All remaining errors are ours. Address Correspondence o Liran Einav, Deparmen of Economics, Sanford Universiy, Sanford, CA ; Tel: (650)

3 1 Inroducion Economiss ofen model sraegic ineracions using simulaneous one-sho games. I is as if decisions were aken in he blink of an eye and realized insananeously. This is, of course, a simplificaion. Complex decisions, such as enry, exi, or producion are normally he resul of a long preparaion process. If plans canno be hidden from compeiors and changing hem is cosly, incenives o behave sraegically during he preparaion sage should be explicily considered, as hey may be an imporan deerminan of he final equilibrium oucomes. Consider, for example, he auomobile indusry. Suppose ha, ahead of ime, an auo manufacurer has planned a cerain producion arge. In order o achieve i, he firm needs o ake cerain acions, such as hiring labor, canceling vacaions, purchasing pars from suppliers, ec. If he firm hen decided o change is desired producion level, i would likely need o incur some coss adjusing he previous acions. To he exen ha such preparaions are no or canno be fully hidden from compeiors, hey may play a sraegic role. Given he cosly naure of hese adjusmens, he preparaion sage acs as a gradual commimen device. Firms realize ha heir planned producion levels affec heir rivals producion plans, and use his o heir advanage, adjusing heir own inenions sraegically. The main goal of he paper is o develop his argumen in he conex of a quaniy seing game, and o esablish is empirical relevance using daa from he U.S. auo indusry. The firs par of he paper consrucs a dynamic quaniy seing game wih a planning phase and adjusmen coss. In he second par, we use daa on monhly producion arges by he Big Three auo manufacurers General Moors, Ford, and Chrysler and show ha he empirical paern is consisen wih he heoreical predicion. The paper makes hree separae conribuions. Firs, we presen new heoreical predicions for quaniy seing games regarding he non-monoone evoluion of producion argeing. Since he framework is fairly simple and general, hese predicions may be relevan in a wide range of sraegic ineracions. Second, we presen empirical evidence ha shows a similar non-monoonic paern of producion arges in he U.S. auo indusry. Since his is one of he larges indusries in he U.S., we hink ha documening his paern is of ineres, even in he absence of he underlying heoreical framework. Finally, he mach beween he heory and he daa suggess wo imporan implicaions for he auo indusry: (i) adjusmen coss and sraegic consideraions may play an imporan role in he planning phase of producion; and (ii) saic models may underesimae he compeiiveness of he indusry. Secion 2 conains he heoreical par of he paper. We firs presen a benchmark model. A some specified dae in he fuure wo symmeric firms engage in Courno compeiion. A dae zero, each firm inheris a producion srucure, which serves as is iniial producion arge. From ha poin onwards, each firm can make coninuous adjusmens o is fuure producion srucure, bu incurs convex adjusmen coss every ime i does so. When inheried producion arges are no oo high, boh firms begin by gradually increasing heir producion plans. Firms 1

4 use hese inended plans as a commimen device; hey wan o commi o high producion levels in order o obain a Sackelberg leadership posiion in he indusry. In equilibrium, however, boh firms are provided wih similar commimen opporuniies, and hereby engage in a Sackelberg warfare, each rying no o become a Sackelberg follower. As he horizon ges closer, however, boh firms become sufficienly commied o producing high quaniies. Thus, a a cerain poin before he final dae, he (dynamic) commimen effec becomes less imporan, while he (saic) incenive o bes respond o he opponen s high producion arge increases and becomes dominan. Therefore, from ha poin on boh firms sar o gradually decrease heir producion plans in he direcion of heir saic bes-response levels. The evenual equilibrium oucome sill remains more compeiive han is saic analog. The res of Secion 2 exends he benchmark model along several dimensions and shows ha all hese exensions reain he same qualiaive predicions. We allow for more han wo players, various forms of asymmeries beween players, ime-varying adjusmen coss, and uncerainy (common across players). We hen nes he benchmark model as he sage game of an infiniely repeaed game. We solve for he Markov Perfec Equilibrium of his game, and show ha is saionary equilibrium pah exhibis he same non-monoonic paern. Moreover, he repeaed game provides a naural way o endogenize he iniial producion plans, which are aken as given in he benchmark model. I also akes he model one sep closer o he realiy of he empirical applicaion we sudy laer in he paper. There are hree key assumpions ha are imporan for our resuls. Firs, conrol variables are sraegic subsiues, leading o a commimen incenive. Second, adjusmen coss are convex, so commimen advanage monoonically increases wih planned producion levels. Third, all he payoffs (ne of adjusmen coss) are colleced in he end, leading o srong compeiive effecs once he producion dae is sufficienly close. Oher assumpions, we believe, are less imporan. For example, all he resuls are obained using a linear-quadraic srucure. Namely, wih linear demand, consan marginal coss, and quadraic adjusmen coss. This is done for racabiliy, as solving for he equilibrium ouside of he linear-quadraic framework is no feasible. Moreover, linear-quadraic games can be viewed as second-order approximaions o more general games. We could also accommodae asymmeric coss, upwards and downwards, wihou affecing he resuls, bu his again would ake us ou of he linear-quadraic framework. 1 The model we presen is a model of endogenous commimen and is herefore relaed o Caruana and Einav (2005), in which we mainly focus on discree decisions, such as enry and exi. The curren work is also close o he dynamic quaniy compeiion lieraure (Cyer and DeGroo, 1970; Hanig, 1986; Fershman and Kamien, 1987; Maskin and Tirole, 1987; Reynolds, 1987 and 1991; Lapham and Ware, 1994; and Jun and Vives, 2004). These papers focus on he saionary equilibrium of an infinie-horizon model (or on he limi of a finie-horizon one, as 1 Saloner (1987) and Romano and Yildirim (2005) sudy an exreme wo-period version of such a model, in which adjusemen coss upwards are free while adjusmen coss downwards are infiniely cosly. Unforunaely, his exreme version gives rise o a wide range of equilibria, and herefore does no provide sharp predicions. 2

