Nominal Rigidities, Asset Returns and Monetary Policy

Transcription

1 Nominal Rigidiies, Asse Reurns and Moneary Policy Erica X.N. Li and Francisco Palomino November 4, 211 Absrac We sudy he asse pricing implicaions of price and wage rigidiies in a quaniaive general equilibrium model. Nominal rigidiies and recursive preferences improve he model abiliy o capure high expeced excess reurns on producion claims. The increased premium is mainly a compensaion for permanen produciviy shocks, since nominal rigidiies generae oupu and profi disorions ha are a source of long-run macroeconomic risk. In he cross secion, heerogeneiy in indusry price rigidiy generaes differences in expeced indusry asse reurns ha are deermined by produc subsiuabiliy wihin and across indusries. Moneary policy affecs expeced excess reurns under nominal rigidiies hrough is response o economic condiions and policy shocks. The model calibraion capures he observed price and wage rigidiies in US daa, he volailiy of key macroeconomic variables, he low volailiy of he risk-free rae, and he high Sharpe raios on financial asses. Wage rigidiies have significanly larger effecs han price rigidiies on asse reurns. Policies ha reac more aggressively o inflaion or less aggressively o oupu increase expeced asse reurns. Policy shocks increase asse reurn volailiy bu have a small effec on expeced excess reurns. JEL Classificaion: D51, E44, E52, G12. Keywords: General equilibrium, asse pricing, moneary policy, nominal rigidiies, expeced sock reurns, cross-secion of sock reurns. The Universiy of Michigan, Ross School of Business, Ann Arbor, MI 4819; Tel: 734) ; xuenanli@umich.edu; hp://webuser.bus.umich.edu/xuenanli/. The Universiy of Michigan, Ross School of Business, Ann Arbor, MI 4819; Tel: 734) ; fpal@umich.edu; hp://webuser.bus.umich.edu/fpal/.

2 1 Inroducion Explaining boh asse reurn and aggregae business cycle flucuaions in a unified framework remains an imporan challenge in financial economics. Sandard real business cycle models imply a counerfacually low compensaion for risk in asse reurns since producion facors can be freely adjused o help smooh households consumpion risk. 1 This has moivaed he inroducion of fricions o hese models, such as invesmen adjusmen coss and imperfec facor mobiliy, 2 ha provide subsanial improvemens in capuring he join dynamics of aggregae quaniies and asse reurns. In his paper, we explore an equilibrium model wih a paricular fricion, rigidiies in nominal produc prices and wages, o address i) how hese rigidiies affec he expeced reurns of producion claims, and ii) how moneary policy affecs he pricing of hese claims. The inroducion of nominal rigidiies o he analysis of asse reurns is moivaed firs by ample evidence of heir exisence in he daa. For insance, Nakamura and Seinsson 27) repor a median duraion of prices beween 8 and 11 monhs, and Taylor 1999) suggess an average wage duraion of 12 monhs. 3 Second, nominal rigidiies play a criical role in generaing consisen business cycle dynamics in equilibrium models such as Chrisiano, Eichenbaum and Evans 25) or Smes and Wouers 27). Third, nominal rigidiies generae real effecs of moneary policy and allow us o explore he link beween moneary policy and asse prices. Undersanding his link is of significan imporance o policymakers and, up o our knowledge, i has no been sudied in he heoreical lieraure. Nominal rigidiies are hen a naural candidae o inroduce fricions in he producion secor o undersand he dynamics of asse reurns and heir link o moneary policy. 1 Campbell and Cochrane 1999) and Bansal and Yaron 24), among ohers, have made significan progress in capuring asse pricing dynamics in endowmen economies. The success of hese models, however, is limied in a producion economy framework as shown by Boldrin, Chrisiano and Fisher 21) and Kalenbrunner and Lochsoer 21), respecively. 2 See Boldrin, Chrisiano and Fisher 21), for insance. 3 Blinder e al. 1998) conducs surveys on firms pricing policies and summarize differen heories for he exisence of price rigidiies based on he naure of coss, demand, conracs, marke ineracions, and imperfec informaion. 1

