Estimating Contract Indexation in a Financial Accelerator Model Charles T. Carlstrom Timothy S. Fuerst Alberto Ortiz Matthias Paustian June 2013

Transcription

1 Esimaing Conrac Indexaion in a Financial Acceleraor Model Charles T. Carlsrom Timohy S. Fuers Albero Oriz Mahias Pausian June 23 Research Papers

2 Esimaing Conrac Indexaion in a Financial Acceleraor Model Research Papers Auhors: Charles T. Carlsrom Timohy S. Fuers Albero Oriz Mahias Pausian 23 Cener for Lain American Moneary Sudies (cemla) Durango 54, Colonia Roma Nore, Delegación Cuauhémoc, 67 México D.F., México. publicaciones@cemla.org hp:// The views expressed in his paper are hose of he auhors, and no necessarily hose of he Bank of England, he Cener for Lain American Moneary Sudies, he Federal Reserve Bank of Cleveland, or of he Board of Governors of he Federal Reserve Sysem or is saff.

3 Esimaing Conrac Indexaion in a Financial Acceleraor Model C h a r l e s T. Carlsrom, Federal Reserve Bank of Cleveland T i m o h y S. Fuers, Universiy of Nore Dame Federal Reserve Bank of Cleveland A l b e r o Oriz, c em l a and eg ade Business School M a h i a s Pausian, Bank of England charles..carlsrom@clev.frb.org fuers@nd.edu oriz@cemla.org mahias.pausian@bankofengland.co.uk absrac This paper addresses he posiive implicaions of indexing risky deb o observable aggregae condiions. These issues are pursued wihin he conex of he celebraed financial acceleraor model of Bernanke, Gerler and Gilchris (999). The principle conclusions include: () he esimaed level of indexaion is significan, (2) he business cycle properies of he model are significanly affeced by his degree of indexaion, (3) he imporance of invesmen shocks in he business cycle depends upon he esimaed level of indexaion, and (4) alhough he daa prefers he financial model wih indexaion over he fricionless model, hey have remarkably similar business cycle properies for non-financial exogenous shocks. JEL Codes: E32, E44. Keywords: Agency coss; financial acceleraor; business cycles. cemla Research Papers June 23 CEMLA Documenos de Coyunura 5 Febrero de 25

4 . INTRODUCTION T he fundamenal funcion of credi markes is o channel funds from savers o enrepreneurs who have some valuable capial invesmen projec. These effors are hindered by agency coss arising from asymmeric informaion. A sandard resul in a subse of his lieraure, he cosly sae verificaion (CSV) framework, is ha risky deb is he opimal conrac beween risk-neural lenders and enrepreneurs. The modifier risky simply means ha here is a non-zero chance of defaul. In he CSV model exernal paries can observe he realizaion of he enrepreneur s idiosyncraic producion echnology only by expending a monioring cos. Townsend (979) demonsraes ha risky deb is opimal in his environmen because i minimizes he need for verificaion of projec oucomes. This verificaion is cosly bu necessary o align he incenives of he firm wih he bank. Aggregae condiions will also affec he abiliy of he borrower o repay he loan. Bu since aggregae variables are observed by boh paries, i may be advanageous o have he loan conrac indexed o he behavior of aggregae variables. Therefore, even when loan conracs canno be designed based on privae informaion, we can exploi common informaion o make hese financial conracs more sae-coningen. Tha is, why should he loan conrac call for cosly monioring when he even ha leads o a poor reurn is observable by all paries? Carlsrom, Fuers, and Pausian (23) examine quesions of his ype wihin he financial acceleraor of Bernanke, Gerler, and Gilchris (999), hereafer BGG. Carlsrom e al. (23) demonsrae ha he privaely opimal conrac in he BGG model includes indexaion o: i) he aggregae reurn o capial (which we will call R k -indexaion), ii) he marginal uiliy of wealh (which we will call - indexaion), and iii) he shadow cos of exernal financing. This is he logic behind Shiller and Weiss s (999) suggesion of indexing home morgages o movemens in aggregae house prices. 2 CEMLA Research Papers June 23

5 In his paper we explore he business cycle implicaions of indexing he BGG loan conrac o he aggregae reurn o capial and o he marginal uiliy of wealh. There are a leas wo reasons why his is an ineresing exercise. Firs, as noed above, Carlsrom e al. (23) demonsrae ha he privaely opimal conrac in he BGG framework includes indexaion of his very ype. Second, indexaion of his ype is no so far removed from some financial conracs we do observe. For example, indexing repaymen o innovaions in he marginal uiliy of wealh is a close approximaion o indexaion o movemens in he risk free rae of ineres. There are many deb insrumens ha are direcly linked o marke ineres rae of his ype, e.g., adjusable rae morgages. More generally, since we are assuming ha he CSV framework proxies for agency cos effecs in he enire US financial sysem, i seems reasonable o include some form of indexaion o mimic he myriad ex pos reurns on exernal financing. For example, in conras o he model assumpion where enrepreneurs ge zero in he even of bankrupcy, his is clearly no he implicaion of Chaper bankrupcy. In any even, we use familiar Bayesian mehods o esimae he degree of conrac indexaion o he reurn o capial and he marginal uiliy of wealh. To avoid misspecificaion problems in he esimaion we need a complee model of he business cycle. We use he recen conribuion of Jusiniano, Primiceri, and Tambaloi (2), hereafer JPT, as our benchmark. A novely of he JPT model is ha i includes wo shocks o he capial accumulaion echnology. The firs shock is a nonsaionary shock o he relaive cos of producing invesmen goods, he invesmen specific echnology shock (IST). The second is a saionary shock o he ransformaion of invesmen goods ino insalled capial, he marginal efficiency of invesmen shock (MEI). For business cycle variabiliy, JPT find ha he IST shocks are irrelevan, while he MEI shocks accoun for a subsanial porion of business cycle flucuaions. Our principle resuls include he following. Firs, he esimaed level of R k -indexaion significanly exceeds uniy, much higher han he assumed BGG indexaion of approximaely zero. A model wih R k -indexaion fis he daa significanly beer when compared o BGG. This is because he BGG model s predicion for he risk premium in esimaing conrac indexaion in a financial acceleraor model 3

