Credit Constraints and Self-fulfilling Business Cycles

Transcription

1 Credi Consrains and Self-fulfilling Business Cycles By Zheng Liu and Pengfei Wang We argue ha credi consrains no only amplify fundamenal shocks, hey can also lead o self-fulfilling business cycles. We sudy a model wih heerogeneous firms, in which imperfec conrac enforcemen implies ha producive firms face binding credi consrains, wih he borrowing capaciy limied by expeced equiy value. A drop in equiy value ighens credi consrains and reallocaes resources from producive o unproducive firms. Such reallocaion reduces aggregae produciviy, furher depresses equiy value, generaing a financial muliplier. Aggregae dynamics are isomorphic o hose in a represenaive-agen economy wih increasing reurns. For sufficienly igh credi consrains, he model generaes self-fulfilling business cycles. JEL: D24, E32, E44. Keywords: Credi consrains, reallocaion, financial muliplier, aggregae produciviy, increasing reurns, indeerminacy. In he presence of credi consrains, financial facors can play an imporan role in macroeconomic flucuaions. For insance, if i is cosly o enforce loan conracs or monior projec oucomes, hen borrowing capaciy will be limied by he value of he borrower s collaeral asses or ne worh. When credi consrains are binding, an increase in asse prices eases he consrains and hus helps expand producion and invesmen. Expanded producion and invesmen in urn raise he borrower s collaeral value and ne worh, furher easing he consrains. This financial acceleraor can, in principle, amplify macroeconomic flucuaions by ransforming small economic shocks ino large business cycles (Kiyoaki and Moore, 1997; Bernanke, Gerler and Gilchris, 1999). Recen sudies show ha he financial acceleraor is empirically imporan, especially for amplifying and propagaing financial shocks (Chrisiano, Moo and Rosagno, 2010; Liu, Wang and Zha, forhcoming). The goal of his paper is o poin ou ha credi consrains no only amplify fundamenal shocks; sufficienly igh credi consrains can also lead o self-fulfilling business cycles. Liu: Federal Reserve Bank of San Francisco, 101 Marke Sree, San Francisco, CA ( zheng.liu@sf.frb.org); Wang: Hong Kong Universiy of Science and Technology, Clear Waer Bay, Hong Kong ( pfwang@us.hk). We are graeful o he edior (John Leahy) and wo anonymous referees for consrucive commens. We also hank Susano Basu, John Fernald, Reuven Glick, Chad Jones, Kevin Lansing, Sylvain Leduc, Mark Spiegel, Chris Waller, Yi Wen, and seminar paricipans a he Bank of France, he Federal Reserve Bank of San Francisco, UC Dan Diego, he 2010 Tsinghua Workshop in Macroeconomics, and he 2011 Shanghai Macroeconomics Workshop for helpful discussions. The views expressed herein are hose of he auhors and do no necessarily reflec he views of he Federal Reserve Bank of San Francisco or he Federal Reserve Sysem. Wang acknowledges he financial suppor form Hong Kong Research Gran Council (Projec ). 1

2 2 AMERICAN ECONOMIC JOURNAL MONTH YEAR A. The key mechanism To make his poin, we sudy a business cycle model wih financial fricion. The model feaures a represenaive household who consumes a homogeneous good and supplies labor o firms. The household invess he good o accumulae capial, which is rened o firms in a compeiive marke. Firms have access o a consan reurns echnology ha ransforms capial and labor ino goods. In each period, firms draw an idiosyncraic produciviy ha is independenly and idenically disribued (i.i.d.) across firms and across ime. If a firm chooses o produce, i hires workers and rens capial from he household bu makes no upfron paymens. In his sense, he household is providing working capial loans o finance he firm s operaion. Afer producion is compleed, firms decide wheher or no o pay heir workers and capial rens. A he end of he period, a firm needs o pay a fixed cos if i would like o say in business for he nex period. Imperfec conrac enforcemen creaes incenive for firms o defaul on loan repaymens. Recognizing he possibiliy of defaul, he household does no lend freely, so ha he amoun of loans available o operaing firms is bounded above by a fracion of firms expeced equiy value. Under opimal incenive-compaible conracs, no firms defaul in equilibrium. Firms wih produciviy above a cu-off level operae and face binding credi consrains. Firms wih produciviy below he cu-off level remain idle in he curren period and say in business expecing ha fuure produciviy may improve. 1 The model generaes a financial muliplier ha amplifies macroeconomic shocks. Since borrowing capaciy is bounded above by firms expeced equiy value, an increase in equiy value raises credi limis for producive firms. Wih more credi available, producive firms expand producion by hiring more workers and rening more capial. Thus, facor prices rise and low-produciviy firms are crowded ou. Since resources are reallocaed o producive firms, aggregae produciviy rises. This leads o furher increases in equiy value and furher expansions in credi, generaing a ripple effec. The reallocaion effec semming from credi consrains leads o procyclical aggregae produciviy. In our model, measured oal facor produciviy (TFP) conains a financial facor, which is a funcion of aggregae leverage measured by he raio of working capial loans o aggregae oupu. An increase in leverage raises firms borrowing capaciy and shifs resources o producive firms. Thus, aggregae produciviy increases wih leverage. Since aggregae produciviy is procyclical, increases in inpu facors lead o more han proporional increases in aggregae oupu. This implies aggregae increasing reurns. The degree of increasing reurns in he aggregaed version of our model corresponds o he size of he financial muliplier. Wih a sufficienly large financial muliplier, he model generaes self-fulfilling, sunspo-driven business 1 Since firms are ex ane idenical, all firms choose o say in business in equilibrium.

