OUTPUT CONTINGENT SECURITIES AND EFFICIENT INVESTMENT BY FIRMS

Transcription

1 INTERNATIONAL ECONOMIC REVIEW Vol. 59, No. 2, May 2018 DOI: /ere OUTPUT CONTINGENT SECURITIES AND EFFICIENT INVESTMENT B FIRMS B LUIS H. B. BRAIDO AND V. FILIPE MARTINS-DA-ROCHA 1 Getulo Vargas Foundaton FGV/EPGE, Brazl; Sao Paulo School of Economcs FGV, Brazl, and Ceremade, CNRS, Unversté Pars-Dauphne, PSL Research Unversty, France We analyze compettve economes wth rsky nvestments. Unlke the classc Arrow Debreu framng, frms and agents cannot contract upon the exogenous states underlyng producton rsks. They can trade equtes and any securty wrtten on the endogenous aggregate output. Ths fnancal structure s rch enough to promote effcent rsk sharng among consumers. However, markets are ncomplete from the producton perspectve, and the absence of prces for each prmtve state of nature rases the queston about the objectve of frms. We show that output-contngent asset prces convey suffcent nformaton to compute the compettve shareholder value that leads to effcent nvestment by frms. 1. INTRODUCTION Followng the work by Léon Walras n the nneteenth century, the general equlbrum lterature focused on understandng how anonymous markets coordnate the producton and consumpton of goods n compettve economes. In ths settng, frms productve decsons and agents consumpton choces are taken ndependently, and market prces are the only nstruments avalable to coordnate dfferent wshes. Hayek 1945) supported the vew that compettve prces have the capacty of aggregatng the necessary socal knowledge to nduce effcency of self-nterested decsons. Ths dea was rgorously formulated and ndependently proven by Kenneth J. Arrow, Gerard Debreu, and Lonel W. McKenze durng the 1950s. They lsted condtons for exstence of a compettve equlbrum and proved that, n the absence of externaltes and other market frctons, compettve markets lead proft-maxmzng frms and utlty-maxmzng agents to acheve a Pareto optmal allocaton of resources. The nformaton embedded n market prces s suffcent to promote effcent socal coordnaton across decson makers. Arrow 1953) and Debreu 1959) extended the general equlbrum analyss to economes n whch random states of nature affect producton. They showed that the classc results carry over to envronments wth uncertanty whenever decson makers are able to trade a complete set of contngent clams each of them promsng to delver goods n the future contngent to the verfcaton of a gven state of nature. However, the descrpton and verfcaton of prmtve states are not smple matters, and most securtes traded n modern fnancal markets are contngent on observed outcomes nstead of prmtve states of nature. We analyze compettve fnancal economes n whch frms make rsky nvestments and consumers trade frms equtes and securtes wrtten on frms endogenous producton. Snce Manuscrpt receved August 2015; revsed November Ths artcle has benefted from nsghtful nteractons wth Pero Gottard, Mchael Magll, Herakles Polemarchaks, Martne Qunz, and Paolo Sconolf. We are also thankful to comments from ves Balasko, Alessandro Ctanna, Jan Eeckhout, John Geanakoplos, Jayant Gangul, Chrstan Ghglno, hguo He, Felx Kubler, Karl Shell, Stephen Spear, Jan Werner, and semnar partcpants at the Cowles Foundaton, Exeter Busness School, Sao Paulo School of Economcs, Unversté Pars-Dauphne, Unversty of ork, Econometrc Socety Meetngs at Malaga and Evanston, and NSF/NBER/CEME Conference at Indana Unversty. Fnancal support from CNPq s gratefully acknowledged. Please address correspondence to: Lus Brado, Fundação Getulo Vargas, Escola Braslera de Economa e Fnanças, Praa de Botafogo 190, Ro de Janero, RJ , Brazl. Phone: Fax: E-mal: Lus.Brado@fgv.br. 989 C 2018) by the Economcs Department of the Unversty of Pennsylvana and the Osaka Unversty Insttute of Socal and Economc Research Assocaton

2 990 BRAIDO AND MARTINS-DA-ROCHA our goal s to analyze effcency of frms nvestment decsons, we assume that fnancal markets allow consumers to nsure each other aganst dosyncratc rsks and make consumpton plans contngent on aggregate output. It s well known that ths market structure s suffcent to mplement Pareto optmal allocatons n exchange economes. We analyze the condtons under whch ths also holds for producton economes. Ths topc was ntroduced by Magll and Qunz 2009, 2010). These papers develop a concept for computng the shareholder value of large corporatons and show that Pareto effcency does not always follow from shareholder value maxmzaton. We argue that proft maxmzaton can stll be socally justfed as a decson crteron f frms are assumed to behave as perfect compettors. Although output-contngent prces do not a pror convey all the requste nformaton to coordnate consumpton and nvestment decsons, we show that ths ssue can be overcome f frms and agents antcpate out-of-equlbrum scenaros n a compettve fashon. In the standard Arrow Debreu approach, all relevant nformaton for an effcent nvestment decson s embedded n the state-contngent prces. In the output-contngent framng, however, fnancal markets do not necessarly dstngush across states of nature that lead to the same equlbrum aggregate output. Ths dstncton s stll relevant for frms decsons. We accommodate the compettve prce-takng paradgm by assumng that frms and agents combne market prces wth compettve belefs about how alternatve nvestment plans would affect the condtonal expectaton of each frm s output gven the economy s aggregate producton. 2 We start by stressng the man nsghts n a smple example wth only two possble producton levels. We descrbe the general model and formally defne the compettve fnancal equlbrum n Secton 3. Agents n ths economy can only trade securtes wrtten on endogenous producton levels. Ths fnancal structure s ncomplete wth respect to the underlyng prmtve states of nature but allows consumers to sell ther endowment rsks and make consumpton plans that are contngent on the equlbrum aggregate output. In Secton 4, we ntroduce our vew on how compettve frms should compute the net present value of out-of-equlbrum nvestment plans. We argue that the fnancal equlbrum n whch frms maxmze our defnton of market value s the counterpart of the Arrow Debreu concept. In partcular, we show that the assocated consumpton and nvestment allocaton s Pareto optmal. In Secton 5, we dscuss some mportant topcs related to our contrbuton. We stress how the results depend on our assumptons and then dscuss exstence of a compettve equlbrum for dfferent specfcatons of the productve technology. We also use ths secton to compare our compettve noton of shareholder value to the alternatve concepts n whch frms exert market power by nternalzng parts of the mpact of ther nvestment decsons. Next, n Secton 6, we descrbe our man contrbuton usng the captal asset prcng model CAPM) framework. Fnancal markets trade only stocks and a rsk-free bond. Snce agents hold mean-varance preferences, effcent rsk sharng among consumers s acheved under ths ncomplete fnancal structure. Lke before, ths guarantees that frms can compute the compettve market value for dfferent out-of-equlbrum nvestments and that proft maxmzaton generates Pareto effcency. Ths secton s self-contaned and s partcularly nterestng for the reader who s famlar wth the fnance lterature and s not concerned wth general equlbrum detals. Concludng remarks appear n Secton 7. The Appendx s reserved for techncal arguments. 2 Ths compettveness assumpton on frms s consstent wth the lterature on the objectve of corporatons under ncomplete markets see, for nstance, Drèze 1974), Ekern and Wlson 1974), Leland 1974), Ekern 1975), Grossman and Hart 1979), Hart 1979), Makowsk 1983), and Bsn et al. 2014). Our markets are also ncomplete wth respect to exogenous uncertanty, and the spannng condton of Ekern and Wlson 1974) does not hold here n the sense that nvestments do affect the ndvdual ablty to transfer wealth across prmtve states of nature). However, dfferently from the aforementoned lterature, agents share the same margnal rates of substtuton for each gven equlbrum aggregate output. Our contrbuton s to show that f managers nternalze ths equlbrum feature when formng ther conjectures on frms out-of-equlbrum market values, then effcency s restored.