6 he wake of a MEI shock is counerfacual. A MEI shock lowers he price of capial and hus leads o a sharp decline in enrepreneurial ne worh in he BGG model. Bu under R k - indexaion, he required repaymen falls also so ha ne worh moves by significanly less. Second, wih R k -indexaion, his financial model and JPT have remarkably similar business cycle properies for non-financial exogenous shocks. For example, for he case of MEI shocks, he esimaed level of indexaion leads o ne worh movemens in he financial model ha accommodae real behavior quie similar o he response of JPT o a MEI shock. We also nes financial shocks ino he JPT model by reaing flucuaions in hese wo financial variables as serially correlaed measuremen error. This model horse race resuls in he R k -indexaion model dominaing BGG, which in urn significanly dominaes JPT. The financial models are improvemens over JPT in wo ways. The financial models make predicions for he risk premium and leverage on which JPT is silen, and he financial models inroduce oher exogenous shocks, e.g., shocks o ne worh or idiosyncraic variance, ha are irrelevan in JPT. Third, we find ha wheher financial shocks or MEI shocks are more imporan drivers of he business cycle depends upon he level of indexaion. Under BGG, financial shocks accoun for a significan par of he variance of invesmen spending. Bu under he esimaed level of R k -indexaion, financial shocks become much less imporan and he MEI shocks are again of paramoun imporance. Two prominen papers closely relaed o he curren work are Chrisiano, Moo, and Rosagno (2), and DeGraeve (28). They each use Bayesian mehods o esimae versions of he BGG framework in medium-scale macro models. Boh papers conclude ha he model wih financial fricions provides a beer fi o he daa when compared o is fricionless counerpar. The chief novely of he curren paper is o inroduce conrac indexaion ino he BGG framework, and demonsrae ha i is empirically relevan, alering he business cycle properies of he model. Neiher of he previous papers considered indexaion of his ype. 4 CEMLA Research Papers June 23

7 The paper proceeds as follows. Secion 2 presens a simple example ha illusraes he imporance of conrac indexaion o he financial acceleraor. Secion 3 develops he DSGE model. Secion 4 presens he esimaion resuls. Secion 5 concludes. 2. WHY DOES INDEXATION MATTER? A SIMPLE EXAMPLE T his secion presens a simple inuiive example ha demonsraes he imporance of indexaion in deermining he size of he financial acceleraor. Consider a world wih agency coss in which he porion of ne worh owned by enrepreneurs ( nw ) has a posiive effec on he value of capial ( q ): q p* nw d () d where he expression is in log deviaions and is an exogenous shock o capial prices, e.g., a shock o MEI in he general equilibrium model below. Equaion is a manifesaion of agency coss in ha he disribuion of ne worh across lenders and borrowers affecs asse prices. The idea is ha higher ne worh in he hands of enrepreneurs makes i easier for hem o access a loan wih which o buy capial, so ha higher levels of ne worh ac like a demand channel on asse prices. In he general equilibrium model below, he value of p is a funcion of he agency cos and (insalled) capial adjusmen cos parameers. The enrepreneur accumulaes ne worh o miigae he agency problems involved in direc lending. The agency problem arises from a CSV problem in he enrepreneur s producion echnology. The enrepreneur akes one uni of inpu and creaes unis of capial, where he uni-mean random variable is privaely observed by he enrepreneur bu can be verified by he lender only by paying a cos. This CSV problem makes equiy finance problemaic, so ha he opimal conrac is given by a risky deb conrac wih a promised repaymen of p r. The repaymen p r canno be indexed o because i is privaely observed. Bu i can be indexed o he aggregae price of capial: p r q. (2) esimaing conrac indexaion in a financial acceleraor model 5

8 This form of indexaion is similar o indexing o he rae of reurn on capial in he general equilibrium model developed below. Enrepreneurial ne worh accumulaes wih he profi flow from he invesmen projec, bu decays via consumpion of enrepreneurs (which is a consan fracion of ne worh). Log-linearized his evoluion is given by: nw q r p nw r p n (3) where denoes leverage (he raio of projec size o ne worh) and n is an exogenous shock o ne worh. Using he indexaion assumpion (2), we can express (3) as nw q [ ( )] nw n (4) Noe ha since κ >, he slope of he ne worh equaion is decreasing in he level of indexaion. Equaions and 4 are a simulaneous sysem in ne worh and he price of capial. We can solve for he wo endogenous variables as a funcion of he pre-deermined and exogenous variables: nw n d nw [ ( )] { p[ ( )]} (5) ( n ) d pnw q (6) { p[ ( )]} The inverse of he denominaor in Equaions 5-6 is he familiar muliplier arising from wo endogenous variables wih posiive feedback. This hen implies ha exogenous shocks are muliplied or financially acceleraed, and ha he degree of his muliplicaion depends upon he level of indexaion. The effec of indexaion on he financial muliplier is highly nonlinear. Figure plos he muliplier for κ = 2, and p =.45, boh of hese values roughly correspond o he general equilibrium analysis below. Noe ha moving from χ = o χ =, has an enormous effec on he muliplier. Bu here are sharp diminishing reurns so he muliplier is lile changed as we move from χ = o χ = 2. This suggess, and we confirm below, ha he daa can disinguish χ = from, say, χ =, 6 CEMLA Research Papers June 23

9 bu ha indexaion values in excess of uniy will have similar business cycle characerisics and hus be difficul o idenify. Consider hree special cases of indexaion:,, and The firs is. he implici assumpion in BGG; he second implies complee indexaion; he hird eliminaes he financial acceleraor alogeher. In hese cases, ne worh and asse prices are given by: Indexaion Ne worh Capial price n d nw ( n ) d pnw p p n d nw ( n ) d pnw p p Muliplier (p=.45, κ =2).82 n nw ( n ) d pnw For boh and, exogenous shocks o asse prices and ne worh have muliple effecs on he equilibrium levels of ne worh and capial prices. Since >, his effec is much larger under BGG s assumpion of no indexaion ( ) p p. Furher, under he BGG assumpion, exogenous shocks o asse prices ( d ) have an added effec as hey are weighed by leverage. Bu for all levels of indexaion, here are always agency cos effecs in ha he price of capial is affeced by he level of enrepreneurial ne worh. The financial muliplier effecs are raced ou in Figure 2: an exogenous shock o asse prices has a much larger effec on boh ne worh and asse prices in he BGG framework. Finally, since 2 he financial acceleraor largely disappears when 2. Before proceeding, i is helpful o emphasize he wo parameers ha are crucial in our simple example as hey will be manifesed below in he richer general equilibrium esimaing conrac indexaion in a financial acceleraor model 7