5 he horizon ends o infiniy); hey ypically find ha (when acions are sraegic subsiues) he saionary equilibrium is more compeiive han is saic analog, as players engage in a Sackelberg warfare. 2 Our model shares his feaure, bu unlike his lieraure our main focus is on he non-saionary dynamic paern of he planning phase. One advanage in sudying he dynamics of he planning phase is is srong non-saionariy; i provides clear esable predicion wih respec o an observed and exogenous sae variable, namely ime. Saionary dynamic models are much harder o es, as he saic benchmark is ypically no available (for example, marginal coss are ypically no observed). Secion 3 ess he predicions of he model using daa on monhly producion arges by he Big Three auo manufacurers in he U.S. during These producion arges are published in a rade journal approximaely every monh saring as early as six monhs before producion. We normalize producion arges by subsequen producion, pool producion arges from differen producion monhs, and esimae a kernel regression in order o describe he evoluion of hese arges as he producion dae ges closer. The resuls show ha, on average, producion arges exhibi a non-monoonic paern, which is consisen wih he heoreical predicion. Early arges, abou six monhs prior o producion, oversae evenual producion by abou five percen. Then hey sar o slowly increase, unil hey peak a en percen abou 2-3 monhs before producion. A his poin, hey sar o gradually decline owards he evenual producion levels. This resul is robus o alernaive measuremens and across differen subsamples. The end of Secion 3 is devoed o a careful discussion of he relaionship beween he daa analyzed and he heory previously developed. Firs, we discuss poenial sources of adjusmen coss in he producion planning phase of he indusry. In paricular, we emphasize he naure and iming of conracs wih suppliers of pars. Second, we discuss he link beween he real producion plans held by firms and he published figures in he sudy. We argue ha hese are likely o be very relaed. Finally, we discuss some relevan differences beween he sylized heoreical model and he naure of compeiion in he indusry (e.g. invenories and produc differeniaion), and argue ha hese gaps are unlikely o change he qualiaive resuls. Thus, esablishing he relaionship beween he empirical paern and he heoreical predicions allow us o conclude ha adjusmen coss and sraegic consideraions play an imporan role in he planning phase of producion and ha saic models may herefore under-esimae he compeiiveness of he indusry. A some general level, his work can be classified wihin he recen empirical sudies of dynamic oligopolies (e.g. Benkard, 2004; and Ryan, 2004). In conras o hese sudies, which primarily focus on esimaing he parameers associaed wih a given heoreical framework, which is assumed, our heoreical framework provides esable implicaions. Therefore, he primary objecive here is esing he qualiaive predicion of he heoreical framework. Once validaed, he nex obvious sep, which is ouside of he scope of his paper, is o parameerize he model and esimae 2 This can also be viewed as a dynamic exension of a op dog sraegy wihin he Fudenberg and Tirole (1984) axonomy of sraegic behavior. 3

6 srucural parameers. The daa we use in his work is also used in Doyle and Snyder (1999), who invesigae he role of he published producion arges as an informaion sharing device by focusing on he posiive correlaion among manufacurers in he revisions o heir producion arges. Our resuls are consisen wih heir heoreical framework, which provides no resricions on he way producion arges evolve over ime. Their resuls are also consisen wih ours, as he model of his paper predics ha manufacurers would follow similar paerns over ime, hereby creaing posiive correlaion in revisions of producion arges. Therefore, we view he wo sudies as complemenary; he observed paern of producion plans may well be driven by boh informaion-sharing moives as well as sraegic commimen consideraions. In fac, we pool observaions from differen periods in order o average ou he period-specific noise. The period-specific paerns vary quie subsanially and may be driven by differen realizaions of uncerainies. Our framework is herefore more relevan for he average paern raher han for he period-by-period paern, while informaion-sharing moives are more likely o be imporan and observed wihin producion periods. We believe ha any aemp o quanify eiher effec, by, for example, esimaing srucural parameers, should ake boh effecs of sraegic consideraions and uncerainy ino accoun. 2 Theory 2.1 The benchmark model There are wo players. A ime =0, hey sar wih exogenously inheried iniial producion plans of (q 1 (0),q 2 (0)). Aallpoins [0,T] each player i chooses x i R, which conrols he rae a which she changes her producion plan, i.e. qi 0() =x i.noehax i can be eiher posiive or negaive. If a player changes her plans a a rae of x i, she has o pay adjusmen coss of c i (x i,). A ime T, and given heir final plans, q 1 (T ) and q 2 (T ), players compee in quaniies and collec final payoffs ofπ i (q i (T ),q j (T )). In order o make he model more racable, we use a linear-quadraic srucure. Thus, we assume ha inverse demand is linear, given by p = a b(q 1 + q 2 ), and marginal coss are consan and given by c. Thus, we have ha π i (q i (T ),q j (T )) = (a bq i (T ) bq j (T ))q i (T ) cq i (T )= (1) = (a c)q i (T ) bqi 2 (T ) bq i (T )q j (T ) In addiion, we assume ha adjusmen coss are quadraic and ake he form of c i (x i,)= θ 2 x2 i (2) Noe ha adjusmen coss are consan over ime, 3 symmeric across players, and symmeric for 3 For simpliciy, here is no ime discouning. Time discouning is a special case of he exension of he model o ime-varying adjusmen coss, which we analyze laer. 4