3 OUTPUT CONTINGENT SECURITIES AN ILLUSTRATION We borrow from Magll and Qunz 2009) the followng smple example wth two perods t {0, 1}, one good, one frm, and one agent. At date 0, the frm undertakes one of two possble nvestment levels a {0, 1}. Ths determnes the probablty over two possble date-1 output levels, namely, y L > 0ory H > y L. The transton a Q H a) represents the probablty of producng y H success) under nvestment a. Investment s productve n the sense that Q H 1) > Q H 0). Ths technology has an alternatve representaton wth three exogenous states of nature {ω H, ω M, ω L } and a random producton functon f a) := f ω H, a), f ω M, a), f ω L, a)), such that f 1) := y H, y H, y L ) and f 0) := y H, y L, y L ). The output s always hgh when the exogenous state s ω H, regardless of the nvestment level. Analogously, t s always low n state ω L. When the state s ω M, the frm s producton s hgh f, and only f, t has nvested a = 1. State probabltes Pω) are exogenous and satsfy Pω H ) = Q H 0), Pω H ) + Pω M ) = Q H 1) and, hence, Pω L ) = 1 Q H 1). 3 At date 0, the agent owns the frm and receves an ntal endowment e 0 > 1. Preferences are represented by the expected utlty u 0 x 0 ) + Eu 1 x 1 )), where the Bernoull utlty functons u 0 and u 1 are strctly ncreasng, contnuously dfferentable, and strctly concave and satsfy the Inada condton at zero. 4 We also assume that 1) u 0 e 0 1) < u 1 y H)y H y L )[Q H 1) Q H 0)]. Snce Bernoull functons are strctly concave, ths condton ensures that nvestng s Pareto optmal. 5 Two assets are traded n the fnancal markets: the frm s equty and a rskless bond. Ths market structure s ncomplete wth respect to exogenous uncertanty, as we have three states of nature but only two assets. Moreover, nvestment plans affect the ndvdual ablty to transfer wealth across prmtve states; that s, the spannng property of Ekern and Wlson 1974) s not satsfed. 6 Therefore, we cannot follow the standard approach of general equlbrum theory to defne unambguously shareholder value for out-of-equlbrum nvestment decsons by means of a nonarbtrage argument. Markets are not complete wth respect to exogenous uncertanty, but they are complete wth respect to endogenous uncertanty. Formally, n the pure exchange economy defned by some arbtrarly gven nvestment level a, trade on the equty and bond markets allows the agent to perfectly nsure aganst producton rsks snce we only have two output levels. In partcular, the consumpton allocaton assocated to any compettve equlbrum s effcent gven the arbtrary nvestment decson a). In ths smple example, the entre producton s 3 The probablty space, P) wth := {ω H,ω M,ω L } represents the exogenous uncertanty snce the probablty measure P does not depend on endogenous actons of the frm or the agent. The probablty space, Qa)) wth := {y H, y L } represents the endogenous uncertanty snce the dstrbuton Qa) depends on the nvestment level a. 4 By that we mean lm xt 0 u t x t) =,fort = 0, 1. 5 Strct concavty mples u 0 e 0 ) u 0 e 0 1) < u 0 e 0 1) and u 1 y H)y H y L ) < u 1 y H ) u 1 y L ). By combnng these nequaltes wth Equaton 1), we obtan that u 0 e 0 ) u 0 e 0 1) < u 1 y H ) u 1 y L ))[Q H 1) Q H 0)]. 6 When the equlbrum nvestment s ā = 1, the equty and bond payoff vectors are respectvely y H, y H, y L ) and 1, 1, 1). The producton vector y H, y L, y L ) assocated wth the out-of-equlbrum nvestment a = 0 does not belong to the marketed space, n the sense that t cannot be generated by the assets payoff vectors y H, y H, y L )and1, 1, 1), respectvely.

4 992 BRAIDO AND MARTINS-DA-ROCHA consumed by the sngle agent at equlbrum. Snce there s only one frm, the aggregate output s ether z H := y H or z L := y L. The lmts of valuaton by nonarbtrage. Consder the pure exchange economy where the frm s nvestment s ā = 1 and denote by Ē ts equlbrum equty prce and by r ts equlbrum rsk-free nterest rate. Snce markets are complete wth respect to producton rsks, the contract payng one unt of the good contngent to aggregate output beng z can be mplemented by tradng the two avalable assets. Denotng by ρz) the nonarbtrage prce of ths contract, we have 1 Ē and ρz L ) = 1 ρz H ) = z H z L z ) L 1 + r z H z L ) zh 1 + r Ē. The standard defnton of the compettve market value of the frm for the out-of-equlbrum nvestment a = 0 s 0) := ω pω)f ω, 0) = pω H )y H + [ pω M ) + pω L )]y L, where pω) s the nonarbtrage prce of the contract payng one unt of the good contngent to the realzaton of the prmtve state ω. However, snce markets are ncomplete wth respect to prmtve states of nature, the valuaton of these contracts s ambguous. Formally, we cannot recover all the prces pω) from the market prces Ē and r or, equvalently, from ρz H ) and ρz L ). Indeed, nonarbtrage only tells us that 2) pω L ) = ρz L ) and pω M ) + pω H ) = ρz H ). The value of a large corporaton. Buldng on the market completeness wth respect to endogenous uncertanty, Magll and Qunz 2009) suggested a defnton for the out-ofequlbrum market value of the frm. They started from the followng descrpton of the equlbrum market value V := Ē ā of the frm, that s, V = y H ρz H ) + y L ρz L ) ā. To make explct the frm s nvestment decson n the above formula, we denote by χz) the equlbrum stochastc dscount factor defned by the equaton 7 ρz s ) = χz s )Q s ā), for all s {L, H}. Therefore, the equlbrum value of the frm takes the followng form: V = y H χz H )Q H ā) + y L χz L )1 Q H ā)) ā. When analyzng the decson of a nonmargnal frm, Magll and Qunz 2009, 2010) suggest replacng ā = 1 n the above formula by a = 0. Ths leads to the followng conjecture: 3) Ma) := y H χz H )Q H a) + y L χz L )1 Q H a)) a of the frm s market value for the out-of-equlbrum nvestment a ā. They show that the Pareto effcent level ā = 1 does not necessarly maxmze M. In our example, ths result follows 7 Consumpton optmalty mples χz s ) = u 1 y s)/u 0 e 0 ā), for s {L, H}.