3 We summarize our main findings as follows. Firs, boh price and wage rigidiies improve he abiliy of real business cycle models o generae a large and posiive equiy premium. The increased premium is mainly a compensaion for permanen produciviy shocks. As illusraed by Kalenbrunner and Lochsoer 21), hese shocks endogenously generae long run consumpion risk ha may lead o a higher or lower price of risk under Epsein and Zin 1989) recursive preferences. We show ha he absence or presence of nominal rigidiies in he model can deermine wheher he premium is negaive or posiive, respecively. Second, he model calibraion shows ha he quaniaive impac of wage rigidiies on he premium is much larger han he impac of price rigidiies. Third, nominal rigidiies generae a posiive equiy premium for moneary policy shocks. This premium, however, is much smaller han he premium for permanen produciviy shocks. Fourh, price rigidiies generae ineresing markup dynamics ha are refleced in differences in he riskiness of oupu and profi claims. Fifh, he effec of heerogeneous price rigidiy across indusries on he cross secion of indusry asse reurns depends on he degree of produc subsiuabiliy wihin and across indusries. Finally, as a resul of wage rigidiies, economies wih moneary policies ha reac more aggressively o inflaion and/or less aggressively o oupu variaion have larger equiy premiums. We model a wo-secor producion economy wih four main ingrediens. Firs, a represenaive household wih Epsein and Zin 1989) recursive preferences over consumpion and leisure. Recursive preferences disenangle he elasiciy of ineremporal subsiuion from risk aversion. As in Tallarini 2), his separaion is useful o keep reasonable values for he elasiciy of subsiuion o mach macroeconomic dynamics, while having values for risk aversion ha mach empirical Sharpe raios of financial asses. The high Sharpe raios resul mainly as a compensaion for long run macroeconomic risk. Second, nominal rigidiies are modeled following Calvo 1983) saggered price and wage seing. The represenaive household provides differeniaed labor ypes o he producion secors and has monopolisic power o se is wages. However, a each poin of ime he household can only adjus he wage opimally for a fracion of labor ypes. Similarly, 2

4 firms provide differeniaed producs and have monopolisic power o se heir prices. However, a each poin of ime a firm can only adjus he price opimally wih some posiive probabiliy. We allow for differen probabiliies for he wo secors o analyze implicaions of heerogeneous price rigidiies on cross-indusry asse reurns. Third, moneary policy is modeled as a Taylor 1993) policy rule o se he level of a nominal ineres rae. This rule responds o economic condiions and is affeced by policy shocks. I allows us o quanify he effec on asse reurns of changes in he policy response o he sae of he economy. Fourh, he model incorporaes permanen and ransiory shocks. As shown by Campbell 1994), permanen and ransiory shocks have differen effecs on opimal consumpion and hen differen implicaions on asse reurns. By incorporaing he wo ypes of shocks, we can deermine how nominal rigidiies affec individual compensaions for hese shocks in financial asses. We calibrae he model o capure he mean and dispersion of price duraion and he mean wage duraion in U.S. daa. The model is able o mach he volailiy of consumpion growh and he Sharpe raio implied in equiy reurns, while also maching he variabiliy explained by he model shocks of inflaion, de-rended consumpion, and he ineres rae. The calibraion relies on a reasonable elasiciy of ineremporal subsiuion EIS) of around.2 and a relaive risk aversion coefficien of 26. Risk aversion is high wih respec o empirical and experimenal evidence, bu significanly lower han in sandard business cycle models such as Tallarini 2). This improvemen is he resul of inroducing permanen produciviy shocks and leisure preferences. We quanify he effec of nominal rigidiies as an increase in 85 bps in he equiy premium relaive o an economy wih no rigidiies. This increase is mainly a compensaion for persisen permanen produciviy shocks. To undersand why, consider firs an economy wih flexible prices and wages. Our calibraion shuing down he rigidiies) implies a negaive equiy premium in his economy. A negaive persisen permanen produciviy shock has a negaive persisen effec on consumpion and profis, and a persisen increase in he marginal uiliy of consumpion. While lower profis decrease he reurn on profi claims subsiuion effec), he persisen increase in 3