7 posiive and negaive raes. None of hese properies is imporan for he main resuls. We solve for he Markov Perfec Equilibrium of he model. Thus, sraegies only depend on he sae variables, q 1 and q 2 and ime. LeVi (q i,q j ) be he value funcion for player i a ime, wih sae variables q i and q j.ifvi (q i,q j ) exiss and is coninuous and coninuously differeniable in is argumens, hen i saisfies he following Bellman equaion µ max θ x 2 V x 2 i + i x i + V i x j + V i =0 (3) q i i q j The firs order condiion for x i implies ha x i = 1 θ V i q i (4) We can now subsiue his back ino equaion (3), and obain he following differenial equaion µ 1 V 2 i + 1 µ Ã! V i Vj + V i =0 (5) 2θ q i θ q j q j The linear-quadraic srucure is aracive. I is known ha in his case, if one resrics he sraegies o be analyic funcions of he sae variables, here exiss a unique equilibrium of he game, which is also he limi of is discree-ime analog. Moreover, in such a case he unique value funcion is a quadraic funcion of he sae variables. 4 Noe ha due o he inheren non-saionariy of he model, he parameers of his quadraic equaion will depend on in an unspecified way. We can express he value funcion as which, using equaion (4), implies ha V i (q i,q j )=A + B q i + C q j + D q 2 i + E q 2 j + F q i q j (6) x i(q i,q j )= 1 θ (B +2D q i + F q j ) (7) Given ha players are symmeric, we can subsiue equaions (6) and (7) ino equaion (5) and obain 0 = 1 2θ (B +2D q i + F q j ) θ (C +2E q j + F q i )(B +2D q j + F q i )+ (8) + A 0 + Bq 0 i + Cq 0 j + Dq 0 i 2 + Eq 0 j 2 + Fq 0 i q j This is a polynomial in q i and q j. Since i has o be saisfied for all values of q i and q j, all is six coefficiens (which are funcions of ) haveobeequalozero.thisgiveshefollowingseof ordinary differenial equaions. To ease noaion, we can jus hink of ime as going backwards. 4 See Kydland (1975), who shows uniqueness for a discree-ime version, and Lukes (1971), Papavassilopoulos and Cruz (1979), and Papavassilopoulos and Olsder (1984) for analysis of exisence and uniqueness in finie-horizon linear-quadraic differenial games. 5

8 This is convenien as our boundary condiion is for = T. Thus, all derivaives wih respec o ime (A 0, B 0, ec.) reverse signs, and he law of moion for he parameers is given by A B 0 B2 + BC 2BD + BF + CF C 0 D 0 = 1 BF +2BE +2CD θ 2D 2 + F 2 (9) E F 2 +4DE 4DF +2EF F 0 wih boundary condiion (for = T ) A T B T C T D T E T F T = 0 a c 0 b 0 b (10) which is provided by he profi funcion in equaion (1). 2.2 Illusraion The sysem of ordinary differenial equaions given by equaion (9), wih is boundary condiion, defines he soluion. I defines he value funcion a any poin in ime, which in urn allows us o compue he equilibrium sraegies using equaion (7). Unforunaely, he sysem canno be solved analyically, so we approximae he equilibrium hrough he soluion of he discree-ime analog of he game for very small ime inervals. Throughou his secion, unless oherwise specified, we se a = b =1, c =0, θ =1,andT =10. This implies ha marginal coss are zero and ha inverse demand is given by p =1 q 1 q 2. Adjusmen coss are c i (x i,)= 1 2 x2 i.5 For laer comparison, i is useful o observe ha, for his choice of parameers, he saic Nash equilibrium of his game involves each player producing her Courno quaniy of q = 1 3, while he Sackelberg leader and follower producion levels are q = 1 2 and q = 1 4, respecively. Figure 1 shows how he parameers of he (symmeric) value funcion, as given in equaion (6), evolve over ime. As he horizon becomes longer (i.e. as T ) A 0 converges o approximaely and all oher parameers approach zero. Thus, for games wih long horizon he equilibrium profis converge o , which are approximaely 17% lower han he saic Courno profis of 1 9 (14% is due o higher producion and lower equilibrium prices, while 3% is due o adjusmen 5 One should noe ha some of hese resricion are no imporan. The effec of a and c only eners hrough heir difference a c, so seing c =0is only a normalizaion. Similarly, opimal sraegies are invarian o monoone ransformaions of he objecive funcion, so, for example, seing b =1is a normalizaion. 6

9 coss). Thisishefirs illusraion of how he dynamic ineracion leads o a reducion in profis. If hey could, he wo paries would have liked o avoid he preparaion race and commi o he saic Courno oucome hroughou. Figure 2 presens he symmeric equilibrium pah for he game in which boh players inheri an iniial producion plan a he saic Courno level. The wo paries begin by increasing heir arges, each rying o become a Sackelberg leader, or a leas no o fall behind and become a Sackelberg follower. As he deadline ges closer, boh firms realize ha hey are sufficienly commied o high oupu, bu ha hey are much above heir saic bes responses, and opimally decide o gradually adjus owards i. Given ha adjusing is cosly, he paries do no adjus all he way o he saic Nash equilibrium. 6 In his paricular example, he equilibrium oucome is abou 0.37, compared o he saic oucome of 1 3. Finally, we also depic one off-equilibrium-pah sraegy for each player. Suppose ha player i receives an unexpeced shock o her inended plan a = T 4 and has her plan revered o he Courno level. Boh players realize ha player j has achieved a leader posiion in he marke. Player j capializes on his advanage by increasing her own plans even furher. Meanwhile, player i s bes response is o rebuild is size. Neverheless, he advanageous posiion acquired by player j never fully diminishes and is kep unil he producion dae. Figure 3 presens he symmeric equilibrium pah for differen iniial producion plans. If hese are no oo high, one observes he same paern as in he previous figure. If iniial producion plans are sufficienly high (greaer han abou 0.44 in his paricular example), boh paries are sufficienly commied o high producion from dae zero and do no need o engage in furher increases of producion arges. The rae a which hey decrease heir producion arges over ime is no consan, however, due o he commimen effec. They firs decrease quaniies slowly, so hey remain commied o high quaniies, and only laer hey speed up adjusmens in he direcion of heir saic bes response levels. 7 Figures 4 and 5 presen comparaive saics wih respec o he lengh of he horizon and wih respec o he size of he adjusmen cos parameer. An inspecion of equaion (9) reveals ha hese wo exercises are similar. A proporional increase in he adjusmen cos can be viewed as a slowdown in he evoluion of he value funcion. Loosely speaking, i is a horizonal srech of Figure 1. Thus, changes in he adjusmen cos parameer are similar o a rescaling of ime. 8 Figure 4 shows how he lengh of he horizon affecs he equilibrium pah. As he horizon 6 Wih convex adjusmen coss, he opimal sraegy always leads o parial adjusmens. This is because he saic profi funcionisfla a he saic bes response level. Thus, he marginal cos of adjusmen is zero for small adjusmens and higher for greaer ones, while he marginal benefi is sricly posiive for small adjusmens bu zero for full adjusmens. 7 Noe ha if he iniial arges were very low and he adjusmen parameers high, one could also see a fully increasing equilibrium pah. 8 I is similar bu no idenical. Think of he game in discree ime. A lower θ is similar o increasing he lengh of a period, wihou changing he number of periods. Increasing T is similar o increasing he number of periods, wihou changing heir lengh. Thus, loosely speaking, sreching of ime allows for more opporuniies o adjus 7