5 OUTPUT CONTINGENT SECURITIES 993 from the fact that the condton 1) whch mples Pareto effcency of nvestment) may hold smultaneously wth M0) > M1) or, equvalently, 8 4) u 0 e 0 1) > [y H u 1 y H) y L u 1 y L)][Q H 1) Q H 0)]. Ths neffcency s due to a partcular type of market power exerted by the frm. When choosng M0) as ts conjecture for the out-of-equlbrum value, the frm takes nto consderaton that ts nvestment decson affects the dstrbuton of aggregated varables. REMARK 1. Magll and Qunz 2009) analyze the decson of a nonneglgble frm that s aware of the mpact of ts own producton on the level of aggregate output. They assume that the frm s manager takes the stochastc dscount factor functon z χz) as gven. They motvate ths choce by argung that n the settng of captal markets, takng securty prces as gven s wdely regarded as a good approxmaton, even for large corporatons. We take the vew that when corporatons make ther nvestment decsons, they take as gven not only securty prces but also the aggregate output. A compettve conjecture for market value. Defnng a compettve conjecture for the outof-equlbrum market value of the frm s not straghtforward for an economy wth outputcontngent fnancal markets. Securtes wrtten on the aggregate output do not necessarly dstngush across prmtve states of nature. Agents are aware that the equlbrum prces ρz) and the theoretcal state prces pω) are related to each other as stated n condton 2). However, market prces alone do not convey suffcent nformaton for splttng ρz H ) and ρz L )nto pω H ), pω M ), and pω L ) n an unambguous way. To overcome ths ssue, we ask frms and agents to antcpate the followng mportant equlbrum feature when analyzng nvestment plans out of the equlbrum. Markets are complete wth respect to endogenous uncertanty. For any gven nvestment level, the equlbrum rato of margnal utltes s constant across prmtve states of nature assocated wth the same aggregate output. Ths s to say that the agent s valuaton of ncome streams contngent to a specfc aggregate output s rsk neutral. We propose to use ths equlbrum feature to construct the shadow prces pω) from the market prces ρz). When the nvestment s ā = 1, we have pω L ) = ρz L ) and pω H ) + pω M ) = ρz H ). Under ā = 1, the condtonal state probabltes gven the aggregate output z H are Pω H {ω H,ω M }) = Pω H ) Pω H ) + Pω M ) and Pω M {ω H,ω M }) = Pω M ) Pω H ) + Pω M ). Therefore, f we use a rsk-neutral prcng condtonal on z H, we obtan pω H ) = ρz H )Pω H {ω H,ω M }) and pω M ) = ρz H )Pω M {ω H,ω M }). Ths lead us to consder the conjecture a) for the frm s market value under nvestment a {0, 1} defned by 0) := y H pω H ) + y L pω M ) + y L pω L ) 0 = ρz H )[y H Pω H {ω H,ω M }) + y L Pω M {ω H,ω M })] + ρz L )y L 8 Snce u 1 s strctly concave, we have u 1 y H) < u 1 y L). Ths mples that [ yh u 1 y H) y L u 1 y L) ] [Q H 1) Q H 0)] < u 1 y H)y H y L )[Q H 1) Q H 0)]. Therefore, we can always choose u 0 and e 0 such that u 0 e 0 1) les between the rght-hand sde and the left-hand sde of the above nequalty.

6 994 BRAIDO AND MARTINS-DA-ROCHA and 1) := y H pω H ) + y H pω M ) + y L pω L ) 1 = y H ρz H ) + y L ρz L ) 1 = V. It s then smple to verfy that the Pareto optmal nvestment level ā = 1 does n fact maxmze the functon a a). We just need to notce that 1) 0) = u 1 y H) u 0 e 0 1) y H y L )Pω M ) 1 > 0, where the last nequalty follows from assumpton 1) and the fact that Pω M ) = Q H 1) Q H 0). There s an equvalent way to defne prce conjectures wthout relyng explctly on shadow or conjectured) prces pω) for the exogenous states of nature. Let ỹa z) := E[f a) f ā) = z] be the frm s average producton, under nvestment a, across states of nature for whch the equlbrum aggregate output s z. We defne the compettve conjecture for frm s value as V a) := ỹa z H )ρz H ) + ỹa z L )ρz L ) a. In ths llustraton, we have that ỹ0 z H ) = y H Pω H {ω H,ω M }) + y L Pω M {ω H,ω M }), ỹ0 z L ) = y L, and then V 0) = ρz H )[y H Pω H {ω H,ω M }) + y L Pω M {ω H,ω M })] + ρz L )y L = 0). Moreover, we also have ỹ1 z H ) = y H, ỹ1 z L ) = y L, and then V 1) = V = 1). 3. GENERAL MODEL Consder an economy wth two perods t {0, 1}, a sngle good, a fnte set K of frms, and a fnte set I of consumers. At the ntal date t = 0), each frm k selects an nvestment level a k from a set A k R +. Makng no nvestment s always a possblty that s, 0 A k, for every k. At date 1, they are exposed to exogenous shocks ω drawn from a probablty space, F, P). Events B F represent prmtve causes, whch odds are represented by the exogenous probablty PB). Ths probablty s ndependent of consumers and frms actons Technology. The ntal nvestment a k and the exogenous shock ω determne frm k s producton y k = f k ω, a k ) at date 1 from a set k R +. The producton possbltes of the economy are represented by the famly f := f k ) k K of nondecreasng random producton functons f k ω, ) :A k k. We assume that, for each nvestment a k, the functon ω f k ω, a k ) s measurable and essentally bounded on, F, P). 9 From ths standard producton functon framework, we derve the followng alternatve representaton of the productve sector. Defne the sets A := k K Ak and := k K k, ther 9 To fx deas, we can take to be the product space k K k ) Ak and F to be the product k K B k of each Borelan σ-algebra B k defned by the product topology of the space k ) Ak. The support of the probablty P s then assumed to be a subset of k K NAk, k ), where NA k, k ) s the set of nondecreasng functons from A k to k.