10 environmen. Our reduced form parameer p in Equaion is he agency cos parameer. In a Modigliani-Miller world we would have p =, as he disribuion of ne worh would have no effec on asse prices or real aciviy. Second, he indexaion parameer χ deermines he size of he financial acceleraor, i.e., how do unexpeced movemens in asse prices feed in o ne worh? These are wo relaed bu logically disinc ideas. Tha is, one can imagine a world wih agency coss (p > ), bu wih very modes acceleraor effecs ( ). To anicipae our empirical resuls, his is he. parameer se ha wins he model horse race. Tha is, he daa is consisen wih an agency cos model bu wih rivial acceleraor effecs. In such an environmen, financial shocks (e.g., shocks o ne worh) will affec real aciviy, bu oher real shocks (e.g., MEI shocks) will no be acceleraed. 3. THE MODEL T he benchmark model follows he JPT framework closely. The model of agency coss comes from BGG wih he addiion of exogenous conrac indexaion. The BGG loan conrac is beween lenders and enrepreneurs, so we focus on hese wo agens firs before urning o he familiar framework of JPT. LENDERS The represenaive lender acceps deposis from households (promising a sure real reurn d R ) and provides loans o he coninuum of enrepreneurs. These loans are ineremporal, wih he loans made a he end of ime being paid back in ime +. The realized gross real reurn on hese loans is denoed by R L. Each individual loan is subjec o idiosyncraic and aggregae risk, bu since he lender holds an enire porfolio of loans, only he aggregae risk remains. The lender has no oher source of funds, so he level of loans will equal he level of deposis. Hence, real dividends are given by Div ( R D R D ). The inermediary seeks o maximize is equiy value which is L d given by: 8 CEMLA Research Papers June 23

11 Λ Q E Div (7) j j L j j Λ where Λ is he marginal uiliy of real income for he represenaive household ha owns he lender. The FOC of he lender s problem is: Λ E R R (8) L d Λ The firs-order condiion shows ha in expecaion, he lender makes zero profis, bu ex pos profis and losses can occur. 2 We assume ha losses are covered by households as negaive dividends. This is similar o he sandard assumpion in he dynamic new-keynesian (DNK) model, e.g., Woodford (23). Tha is, he sicky price firms are owned by he household and pay ou profis o he household. These profis are ypically always posiive (for small shocks) because of he seady sae mark-up over marginal cos. Similarly, one could inroduce a seady-sae wedge (e.g., monopolisic compeiion among lenders) in he lender s problem so ha dividends are always posiive. Bu his assumpion would have no effec on he model s dynamics so we dispense from i for simpliciy. ENTREPRENEURS AND THE LOAN CONTRACT Enrepreneurs are he sole accumulaors of physical capial. A he beginning of period, he enrepreneurs sell all of heir accumulaed capial o capial agencies a beginningof-period capial price enire capial sock beg Q. A he end of he period, he enrepreneurs purchase he K, including any ne addiions o he sock, a end-of-period price Q. This re-purchase of capial is financed wih enrepreneurial ne worh ( NW ) and exernal financing from a lender. The exernal finance akes he form of a one period loan conrac. The gross reurn o holding capial from ime- o ime + is given by: 2 In conras, BGG assume ha bank profis are always zero ex pos so ha he lender s reurn in pre-deermined. This is no a feaure of he opimal conrac. See Carlsrom e al. (23) for deails. esimaing conrac indexaion in a financial acceleraor model 9

12 R Q Q. (9) k beg beg Below we show ha ( ) Q Q u a u Q, he laer erm coinciding wih he expression in BGG. Variaions in R k are he source of aggregae risk in he loan conrac. The exernal financing is subjec o a cosly-saeverificaion (CSV) problem because of idiosyncraic risk. In paricular, one uni of capial purchased a ime- is ransformed ino unis of capial in ime +, where is an idiosyncraic random variable wih densiy and cumulaive disribuion Φ. The realizaion of is direcly observed by he enrepreneur, bu he lender can observe he realizaion only if a monioring cos is paid, a cos ha is fracion mc of he projec oucome. Assuming ha he enrepreneur and lender are risk-neural, Townsend (979) demonsraes ha he opimal conrac beween enrepreneur and inermediary is risky deb in which monioring only occurs if he promised payoff is no forhcoming. Payoff does no occur for sufficienly low values of he idiosyncraic shock,. Le p R denoe he promised gross rae-of-reurn so ha R p is defined by R ( QK NW) R QK. () p k QK We find i convenien o express his in erms of he leverage raio NW Equaion becomes Wih ( ) f and p k so ha R R () g denoing he enrepreneur s share and lender s share of he projec oucome, respecively, he lender s ex pos realized + reurn on he loan conrac is defined as: where R g QK R R g k L k QK NW (2) CEMLA Research Papers June 23

13 f d [ Φ ] (3) Φ ( ) g mc d (4) Recall ha he lender s sochasic discoun facor comes from he household, and he lender s reurn is linked o he reurn on deposis via 8: L d ER Λ R E Λ (5) As for he enrepreneur, Carlsrom, Fuers and Pausian (23) show ha he enrepreneur s value funcion is linear wih a ime-varying coefficien we denoe by V, where V saisfies: k V ( ) EV R f (6) Using his valuaion and expression 5, he end-of-ime- conracing problem is hus given by: subjec o k max, EV R f ( ) (7) ER Λ g R E Λ (8) k d An imporan observaion is ha he choice of can be made coningen on public informaion available in ime +. Indexaion of his ype is opimal. Afer some rearrangemen, he conrac opimizaion condiions include: ' ' ' Λ ' EΛ g EV f V f g EV f ( ) EV R f EΛ R g k k ' EΛ g ' k d (9) (2) EΛ R g R E Λ (2) esimaing conrac indexaion in a financial acceleraor model