10 ges longer, here is more ime o build up commimen. Similarly, Figure 5 shows ha as he adjusmen coss decrease, building commimen becomes cheaper. In boh cases his leads o higher arges and an ulimae faser decline. 2.3 Inuiion from a wo-period model The key qualiaive predicion of he model, namely ha players have an incenive o exaggerae heir producion inenions as a way o achieve commimen, can be obained wihin he conex of a simple wo-period model. Suppose ha firms sar wih inheried producion arges of y. A =1hey can revise heir plans o z 1 and z 2, bu pay a quadraic adjusmen cos when hey do so. Then, in period =2firms have a final opporuniy o revise he quaniies hey wan o produce and se hem o q 1 and q 2, paying he corresponding adjusmen coss. Given hese producion levels, marke price is given by p =1 q 1 q 2. There is no discouning, so payoffs are he final Courno profis (wih zero marginal coss) minus any adjusmen coss incurred in he process. We can solve for he Subgame Perfec Equilibrium of he game using backward inducion. In period =2each player i chooses q i o solve max(1 (q i + q j ))q i θ q i 2 (q i z i ) 2 (11) Bes response funcions are q i = 1 q j + θz i (12) 2+θ and he second period equilibrium sraegies are q i (z i,z j )= 1+θ (1 + (2 + θ)z i z j ) (13) (θ +3)(θ +1) Onecaneasilyobservehaiffirms arge he Courno quaniies, z i = z j = 1 3, hen seing q i = z i for each i is an equilibrium. In general, he firs order condiions define a bes-response funcion which is a roaion of he saic bes-response a he previously argeed producion level (see Figure 6). Each player s response o a change in her opponen s quaniy is no as srong as in he absence of adjusmen coss. Thus, if z i = z j are greaer (less) han 1 3 he players end up adjusing in he direcion of heir saic bes responses, bu no fully, hereby ending up in a more (less) compeiive equilibrium. In period =1firms choose z i and z j, accouning for he equilibrium sraegies a =2. Thus, each player i chooses z i o solve max x i (1 q i (z i,z j ) q j (z i,z j ))q i (z i,z j ) θ 2 (q i(z i,z j ) z i ) 2 θ 2 (z i y) 2 (14) implying he following firs order condiion for each player: µ q i qi (1 q i q j ) q i + q µ j qi θ(q i z i ) 1 z i z i z i z i behavior. θ(z i y) =0 (15) 8

11 This yields a soluion z(y, θ) and q(y, θ). 9 For example, if y = 1 3,i.e. firms inheried arges are a he Courno level, heir final producions would be q( 1 3,θ)=1 3 + θ 3θ 3 +30θ 2 +78θ +54 (16) which are always above 1 3 for any θ>0. When θ =1, for example, equilibrium arges a =1 are z and final producions are q Thus, he qualiaive conclusions are he same as in he coninuous ime case: planned producion levels increase firs, and decrease laer. 2.4 Exensions o he benchmark model Here we presen some of he mos naural exensions o he benchmark model. The main message is ha all of hem reain he same qualiaive predicions of he model. The derivaions are provided in he appendix. N players: The benchmark model is consruced for wo players only for convenience. Resuls remain unchanged wih more han wo players. The value funcion has one addiional elemen, P P j6=i k6=i,j q jq k, which resuls in an addiional equaion in he sysem of differenial equaions. We compued he equilibrium for differen ses of parameers and he equilibrium paerns are qualiaively idenical o hose obained for he wo-player model. Asymmeric players: Asymmeries among firms can be inroduced eiher hrough he final payoff funcion (for example, firms may vary in heir marginal coss) or hrough he adjusmen coss (for example, labor may be more unionized in one firm han he oher). In he appendix we rea hem joinly, bu we do comparaive saics on each dimension separaely. Figure 7 illusraes he case of asymmeric marginal coss. In paricular, i uses he same parameer values as in Secion 2.2, bu inroduces a (consan) marginal cos of 0.2 for player 2. The figure presens he equilibrium pahs for differen (bu symmeric) iniial condiions. The general paern is similar o he benchmark case. Now he more efficien player produces more han her opponen, and more han her saic Nash equilibrium quaniy (q 1 =0.4 and q 2 =0.2). In his case he less efficien player may produce less han her saic Nash quaniy. This is shown in he hin solid line. The reason for his is ha asymmeric marginal coss inroduce asymmeries in he commimen opporuniies. Given ha he more efficien player is producing more, her saic payoff funcion is seeper around he equilibrium. This allows her o enjoy higher levels of commimen and aain a Sackelberg advanage. In all cases, however, overall quaniy is higher (more compeiive) han he saic equilibrium level of 0.6. This migh hin a welfare improvemen, due o boh higher consumer surplus and more efficien allocaion of resources among he firms, bu one has o include he adjusmen coss in he analysis o obain a definiive answer. 9 The soluion is z(y, θ) = 4+4θ+θ2 +y(θ+1)(θ+3) 2 (26θ+10θ 2 +θ 3 +18) and q(y, θ) = (yθ+2)(θ+1)(θ+3) (26θ+10θ 2 +θ 3 +18). 9