7 OUTPUT CONTINGENT SECURITIES 995 respectve elements a := a k ) k K and y := y k ) k K, and the transton probablty a Qa) gven by QB, a) := P{f a) B}), for every Borel set B. 10 The nvestment profle a A undertaken at date 0 determnes the jont probablty Qa) of frms random outcomes at date 1. To represent aggregate producton, we defne the σ-operator to be σy := k K y k, for all y. The random aggregate producton s then represented by the functon ω σf ω, a). We let := k K k denote the set of all possble aggregate outputs and derve the transton probablty a μa) by posng μb, a) := P{σf a) B}), for every Borel set B. We assume that date-1 output s bounded away from zero, n the sense that there exsts ε>0 such that μ[ε, ), a) = 1, for every a A Agents. Each agent has ntal resources consstng of an endowment e 0 > 0 at date 0 and the ownershp shares δ k [0, 1] of each frm k, where I δ k = 1. Agents have no ntal endowment at date 1, so that all consumpton n that perod comes from the frms output. Preferences are represented by a utlty functon that s separable across tme and has the expected utlty form for future rsky consumpton. Let x 0 0 denote agent s consumpton at date 0 and γ be a probablty measure on R + that represents random consumpton at date 1. Agent s expected utlty functon s gven by u 0 x 0 ) + R + u 1 x 1 ) γ dx 1), where u 0 and u 1 are strctly ncreasng, contnuously dfferentable, and strctly concave functons whch map R + nto [, ) and satsfy the Inada condton at zero. REMARK 2. The tme separablty of the expected utlty and the cross-agent homogenety of the Bernoull utlty functons are only assumed to smplfy the notaton. It s straghtforward to extend all results n ths artcle to the case wth heterogeneous Bernoull utlty functons x 0, x 1 ) ν x 0, x 1 ) Compettve Equlbrum for a Gven Investment. For the sake of expostonal clarty, we frst defne a compettve equlbrum for our output-contngent envronment by takng the nvestment level of each frm as gven. After understandng how agents nsure each other, we analyze the problem of how frms choose ther nvestments. Unlke the tradtonal Arrow Debreu model, we do not consder contracts contngent on the realzaton of the prmtve states of nature ω. We assume that the probabltes and the economc consequences of the events n F are well understood by frms and agents, but the costs of descrbng ex ante each prmtve state of nature and enforcng ex post state-contngent contracts are too large. The only traded contracts are those based on frms output. We consder two types of assets: the equty of each frm k K traded n postve net supply and securtes n zero net supply representng bonds and all possble output-contngent dervatves. 10 The set {f a) B} stands for {ω : f ω, a) B}. Smlar notaton omttng ω s used throughout the artcle.

8 996 BRAIDO AND MARTINS-DA-ROCHA Uncertanty only derves from producton rsks. Hence, an effcent allocaton of rsks among consumers for a gven vector of frms nvestments) only requres that agents trade frms equtes and clams contngent on date-1 aggregate output. At date 0, for a gven nvestment profle a, each agent chooses current consumpton x 0 R +, new equty holdngs η R K, and a Borel-measurable) contract θ : R contngent on aggregate output such that 5) x 0 + θ z)ρdz) + E η e 0 + E a) δ, where E stands for the vector of equty prces, and ρ s a postve measure on the Borel sets of such that ρ[0, z]) represents the date-0 prce of the contract delverng one unt of consumpton good contngent on the aggregate output beng lower than or equal to z. At date 1, contngent on output profle y, agent consumes 6) x 1 y) := θ σy) + y η 0. Each agent maxmzes the expected utlty ) u 0 x 0 + u 1 x 1 y) ) Qdy, a) among all ndvdual plans x 0, x 1,η,θ ) satsfyng the budget constrants 5) and 6), for Qa)- almost every y. A fnancal equlbrum assocated wth a gven nvestment profle ā s a lst Ē, ρ, x 0, x 1, η, θ)), where x 0, x 1, η, θ) := x 0, x 1, η, θ ) I s a consumpton-portfolo allocaton such that: ) for every I, the plan x 0, x 1, η, θ ) solves agent s optmzaton problem gven Ē, ρ, ā); ) the consumpton markets clear, that s, e 7) 0 x 0) = ā k I k K and 8) I x 1 y) = σy, for Qā)-almost every y; ) the fnancal markets clear, that s, 9) η = 1 and θ z) = 0, for μā)-almost every z. 11 I I An allocaton x 0, x 1 ), a) s a par composed of a consumpton allocaton x 0, x 1 ) = x 0, x 1 ) I and an nvestment profle a = a k ) k K. An allocaton s sad to be feasble f the consumpton markets clear. An allocaton x 0, x 1 ), a) s sad to Pareto domnate the allocaton x 0, x 1 ), ā) whenever ) u 0 x 0 + u 1 x 1 y) ) ) Qdy, a) u 0 x 0 + u 1 x 1 y) ) Qdy, ā), for every agent, wth strct nequalty for at least one agent. An allocaton x 0, x 1 ), ā) s Pareto optmal f x 0, x 1 ), ā) s feasble and there s no other feasble allocaton x 0, x 1 ), a) that 11 The term 1 represents the K-dmensonal vector of ones.

9 OUTPUT CONTINGENT SECURITIES 997 Pareto domnates x 0, x 1 ), ā). A consumpton allocaton x 0, x 1 )spareto optmal for a gven nvestment ā f x 0, x 1 ), ā) s feasble and there s no other feasble allocaton x 0, x 1 ), ā) wth the same nvestment profle that Pareto domnates x 0, x 1 ), ā). The next result shows that our market structure mplements a Pareto optmal dstrbuton of resources among consumers for any gven nvestment profle. 12 PROPOSITION 1. Fx a fnancal equlbrum Ē, ρ, x 0, x 1, η, θ)) assocated wth an arbtrary nvestment vector ā. The correspondng consumpton allocaton x 0, x 1 ) s Pareto optmal gven ā. The equlbrum measure ρdz) s absolutely contnuous wth respect to μdz, ā). Ths s to say that there s a Borel-measurable functon χ : R + called the stochastc dscount factor) such that 10) ρdz) = χz)μdz, ā). Pareto optmalty of the consumpton allocaton x 0, x 1 ) gven ā mples the ndvdual consumpton x 1 y) to be constant across output vectors y generatng the same aggregate output σy. Therefore, there exst Borel-measurable functons c 1 : R + such that x 1 y) = c 1 σy) for all y and. Snce date-1 aggregate producton s bounded away from zero, our assumptons on the Bernoull utltes mply that the rato of margnal utltes equals the stochastc dscount factor 11) u 1 u 0 x 1 y) ) x 0 ) = u 1 c 1 σy) ) u 0 ) = χσy), x 0 for every and y. As a consequence, the market s rsk neutral condtonal on the aggregate output, and the equlbrum equty prces can be wrtten as 12) Ē = χσy)yqdy, ā). 4. THE MARKET VALUE OF EACH FIRM We now analyze how nvestment levels are chosen by frms n equlbrum. Each frm s assumed to be small relatve to the aggregate economy and does not seek to manpulate prces. Under these compettve condtons, a natural objectve functon for a frm to maxmze s ts market value. When markets are complete wth respect to prmtve states lke n the standard Arrow Debreu framework), the Arrow prces pb) assocated wth any F-measurable) prmtve event B are quoted n the market. Frm k s manager can take these prces as gven and use them to compute the followng conjectured equty value E k a k ):= f k ω, a k ) pdω) assocated wth any out-of-equlbrum nvestment a k. In that case, maxmzng the standard compettve market value 13) k a k ):= E k a k ) a k = f k ω, a k ) pdω) a k leads to Pareto optmalty. 12 The proof follows from standard arguments and the detals are postponed to Appendx A.1.