14 A key resul is given by (9). The opimal monioring cu-off is independen o K k K innovaions in R. The second-order condiion implies ha he raio increasing in. Hence, anoher implicaion of he privaely opimal conrac is ha p (and hus he opimal repaymen rae R ) is an increasing funcion of (innovaions in) he marginal uiliy of wealh Λ, and a decreasing funcion of (innovaions in) he enrepreneur s valuaion V. The monioring cu-off implies he behavior of he repaymen rae (see ). In log deviaions, Carlsrom, Fuers, and Pausian (23) show ha he repaymen rae under he opimal conrac is given by: g Θ ˆ ˆ rˆ r ( rˆ E rˆ ) (λ E λ ) ( vˆ E v ˆ ) (22) p d k k Θ g ( ) Ψ j Ξ ˆ ˆ k j j j f vˆ E r (23) where he haed lower case leers denoe log deviaions, and he posiive consans Ξ and are defined in he Appendix. Innovaions in v ˆ will be driven by innovaions in r ˆk. Hence, for parsimony we will esimae an indexaion rule of he form: rˆ Ω ( rˆ E rˆ ) (λ E λ ) (24) p k k k where Ω are he pre-deermined variables ha affec he repaymen rae, e.g., noed in he example skeched in Secion 2, differen indexaion values will have dramaic effecs on he financial acceleraor. The original BGG model assumes ha he lender s reurn was pre-deermined. From Equaions -2, his implies ha and k, where k is modesly negaive ( k., in our benchmark calibraion). Enrepreneurs have linear preferences and discoun he fuure a rae β. Given he high reurn o inernal funds, hey will pospone consumpion indefiniely. To limi ne worh accumulaion and ensure ha here is a need for exernal finance in he long run, we assume ha fracion (-γ) of he enrepreneurs die each period. Their g ' ' is. As 2 CEMLA Research Papers June 23

15 accumulaed asses are sold and he proceeds ransferred o households as consumpion. Given he exogenous deah rae, aggregae ne worh accumulaion is described by NW γ NW R f ( ) (25) k nw, where nw, is an exogenous disurbance o he disribuion of ne worh. We assume i follows he sochasic process log log, (26) nw, nw nw, nw, 2 where nw, is i.i.d. N(, nw ). Equaion 25 implies ha NW is deermined by he realizaion of R k and he response of o hese realizaions. NW hen eners he conracing problem in ime so ha he realizaion of k R is propagaed forward. As in Chrisiano, Moo, and Rosagno (2), and Gilchris, Oriz and Zakrajšek (29), we also consider ime variaion in he variance of he idiosyncraic shock. The variance of is denoed by and follows he exogenous sochasic process given by log log,, Shocks o his variance will aler he risk premium in he model. (27) FINAL GOOD PRODUCERS Perfecly compeiive firms produce he final consumpion good Y combining a coninuum of inermediae goods according o he CES echnology: λp, /( λ p,) Y Y() i di (28) The elasiciy p, follows he exogenous sochasic process log log log, (29) p, p p p p, p, p p, 2 where ε p, is i.i.d. N(, p ). Flucuaions in his elasiciy are price markup shocks. Profi maximizaion and he zero profi condiion imply ha he price of he final good, P, is he familiar CES aggregae of he prices of he inermediae goods. esimaing conrac indexaion in a financial acceleraor model 3

16 INTERMEDIATE GOODS PRODUCERS A monopolis produces he inermediae good i according o he producion funcion α α Y i max{ A K ( i) L( i) AΥ F ;}, (3) where K (i) and L (i) denoe he amouns of capial and labor employed by firm i. F is a fixed cos of producion, chosen so ha profis are zero in seady sae. The variable he exogenous non-saionary level of TFP progress. Is growh rae (z ΔlnA ) is given by, z z, (3) z z z z 2 where ε z, is i.i.d.n(, z ). The oher non-saionary process secor and is discussed below. A is is linked o he invesmen Every period a fracion p of inermediae firms canno choose is price opimally, bu reses i according o he indexaion rule p p, P i P i (32) where π P /P - is gross inflaion and π is is seady sae. The remaining fracion of firms chooses is price P (i) opimally, by maximizing he presen discouned value of fuure profis s s / s s Ps E p p p P i k Y s i W sl s i PssKs() i s / P k where he demand funcion comes from he final goods producers, / (33) P is he marginal uiliy of nominal income for he represenaive household, and W is he nominal wage. EMPLOYMENT AGENCIES Firms are owned by a coninuum of households, indexed by j,. Each household is a monopolisic supplier of specialized labor, L (j), as in Erceg e al. (2). A large number of compeiive employmen agencies combine his specialized labor ino a homogenous labor inpu sold o inermediae firms, according o 4 CEMLA Research Papers June 23

17 w, /(, ) w L L( j) dj (34) As in he case of he final good, he desired markup of wages over he household s marginal rae of subsiuion,, w, follows he exogenous sochasic process where w, is i.i.d. N (, log log log, (35) w, w w w w, w, w w, 2 w ). This is he wage markup shock. Profi maximizaion by he perfecly compeiive employmen agencies implies ha he wage paid by inermediae firms for heir homogenous labor inpu is w, / w, W W( j) dj (36) CAPITAL AGENCIES The capial sock is managed by a collecion of perfecly compeiive capial agencies. These firms are owned by households and discoun cash flows wih Λ, he marginal uiliy of real income for he represenaive household. A he beginning of period, hese agencies purchase he capial sock K from he enrepreneurs a beginning-ofperiod price beg Q. The agencies produce capial services by varying he uilizaion rae u which ransforms physical capial ino effecive capial according o K u (37) K. Effecive capial is hen rened o firms a he real renal rae. The cos of capial uilizaion is au ( ) per uni of physical capial. The capial agency hen re-sells he capial o enrepreneurs a he end of he period a price Profi maximizaion implies ( ) Q. The profi flow is hus given by: Q K u a u K Q beg K (38) ( ) beg Q Q u a u (39) a'( u ) (4) esimaing conrac indexaion in a financial acceleraor model 5

18 In seady sae, u =, a() = and a'' ()/a'(). Hence, in he neighbourhood of he seady sae beg Q Q (4) which is consisen wih BGG s definiion of he ineremporal reurn o holding capial R k Q Q. NEW CAPITAL PRODUCERS New capial is produced according o he producion echnology ha akes I invesmen goods and ransforms hem ino ime- profi flow is hus given by I -S I - I new capial goods. The where I Q -S I -P I I I- (42) I P is he relaive price of he invesmen good. The funcion S capures he presence of adjusmen coss in invesmen, as in Chrisiano e al. (25). The funcion has he following convenien seady sae properies: S = S' = and S'' >. These firms are owned by households and discoun fuure cash flows wih Λ, he marginal uiliy of real income for he represenaive household. JPT refer o he invesmen shock μ as a shock o he marginal efficiency of invesmen (MEI) as i alers he ransformaion beween invesmen and insalled capial. JPT conclude ha his shock is he primary driver of oupu and invesmen a business cycle frequencies. The invesmen shock follows he sochasic process where, is i.i.d. N 2,. log log (43),, 6 CEMLA Research Papers June 23