12 Figure 8 presens he case of asymmeric adjusmen coss for differen values of he θ coefficiens. The shape of he equilibria is he same as before. I is ineresing o noice ha i is he more flexible player who is able o end up producing more. When adjusmen coss are high (θ 1 =1and θ 2 =5) his is simply because player 2 canno afford o increase her plans so rapidly (recall ha iniial plans and he lengh of he horizon are fixed in his exercise). When he coss are lower he leadership posiion is achieved hrough he higher abiliy of he flexible player o increase her plans furher as a way o commi o high oupu. 10 Time-varying adjusmen coss: One may argue ha adjusmen coss may vary over ime. One reason may be discouning, which would resul in declining adjusmen coss. I is also reasonable o consider ha adjusmens become more expensive as he producion dae ges closer. As an example, hiring emporary labor hree monhs before producion may be cheap, while labor availabiliy one day before producion is scarce, and will require higher wages or higher search coss on he employer par. 11 I is sraighforward o incorporae such effecs ino he benchmark model. The adjusmen cos funcion would be c i (x i,)= θ() 2 x2 i (17) where no resricions are imposed on θ(). The derivaion of he sysem of ordinary differenial equaions is he same as in equaion (9), wih θ replaced by θ(). Noice ha θ eners ino he sysem in a proporional way. Therefore, replacing i by θ() is similar o a rescaling of ime. When θ() is low he coefficiens on he value funcion change fas, and when θ() is high he coefficiens change slow. Qualiaively, he predicions of he model remain unchanged. Uncerainy: In he presence of uncerainy, here is a general rade-off beween commimen and flexibiliy, as remaining flexible would allow firms o adjus o unexpeced evens. The precise impac of considering uncerainy wihin he conex of his work will depend on he ype of uncerainy explored. In he appendix we consider a model wih a naural source of common uncerainy wihin he linear-quadraic framework. Suppose ha final demand can be high or low depending on wheher he sae of he economy is eiher high (H) orlow(l). The economy (symmerically) flucuaes beween he wo saes following a Poisson process: a each poin, a hazard rae λ he sae changes. Iniially, wih he horizon far enough in he fuure, he curren sae is no paricularly informaive abou he final sae of demand. Given ha firms only care abou he evenual realizaion of demand, on equilibrium hey sar by having a similar behavior independenly of he acual sae. As he producion dae draws near, however, firms become more responsive o changes in 10 Noe ha if he iniial inheried posiions were higher, say q 0 =0.4, and he adjusmen coss high as well, he previous resul could be reversed. In his case he non-flexible player would be a a credible posiion no o change her plans far away from 0.4, which would force he flexible player o adjus downwards. 11 This second case is closer o he framework sudied in Caruana and Einav (2005). 10

13 he sae of he economy. This ypically resuls in upwards (downwards) adjusmens o producion arges in response o changes ino he high (low) sae. As firms foresee his happening, hey are more relucan o adjus early, compared o he benchmark model, and herefore build up commimen more slowly. While he equilibrium pah is random as i depends on he realizaion of uncerainy, he expeced equilibrium pah (compued numerically) exhibis a non-monoonic paern as in he benchmark model. 2.5 Repeaed ineracion Many real-world siuaions, like he monhly producion decisions in he auo indusry we sudy laer, are repeaed in naure. Here we consider an infiniely repeaed game in which he benchmark model is he sage game and here are adjusmen coss beween sages. These coss beween sages capure he fac ha firms are consrained in heir fuure plans by heir acual producion infrasrucure. Formally, each sage of he game is played as follows. Given las period producion of (y 1,y 2 ), players firs decide simulaneously on heir iniial producion plans q 1 (0) and q 2 (0) for nex period, bu pay a cos of ϕ 2 (q i(0) y i ) 2 when hey do so. For he nex T unis of ime hey play he benchmark model wih inheried iniial plans of (q i (0),q j (0)) and quadraic adjusmen coss wih parameer θ. Tha is, hey can coninuously adjus heir producion arges, paying an adjusmen cos of θ 2 (q0 i ())2 if hey do so (where is he ime elapsed since he beginning of he period). A he end of each sage, producion akes place and he sage payoffs are colleced. Players discoun profis wih a common discoun facor β per period. For simpliciy we assume ha players do no discoun payoffs wihinaperiod. We solve for a symmeric Markov Perfec Equilibrium (MPE). Thus, he sae variables are he mos recen producion plans and he elapsed ime. Given ha he game has a linear-quadraic srucure, we guess ha he value funcion is quadraic in he sae variables. We search for an equilibrium saisfying his assumpion and find one, jusifying he iniial guess. The soluion o he value funcion wihin each sage follows he same law of moion as in he benchmark model and hus saisfies equaion (9). The boundary condiion is differen: in his case, i is deermined endogenously as par of he equilibrium. In paricular, here is a relaionship beween he value funcion a he beginning of he sage game and he value funcion a he end of i. We esablish his relaionship below. In equilibrium, players se iniial producion plans o saisfy max A0 + B 0 q i + C 0 q j + D 0 q q i 2 + E 0 q 2 ϕ j + F 0 q i q j i 2 (q i q i (T )) 2 (18) which leads o he following firs order condiion: B 0 +2D 0 q i + F 0 q j ϕ (q i q i (T )) = 0 (19) Equaion (19), ogeher wih is analog for q j, provides a closed-form relaionship beween (q 1 (0), 11