3 VOL. VOL NO. ISSUE LIU AND WANG: CREDIT CONSTRAINTS 3 cycles. To undersand our model s mechanism hrough which equilibrium indeerminacy arises, consider he effec of expecaions of higher fuure oupu. Such expecaions shif he labor supply curve upward hrough a wealh effec, as in he sandard real business cycle (RBC) model. In he sandard RBC model, however, he wealh effec raises equilibrium real wage and lowers equilibrium hours and oupu, which would invalidae he iniial expecaions. However, in our model wih credi fricions, he rise in he real wage makes i unprofiable for low-produciviy firms o operae. Thus, labor and capial are reallocaed o high-produciviy firms and aggregae produciviy rises. The improvemen in aggregae produciviy raises aggregae labor demand. Thus, he labor demand schedule shifs upward, offseing he negaive effecs on equilibrium hours from he shif in he labor supply curve. Since a produciviy improvemen also raises firm value, he credi limi for producive firms expands, reinforcing he reallocaion effecs hrough a financial muliplier. The greaer he size of he financial muliplier, he larger he rise in labor demand, and he more likely equilibrium hours would rise. If hours do rise, oupu would rise as well, rendering he iniial expecaions self-fulfilling. B. Empirical evidence Our model s implicaions for he allocaions of credi and producion over business cycles are consisen wih empirical evidence. The model has hree key implicaions. Firs, credi allocaions (leverage) are procyclical, wih credi-o-oupu raio rising in business cycle booms. Second, producive facors are reallocaed from low-produciviy o high-produciviy firms in business cycle booms, implying procyclical aggregae produciviy. Third, he degree of reurns o scale increases wih he level of aggregaion. Each of hese implicaions is consisen wih empirical evidence. Procylical leverage is consisen wih empirical evidence. For example, Jermann and Quadrini (2012) presen evidence ha, in aggregae U.S. daa, deb repurchases relaive o business secor oupu is counercyclical. Their deb repurchases are defined as he reducions in oal non-financial business deb. Thus, heir evidence implies ha deb ousanding relaive o oupu is procyclical. Covas and den Haan (2011) sudy deb and equiy issuances by firms of differen sizes. They repor ha, alhough he cyclical behavior of equiy issuance differs across firms (especially for very large firms), deb issuance scaled by oal asses is consisenly procyclical for firms in all size caegories. These sudies are consisen wih our model s predicions. Empirical sudies also suppor our model s implicaion ha reallocaions help explain procyclical aggregae produciviy. For example, Basu and Fernald (2001) repor ha, in business cycle booms, producive facors are reallocaed from lowmarkup firms o high-markup firms. They furher argue ha, if facor prices are idenical across firms, hen high-markup firms are also hose wih high so-

4 4 AMERICAN ECONOMIC JOURNAL MONTH YEAR cial marginal produciviy. Thus, reallocaions from low-produciviy o highproduciviy firms in business cycle booms raise aggregae produciviy, as our model predics. 2 In our model, firms operae a consan-reurns producion echnology in variable facors, alhough he presence of he fixed cos (i.e., he cos for saying in business) implies firm-level increasing reurns. We show ha, despie a small degree of increasing reurns a he firm level calibraed o mach microeconomic evidence (Basu and Fernald, 1997), procyclical reallocaions of credi and producion generae large increasing reurns a he aggregae level, wih he magniude of increasing reurns corresponding o he size of he financial muliplier. The model s implicaion ha he magniude of reurns o scale increases wih he level of aggregaion is consisen wih he evidence provided by Basu and Fernald (1997). The financial ransmission mechanism in our model is also consisen wih some cross-counry evidence documened in he lieraure. For example, Aghion e al. (2010) presen evidence from a panel of counries ha suggess ha igher credi consrains lead o higher volailiy and lower mean growh. 3 This evidence is consisen wih our model s predicion. In our model, poorer conrac enforcemen implies igher credi consrains, which lead o greaer misallocaion and hus lower levels of aggregae produciviy and oupu. Poorer conrac enforcemen also leads o greaer volailiy of aggregae oupu because i implies a larger financial muliplier ha amplifies fundamenal shocks and, wih sufficienly igh credi consrains, i may also lead o sunspo-driven business cycle flucuaions. C. Relaion o lieraure Our work builds on a large srand of lieraure ha examines he possibiliy of indeerminae equilibria in RBC models. In an influenial sudy, Benhabib and Farmer (1994) firs poin ou ha a sandard one-secor RBC model wih increasing reurns o scale can generae indeerminacy. Farmer and Guo (1994) show ha, in such an economy, sunspo shocks are quaniaively imporan for business cycles. The degree of increasing reurns required o generae indeerminacy in his class of models, however, is considered oo large o be consisen wih empirical evidence (Basu and Fernald, 1995, 1997). Subsequen conribuions by Benhabib and Farmer (1996) and Benhabib and Nishmura (1998) show ha, in muli-secor RBC models, he required exernal- 2 There is also evidence ha capial reallocaions are procyclical (Eisfeld and Rampini, 2006; Gilchris, Sim and Zakrajsek, 2010). To he exen ha a higher volume of capial reallocaions improve he qualiy of maching beween users and producers of capial, such reallocaions can also improve aggregae produciviy in business cycle booms and hus lead o procyclical produciviy. However, unlike Basu and Fernald (2001), hese sudies do no explicily esablish he direcions of reallocaions over he business cycles (e.g., Are resources shifed from low-produciviy firms o high-produciviy firms in booms?). 3 See also Ramey and Ramey (1995), who find ha counries wih lower mean growh experience larger growh volailiy.

5 VOL. VOL NO. ISSUE LIU AND WANG: CREDIT CONSTRAINTS 5 iy o generae indeerminacy is subsanially smaller. Wen (1998) exends he one-secor model in Benhabib and Farmer (1994) by inroducing variable capaciy uilizaion and shows ha he model can generae indeerminacy wih empirically plausible increasing reurns. Benhabib and Wen (2004) sudy a version of he Wen (1998) model and find ha, under parameer configuraions ha allow for indeerminacy, he RBC model driven by demand shocks performs well in maching he business cycle facs along several imporan dimensions. Schmi-Grohe (1997) compares four differen models and finds ha models wih counercyclical markups rely on a lower degree of increasing reurns o generae indeerminacy han hose wih consan markups. Galí (1994) and Wang and Wen (2008) show ha variaions in he composiion of aggregae demand help generae counercyclical markups and indeerminacy. Jaimovich (2007) shows ha a model wih endogenous enry and exi of firms and hus counercyclical markups can generae indeerminacy even wihou increasing reurns. 4 Our model complemens his lieraure by providing a microeconomics foundaion for aggregae increasing reurns hrough financial fricions. We show ha, in a model wih heerogenous firms, credi consrains no jus help amplify fundamenal shocks, hey may also lead o aggregae increasing reurns and self-fulfilling equilibria. Indeed, aggregae dynamics in our model wih credi consrains are isomorphic o hose in a represenaive-agen economy wih increasing reurns, such as he one sudied by Benhabib and Farmer (1994). However, he mechanism hrough which muliple equilibria can be obained is quie differen. In he represenaiveagen model, as shown by Benhabib and Farmer (1994), obaining indeerminacy requires sufficienly large increasing reurns so ha he labor demand curve is upward sloping and has a slope seeper han he labor supply curve. If his condiion is me, hen an upward shif in he labor supply curve (hrough a wealh effec caused by expecaions of higher fuure oupu) would lead o an increase in equilibrium hours as he real wage rises. The increase in hours and oupu would make he iniial expecaion self-fulfilling. A well-known criique of his paricular mechanism is ha he large increasing reurns required o generae indeerminacy in he represenaive agen model are no suppored by empirical evidence (Basu and Fernald, 1997). Our findings sugges ha, in an economy wih heerogeneous firms and credi consrains, obaining muliple equilibria does no require implausibly large increasing reurns. Thus, our model provides a microeconomic foundaion for he sandard model wih increasing reurns hrough financial fricions. The idea ha financial fricions migh lead o muliple equilibria is no new. I has been explored, for example, by Azariadis and Smih (1998). The idea has been revisied in he recen lieraure on raional asse-price bubbles. Examples of his 4 The indeerminacy in his class of business cycle models are dynamic examples of he sunspo equilibria iniially sudied by Azariadis (1981) and Cass and Shell (1983). See Benhabib and Farmer (1999) for a comprehensive survey of he lieraure.