19 INVESTMENT PRODUCERS A compeiive secor of firms produces invesmen goods using a linear echnology ha ransforms one consumpion good ino produciviy invesmen goods. The exogenous level of is non-saionary wih a growh rae ( Δlog Υ ) given by,. (44) The consan reurns producion funcion implies ha he price of invesmen goods (in consumpion unis) is equal o. HOUSEHOLDS Each household maximizes he uiliy funcion L ( j) E s ln s b s C s hc s, (45) s where C is consumpion, h is he degree of habi formaion and b is a shock o he discoun facor. This ineremporal preference shock follows he sochasic process logb logb (46) b b,, 2 where b, is i.i.d. N(, b ). Since echnological progress is nonsaionary, uiliy is logarihmic o ensure he exisence of a balanced growh pah. The exisence of sae coningen securiies ensures ha household consumpion is he same across all households. The household s flow budge consrain is where j B R B W C T D L j D R profis d, P P P (47) D denoes real deposi a he lender, T is lump-sum axes, and B is holdings of nominal governmen bonds ha pay gross nominal rae R. The erm profis denoes he combined profi flow of all he firms owned by he represenaive agen including lenders, inermediae goods producers, capial agencies, and new capial producers. Every period a fracion ξ w of households canno freely se is wage, bu follows he indexaion rule esimaing conrac indexaion in a financial acceleraor model 7

20 ( )( z ) ( w ) w W j W j e e, (48) z The remaining fracion of households chooses insead an opimal wage W (j) by maximizing L ( j) Λ E s s S s w b s W jls( j) s P s subjec o he labor demand funcion coming from he firm. (49) THE GOVERNMENT A moneary policy auhoriy ses he nominal ineres rae following a feedback rule of he form R π x R dx X / * * * X /X R R X X R R X where R is he seady sae of he gross nominal ineres rae. The ineres raes respond o deviaions of inflaion from is seady sae, as well as o he level and he growh rae of he GDP gap ( X / X * ). The moneary policy rule is also perurbed by a moneary policy shock, mp,, which evolves according o mp, mp mp, mp, mp,, (5) log log, (5) where mp, is i.i.d. N 2 (, mp ). Public spending is deermined exogenously as a imevarying fracion of oupu. G - Y, g where he governmen spending shock g follows he sochasic process (52) 2 wih g, ~... iid N, g, log g log g logg, (53) g g g. The spending is financed wih lump sum axes. MARKET CLEARING The aggregae resource consrains are given by: 8 CEMLA Research Papers June 23

21 K I C G a u Y (54) - v I mcò f d K S I, (55) I- This complees he descripion of he model. We now urn o he esimaion of he linearized model. 4. ESTIMATION T he linearized version of he model equaions are colleced in he appendix. The hree fundamenal agency cos parameers are he seady sae idiosyncraic variance ( ss ), he enrepreneurial survival rae (γ), and he monioring cos fracion ( mc ). In conras o DeGraeve (28), we follow Chrisiano e al. (2), and calibrae hese parameers o be consisen wih long run aspecs of US financial daa. We follow his calibraion approach because hese parameers are pinned down by long run or seady sae properies of he model, no he business cycle dynamics ha he Bayesian esimaion is rying o mach. In any even, hese hree parameers are calibraed o p d mach he seady sae levels of he risk premium ( R R ), leverage raio ( ), and defaul rae ( Φ( ss )). In paricular, hey are chosen o deliver a 2 bp annual risk premium (BAA-Treasury spread), a leverage raio of =.95, and a quarerly defaul rae of.3/4. These imply an enrepreneurial survival rae of γ =.98, a sandard deviaion of ss =.28, and a monioring cos of mc =.2. A key expression in he loglinearized model is he reduced-form relaionship beween he risk premium and leverage: k d Erˆ rˆ qˆ kˆ n ˆ ˆ (56) esimaing conrac indexaion in a financial acceleraor model 9

22 (See he Appendix for deails.) The value of ν implied by he previous calibraion is ν =.4. This is hus imposed in he esimaion of he financial models. 3 For JPT we have ν =. Seady sae relaionships also imply ha we calibrae δ =.25, and (-/g) =.22. The remaining parameers are esimaed using familiar Bayesian echniques as in JPT. For he non-financial parameers of he model we use he same priors as in JPT. We rea as observables he growh raes of real GDP, consumpion, invesmen, he real wage, and he relaive price of invesmen. The oher observables include employmen, inflaion, he nominal rae, leverage, and he risk premium. Employmen is measured as he log of per capia hours. Inflaion is he consumpion deflaor, and he nominal rae is he federal funds rae. The series for leverage comes from Gilchris, Oriz and Zakrajsek (29). The risk premium is he spread beween he BAA and en year Treasury. The ime period for he esimaion is 954:3-29:. We choose he end of he sample period o avoid he observed zero bound on he nominal rae. We esimae four versions of he model. Along wih all he exogenous shocks oulined in he paper, we also include auocorrelaed measuremen error beween he model s risk premium and he observed risk premium. Auocorrelaed measuremen error is also included for leverage. The firs model we label JPT as i corresponds o he model wihou agency coss (ν = ). Noe ha o mach he observed financial variables, he JPT model will assign all risk premium and leverage variaion o auocorrelaed measuremen error. The remaining hree models have operaive agency coss (ν =.4). Recall ha he opimal conrac has he form given in Equaion 24. Our hree esimaes consider variaions on his basic form. In he model labeled BGG we impose he level of indexaion implicily assumed by BGG: k.,. For he model labeled R k -indexaion, we se and esimae he value of k. For he model labeled R k & λ- indexaion, we esimae boh indexaion parameers. We use diffuse priors on he 3 As a form of sensiiviy analysis, we also esimaed in he financial models. We found ha he esimaion is quie sensiive o priors, again suggesing ha i is no well idenified by business cycle dynamics. 2 CEMLA Research Papers June 23