10 998 BRAIDO AND MARTINS-DA-ROCHA In the absence of securtes whose payoffs are contngent on exogenous events, a deep ssue arses: How should frms assess ther equty value for producton plans dfferent from the equlbrum ones? Or equvalently, how can we defne the conjecture E k a k ) when the prces pdω) are not avalable? 4.1. Compettve Conjectures Condtonal on Aggregate Producton. The equlbrum prces ρdz) allow us to prce by nonarbtrage any bounded) contngent clam wrtten on the aggregate output. The nonarbtrage prce of a clam represented by a bounded functon h : R s hz) ρdz). However, we are also nterested n prcng random varables that are not measurable wth respect to the equlbrum aggregate output. 13 The queston at ssue s then how to extend ths prcng formula to the space of bounded random varables. We recall that markets are ncomplete and, then, there are nfntely many stochastc dscount factors on that are consstent wth the equlbrum prces ρ. These dscount factors, however, do not generate the same value for random varables that are not σf ā)-measurable. We propose to assume that frms and agents ratonally antcpate the equlbrum property that the market s rsk neutral condtonal on aggregate output as follows from Proposton 1 and Equaton 11). Combnng ths wth standard nonarbtrage valuaton leads to the followng defnton of condtonal rsk-neutral valuaton S. For any bounded random varable g : R on, F, P), Sg) := hz) ρdz), where hz) := E[g σf ā) = z]. 14 In other words, among the many possble stochastc dscount factors, we take the one that s constant across states ω that are assocated at equlbrum) wth the same aggregate output z. Under ths condtonal rsk-neutral valuaton, frms and agents hold the followng conjecture for the value of the out-of-equlbrum clam f k a k ): 14) where Ẽ k a k ):= Sf k a k )) = ỹ k a k z) ρdz), 15) ỹ k a k z) := E[f k a k ) σf ā) = z] s the condtonal expected output assocated wth the out-of-equlbrum nvestment a k gven the equlbrum aggregate output z. The value of the frm consstent wth ths prcng rule s gven by 16) V k a k ):= Ẽ k a k ) a k = ỹ k a k z) ρdz) a k. The key behavoral assumpton behnd Equatons 14), 15), and 16) s that frms and agents take prces as gven and form compettve belefs about the condtonal expected producton under dfferent out-of-equlbrum nvestment levels. They understand that frm k s output 13 For nstance, snce the spannng property of Ekern and Wlson 1974) s not necessarly satsfed, the out-ofequlbrum producton clam f k a k ) may not be σf ā)-measurable when a k ā k. 14 By ths we mean that h : R s a Borel functon such that E[g σf ā)] = hσf ā)), almost everywhere.

11 OUTPUT CONTINGENT SECURITIES 999 becomes the random varable ω f k ω, a k ) whenever t nvests a k. However, they also beleve that frm k s decsons do not affect the lkelhood of aggregate producton and, therefore, compute expected producton ỹ k a k z) condtonal on the event {σf ā) = z}. The condtonng event s evaluated at the equlbrum nvestment vector, whch ncludes the nvestment choce of frm k. One could metaphorcally thnk about ths as f there was a contnuum of frms so that the term ỹ k a k z) represented the condtonal expected output when a frm nvested a k whle all other nfnte frms nvested the equlbrum level. REMARK 3CORRECTNESS AT EQUILIBRIUM). Our compettve prce conjectures concde wth the equlbrum prces. Formally, we have Ẽ k ā k ) = ỹ k ā k z) ρdz) = ỹ k ā k z) χz)μdz, ā) = χσy)ỹ k ā k σy)qdy, ā) = χσy)y k Qdy, ā) = Ē k, where these equaltes follow from Equatons 12), 14), and 15) Effcency. We turn now to show the Pareto optmalty of any allocaton x 0, x 1 ), ā) derved from a fnancal equlbrum n whch each nvestment ā k maxmzes the compettve market value V k. We have argued before that the mpossblty to trade assets contngent on prmtve states s an essental ncompleteness of markets from the perspectve of frms. Although consumers do not need to trade securtes contngent on prmtve states to perfectly share dosyncratc rsks, frms need the nformaton embedded n the state-contngent prces p n order to compute the compettve present value k a k ) assocated wth out-ofequlbrum nvestment plans a k ā k, as defned n Equaton 13). We show that ths ncompleteness can be overcome f frms beleve the market s rsk neutral condtonal on aggregate output or, equvalently, f they hold the compettve prce conjectures Ẽ k a k ). To prove ths, we frst show that our defnton of compettve conjectures provdes a connecton between the standard Arrow Debreu concept of compettve equlbrum and our defnton of fnancal equlbrum. We recall that an Arrow Debreu equlbrum s a lst p, ξ 0, ξ 1 ), ā) composed of ) a postve measure p on, F) representng state-contngent prces; ) an allocaton ξ 0, ξ 1 ):= ξ 0, ξ 1 ) I of consumpton plans, where ξ 0 0 and ξ 1 : R + s a random varable; and ) an nvestment vector ā := ā k ) k K such that a) the allocaton ξ 0, ξ 1 ), ā) s feasble, n the sense that 17) e 0 ξ 0 = I k K ā k and 18) ξ 1 ω) = f k ω, ā k ), I k K for P-almost every ω ;