6 6 AMERICAN ECONOMIC JOURNAL MONTH YEAR lieraure include Farhi and Tirole (forhcoming), Marin and Venura (2011), Miao and Wang (2011), and Miao and Wang (2012). The bubble lieraure shows ha, in he presence of financial fricions, asse price bubbles help provide liquidiy and improve capial allocaions. However, he noion of muliple equilibria associaed wih asse price bubbles is differen from ha in our model. While bubbly equilibria represens muliple saddle poins, we obain muliple equilibria around a unique seady sae, which is a sink insead of a saddle poin. Our mechanism is also relaed o Woodford (1986), who recognizes he possibiliy ha borrowing consrains migh generae saionary sunspo equilibria. Woodford (1986) considers an economy wih wo classes of represenaive agens and wih no heerogeneiy wihin each class. Thus, he Woodford model does no generae he reallocaion effec of credi consrains which, as we show, is a cenral ingredien in our model s amplificaion mechanism and crucial for generaing indeerminacy. Our paper adds o he rapidly growing lieraure on he role of financial fricion for macroeconomic flucuaions. A comprehensive survey of ha lieraure is beyond he scope our paper. 5 The credi amplificaion mechanism in our model is closely relaed o he financial acceleraor sudied by Carlsrom and Fuers (1997), Bernanke, Gerler and Gilchris (1999), and Kiyoaki and Moore (1997, 2008), Jermann and Quadrini (2012), among ohers. We follow Kiyoaki and Moore (1997, 2008) and Jermann and Quadrini (2012) and sudy financial fricions ha arise from limied conrac enforcemen. We build on his lieraure and inroduce firm-level heerogeneiy o simplify soluion mehods for his class of models. Since i is difficul o solve a model wih occasionally binding credi consrains, exising lieraure ypically follows Kiyoaki and Moore (1997) and assumes ha borrowers are less paien han lenders so ha credi consrains are binding in he seady sae equilibrium (and also around he seady sae). In our model, only high-produciviy firms are acive and face credi consrains. A he aggregae level, credi fricions are summarized by he cu-off level produciviy ha depends only on aggregae economic condiions. Thus, when we solve for aggregae dynamics, we do no need o deal wih occasionally binding consrains. I. The model The model economy is populaed by wo ypes of infiniely lived agens households and enrepreneurs wih a coninuum of each. The represenaive household consumes and invess a homogenous good and supplies labor and capial o he enrepreneur. The enrepreneur family has a large number of managers, each managing a firm wih a consan reurns echnology ha ransforms labor and capial ino consumpion goods. To incorporae financial fricion, we assume ha firms face idiosyncraic produciviy shocks. If i is profiable o produce, 5 For surveys of he lieraure on financial fricion in DSGE models, see Bernanke, Gerler and Gilchris (1999) and Gerler and Kiyoaki (2010).

7 VOL. VOL NO. ISSUE LIU AND WANG: CREDIT CONSTRAINTS 7 acive firms need o hire workers and ren capial from he household, bu canno make upfron paymens o hese inpu facors unil producion is compleed. Since conrac enforcemen is cosly, firms may defaul on facor paymens. This enforcemen problem gives rise o credi consrains. A. The represenaive household and he represenaive enrepreneur The represenaive household has he uiliy funcion (1) E { } β ln C h N 1+χ a L, 1 + χ =0 where β (0, 1) denoes he household s subjecive discoun facor, C h denoes he household s consumpion, N denoes he hours worked, a L > 0 is he uiliy weigh for leisure, χ > 0 is he inverse Frish elasiciy of labor supply, and E is he expecaion operaor. The household chooses consumpion C h, labor supply N, and new capial sock K+1 h o maximize he uiliy funcion (1) subjec o he sequence of budge consrains (2) C h + K h +1 w N + (1 + r δ)k h 0, and he non-negaiviy consrains C h 0 and K+1 h 0, aking as given he labor wage rae w and he capial renal rae r. The parameer δ (0, 1) denoes he capial depreciaion rae. The represenaive enrepreneur has he uiliy funcion (3) E β ln C e, =0 where β (0, 1) denoes he enrepreneur s subjecive discoun facor and C e denoes he enrepreneur s consumpion. The enrepreneur chooses consumpion C e and new capial sock K+1 e o maximize (3) subjec o he sequence of budge consrains (4) C e + K e +1 (1 + r δ)k e + D, 0, along wih he non-negaive consrains C e 0 and K+1 e 0, aking as given he renal rae r. Enrepreneurs are shareholders of firms. The erm D in he budge consrain denoes he dividend paymens ha he represenaive enrepreneur receives from firms. We assume ha he enrepreneur is sufficienly less paien han he household