23 indexaion parameers wih a uniform disribuion cenered a and wih a sandard deviaion of 2. The agency cos models also include wo financial shocks: i) ime-varying movemens in idiosyncraic risk, and ii) exogenous redisribuions of ne worh. Boh of hese shocks are irrelevan in he JPT model in which lending is no subjec o he CSV problem. We posi priors for he sandard deviaion and auocorrelaion of hese financial shocks in a manner symmeric wih he non-financial exogenous processes in JPT. The esimaion resuls are summarized in Table. The BGG, R k -indexaion, and R k & λ-indexaion agency cos models dominae he JPT model as he JPT model canno capure he forecasabiliy of leverage and he risk premium ha is in he daa. Comparing BGG and R k -indexaion, he daa rejecs he BGG level of indexaion preferring a level of conrac indexaion ha is economically significan: k =.7 wih a 9% confidence inerval beween.36 and 2.5. Resuls are on a similar range in he case of he R k & λ-indexaion esimaion wih k = 2.23 wih a 9% confidence inerval beween.52 and As suggesed by he example in Secion 2, his level of indexaion will imply significanly differen responses o shocks compared o he BGG assumpion. We will see his manifesed in he IRF below. The esimaed level of indexaion o he marginal uiliy of wealh is =.67 wih a 9% confidence inerval beween and.38. The combinaion of he wo indexaion parameers under he R k & λ-indexaion specificaion generaes dynamics ha are similar o hose of he R k -indexaion sp ecificaion alone. Two oher differences in parameer esimaes are worh some commen. Firs, he BGG model esimaes a significanly smaller size for invesmen adjusmen coss ( S ") in he able: S " =.68 for BGG, bu 2.99 for R k -indexaion, 2.5 for R k & λ-indexaion, and 2.92 for JPT. The level of adjusmen coss has wo conrasing effecs. Firs, lower adjusmen coss will increase he response of invesmen o aggregae shocks. Second, lower adjusmen coss imply smaller movemens in he price of insalled capial (Q ) and hus smaller financial acceleraor effecs in he BGG model. esimaing conrac indexaion in a financial acceleraor model 2

24 A second imporan difference in parameer esimaes is in he sandard deviaion of he shocks. Compared o JPT, he BGG model esimaes a significanly smaller volailiy in he MEI shocks, and insead shifs his variance on o ne worh shocks. Recall ha he principle conclusion of JPT is he imporance of he MEI shocks in he business cycle. Bu we once again end up wih he JPT conclusion wih regards o he imporance of MEI shocks in he R k -indexaion and R k & λ-indexaion models. An ineresing quesion we ake up below is why he BGG model downplays hese shocks so significanly. Table 2 repors he variance decomposiion of hree key variables: GDP, invesmen, and he risk premium. The JPT resuls are replicaed here: he MEI shocks accoun for a subsanial amoun of business cycle variabiliy in GDP (6% a he 8-quarer horizon) and invesmen (77% a he 8-quarer horizon). This conclusion is largely unchanged wih R k -indexaion and R k & λ-indexaion. Evidenly he esimaed level of indexaion resuls in real behaviour similar o a model wihou agency coss. This is paricularly clear in he IRFs presened in Figure 3 ha we discuss below. In conras o he R k -indexaion and R k & λ-indexaion models, BGG places much less weigh on he MEI shocks and insead shifs his variance o he financial shocks (he idiosyncraic variance and ne worh shocks) and he moneary policy shock. For he case of invesmen a he 8-quarer horizon, he BGG model places 5% of he variance on he MEI shocks (compared o 68% for he R k -indexaion and R k & λ-indexaion models, and 77% for JPT). The imporance of he wo financial shocks increases from 3% under R k & λ-indexaion and 4% under R k -indexaion, o 44% for BGG. The esimaed level of financial shocks depends criically upon he esimaed level of indexaion. The advanage of he financial models is showcased in he variance decomposiion of he risk premium. By assumpion, JPT assigns % of his variaion o measuremen error. In conras, he financial models explain large porions of he risk premia movemen by forces wihin he model. For example, a he 8-quarer horizon, he R k - indexaion model assigns less han 5% o measuremen error, and he R k & λ- indexaion assigns less han % o measuremen error. This predicabiliy of he risk premium is echoed by De Graeve (28). 22 CEMLA Research Papers June 23

25 Why does he BGG model downplay he MEI shocks and hus shif variance o he oher shocks? The answer is quie apparen from Figure 3a. The figure ses all parameer values o hose esimaed in he R k -indexaion model, excep for he levels of indexaion ( k for BGG and k = 2.23 and =.67 for he R k & λ-indexaion model), and he level of agency cos effecs (ν = for JPT). 4 A posiive innovaion in MEI leads o a fall in he price of capial. Since he BGG conrac is no indexed o he reurn o capial, he shock leads o a sharp decline in enrepreneurial ne worh, and hus a sharp increase in he risk premium. This procyclical movemen in he risk premium is in sharp conras o he daa. Hence, he Bayesian esimaion in he BGG model esimaes only a small amoun of variabiliy coming from he MEI shocks. Noice ha in he R k -indexaion and R k & λ-indexaion models ne worh is almos unchanged in response o an MEI shock, so ha he impac effec on he risk premium is counercyclical. The main difference among models is he behavior of he repaymen ha under credi conrac indexaion is lowered in response o he drop in he reurn on capial, while in BGG i remains unchanged. The R k -indexaion and R k & λ-indexaion models are hus consisen wih MEI shocks driving he cycle, and he risk premium being counercyclical. The similariy of he R k -indexaion and JPT model is also apparen: he wo IRFs o an MEI shock largely lie on op of one anoher. Since he BGG model downplays he imporance of MEI shocks, and shifs his variance o oher shocks. Figures 3b-3c plo he IRFs o he wo financial shocks. The good news wih he wo financial shocks is ha he spread is now counercyclical. Bu he difficuly wih he financial shocks is ha hey resul in counercyclical consumpion. This is he familiar co-movemen puzzle ha arises when a posiive shock in one secor (e.g., higher ne worh miigaes agency coss in capial accumulaion) leads o a downward producion movemen in he oher secor. However, his comovemen problem does no arise wih risk premium shocks in he R k -indexaion and R k & λ-indexaion models. As an 4 Alernaively we could have considered he IRFs for each model a each model s parameer esimaes. These IRFs are similar o hose repored here, bu we find Figure 3 more inuiive as i is holding all oher parameers fixed excep for he degree of indexaion and he presence of agency coss. esimaing conrac indexaion in a financial acceleraor model 23