5 marginal uiliy makes claims on fuure consumpion more valuable and have a posiive effec on reurns wealh effec). The wealh effec dominaes he subsiuion effec leading o a negaive premium in profi claims. In he presence of wage and price rigidiies, he subsiuion effec ouweighs he wealh effec. The subsiuion effec is sronger because he negaive response of consumpion and profis o a negaive shock is amplified relaive o an economy wih no rigidiies. Under wage rigidiies, households canno adjus some wages downwards and producers keep higher prices o obain an opimal markup over marginal coss. Higher prices reduce he produc demand and imply lower oupu and profis han under flexible wages. Similarly, under price rigidiies, firms ha canno adjus prices downwards face a lower demand, and oupu and profis are lower han under flexible prices. Simulaneously, he wealh effec is weaker under nominal rigidiies since he parial adjusmen of prices and wages induces less persisen effecs of permanen produciviy shocks on he marginal uiliy of consumpion. Quaniaively, he effec on reurns of wage rigidiies is significanly larger han he effec of price rigidiies and generaes a larger premium. Price rigidiies also generae differences beween he expeced reurns of oupu and profi claims. The difference is he resul of ime variaion in producion markups and depends on wheher or no wages are flexible. If wages are flexible sicky), prices following a negaive shock are high low) relaive o marginal coss which increases reduces) he markup. The counercyclical procyclical) markup reduces increases) he riskiness and expeced reurns of profi claims relaive o oupu claims. Our wo-secor model allows us o analyze he link beween indusry price rigidiy and indusry expeced asse reurns. High prices in an indusry wih respec o anoher one lead o wo opposie effecs on profis: a low indusry oupu demand low profis) and a high indusry markup high profis) relaive o he oher indusry. The subsiuabiliy of goods across indusries deermines he magniude of he difference in indusry oupu demands. The subsiuabiliy of goods wihin indusries deermines he magniude of he difference in indusry markups. If he wo elasiciies are he same, relaive oupu and markup effecs exacly cancel each oher and profi claims in 4

6 he wo indusries have he same expeced reurns. If he elasiciies are differen, he difference in expeced reurns is deermined by he join dynamics of he marginal uiliy of consumpion and indusry prices. We consider a nominal ineres rae policy rule ha reacs o inflaion and he oupu level, and is affeced by an ineres rae smoohing componen and policy shocks. Under nominal rigidiies, his rule affecs real ineres raes, consumpion and producion decisions, and asse reurns. A sronger response o inflaion in he rule implies a higher real ineres rae and lower oupu, which increases expeced asse reurns. A sronger response o oupu in he rule implies a more sable oupu and, herefore, lower expeced asse reurns. A larger weigh on ineres rae smoohing has a similar effec as a reducion in he response o inflaion. Finally, policy shocks require a posiive compensaion in expeced asse reurns since conracionary shocks increase he marginal uiliy of consumpion and reduce oupu and profis. Quaniaively, he premium for policy shocks is significanly smaller han he premium for permanen produciviy shocks. Relaed lieraure Our paper belongs o he lieraure ha links he real economy o financial markes in a unified framework. 4 I builds on he pioneer work of Kydland and Presco 1982), and is mosly relaed o Boldrin, Chrisiano and Fisher 21) and Chrisiano, Eichenbaum and Evans 25). Boldrin, Chrisiano and Fisher 21) show ha fricions in he producion secor are criical for real business cycle models o capure salien asse pricing dynamics. They find ha fricions in inersecoral facor mobiliy and habi formaion in preferences can simulaneously reproduce imporan business cycle properies, a high price of risk, and he observed equiy premium. However, habi formaion in heir model also leads o a counerfacual high volailiy in he risk-free rae. Our model insead relies on Epsein and Zin 1989) recursive preferences and permanen produciviy shocks o achieve boh a high price of risk and low volailiy in he risk-free rae. As in Chrisiano, Eichenbaum and Evans 25), fricions in our model resul from nominal price and 4 Cochrane 25) provides an exensive summary of he main findings and challenges in his lieraure. 5