12 1000 BRAIDO AND MARTINS-DA-ROCHA b) for each frm k, the nvestment ā k maxmzes the present-value functon k a k ):= f k ω, a k ) pdω) a k ; c) for each agent, the consumpton plan ξ 0, ξ 1 ) maxmzes the expected utlty u 0 ξ 0 ) + subject to the present-value budget constrant 19) where ā) := ā k )) k K. ξ 0 + u 1 ξ 1 ω) ) Pdω) ξ 1 ω) pdω) e 0 + ā) δ, PROPOSITION 2. There exsts an Arrow Debreu equlbrum p, ξ 0, ξ 1 ), ā) f, and only f, there exsts a fnancal equlbrum Ē, ρ, x 0, x 1, η, θ)) assocated wth the nvestment vector ā such that, for each k, ā k maxmzes the compettve conjecture V k. The proof of ths proposton s gven n Appendx A.2. We sketch the arguments here. It conssts n constructng the elements of a gven equlbrum concept from the elements descrbng the alternatve equlbrum. If p, ξ 0, ξ 1 ), ā) s an Arrow Debreu equlbrum, t follows from standard arguments that the consumpton allocaton at t = 1 only depends on aggregate resources, that s, there are Borel-measurable functons c 1 : R + such that ξ 1 ω) = c 1 σf ω, ā)), for all ω and. We defne x 0, x 1 ) by posng x 0 := ξ 0 and x 1 y) := c 1 σy) for each y and. Snce ξ 1 s σf ā)-measurable, the contngent consumpton x 1 can be mplemented by some portfolo θ, η ). Asset prces are defned usng the standard present value prcng rule: ρdz) := χz)μdz, ā) and Ē := f ω, ā) pdω), where χz) = u 1 c 1 z))/u 0 ξ 0 ) s the equlbrum stochastc dscount factor. Fnally, by the law of terated expectatons, we get that the Arrow Debreu present value k a k ) of the out-ofequlbrum equty concdes wth our defnton V k a k ) of compettve conjecture: 20) k a k ):= f k a k ) pdω) a k = = =: V k a k ). E[f k a k ) σf ā)] pdω) a k ỹ k a k z) ρdz) a k Conversely, let Ē, ρ, x 0, x 1, η, θ)) be a fnancal equlbrum assocated wth the nvestment vector ā such that, for each k, ā k maxmzes the compettve conjecture V k. Recall from Proposton 1 that there are Borel-measurable functons c 1 : R + such that x 1 y) = c 1 σy), for all y and. We then defne the state-contngent consumpton allocaton ξ 0, ξ 1 ) by posng ξ 0 := x 0

13 OUTPUT CONTINGENT SECURITIES 1001 and ξ 1 := c 1 σf ā)). The novelty of our approach s to use condtonal rsk-neutral valuaton n order to defne the state prces p by posng d p ω) := d ρ σf ω, ā)) dp dμā) or, equvalently, pdω) = χσf ω, ā))pdω), where χ ) s the equlbrum stochastc dscount factor defned by the equaton ρdz) = χz)μdz, ā). Ths defnton of p allows us to go from the bottom to the top n Equaton 20). The Frst Welfare Theorem holds true n the Arrow Debreu framework, gven the baselne assumptons of our model. It then follows from Proposton 2 that the allocaton composed of an nvestment vector that maxmzes each frm s compettve conjecture for out-of-equlbrum market value and the consumpton profle from the correspondng fnancal equlbrum s Pareto optmal. Ths s proven n our next result. THEOREM 1. Let Ē, ρ, x 0, x 1, η, θ)) be a fnancal equlbrum assocated wth an nvestment vector ā.ifā k maxmzes V k, for all k, then the allocaton x 0, x 1 ), ā) s Pareto optmal. PROOF. Take the fnancal equlbrum Ē, ρ, x 0, x 1, η, θ)) assocated wth ā. Consder the correspondng Arrow Debreu equlbrum p, ξ 0, ξ 1 ), ā) defned by Proposton 2. Assume then, by way of contradcton, that there s an alternatve output-contngent feasble allocaton x 0, x 1 ), a) that Pareto domnates x 0, x 1 ), ā). For each, defne the state-contngent consumpton plan ξ0,ξ 1 ) by posng ξ 0 := x 0 and ξ 1 ω) := x 1 f ω, a)), for every ω. The statecontngent allocaton ξ 0,ξ 1 ), a) satsfes the feasblty constrants 17) and 18). Moreover, we have u 0 ξ0 ) + u 1 ξ 1 ω) ) ) Pdω) = u 0 x 0 + u 1 x 1 y) ) Qdy, a) ) u 0 x 0 + u 1 x 1 y) ) Qdy, ā) = u 0 ξ 0 ) + u 1 ξ 1 ω)) Pdω), where the nequalty s strct for at least one agent. Ths means that the allocaton ξ 0,ξ 1 ), a) Pareto domnates ξ 0, ξ 1 ), ā), whch contradcts the Frst Welfare Theorem appled to the Arrow Debreu equlbrum p, ξ 0, ξ 1 ), ā). 5. SOME IMPORTANT DISCUSSIONS We use ths secton to dscuss the lmts of our results, comment on the exstence of a compettve equlbrum, and compare our defnton of compettve valuaton wth an alternatve concept used n the lterature. We start by stressng the mportance of our key assumptons About Our Assumptons. Snce the focus of ths artcle s the effcency of frms nvestment decsons, we restrct attenton to securty markets that mplement an effcent consumpton allocaton n a compettve exchange equlbrum for fxed and known nvestment decsons by frms effcency of exchange). We have assumed that a) agents preferences are represented by dscounted expected utlty wth state-ndependent Bernoull functons, b) ther endowments at the second date are determnstc, and c) agents agree on the probablty dstrbuton over the exogenous shocks ω. Under these assumptons, effcency of exchange follows through f fnancal markets allow consumers to trade frms equtes and make consumpton plans