8 8 AMERICAN ECONOMIC JOURNAL MONTH YEAR Observe ω, producion decsions Producion compleed Defaul Caugh wih prob θ No caugh Pay fixed coss - 1 No defaul +1 Don pay fixed coss Figure 1. Timing of evens and decisions (i.e., β is sufficienly smaller han β) so ha, in equilibrium, he household is he only saver and he enrepreneur does no hold any capial (i.e., K+1 e = 0 for all 0). Thus, he enrepreneur s consumpion equals dividend paymens from firms. 6 B. The firms The enrepreneur owns a large number of firms wih he mass normalized o one. Each firm has access o a consan reurns echnology ha ransforms capial and labor ino goods. Timing of evens. Timing is essenial for incorporaing financial fricions in he model economy. Figure 1 illusraes he iming of evens. In he beginning of period, each firm observes aggregae shocks and draws an idiosyncraic produciviy ω from a Pareo disribuion defined over he inerval [1, ), wih he disribuion funcion (5) F (ω) = 1 ω σ, where σ > 1 is he shape parameer, which deermines he dispersion of he disribuion.. To simplify analysis, we follow Carlsrom and Fuers (1997) and Bernanke, Gerler and Gilchris (1999) by assuming ha he idiosyncraic produciviy is i.i.d across firm and across ime. Thus, firms are ex ane idenical. 6 The assumpion ha he represenaive enrepreneur is less paien han he represenaive household helps simplify he analysis. The derivaions of firm-level credi consrains below do no rely on his assumpion. In Appendix A.A8, we show ha his assumpion alone is no sufficien o ensure ha firms do no have incenive o accumulae capial. There, we also discuss condiions under which firms choose no o accumulae capial. In solving he model, we focus on equilibria in which firms do no accumulae capial. We laer verify ha hese assumpions are saisfied under our parameer calibraion.

9 VOL. VOL NO. ISSUE LIU AND WANG: CREDIT CONSTRAINTS 9 Afer drawing produciviies, firms decide wheher or no o operae. A firm wih produciviy ω has he producion funcion (6) y (ω) = ωa k (ω) α n (ω) 1 α, where y (ω) denoes oupu, k (ω) and n (ω) denoe capial and labor inpus, α (0, 1) denoes he elasiciy of oupu wih respec o capial inpu, and A denoes aggregae produciviy, which follows a saionary sochasic process. 7 An operaing firm hires workers and ren capial from he represenaive household a he compeiive real wage w and he renal rae r. Since firms do no make upfron paymens o he household, he household is effecively providing credi o firms. We inerpre his credi as working capial loans. When producion is compleed, firms decide wheher or no o repay he wages and rens hey owe o he household. We assume ha conrac enforcemen is imperfec so ha firms have an incenive o defaul. If a firm defauls, i can be caugh wih probabiliy θ, in which even he firm would be perpeually excluded from fuure access o credi. If a defauling firm is no caugh (wih probabiliy 1 θ), however, i appears no differen from a non-defauling firm and coninues o reain access o credi in fuure periods. 8 A he end of period, firms decide wheher or no o say in business for fuure periods. To say in business, a firm needs o pay a fixed cos of φ. Such fixed coss are deadweigh losses o he sociey. The fixed cos is required no jus for firms ha are acive in period, bu also for inacive firms ha choose no o operae afer drawing heir produciviy in he beginning of he period. If a firm has no produced and decides o say in business for he nex period, hen i finances he paymen of he fixed cos by issuing equiy o he represenaive enrepreneur he owner of all firms. Or equivalenly, he fixed cos represens a lump-sum reducion of dividend paymens from firms o he enrepreneur. Since firms are ex ane idenical, hey all decide o say in business if he fixed cos does no exceed he expeced coninuaion value. Afer making he decision for coninuaion in business, firms pay dividends o he enrepreneur (if a firm has no produced in period, hen i pays a negaive dividend ha equals he fixed cos for saying in business). The enrepreneur consumes. The economy hen eners period + 1, wih he same sequence of evens repeaed. 7 The specific sochasic process for A (e.g., wheher i includes a rend or how persisen i is) is no crucial for our analysis. Our goal is o show ha credi fricions a he microeconomic level can generae increasing reurns a he aggregae level and hey can hus poenially generae equilibrium indeerminacy. For his purpose, we do no need o ake a sand on he sochasic processes of fundamenal shocks. We allow A o be ime varying mainly o illusrae how credi consrains amply fundamenal shocks (such as produciviy shocks). 8 The parameer θ can also be inerpreed as he fracion of firm value ha he lender can recover in he even of a defaul (e.g., Jermann and Quadrini (2012)).

10 10 AMERICAN ECONOMIC JOURNAL MONTH YEAR The credi consrains. Since conrac enforcemen is imperfec, firms have an incenive o defaul on heir loans (i.e., wages and rens). Opimal conracs solve he firm s problem subjec o an incenive consrain, so ha no defaul occurs in equilibrium. We now show ha such incenive consrains give rise o a credi consrain for all firms, under which he amoun of loans canno exceed a credi limi deermined by he expeced coninuaion value of he firms. If a firm wih produciviy ω does no defaul a he end of he period, i pays wages and rens o he household and keeps he remaining revenue. Furher, by paying he fixed cos, he firm can say in business in he nex period and obain he coninuaion value. The non-defaul value for he firm is hus given by { (7) V N Λ (ω) = y (ω) w n (ω) r k (ω) + max βe e } +1 Λ e V +1 φ, 0, where Λ e denoes he marginal uiliy of he enrepreneur (who owns he firm) and V V (ω)f(ω)dω denoes he ex ane value of he firm, wih idiosyncraic produciviy inegraed ou. The max operaor on he righ hand side of equaion (7) represens he firm s enry and exi decision. If he expeced coninuaion value exceeds he fixed cos, hen he firm decides o pay he fixed cos and say in business. Oherwise, i exis. Since he idiosyncraic shocks are i.i.d., firms are ex ane idenical so ha hey face idenical enry and exi decisions. We impose parameer resricions such ha all firms choose o say in business. We laer verify ha such parameer resricions are plausible. 9 If he firm chooses o defaul, hen i can keep all he revenue y (ω). If i is caugh (wih probabiliy θ), hen he firm would be excluded from fuure access o credi and i would lose he coninuaion value. If i is no caugh (wih probabiliy 1 θ), however, he firm can say in business and enjoy he coninuaion value provided ha i pays he fixed cos a he end of he period. Thus, he expeced value of defaul is given by { (8) V D Λ (ω) = y (ω) + (1 θ) max βe e } +1 Λ e V +1 φ, 0. The firm chooses no o defaul if and only if V D (ω) V N (ω). Therefore, by comparing (7) and (8), we obain he incenive consrain { Λ (9) w n (ω) + r k (ω) θ max βe e } +1 Λ e V +1 φ, 0. Since we focus on he case in which he coninuaion value ne of fixed coss is 9 We absrac from more ineresing enry and exi decisions in his model, no because we view hese decisions as unimporan bu we would like o gain analyical racabiliy o illusrae he role of financial fricions in generaing indeerminacy.