26 aside, noe ha a shock o ne worh has a larger effec on ne worh and capial prices in he BGG model. This is jus a manifesaion of he muliplier inuiion oulined in Secion 2. Figure 3d plos he IRF o a moneary shock. In he case of BGG, he IRFs exhibi plausible comovemen and counercyclical spreads. The BGG esimaion does no pu more weigh on hese policy shocks because he funds rae is an observable, and hus limis possible ineres rae variabiliy. In conras, i is quie clear why he Indexaion model pus so lile weigh on moneary policy shocks. In he case of Indexaion, he spread is procyclical, a clear counerfacual predicion. As a form of sensiiviy analysis, Table 3 presens he esimaion resuls for i.i.d. measuremen error in he financial variables. The R k & λ-indexaion now wins he model horse race, wih JPT coming in significanly worse han he hree financial models. The degree of R k -indexaion is much larger han he case wih auocorrelaed measuremen error. Furher he level of indexaion o he marginal uiliy of wealh is esimaed o be significanly posiive, in line wih he heory oulined above. 5. CONCLUSION T his paper began as an empirical invesigaion of he imporance of agency coss and conrac indexaion in he business cycle. To reierae, our principle resuls include he following. Firs, he financial models appear o be an improvemen over he financial-fricionless JPT. Second, R k -indexaion appears o be an imporan characerisic of he daa. Third, he imporance of financial shocks (ne worh and idiosyncraic variance) in explaining he business cycle is significanly affeced by he esimaed degree of R k -indexaion. In shor, we find evidence for he imporance of financial shocks in he business cycle. Bu he evidence also suggess ha he effec of non-financial shocks on real aciviy is unaffeced by he inclusion of financial forces in he model. Tha is, he resuls sugges he imporance of financial shocks, bu no he exisence of a financial acceleraor. This analysis hus implies ha Bayesian esimaion of 24 CEMLA Research Papers June 23

27 financial models should include esimaes of conrac indexaion. Empirical analyses ha impose zero conrac indexaion likely disor boh he source of business cycle shocks and heir ransmission mechanism. esimaing conrac indexaion in a financial acceleraor model 25

28 APPENDIX. Linearized Sysem of Equaions: y F yˆ kˆ ( ) Lˆ y ˆ wˆ Lˆ k ˆ (A2) sˆ ˆ ( ) w ˆ (A3) p p ˆ E ˆ ˆ s ˆ ˆ (A4) p p, p p p p z 2z 2 z ˆ he e h he Ecˆ cˆ cˆ z z z z z z e he h e he h e he h z z z z z h e ˆ ˆ z he e h b h e he z ˆ b z z z z z e h( e h) e h ( ) e h e h (A5) ˆ ˆ ˆ R E ( ˆ ˆ z ˆ ) (A6) ˆ u ˆ k ˆ ˆ Erˆ ˆ E Ez ˆ E (A8) 2( ) " ˆ ˆ ˆ ˆ z q ˆ e S i i z ˆ ˆ ˆ e S E i i z ˆ ˆ (A9) 2( z ) " ˆ k ˆ ˆ ˆ u k z (A) ) z ˆ ˆ z k e k z ) ( ˆ ˆ ) e i (A) wˆ wˆ Ewˆ g ˆ w w w w, w w ˆ (A) (A7) 26 CEMLA Research Papers June 23

29 ˆ E ˆ zˆ ˆ z ˆ w w w z w ˆ ˆ w, w, (A2) gˆ wˆ ( Lˆ b ˆ ˆ ) (A3) ˆ ˆ ˆ * ˆ ˆ ) ( ˆ * ˆ * Rˆ ˆ ˆ RR R x x x dx x x x x mp, (A4) k xˆ ˆ ˆ y u y yˆ ˆ ˆ ˆ ˆ g c c i k i u g g y y y (A5) (A6) d rˆ ˆ R E ˆ (A7) k ( z ) ( z ) rˆ ( ) ˆ ( ) ˆ ˆ e q e q (A8) For he agency cos model, we replace (A8) wih k ˆ ˆ Erˆ ˆ ˆ ˆ ˆ ˆ E Ez ˆ E qkn (A8 ) And add he following equaions: ˆ nˆ ˆ ˆ ˆ ˆ ˆ zˆ ˆ ˆ r r r n k q r nw, (A9) k l l rp k ˆ rˆ rˆ [ ]( rˆ E rˆ ) ( λ E λ ) (A2) l d k k g k g 2. The Derivaion of A8 and A2 The opimal conrac (9)-(2) can be expressed as ' ' ' Λ ' EΛ g EV f V f g ' EV f ER V f R EΛ k d ' EΛ g k d (A2) (A22) EΛ R g R E Λ (A23) esimaing conrac indexaion in a financial acceleraor model 27

30 I is convenien o define ' f F( ) ' g condiion. Linearizing (A2)-(A23) we have where, where ' ssf ss Ψ F( ) ss, by he second order Ψ E λ Eλ ( v Ev ) (A24) E rˆ rˆ (Ψ ) E (A25) k d f k d E ˆ ˆ r r ge (A26) ssg ' ss g g ss, wih Ψ f g g, and ss f ' ss f f ss. Solving (A25)-(A26) we have: E (A27) Ψ f gg Ψ f g k d E ˆ ˆ r r (A28) Using he definiion of leverage and he deposi rae, (A28) is he same as (A8 ). The linearized lender reurn and promised paymen are given by: l k rˆ ˆ r g (A29) p k rˆ ˆ r (A3) Combining (A24) and (A3) we have: g Θ ˆ ˆ rˆ r ( rˆ E rˆ ) (λ E λ ) ( vˆ E v ˆ ) (A3) p d k k Θ g ( ) Ψ where from (6) we have 28 CEMLA Research Papers June 23

31 j Ξ ˆ ˆ k j j j vˆ E r (A32) where Ξ ν. Innovaions in v ˆ are dominaed by innovaions in r ˆk, so for parsimony we will esimae a promised paymen of he form rˆ E rˆ ( rˆ E rˆ ) (λ E λ ) (A33) p p k k k Combining his wih (A29)-(A3) we have: rˆ rˆ [ ]( rˆ E rˆ ) ( λ E λ ) (A34) l d k k g k g This is jus (A2). esimaing conrac indexaion in a financial acceleraor model 29