14 1002 BRAIDO AND MARTINS-DA-ROCHA contngent on aggregate output. Consumpton rsks become assocated only wth the varablty of the aggregate output. Effcency of exchange s mantaned f we relax these three assumptons and assume that fnancal markets are suffcently rch to span the uncertanty n the outcomes of the frms, for each possble nvestment profle. 15 However, our defnton of the compettve conjecture for the value of the frm depends on assumpton c) and cannot be easly extended to envronments n whch agents have heterogeneous expectatons over the exogenous shocks. Ths s because usng the condtonal rsk-neutral valuaton to nfer shadow prces for exogenous states may lead to heterogeneous out-of-equlbrum conjectures among agents wth heterogeneous expectatons. We have also focused on a model wth a sngle good and two dates. These modelng choces are only for the sake of smplcty. Our results can be extended to envronments wth fntely many goods and perods as far as the fnancal structure ensures effcency of exchange About Exstence. The objectve of ths artcle s to nvestgate whether the maxmzaton of a sutably defned conjecture of compettve market value leads to effcent nvestment decsons by frms, even f nvestors can only wrte contracts on observable output whch dstrbuton of rsk s endogenous). An mportant related ssue s exstence of a compettve equlbrum where frms maxmze the correspondng compettve market value. Gven an arbtrary nvestment profle a, exstence of a compettve fnancal equlbrum s assured by our assumptons on preferences and postveness of date-0 endowments and of date-1 producton outcomes. In addton to that, f the sets A k were convex and the producton functons a k f k ω, a k ) were contnuous and concave on A k for every ω), then we could assure exstence of a compettve fnancal equlbrum assocated wth an nvestment profle ā that maxmzes each frm s compettve market value V k. Ths follows from Proposton 2 coupled wth classc theorems on exstence of Arrow Debreu equlbrum see Bewley 1972). REMARK 4FINITEL MAN STATES). When the support of the probablty P s fnte, the nvestment sets A k are convex, and the producton functons f k are concave, then our assumpton that securtes markets are complete wth respect to aggregate output may genercally) yeld complete markets wth respect to the prmtve states. 16 Ths feature s not generc when A k s fnte and also does not appear when we have a contnuum of prmtve states of nature. We refer the reader to Appendx A.3 for a smple technology for whch, gven any output profle y and nvestment vector a, there s a contnuum of states ω satsfyng f ω, a) = y. Contnuum of frms. Among the many technologes that do not satsfy the general assumptons for exstence of an Arrow Debreu equlbrum, there s one that deserves partcular attenton. The benchmark producton model n contract theory s such that each frm ether succeeds or fals to produce a certan amount of output, and the probablty of success ncreases wth the frm s nvestment. Ths success-or-falure technology s represented by a nonconvex producton functon, and nonexstence problems may arse. A tradtonal approach to overcome ths ssue consders a contnuum of ex ante dentcal frms wth..d. producton draws. In ths case, all varaton n output that underles ths partcular technology s elmnated, and the objectve of the frm ceases to be an ssue. In the Supportng Informaton, we present a model wth a contnuum of dentcal frms and perfectly correlated success-or-falure shocks. We explctly compute an equlbrum under specfc assumptons and also derve a general exstence result. Ths llustrates an nterestng way to keep varablty n 15 Ths condton s called complete spannng n Magll and Qunz 2010). It means that t s possble at a cost) to fnd a portfolo of bonds, equty contracts, and dervatves whose payoff s one unt f a gven profle of outcomes for the frms s realzed and nothng otherwse. As Ross 1976) showed, n a two-perod model ths s always possble f a suffcent number of optons are ntroduced. 16 Indeed, concavty of the producton functon leads to a contnuum of possble outcomes. Completeness wth respect to aggregate output then requres nfntely) many more securtes than prmtve states. We thank Martne Qunz and Mchael Magll for pontng ths out.

15 OUTPUT CONTINGENT SECURITIES 1003 the average) aggregate output whle smoothng nonconvextes through a contnuum of frms. Note also that modelng the productve sector wth many small frms s consstent wth the behavoral assumptons used along the artcle Strategc Conjectures. When analyzng the decson of nonmargnal frms, Magll and Qunz 2009, 2010) defne the followng compettve conjecture for the value of a frm: 21) M k a k ):= χσy)y k Qdy, a k, ā k )) a k, where a k, ā k ) represents a vector n whch the kth entry of ā s replaced by a k. Each frm k takes as gven the equlbrum stochastc dscount factor χz) condtonal on each aggregate output z and the nvestment vector ā k of all other frms. We have that V k ā k ) = M k ā k ) at any gven fnancal equlbrum Ē, ρ, x 0, x 1, η, θ)) assocated wth ā. However, M k a k ) typcally dffers from our defnton of compettve market value V k a k ) for out-of-equlbrum nvestments a k ā k. The equlbrum n whch each frm k sets an nvestment a k to maxmze M k a k ) s not necessarly Pareto optmal. Ths may be surprsng snce agents dsplay some compettve behavor by takng as gven stochastc dscount factors. However, by analyzng Equaton 21), one realzes that frm k antcpates the mpact of the nvestment a k over the probablty dstrbuton Q, whch turns out to affect the value of all other frms through the probablty Qdy, a k, ā k )). To assess ths ssue from a dfferent perspectve, we use the exogenous shocks to wrte 22) M k a k ) = f k ω, a k ) p k dω, a k ) a k, where p k dω, a k ):= χσf ω, a k, ā k )))Pdω). Recall that f state-contngent clams were ntroduced n the market, all consumers would just be ndfferent to buyng or sellng them f prces satsfed pdω) = χσf ω, ā))pdω). Therefore, we can nterpret pdω) as the compettve market shadow prce for event dω F. By replacng pdω)wth p k dω, a k ), frm k s manager mplctly antcpates the mpact of her nvestment decsons over the aggregate output dstrbuton and manpulates the underlyng state prces that affect the market values of all frms. The neffcences assocated to the conjecture M k are therefore due to the strategc behavor of the frm and are not related to market ncompleteness as n the models analyzed by Geanakoplos and Polemarchaks 1986). Actually, even f all state-contngent clams were avalable for trade, neffcency would stll arse f frms exerted the same market power through ts nvestment decsons usng the prce measure p k dω, a k ) nstead of the equlbrum prce pdω) to compute prce conjectures. 6. PRESENTING THE MAIN POINT IN THE CAPM SETUP Let us now llustrate our man contrbuton n an envronment that s wdely used n fnancal economcs. For ths, we adapt the CAPM envronment descrbed n Magll and Qunz 1996, Chapter 3). There are two perods t {0, 1}, one good, and fntely many frms and nvestors. Uncertanty s represented by an objectve probablty P on a fnte set of prmtve states of nature. We consder the smple case where only frm 1 has an exposure-to-rsk decson. In the ntal perod, ths frm chooses an nvestment level a [0, 1] that determnes ts output next perod, accordng to a bounded random functon f 1 a). The producton of all other frms

16 1004 BRAIDO AND MARTINS-DA-ROCHA s random and not related wth the nvestment a. Weuseσf a) to represent the sum of all productons. 17 The varance varσ f a)) of the aggregate output s strctly postve, for every a. Investors hold postve ntal endowments and own shares of each frm. The current earnng of nvestor s gven by a real number ξ0, and the future random earnngs are represented by a random varable ξ1. These earnngs are ranked by υ 0 ξ 0 ) + E [ υ1 ξ 1 )], where the functons υ 0 and υ 1 are strctly ncreasng on the relevant part of the doman), contnuously dfferentable, strctly concave, and such that E[υ 1 ξ1 )] has a mean-varance representaton. 18 The only traded assets are a rskless bond n zero net supply and frms equtes n postve net supply. Markets are ncomplete, as there are only three assets and possbly more than three states n. Nonetheless, thanks to the mean-varance preference, for any arbtrary nvestment ā, the compettve equlbrum n the bond-equty markets ensures a Pareto optmal dstrbuton ξ 0, ξ 1 ) I of the avalable resources among nvestors. Moreover, there s also a dscount factor χ that s an affne functon of aggregate output. Formally, there exst two strctly postve constants γ and λ such that, for every nvestor, where υ 1 υ 0 ξ 1) ξ 0) = χσ f ā)), χz) = γ λz, for any possble equlbrum aggregate output z. The market value Ē 1 of frm 1 s then gven by the CAPM asset prcng equaton 23) Ē 1 = E[f 1 ā)] 1 + r λcovf 1 ā),σf ā)), where r s the equlbrum rsk-free rate and λ s nterpreted as the market prce per unt of rsk. 19 In ths context, we ask how frm 1 should make ts nvestment decson. Wll the objectve of ths frm lead to Pareto effcency n the producton sde? If markets were complete, then frms and nvestors could use the nformaton contaned n asset prces to nfer the equlbrum Arrow prce pω) for each exogenous state ω. In ths case, any nvestment that maxmzes the Arrow Debreu present value of frm 1, that s, a 1 a) := pω)f 1 ω, a) a, ω s Pareto effcent and unanmously supported by the stockholders. But how to compute a 1 a) when Arrow prces are not avalable? Snce the bond-equty markets are ncomplete, there are nfntely many stochastc dscount factors leadng to the same equlbrum prces but to dfferent evaluatons about out-of-equlbrum nvestment plans a ā. 17 Formally, f we have K frms, then σf a) := f 1 a) + f 2 + +f K, where f k s a random varable, for any k That s, there exsts a functon φ : R R + R such that E[υ 1 ξ1 )] = φe[ξ 1 ], var[ξ 1 ]), where φ s ncreasng n the frst varable and decreasng n the second. 19 These results follow from Magll and Qunz 1996), Theorem 17.3.