11 VOL. VOL NO. ISSUE LIU AND WANG: CREDIT CONSTRAINTS 11 non-negaive, he incenive consrain can be rewrien as ( Λ (10) w n (ω) + r k (ω) θ E β e ) +1 Λ e V +1 φ b, where b denoes he credi limi. Thus, imperfec conrac enforcemen gives rise o a credi consrain, under which he available credi (and hus he facor paymens ha a firm can afford) is limied by a fracion of he expeced coninuaion value ne of fixed coss. Since all firms are ex ane idenical, he expeced coninuaion value is also idenical, so ha he credi limi b is independen of firms produciviy. C. Allocaions of credi and producion We now examine he allocaions of credi and producion across firms wih differen produciviy draws. Opimal conracs maximize he firm s value subjec o he incenive consrain (10). Formally, he firm wih produciviy ω has he value funcion (11) V (ω) = y (ω) C(y, ω) φ + βe Λ e +1 Λ e V +1, where he erm C(y, ω) denoes he variable cos funcion. In paricular, (12) C(y, ω) min n,k w n + r k, s.. ωa k α n 1 α y. Cos-minimizing implies ha he variable cos funcion is given by (13) C(y, ω) = y ω ω, where he erm ω is given by (14) ω 1 ( ) 1 α (r w ) α. A 1 α α Thus, we can rewrie he incenive consrain (10) as (15) y (ω) ω ω b. The firm chooses y (ω) o maximize he value V (ω) in (11) subjec o (13), (14), he incenive consrain (15), and a non-negaiviy consrain y (ω) 0.

12 12 AMERICAN ECONOMIC JOURNAL MONTH YEAR Solving he opimal conrac problem gives he allocaions of producion and credi, as we summarize in he following proposiion: PROPOSITION 1: There exiss a cu-off level of produciviy ω [1, ) such ha he producion allocaion across firms is given by (16) y (ω) = { bω ω, ifω ω, 0, oherwise, and he credi allocaion across firms is given by (17) b (ω) = { b, ifω ω 0 oherwise. The credi limi b and he cu-off produciviy ω are defined in (10) and (14), respecively. Proposiion 1 shows ha firms wih sufficienly high levels of produciviy choose o operae and less producive firms remain inacive. The marginal firm wih produciviy ω is indifferen in operaing or no. Wihou loss of generaliy, we assume ha he marginal firm operaes. Under opimal conracs, high-produciviy firms are credi-consrained, wih he credi limi given by a fracion θ of he firms expeced equiy value ne of fixed coss (see (10)). Low-produciviy firms say inacive and hey do no borrow. 10 The parameer θ capures he srengh or effeciveness of conrac enforcemen. Sronger conrac enforcemen (i.e., larger θ) implies a higher credi limi. D. Compeiive equilibrium and aggregaion A compeiive equilibrium consiss of sequences of prices {w, r }, allocaions for he household {C h, K+1 h, N }, allocaions for he enrepreneur {C e, K+1 e }, and allocaions for firms {k (ω), n (ω), y (ω)} such ha, aking he prices as given, he allocaions for each ype of agens solve heir opimizing problems and all markes clear. Facor marke clearing implies ha (18) k (ω)df (ω) = K h + K e = K, n (ω)df (ω) = N. 10 As we have discussed in Secion I.B, hese unproducive firms do no have any revenue o cover he fixed cos and hey finance he fixed cos by paying negaive dividends o he enrepreneur, who owns all firms. Thus, he fixed cos is required for all firms and is ulimaely borne by he enrepreneur in he form of a lump-sum reducion in dividend income.

13 VOL. VOL NO. ISSUE LIU AND WANG: CREDIT CONSTRAINTS 13 Goods marke clearing implies ha (19) C h + C e + K +1 = (1 δ)k + Y φ, where Y = y (ω)df (ω) denoes aggregae oupu. 11 Inegraing he producion allocaion in equaion (16) across firms, we obain aggregae oupu (20) Y = ω ω b ω ωdf (ω) = σ σ 1 b ω σ. Wih consan reurns echnology and perfec mobiliy of facors, he capiallabor raio is independen of firms idiosyncraic produciviy. In paricular, cosminimizing implies ha (21) w r = 1 α α k (ω) n (ω) = 1 α α K N. Cos minimizing also implies ha aggregae facor paymens are given by (22) w N + r K = ω A K α N 1 α. As he capial-labor raio for each firm is idenical, inegraing he producion funcion (6) across firms leads o (23) A K α N 1 α = y (ω) df (ω) = ω ω ω b ω df (ω) = b ω σ 1, where he second equaliy follows from he producion allocaion (16) and he final equaliy from he Pareo disribuion funcion. Combining (20) and (23), we obain he aggregae producion funcion (24) Y = σ σ 1 ω A K α N 1 α. II. Credi consrains and aggregae produciviy Since producive firms face binding credi consrains and canno operae a full capaciy, he presence of credi consrains leads o misallocaion and depressed oal facor produciviy (TFP). We now draw a formal connecion beween credi consrains and aggregae produciviy in our model. We show how credi con- 11 Since all firms no jus acive firms need o pay he fixed cos for saying in business (in equilibrium, hey all choose o pay), aggregae oupu available for consumpion and invesmen is reduced by he amoun of he fixed cos φ.

14 14 AMERICAN ECONOMIC JOURNAL MONTH YEAR srains amplify macroeconomic flucuaions hrough reallocaion of resources. A. Credi consrains and misallocaion The aggregae producion funcion (24) reveals ha measured TFP is given by (25) T F P Y K α N 1 α = σ σ 1 ω A. Thus, measured TFP reflecs he join effecs of rue echnology changes (A ) and endogenous variaions in he cuoff level of produciviy (ω ) ha deermines which firms are acive. Absen credi consrains, he only firms ha operae would be hose wih he highes produciviy ω max. 12 In he presence of credi consrains, however, less producive firms (i.e., hose wih produciviy beween ω and ω max ) become acive. In his sense, credi consrains creae misallocaion of resources, he magniude of which is capured by he cuoff level of produciviy ω. The following proposiion esablishes he relaion beween he endogenous componen of TFP (he misallocaion effec) and he average ighness of credi consrains. PROPOSITION II.1: There is a one-o-one and monoonic mapping beween ω and aggregae loan-o-oupu raio. In paricular, (26) ω = [ b Y ] σ 1/σ. σ 1 The loan-o-oupu raio in urn depends on he ighness of he credi consrains: (27) where φ φ/y. [ ] b β = θ Y 1 β ( 1 σ φ ) φ, PROOF: The equaliy in (26) follows immediaely from (20). To obain he relaion in (27), we firs noe ha (10) implies ha he credi limi b is deermined by expeced fuure equiy value. The equiy value V can be solved ou from he recursive relaion V = C e + βe C e C+1 e V +1, which yields (28) V = 1 1 β Ce. 12 Sricly speaking, he noion of he mos producive firms is a limiing concep since he suppor of he Pareo disribuion is no bounded from above (i.e., wih ω max approaching infiniy).