14 q 2 (0)) and (q 1 (T ),q 2 (T )). Since, by consrucion V T i (q i (T ),q j (T )) = π i (q i (T ),q j (T )) β ϕ 2 (q i(0) q i (T )) 2 + βv 0 i (q i (0),q j (0)) (20) we can subsiue he relaionship beween (q 1 (0),q 2 (0)) and (q 1 (T ),q 2 (T )) ino equaion (20). As his has o be saisfied for any q i (T ) and q j (T ) we can equae coefficiens, and obain a sysem of six equaions ha provides a closed-form relaionship beween A 0,...,F 0 and A T,...,F T. This is he boundary condiion ha subsiues equaion (10) of he benchmark game. The soluion o equaion (9) and his new boundary condiion consiues he MPE of he repeaed game. Finally, we focus on he seady sae of he equilibrium, in which he producion decisions (bu no producion plans) are consan a every sage. The equilibrium is compued by numerically searching for a soluion. One sars wih a guess for A T,...,F T, and hen ieraes he law of moion in equaion (9) o obain A 0,...,F 0. Then, using he boundary condiion one obains new values for A T,...,F T. We ierae his procedure unil convergence. Alhough, in general, one canno esablish uniqueness (or even exisence) for his game, he problem seems o be well behaved. The procedure converges exremely rapidly o he same values for a wide range of iniial condiions. Thus, on numerical grounds, we believe ha he repeaed ineracion game has a unique symmeric MPE, or a leas a unique symmeric linear-quadraic MPE. In Figure 9 we show he equilibrium pah for he usual benchmark parameer values (a = b =1, c =0,θ =1,T =10), a discoun facor of β =0.9, andϕ =0.1. As one can see, he equilibrium sage paern exhibis he same hump shape as in he benchmark model. The producion levels are now higher han wha would be produced in he benchmark model if he inheried plans were he ones from he seady sae equilibrium. This is because, in addiion o he commimen effec already described, here is a dynamic effec of commimen hrough he adjusmen coss beween sages. This second effec is he same ha is presen in all dynamic quaniy games wih sicky conrols analyzed in he lieraure (Maskin and Tirole, 1986; Reynolds, 1987 and 1991; Jun and Vives, 2004). Is imporance is diminished in his model by he fac ha he planning phase provides an addiional opporuniy o revise producion levels. Naurally, his addiional dynamic effec increases wih θ and decreases wih T. Figure 10 provides some comparaive saics wih respec o he relaive imporance of he wo ypes of adjusmen coss by varying ϕ and θ. Asone can observe, ϕ primarily affecs he size of he jump beween producion levels and iniial plans for he subsequen producion period, wih high values of ϕ implying small jumps. In conras, θ primarily affecs he shape of he producion plan adjusmens and final equilibrium producion levels. One imporan special case of he repeaed game is he one in which ϕ = 0. Insucha case, here is no link beween consecuive producion periods and he model collapses o he benchmark model wih free iniially chosen plans. Tha is, a =0players decide simulaneously and coslessly on heir iniial plans (q 1 (0),q 2 (0)) and hen coninue playing as in he benchmark model. In he simulaneous-move game played a dae zero players solve equaion (18) (wih 12

15 ϕ =0), implying a unique equilibrium of q j = q i = B 0 2D 0 + F 0 (21) These iniial plans give rise o an equilibrium pah, in which producion plans are fla a =0 and gradually decline hereafer (see also Figure 10). 12 For any ϕ>0, however, he equilibrium pah presens he hump-shaped paern emphasized hroughou. 3 Evidence 3.1 Daa We use daa on domesic producion arges of he major auo manufacurers in he U.S. These are he same daa used by Doyle and Snyder (1999). 13 Therefore, we focus only on he dimensions of he daa ha are relevan for our empirical analysis; Doyle and Snyder (1999) provide descripive saisics and furher deails of he daa. The uni of analysis is a producion monh. Prior o each producion monh, he Big Three U.S. auo manufacurers General Moors (GM), Ford, and Chrysler decide abou heir producion arges for fuure monhs. 14 These arges are posed in a weekly indusry rade journal, Ward s Auomoive Repors, which specializes in indusry daa and saisics. Targes are posed approximaely every monh, saring as early as six monhs prior o acual producion. Producion arges are summarized by he number of cars o be produced by each manufacurer, aggregaed over all models. Thus, variaion across models or he inroducion of new models canno be direcly used. The daa se has a panel srucure and covers he years 1965 o 1995, for a oal of 372 producion monhs. 15 Every ime a producion arge is published, i includes producion arges for all hree manufacurer. Thus, manufacurers do no decide when o pos heir arges, as his is requesed by Ward s. Overall, we observe 1, 621 producion arges for each manufacurer. 16 This amouns o an average of 4.42 producion arges per producion 12 This pah is iniially fla because, in equilibrium, iniial producion plans (qi,qj i0 Vi (qi ) saisfy,q j ) q i =0.From equaion (4), he rae of adjusmen a =0is given by x 0 i (q i,q j )= 1 V0 i (q i,q j ) θ q i, implying x 0 i (qi,qj )=0. 13 We are exremely graeful o Maura Doyle and Chris Snyder for he willingness o share heir daa wih us. 14 These arges are being described by various synonyms: assembly arges, assembly schedules, producion plans, producion forecass, ec. 15 Some of he observaions in he daa include pos-producion revisions. We discard hese observaions. We only focus on arges posed before producion. Five producion monhs have no pre-producion arges, and are herefore omied from he analysis. 16 The daa also include producion arges for American Moors (AMC) unil is exi from he marke in We do no use hese daa for he repored resuls. AMC has a small marke share (2.3% on average) and i exhibis a similar paern o ha of he Big Three, wih he excepion of is las hree years of operaion, during which AMC s marke share, producion, and producion arges rapidly declined. The qualiaive resuls of he paper remain unchanged if we use pre-1984 AMC daa. 13