17 OUTPUT CONTINGENT SECURITIES 1005 We propose to buld on the CAPM asset prcng Equaton 23) to defne conjectures about how the market would value the frm s equty under an alternatve nvestment level a ā. When makng out-of-equlbrum prce conjectures, we assume that the manager acts compettvely n the followng sense: He takes the rsk-free nterest rate r and the premum for aggregate rsk λ as gven, and he does not nternalze the mpact of hs nvestment decson on the equlbrum aggregate producton σ f ā). These behavoral assumptons lead to the followng defnton of compettve prce conjecture: For each a [0, 1], 24) Ẽ 1 a) := E[f 1 a)] 1 + r λcovf 1 a),σf ā)). If the frm 1 maxmzes the assocated compettve market value functon a V 1 a) := Ẽ 1 a) a, then the correspondng soluton ā s Pareto optmal. To see ths, recall from the Second Welfare Theorem that, after approprate transfers, the effcent allocaton of consumpton ξ 0, ξ 1 ) I can be mplemented as the equlbrum of a pure exchange economy wth a complete set of Arrow securtes contngent on prmtve states ω. Ifwelet m : R + be the Arrow Debreu stochastc dscount factor.e., mω)pω) = pω), for all ω), then we have υ 1 υ 0 ξ 1) ξ 0) = m, for every nvestor. Ths mples the followng relaton between the stochastc dscount factor χ from the bond-equty economy and the Arrow Debreu factor m: As a consequence, we obtan χσf ā)) = m. Ẽ 1 a) := E[f 1 a)] λcovf 1 a),σf ā)) 1 + r = E[f 1 a)]e[ χσf ā))] covf 1 a), χσf ā))) = E[f 1 a)]e[ m] covf 1 a), m) = E[ mf 1 a)] = 1 a) + a. Therefore, an nvestment ā maxmzes V 1 a) = Ẽ 1 a) a f, and only f, t maxmzes 1 a). From the Frst Welfare Theorem for Arrow Debreu economes, such an nvestment level s a Pareto effcent productve decson. Relatng to our man framng. The CAPM prce conjecture defned n Equaton 24) could be obtaned from our man framng by developng Equaton 14) as follows: Ẽ 1 a) = ỹ 1 a z) ρdz) = χσy)ỹ 1 a σy)qdy, ā) = E[ χσf ā))f 1 a)] = E[f 1 a)] λcov[f 1 a),σf ā)], 1 + r

18 1006 BRAIDO AND MARTINS-DA-ROCHA where [1 + r] 1 := E[ χσf ā))] s the equlbrum rsk-free dscount factor. Ths s the CAPM verson of our concept of compettve prce conjectures. When makng out-of-equlbrum prce conjectures, frms and agents take the rsk-free dscount factor r and the premum λ for aggregate rsks as gven and do not nternalze the mpact of the nvestment decson on aggregate output. A smlar decomposton for the market-value concept proposed by Magll and Qunz 2009, 2010) leads to Ma k ) a k := χσy)y k Qdy, a k, ā k )) = E[ χσf a k, a k ))f k a k )] = E[f k a k )] 1 + ra k, ā k ) λcov f k a k ), f k a k ) + f k ā k ) k k where [1 + ra k, ā k )] 1 := E[ χσf a k, ā k )] s a modfed rsk-free dscount factor, and f k a k ) + k k f k ā k ) s the out-of-equlbrum aggregate output. In ths approach, frm k s aware of ts mpact over the aggregate output and makes a conjecture about how ts nvestment decson a k would affect the rsk-free dscount factor and the market rsk premum. Alternatve strategc concepts for market value have been analyzed n the corporate fnance lterature. Important references ncludes Fama 1972), Jensen and Long 1972), Stgltz 1972), Leland 1974), Merton and Subrahmanyam 1974), Greenberg et al. 1978, 1981), Baron 1979), and James 1981), among others. These papers use the CAPM framng. They assume that frms take as gven the equlbrum rsk-free dscount but ncorporate n dfferent ways) the mpact of alternatve nvestments over the aggregate output. 7. CONCLUSION There are two alternatve tradtons n economcs to represent the outcome of rsky enterprses. On one hand, the reference model n macroeconomcs, fnance, and general equlbrum uses the state-of-nature approach, whch reles on random producton functons that map nvestments and random prmtve states of nature wth fxed objectve probabltes) nto realzed outputs. On the other hand, the lterature on contract theory reles on the probablty approach n whch producton s modeled through transton functons mappng nvestments nto probablty measures over the set of possble outcomes. As far as the descrpton of producton possbltes s concerned, the two approaches are equvalent. However, the two approaches dffer on the fnancal contracts that are used to share rsks and drect nvestments. By keepng states of nature hdden, the probablty approach remnds us that wrtng contracts on the prmtve states s not realstc. It s sometmes dffcult to descrbe these states n a contract or to verfy them ex post for executon. Ths s why most fnancal contracts avalable n practce are wrtten on observed producton outcomes or profts e.g., stocks and optons). It s then natural to ask whether ths market ncompleteness generated by the lack of state-contngent clams matters for effcency. It s well known that the ablty to contract upon prmtve states of nature s not essental for an effcent allocaton of resources n exchange economes wth endowment rsks only. If agents can sell ther endowed stocks and trade clams wrtten on the aggregate output, then the equlbrum consumpton s effcent and only vares wth aggregate rsks. The nterestng queston concerns the ablty of fnancal markets to effcently drect frms nvestments. We show that the dffculty rased by the lack of state prces can be overcome f frms and agents compettvely antcpate out-of-equlbrum scenaros. The correspondng market value conjecture s then consstent wth an approprate noton of compettve belefs out of the equlbrum. It descrbes the way each frm assesses the mpact of alternatve out-of-equlbrum nvestments wthout antcpatng