15 VOL. VOL NO. ISSUE LIU AND WANG: CREDIT CONSTRAINTS 15 In equilibrium, C e given by is he aggregae dividend paymens from all firms, which is (29) C e = Y w N r K φ = 1 σ Y φ, where he las equaliy follows from (22) and (24). We hen have he credi limi [ (30) b = θ βe C e C e +1 ] [ ] β V +1 φ = θ 1 β ( 1 σ Y φ) φ, where he second equaliy uses (28) o subsiue ou V +1 and (29) o subsiue ou C e. Dividing (30) hrough by Y, we obain he desired equaliy in (27). Proposiion II.1 implies ha more credi would be available for producive firms if conrac enforcemen is sronger (i.e., θ is larger). Wih more available credi, resources are more concenraed in high-produciviy firms, so he economy should have a higher level of TFP (see (26)). Empirical sudies sugges ha misallocaion accouns for a large fracion of cross-counry differences in TFP (Hsieh and Klenow, 2009). Our heory suggess ha financial fricion can be a source of misallocaion ha depresses TFP. Since aggregae oupu increases wih he level of TFP, which in urn increases wih θ, we have COROLLARY II.2: Seady-sae oupu is an increasing funcion of θ. B. Amplificaion hrough reallocaion Credi consrains no only lead o misallocaion ha depresses seady-sae TFP, hey also amplify echnology shocks hrough reallocaion of resources beween firms wih differen levels of produciviy. The following proposiion esablishes he condiions under which a financial muliplier arises. PROPOSITION II.3: Holding inpu facors consan, a 1 percen change in echnology shock leads o µ > 1 percen change in aggregae oupu, where µ is given by (31) µ d log Y d log A = σ σ + 1 ξ, ξ [ 1 φσ ] 1 > 1, β where φ = φ/y measures he seady-sae cos of financial inermediaion as a fracion of GDP. PROOF:

16 16 AMERICAN ECONOMIC JOURNAL MONTH YEAR Subsiuing for ω erms, we obain in (24) by he relaions in (26) and (27) and rearranging ( ) σ 1+σ [ βθ 1 (32) Y 1+σ = σ 1 1 β σ Y φ β ] [A K α N 1 α Taking logarihms on boh sides and applying oal differeniaion, we obain (31). Proposiion II.3 shows ha credi consrains help amplify echnology shocks. Weaker conrac enforcemen (i.e., a smaller value of θ) implies sronger amplificaion (i.e., a greaer value of µ). This is because a lower θ implies a lower seady-sae oupu (see Corollary II.2) and hus a higher average fixed cos for firms o say in business (i.e., a higher φ). From equaion (31), i follows ha a lower θ implies a greaer µ. To undersand how he credi amplificaion mechanism works, consider a posiive echnology shock. The shock raises oupu and herefore firms equiy value and he credi limi (see (30)). In he presence of he fixed cos, he credi limi rises more han proporionaely han does aggregae oupu because he boom in oupu reduces he average fixed cos for firms o say in business (see (27)). In oher words, he loan-o-oupu raio is procyclical. The increased credi limi enables high-produciviy firms o expand producion and forces low-produciviy firms o become inacive. As a consequence, he cuoff level of produciviy shifs up as he loan-o-oupu raio rises (see equaion (26)), which raises measured TFP (equaion (25)) and hereby reinforcing he iniial echnology shock. Following a negaive echnology shock, he same logic applies wih he direcion reversed. Credi consrains hus generae a reallocaion effec ha leads o procyclical aggregae produciviy and amplificaion of macroeconomic flucuaions. ] σ. III. Aggregae Increasing Reurns and Indeerminacy The reallocaion effec represens a source of inefficiency ha sems from credi consrains. In an unconsrained economy, resources are concenraed in he mos producive firms and changes in equiy value do no have direc impac on producion allocaion. In he presence of credi consrains, however, changes in equiy value direcly affec firms borrowing capaciy and lead o procyclical reallocaion. We have esablished ha he reallocaion effec of credi consrains is crucial for amplifying fundamenal shocks (such as echnology shocks). We now show ha credi consrains a he firm level are equivalen o increasing reurns a he aggregae level, despie ha all firms operae a consan reurns echnology wih respec o variable facors. Furher, he economy wih sufficienly igh credi consrains is prone o indeerminae equilibria and sunspo driven flucuaions.

17 VOL. VOL NO. ISSUE LIU AND WANG: CREDIT CONSTRAINTS 17 A. Credi consrains and aggregae increasing reurns To see ha aggregae echnology in he model exhibis increasing reurns, we log-linearize he reduced-form aggregae producion funcion (32) around he (unique) deerminisic seady sae o ge (33) ŷ = µ[â + αˆk + (1 α)ˆn ], where a haed variable denoes he log-deviaions of he corresponding variable from is seady-sae value (e.g., ŷ log(y /Y ), where Y is he seady-sae level of oupu). The erm µ is he financial muliplier given by (31). Equaion (33) shows ha µ also measures aggregae reurns o scale. This resul is formally saed in he following proposiion. PROPOSITION III.1: The reduced-form aggregae echnology in our model exhibis increasing reurns if and only if here is a posiive financial muliplier (i.e., µ > 1). B. The possibiliy of self-fulfilling equilibria I is well known ha an economy wih increasing reurns can be prone o selffulfilling, sunspo driven business cycles (Benhabib and Farmer, 1994). Since credi consrains in our model are observaionally equivalen o aggregae increasing reurns, we now examine he condiions under which sunspo-driven flucuaions can occur. To help exposiion, we follow he lieraure by focusing on local dynamics around he deerminisic seady sae and we absrac from aggregae shocks. We firs esablish analyical condiions for indeerminacy o arise. We hen evaluae he empirical plausibiliy for credi fricions o generae indeerminacy under calibraed parameer values based on numerical simulaions. The following proposiion summarizes he condiions for equilibrium indeerminacy. PROPOSITION III.2: Assume ha he oupu elasiciy of capial in he aggregae producion funcion is less han one (i.e.,µα < 1). Assume also ha 2 β < 1+χ 1 α. The necessary and sufficien condiion for equilibrium indeerminacy in he benchmark economy is given by (34) µ > max{µ 1, µ 2} µ, where (35) µ 1 = β(2 δ)(1 + χ) (1 + β)(1 α)β(1 δ) [1 β(1 δ)] (1 + χ),