32 Figure The Muliplier as a Funcion of Indexaion. Figure 2 A Shock o Asse Demand q "= "= (Asse price is blue line. Ne worh evoluion is red line.) Demand shock shifs up asse price. The new equilibrium in (n,q) space depends upon he level of indexaion. Lower levels of indexaion amplify hese effecs. n 3 CEMLA Research Papers June 23

33 Figure 3.a. Impulse Response Funcions o a One Sandard Deviaion Marginal Efficiency of Invesmen Shock Keeping parameers consan o he R k indexaion model excep R k and lambda indexaion parameers (χk and χλ) Oupu Consumpion Invesmen Inflaion Wages Federal Funds Spread Ne Worh Price of Capial Reurn on Capial (R k ) Promised Repaymen Marginal Uiliy of Wealh (λ) JPT... BGG _ Indexaion o R k _. _. _ Indexaion o R k and λ esimaing conrac indexaion in a financial acceleraor model 3

34 Figure 3.b. Impulse Response Funcions o a One Sandard Deviaion Ne Worh Shock Keeping parameers consan o he R k indexaion model excep R k and lambda indexaion parameers (χk and χλ) Oupu Consumpion Invesmen Inflaion Wages Federal Funds Spread Ne Worh Price of Capial Reurn on Capial (R k ) Promised Repaymen Marginal Uiliy of Wealh (λ) BGG _ Indexaion o R k _. _. _ Indexaion o R k and λ 32 CEMLA Research Papers June 23

35 Figure 3.c. Impulse Response Funcions o a One Sandard Deviaion Idiosyncraic Variance Shock Keeping parameers consan o he R k indexaion model excep R k and lambda indexaion parameers (χk and χλ) Oupu Consumpion Invesmen Inflaion Wages Federal Funds Spread Ne Worh Price of Capial Reurn on Capial (R k ) Promised Repaymen Marginal Uiliy of Wealh (λ) BGG _ Indexaion o R k _. _. _ Indexaion o R k and λ esimaing conrac indexaion in a financial acceleraor model 33

36 Figure 3.d. Impulse Response Funcions o a One Sandard Deviaion Moneary Policy Shock Keeping parameers consan o he R k indexaion model excep R k and lambda indexaion parameers (χk and χλ) Oupu Consumpion Invesmen Inflaion Wages Federal Funds Spread Ne Worh Price of Capial Reurn on Capial (R k ) Promised Repaymen Marginal Uiliy of Wealh (λ) JPT... BGG _ Indexaion o R k _. _. _ Indexaion o R k and λ 34 CEMLA Research Papers June 23

37 Table : Models Esimaions and Models Comparisons wih BAA - T credi spread and leverage daa. All models have auocorrelaed measuremen errors in he credi spread and leverage series. JPT Model a BGG Model b Indexaion o R k Model c Indexaion o R k and λ Model d Log daa densiy Poserior Model Probabiliy % % % % Coefficien Descripion Prior Poseriors f Poseriors Poseriors Poseriors Prior densiy e prior mean psdev pos. mean 5% 95% pos. mean 5% 95% pos. mean 5% 95% pos. mean 5% 95% α Capial share N ι p Price indexaion B ι w Wage indexaion B γ z SS echnology growh rae N γ υ SS IST growh rae N h Consumpion habi B λ p SS mark-up goods prices N λ w SS mark-up wages N log L ss SS hours N (π - ) SS quarerly inflaion N ( β - - ) Discoun facor G Ψ Inverse frisch elasiciy G ξp Calvo prices B ξw Calvo wages B ϑ Elasiciy capial uilizaion coss G S Invesmen adjusmen coss G φp Taylor rule inflaion N φy Taylor rule oupu N φdy Taylor rule oupu growh N ρr Taylor rule smoohing B ρmp Moneary policy B ρz Neural echnology growh B ρg Governmen spending B ρυ IST growh B ρp Price mark-up B ρw Wage mark-up B ρb Ineremporal preference B θp Price mark-up MA B θw Wage mark-up MA B ρσ Idiosyncraic variance B ρnw Ne worh B ρμ Marginal efficiency of invesmen B ρrpme Risk premium measuremen error B ρlevme Leverage measuremen error B ν Elasiciy risk premium N χk Indexaion o R k U BGG χλ Indexaion o Marginal Uiliy U sandard deviaion of shocks Prior densiy prior mean psdev pos. mean 5% 95% pos. mean 5% 95% pos. mean 5% 95% pos. mean 5% 95% σmp Moneary policy I σz Neural echnology growh I σg Governmen spending I συ IST growh I σp Price mark-up I σw Wage mark-up I σb Ineremporal preference I σσ Idiosyncraic variance I σnw Ne worh I σμ Marginal efficiency of invesmen I σrpme Risk premium measuremen error I σlevme Leverage measuremen error I Noe: calibraed coefficiens: δ =.25, g implies a SS governmen share of.22. For he agency cos models (BGG and Indexaion) he following parameers are also calibraed: enrepreneurial survival rae γ =.98, a SS risk premium rp =.2/4, and a SS leverage raio κ =.95. a In JPT model here are no financial (risk premium and ne worh) shocks. The elasiciy of risk premium, ν, is se o and he indexaion parameers, χk and χλ, are irrelevan and se o. b In BGG model here are financial shocks and he elasiciy of risk premium, ν, is calibraed o.4, while he indexaion o he reurn of capial parameer, χk, is se o he implied in BGG, χk = (Θg - )/Θg where Θg =.985, and he indexaion o he marginal uiliy of wealh parameer, χ λ, is se o. c In he Indexaion o R k model here are financial shocks and he elasiciy of risk premium, ν, is calibraed o.4, while he indexaion o he reurn of capial parameer, χ k, is esimaed, and he indexaion o he marginal uiliy of wealh wealh parameer, χ λ, is se o. d In he Indexaion o R k and λ model here are financial shocks an he elasiciy of risk premium, ν, is calibraed o.4, while he indexaion o he reurn of capial parameer, χk, and he indexaion o he marginal uiliy of wealh parameer, χ λ, are boh esimaed. e N sands for Norman, B-Bea, G-Gamma, U-Uniform, I-Invered-Gamma disribuion. f Poserior perceniles are from 2 chains of 5, draws generaed using a Random Walk Meropolis algorihm. We discard he iniial 25, and reain one every 5 subsequen draws. esimaing conrac indexaion in a financial acceleraor model 35