16 monh, ranging from some cases wih a single producion arge o ohers wih up o 12 associaed arges. 17 Figure 11 presens he oal number of published arges made a each 10 day inerval prior o acual producion. 18 I shows ha producion arges are published approximaely once a monh, ypically on he las week of he monh, alhough one can see some densiy beween he monhly peaks. One can also observe ha he number of observaions is quie sable over he 3-4 monhs before producion. There are significanly fewer earlier observaions. 3.2 Empirical Analysis Le us firs inroduce some noaion. Denoe by Q i he acual quaniy produced by manufacurer i during monh. Denoe by A d i he producion arge made by manufacurer i for producion monh, wih d represening he number of days beween he dae of he producion arge and he arge dae. Namely, if a producion arge A d i is made a dae 0 hen d = 0. Thefocus ofheanalysisisonhewayinwhicha d i evolves wih d. In order o make arges comparable over ime and across manufacurers, we normalize all arges by evenual producion. Namely, a (normalized) producion arge is defined as a d i Ad i Q i Q i (22) Thus, a d i is he percenage deviaion of he arge from he evenual producion; i is posiive (negaive) when a producion arge is higher (lower) han evenual producion. 19,20 Our key heoreical predicion concerns he change of a d i wih respec o d. We expec ad i o gradually increase early on, when d is high (in absolue value), and decrease laer, as i ges closer owards he producion dae. Our analysis is based on pooling observaions from muliple producion monhs. The underlying assumpion is ha, up o he normalizaion, he same game is played repeaedly over ime. This enables us o rea differen producion arges in differen games as if hey are made in he 17 The frequency of posed producion arges significanly increased in he 1970s. The average number of producion arges per producion monh was 2.13 during , compared o 5.94 and 5.32 during and , respecively. 18 Since producion decisions reflec oal producion for he monh, we follow Doyle and Snyder (1999) and use he las day of he producion monh as he relevan dae of producion. 19 This ransformaion of he daa is similar o he PPE measure used in Doyle and Snyder (1999). Our measure uses a slighly differen normalizaion o relae i more closely o he heoreical predicions. All he qualiaive resuls are robus o alernaive normalizaion choices, including he PPE measure of Doyle and Snyder. 20 There are six insances of exreme ouliers. Five of hem are due o unexpeced low Q i s, which generae high a d i s, more han hree imes evenual producion (a d i > 2). The sixh insan is of zero announcemens by Chrysler. While hese cases do no affec he general paern in any imporan way, we drop hem o reduce noise. We ake a conservaive approach and also drop all oher producion arges (a differen imes and by oher manufacurers) associaed wih he same producion monh. This leaves us wih 361 producion monhs and 1, 598 arges by each manufacurer for he empirical analysis. 14

17 same conex. We hen use quaric (biweigh) kernel regressions of a d i on d o non-paramerically describe he evoluion of producion arges over ime. In all figures, we use a bandwidh of 30 days. We ³ repea his exercise for each manufacurer i separaely, for he Big Three average, a d Big3, = 1 3 a d GM, + ad Ford, + ad Chrysler,,andfordifferen subsamples of he daa. In his secion we describe our findings; we defer o he nex secion he discussion of he link beween he empirical exercise and he heoreical assumpions. The key evidence is presened in Figure 12, which pools all producion monhs in he daa. The qualiaive picure is of a non-monoonic paern. On average, producion arges sar abou 5 percen above evenual producion levels and gradually increase. They peak 2-3 monhs before producion a abou 10 percen, and hen gradually decline owards acual producion levels. This paern is no uniform across manufacurers. While Ford and Chrysler, he wo smaller firms, follow a similar non-monoonic paern of producion arges, GM exhibis a differen behavior. GM s average iniial producion arge is abou 15 percen above is evenual producion level, and i gradually declines as he deadline ges closer. This is no inconsisen wih he model: if iniial producion arges are high, he model predics a gradual decline over ime. I would be ineresing o explain why GM s (relaive) iniial producion plans are consisenly higher han hose of Ford and Chrysler. In he repeaed game model, for example, such variaion could arise if he ϕ parameer for GM is sufficienly close o zero. The dashed lines in Figure 12 repor 95 percen confidence inervals. These are compued by boosrapping he daa, and running he same kernel regression on each boosrapped sample; he dashed lines in each figure repor he poin-by-poin 2.5 and 97.5 perceniles. These show ha he observed decline in planned producion owards he producion deadline is quie precisely esimaed. This is a paern ha i exremely consisen across manufacurers and for differen subsamples. Figure 12 also shows ha he confidence inervals significanly shrink as he producion deadline ges closer. This happens for wo reasons. Firs, as may be expeced, he variance in he esimaes is lower close o he day of producion. This may be due o informaion shocks, which are likely o be more pronounced when he producion deadline is furher away in he fuure. The second reason is apparen from Figure 11: he number of observed producion arges 3-6 monhs before producion is significanly smaller han he number of observaions 0-3 monhs before producion. Our heoreical predicion concerns a non-monoonic paern of producion arges wih respec o he same producion monh. A poenial concern may be ha while he average paern shown is qualiaively consisen wih he heoreical predicion, i may be generaed by aggregaion over periods, bu is no presen in individual paerns. 21 To address his concern, we repea he same exercise for differen subsamples of he daa. Figure 13 divides he sample ino hree decades. Figure 14 performs he analysis for each calendar monh separaely o accoun for poenial 21 For example, one could imagine an exreme case in which half of he paerns are monoonically increasing and concave and half are monoonically decreasing and concave. In such a case, he average paern may show non-monooniciy even hough none of he individual paerns is such. 15