18 18 AMERICAN ECONOMIC JOURNAL MONTH YEAR and (36) µ 2 = { } β(2 δ) + [1 β(1 δ)] 1 β+βδ(1 α) 2α (1 + χ) [ ]. (1 + β)(1 α)β(1 δ) [1 β(1 δ)] (1 + χ) 1 1 β+βδ(1 α) 2 PROOF: See Appendix A.A2. Proposiion III.2 implies ha, all else equal, equilibrium indeerminacy arises if he size of he financial muliplier (µ) is sufficienly large. To undersand how indeerminacy arises in our model, consider a hypoheical increase in expeced fuure equiy value wihou any fundamenal shocks. If such expecaions can be validaed in an equilibrium, hen indeerminacy and selffulfilling equilibria would arise. If such expecaions are no validaed, however, here is a unique equilibrium. [INSERT FIGURE 2 HERE] We illusrae here wih a labor marke diagram he mechanism hrough which credi consrains can generae indeerminacy. Figure 2 (he op panel) shows he adjusmens in he labor marke in our benchmark model wih credi consrains following a rise in expeced fuure equiy value. We assume ha he labor supply curve is fla, or equivalenly, labor is indivisible (Hansen, 1985; Rogerson, 1988). Suppose ha he iniial equilibrium is a poin A. The expecaion of higher fuure equiy value creaes a posiive wealh effec ha shifs he labor supply curve upward. In a model wihou credi fricions, such as he sandard real business cycle (RBC) model, labor demand does no shif because he capial sock is predeermined and here is no fundamenal echnology shock. Thus, he upward shif in he labor supply curve leads o a rise in equilibrium real wage and a fall in equilibrium employmen (from poin A o poin B). The decline in employmen leads o a recession, which invalidaes he iniial expecaions of a higher equiy value. Thus, he sandard RBC model wihou credi consrains implies a unique equilibrium. In he presence of credi consrains as in our benchmark model, however, he expecaion of a rise in equiy value enables high-produciviy firms o borrow more and produce more and hus leads o a reallocaion ha implies higher aggregae produciviy. Wih higher aggregae produciviy, he aggregae labor demand curve shifs upward. The igher he credi consrains, he greaer he financial muliplier, and he larger he increase in aggregae produciviy following an expecaion shock. Thus, wih a sufficienly large financial muliplier (µ), he upward shif in he labor demand curve can more han offse he recessionary effecs of he upward shif in he labor supply curve; i can poenially lead o

19 VOL. VOL NO. ISSUE LIU AND WANG: CREDIT CONSTRAINTS 19 an expansion in equilibrium employmen and oupu (from poin B o poin C). Such an expansion confirms he iniial expecaions and hus leads o self-fulfilling equilibria. C. Relaion o Benhabib and Farmer (1994) In he coninuous-ime limi wih δ 0 and β 1, we have µ 1 = µ 2 = 1+χ 1 α. In his case, he indeerminacy condiion in equaion (34) reduces o (37) µ(1 α) 1 > χ, which is apparenly idenical o ha obained by Benhabib and Farmer (1994) in he real business cycle model wih increasing reurns, where µ corresponds o he degree of reurns o scale in heir aggregae producion funcion. 13 The mechanism hrough which indeerminacy arises, however, is quie differen. In heir model, indeerminacy requires a large enough increasing reurns so ha he labor demand curve is upward-sloping, wih a slope seeper han he labor supply curve (see he boom panel of Figure 2). Under such condiions, he upward shif in he labor supply curve following a change in expecaions of fuure oupu raises he real wage and moves along he (upward-sloping) labor demand curve, resuling in an increase in equilibrium hours and an expansion in oupu. This expansionary effecs confirm he iniial expecaions, rendering he expecaions self-fulfilling. In conras, our model does no require labor demand curve o be upwardsloping. Insead, he presence of credi consrains inroduces a financial facor in oal facor produciviy, so ha aggregae produciviy increases when equiy value is expeced o rise; he increase in aggregae produciviy shifs up he downward-sloping labor demand curve, which can poenially lead o an expansion in equilibrium hours and oupu. IV. Empirical plausibiliy of self-fulfilling equilibria We have esablished ha, o a firs-order approximaion, he model wih financial fricion is observaionally equivalen o a represenaive-agen economy wih aggregae increasing reurns (Proposiion III.1). We have also shown ha i is possible, a leas in heory, o generae self-fulfilling equilibria in our model, provided ha he financial muliplier is sufficienly large (Proposiion III.2). We now examine he empirical plausibiliy of self-fulfilling business cycles under calibraed parameers in an exended version of he model wih variable capaciy uilizaion. In an imporan conribuion, Wen (1998) shows ha incorporaing variable capaciy uilizaion in he Benhabib and Farmer (1994) model helps o 13 In a working paper version (Liu and Wang, 2010), we provide a formal proof ha he indeerminacy condiions in our model are idenical o hose in he real business cycle model wih increasing reurns sudied by Benhabib and Farmer (1994).

20 20 AMERICAN ECONOMIC JOURNAL MONTH YEAR lower subsanially he required degree of increasing reurns o generae muliple equilibria. We now show ha our model wih credi consrains generalized o incorporae variable capaciy uilizaion can generae muliple equilibria under plausible parameer calibraion. A. The model wih variable capaciy uilizaion Following Wen (1998), we assume ha increases in he capaciy uilizaion rae accelerae capial depreciaion. In paricular, we assume ha he household s budge consrain (2) is replaced by (38) C h + K +1 w N + [1 + r u δ(u )] K, 0, where u denoes he capaciy uilizaion. As in he benchmark model, all capial accumulaion is done by he household. The capial depreciaion rae varies wih capaciy uilizaion according o u 1+η (39) δ(u ) = δ η, where δ 0 (0, 1) is a consan and η > 0 measures he elasiciy of he depreciaion rae wih respec o capaciy uilizaion. The household s opimizing choices now include an addiional endogenous variable he capaciy uilizaion rae u. Wih variable capaciy, he aggregae producion funcion (32) becomes [ (40) Y = σ ( ) ] βθ 1 1 σ 1 1 β σ Y φ σ+1 ( A (u K ) α N 1 α ) σ σ+1. where φ = φ β. Here, unlike in he benchmark model, he quaniy produced depends on he effecive capial services u K raher han he physical unis of capial K. We firs noe ha inroducing variable capaciy uilizaion raises he magniude of he financial muliplier. In paricular, we have PROPOSITION IV.1: In he exended model wih variable capaciy uilizaion, a 1 percen change in TFP holding inpu facors consan resuls in µ > 1 percen change in aggregae oupu, where (41) µ d log Y d log A = µ(1 + η) 1 + η αµ > µ, where µ is he financial muliplier in he benchmark model given by equaion (31).

21 VOL. VOL NO. ISSUE LIU AND WANG: CREDIT CONSTRAINTS 21 PROOF: See Appendix A.A4. Accordingly, inroducing variable capaciy uilizaion makes indeerminacy more likely, as we show in he Proposiion below. PROPOSITION IV.2: Assume ha α η 1+η µ < 1. The necessary and sufficien condiion for equilibrium indeerminacy in he model wih variable capaciy uilizaion is given by (42) µ > max( µ 1, µ 2) µ, where (43) (44) µ 1 = µ 2 = 2(1 + χ) (1 + β)(1 α) 1+η α (1+η) (1 + χ)(β 1 1), 2(1 + χ) η α 2 (1 β) α (1 + χ)δ (1 + β)(1 α) η(1+η α) (1+η) (1 + χ)δ[1 1 2 (1 β)]. PROOF: The proof is similar o ha of Proposiion III.2, wih he financial muliplier µ replaced by µ. In he coninuous-ime limi wih δ 0 and β 1, we again obain µ 1 = µ 2 = 1+χ 1 α. In his case, he necessary and sufficien condiions for indeerminacy are simplified o µ > 1+χ 1 α. B. Parameer calibraion The parameers o be calibraed include he subjecive discoun facors β for he household and β for he enrepreneur, he capial depreciaion rae δ, he capial income share α, he inverse Frisch elasiciy of labor supply χ, he elasiciy η of depreciaion wih respec o capaciy uilizaion rae, he scale parameer of he Pareo disribuion for he idiosyncraic produciviy shocks σ, he parameer measuring he srengh of conrac enforcemen θ, and he seady-sae raio of fixed coss o aggregae oupu φ. Table 1 summarizes he calibraed parameer values. We se β = 0.99, implying an annual risk-free ineres rae of 4 percen. We se δ = 0.025, corresponding o an annual capial depreciaion rae of 10 percen. We se α = 0.3 o mach he labor income share of 70 percen in he U.S. daa. We assume ha labor is indivisible (Hansen, 1985; Rogerson, 1988), which implies ha χ = 0. We follow Wen (1998) and se η = 0.4. The remaining parameers are hose relaed o financial fricion. We se he subjecive discoun facor for he enrepreneur o β = 0.98, implying an average

22 22 AMERICAN ECONOMIC JOURNAL MONTH YEAR Table 1 Calibraed Parameers Parameer Descripion Value β Household discoun facor 0.99 β Enrepreneur discoun facor 0.98 α Capial income share 0.30 δ Capial depreciaion rae η Uilizaion elasiciy of depreciaion 0.40 χ Inverse Frisch elasiciy of labor supply 0 σ Shape parameer of produciviy disribuion 6.00 θ Conrac enforcemen 0.41 φ Seady-sae share of fixed cos in aggregae oupu 0.10 excess reurn of abou 4 percen per year, in line wih he esimaes obained by Liu, Wang and Zha (forhcoming). Since acive firms have higher produciviy han he cu-off level, hese firms σ σ 1. earn economic profis, wih he seady-sae profi rae given by We se σ = 6, which implies a seady-sae profi rae of abou 17 percen for acive firms. 14 We se φ = 0.1, which, ogeher wih he calibraion of σ = 6, implies an average economic profi (ne of fixed coss) of abou 7 percen, in line wih he empirical evidence provided by Basu and Fernald (1997). Given he calibraed values of β, σ, and φ, we se θ = 0.41 so ha he model implies a seady-sae raio of privae credi o quarerly GDP of 2.08, as in he U.S. daa. 15 These calibraed parameers imply a financial muliplier of µ = 1.11 in he benchmark model and µ = 1.45 in he exended model wih variable capaciy uilizaion. The size of he financial muliplier µ implies ha, under he calibraed labor share and Frisch elasiciy of labor supply, he condiion (34) for muliple equilibria is no saisfied and hus he benchmark model has a unique local equi- 14 The economic profi in our model is differen from he sandard monopolisic markup because firms in our model are compeiive. 15 We measure nominal privae credi by he non-farm, nonfinancial business liabiliies based on credi marke insrumens. The daa are aken from he Flow of Funds Tables of he Federal Reserve hrough Haver Analyics (privae credi is he sum of OL10TCR5@FFUNDS and OL11TCR5@FFUNDS). The sample mean of he raios of nominal privae credi o annualized nominal GDP is abou 0.52 for he period from 1960:Q1 o 2011:Q1, implying a quarerly average of This is a broad measure of non-financial business deb, which includes boh long-erm and shor-erm debs. I hus oversaes he imporance of working capial loans (which are shor-erm) in he economy. Wih a narrower measure of credi ha corresponds more closely o working capial loans in our model, he implied value of θ would be smaller. Since a smaller θ implies a greaer financial muliplier and hus a larger degree of increasing reurns, our calibraion wih a high value of θ biases he numerical resuls agains finding indeerminacy (see Secion IV.C). This calibraion also verifies ha all firms choose o say in business a he end of he period because he expeced coninuaion value exceeds he size of he fixed cos. To see his, noe ha equaion (30) implies ha he expeced coninuaion value exceeds he fixed cos if and only if b > 0. Wih our b calibraion, = 2.08 > 0 in he seady sae, so ha all firms choose o say in business. Y