18 seasonal variaion (due, for example, o model-year produc-life-cycle effecs; see Copeland, Dunn, and Hall, 2005). Figure 15 repeas he exercise separaely for monhs in which producion growh is posiive and monhs in which producion growh is negaive. All hese exercises show similar qualiaive paerns. Firs, he declining producion arges during he las 2-3 monhs before producion are presen in every single regression. Second, in he majoriy of he cases one can observe he increase in producion arges early on. This second observaion does no hold in every regression. This may be expeced because, as already menioned, he daa are more noisy for early arges. As already discussed, he non-monoonic paern predics a posiive slope of a d i wih respec o d early on, and a negaive slope owards producion. In order o es direcly for he change in slopes, we perform wo final exercises. Firs, we divide producion arges ino hree caegories Early, Middle, and Lae according o how far in advance hese arges are made. Table 1 repors he frequencies in which (i) early arges are lower han inermediae arges, (ii) inermediae arges are higher han lae arges, and (iii) lae arges are higher han evenual producion. We repor his for each manufacurer, as well as for he Big Three average. All hese 12 frequencies excep one are greaer han 50%. None of hem is significanly lower han 50% and he majoriy of hem are significanly higher. This is all consisen wih he heory, and gives suppor o he non-monoonic paern. Second, we define he percenage change, per day, in producion arges by s d i Ad i Ad0 i (d d 0 )A d0 i (23) where A d0 i and Ad i are wo consecuive producion arges associaed wih he same producion monh. We hen run similar kernel regressions of s d i wih respec o d. Figure 16 repors hese regressions. One can observe ha in all cases he slope of producion arges is posiive beween 130 days and 80 days before producion, and ha he confidence inerval for he slope esimaes lies enirely or almos enirely, depending on he manufacurer, in he posiive region. Laer on, he slope is significanly negaive in all regressions, esablishing he non-monoonic paern. 3.3 Discussion The heoreical model presened absracs from cerain imporan aspecs of he empirical applicaion, such as invenories, produc differeniaion, and muli-produc manufacurers. While any quaniaive analysis ough o accoun for hese effecs explicily, we argue ha he qualiaive predicions should sill hold. The key for he heoreical resuls is ha conrol variables are sraegic subsiues. Thus, as long as producion decisions, raher han sales, operae as sraegic subsiues, he exisence of invenories should no have a qualiaive effec on he heory, and herefore on he inerpreaion of he empirical exercise. Moreover, as long as invenories (and, o a lesser exen, quaniy produced abroad) are roughly sable over ime, i seems difficul for 16

19 invenory flucuaions per se o generae he paern of producion arges we observe. 22 Similarly, produc differeniaion and muli-produc firms are also unlikely o change he mainained assumpion ha conrol variables are sraegic subsiues. These assumpions are also consisen wih earlier works, which use a Courno framework o model compeiion among he Big Three. Bernd e al. (1990) canno rejec he Courno model in his conex, and Doyle and Snyder (1999) use i o es for informaion-sharing. In he previous secion we show ha he paern of producion arges in he U.S. auo indusry is consisen wih he heoreical framework. To complee he analysis, i is imporan o discuss wo key aspecs. Firs, we idenify sources of adjusmen coss in he auo indusry. Second, we quesion he manufacurers incenives o reveal heir producion arges ruhfully. We discuss each aspec in urn. Firs, he model assumes ha producion arges are associaed wih some real acions, which canno be coslessly reversed. Auo manufacurers are coninuously aking acions ha affec heir fuure producion capabiliies. They conrac pars from suppliers, hire emporary labor, cancel vacaions, ec. I seems naural o assume ha such producion-relaed decisions are cosly o change. A lae order of pars may be more expensive, revising previously signed conracs may involve penalies, firing workers resuls in compensaion paymens, and changing promises may have repuaional coss. Moreover, auo manufacurers deal wih many hird paries, boh on he supply and he reail level. If hese paries also organize heir plans according o he manufacurers publicly posed arges, a change in hese arges may cause hem some adjusmen coss which may laer feed back o he manufacurers profis. The conracing channel is one of he main sources of adjusmen coss. Given he magniude and iming of he processes involved in he indusry, forward conracs are widespread. In principle, every change in plans would involve renegoiaing hese conracs. In realiy his does no happen so ofen, as conracs ofen sipulae clauses ha deal wih hese insances. Typical par conracs in he indusry explicily specify minimum and maximum monhly orders, assigning financial penalies o deviaions from his conraced range. Even if hese conracs are never renegoiaed, implici adjusmen coss arise when conracing wih differen paries does no simulaneously ake place. Since pars are complemens in producion (consider, for example, an O-ring producion funcion), once conracs are signed sequenially each new conrac represens a sronger commimen o a cerain producion level. Therefore, signing new conracs, which are no fully consisen wih earlier signed conracs, carries an implici adjusmen cos, as i would have been cheaper if previous conracs had been se differenly. Second, in he analysis we assume ha he manufacurers repored producion arges o Ward s ruhfully represen heir real producion plans. This is an imporan assumpion. If hese announcemens were no anchored o any real decision, hey would consiue pure cheap alk. Our view, which is consisen wih conversaions wih manufacurers and Ward s publishers, 22 See Kahn (1992) and Bresnahan and Ramey (1994) for heory and evidence abou he relaionship beween sales and producion. See also Judd (1996) for a dynamic model of invenories in a framework similar o ours. 17

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