7 wage rigidiies, and allow us o analyze he effecs of moneary policy on asse prices. Chrisiano, Eichenbaum and Evans 25) focus on he business cycle implicaions of moneary policy shocks and do no analyze asse pricing dynamics. We find ha policy shocks have a small effec on asse reurns in comparison o permanen produciviy shocks. Li and Palomino 21) provide a qualiaive analysis of he effecs of price rigidiies and policy shocks on asse reurns. They do no calibrae heir model and do no quanify hese effecs. We find ha wage rigidiies have larger quaniaive effecs han price rigidiies on asse reurns. Our paper is also relaed o Kalenbrunner and Lochsoer 21). They find ha permanen and ransiory produciviy shocks can endogenously generae long run consumpion risk as in he endowmen economy of Bansal and Yaron 24) and hen high compensaions for risk in financial asses. Their model calibraion wih permanen produciviy shocks relies on a high elasiciy of ineremporal subsiuion of consumpion and generaes a low equiy premium. We show ha nominal rigidiies increase he premium for permanen produciviy shocks relying on a low elasiciy of ineremporal subsiuion. Finally, our paper joins recen aemps o undersand he effecs of labor markes on financial asse reurns. Leau and Uhlig 2) find ha adding labor negaively affecs he performance of habi models since labor provides an addiional channel o smooh consumpion. Uhlig 27) shows ha real wage rigidiies can improve he abiliy of habi models o capure a high equiy premium. In he same spiri, Favilukis and Lin 211) analyze he ime series and cross secional asse reurn implicaions of infrequen renegoiaion of wages. We focus on nominal wage rigidiies raher han real wage rigidiies o undersand he implicaions of moneary policy on asse reurns. The paper is organized as follows. Secion 2 presens he model and is opimaliy condiions. Secion 3 explains he mechanism ha links expeced reurns and nominal rigidiies, and shows he quaniaive implicaions of he calibraion. Secion 4 concludes. 6

8 2 The Model We model a producion economy where households derive uiliy from he consumpion of a baske of wo goods and disuiliy from supplying labor for he producion of hese goods. The wo goods are produced in wo differen indusries characerized by monopolisic compeiion and nominal price and wage rigidiies. We allow for heerogenous degrees of price rigidiy in he wo indusries o learn abou he effecs of differen rigidiies on he cross-secion of sock reurns. 5 Nominal rigidiies generae real effecs of moneary policy. If some producers are no able o adjus prices opimally and/or if households are no able o adjus heir wages opimally, inflaion generaes disorions in relaive prices and/or relaive real wages ha affec producion decisions. Since inflaion is deermined by moneary policy, differen policies have differen implicaions for real aciviy, and affec he reurns on financial claims linked o producion e.g., socks). We model moneary policy as an ineres-rae policy rule ha reacs o inflaion and deviaions of oupu from a arge. Risk in he economy is driven by permanen and ransiory produciviy shocks and moneary policy shocks. In secion 3, we analyze how nominal rigidiies and moneary policy affec he compensaion for hese shocks in producion claims. 2.1 Households A represenaive household maximizes is recursive uiliy 1 ψ 1 γ V = U + βq, 1) where U = C1 ψ 1 ψ κ N s ) 1+ω [ 1 + ω, and Q = E 5 Aoki 21) sudies a paricular case for his economy in which one of he indusries has perfecly flexible prices, and wages are perfecly flexible. His analysis focuses on he implicaions for opimal moneary policy in his economy, and does no explore any asse pricing implicaions. 7 V 1 γ 1 ψ +1 ].

9 The parameers ψ and γ characerize he elasiciy of ineremporal subsiuion of consumpion and risk aversion, respecively. The paricular case ψ = γ corresponds o he sandard power uiliy specificaion. C is he consumpion of he final good, and N s is he aggregae supply of labor a ime. The process κ is used o obain balanced growh. Growh is he resul of permanen shocks o produciviy, as described in he producion secor secion. The final good is a baske of wo inermediae goods produced in wo indusries. We refer o hese indusries as I = {H, L}, o indicae indusries wih high and low price rigidiies, respecively. The consumpion of each indusry s good is C I,, and he final good is given by [ C = ϕ 1/η H η 1 C η H, + ϕ 1/η L ] η η 1 C η L, η 1, 2) where ϕ I is he weigh of indusry I in he baske ϕ L 1 ϕ H ), and η > 1 is he elasiciy of subsiuion beween indusry goods. Each indusry good is a Dixi-Sigliz aggregae of a coninuum of differeniaed goods, defined as [ 1 C I, = ] θ C I, j) θ 1 θ 1 θ dj, 3) where he elasiciy of subsiuion across differeniaed goods is θ > 1. The ineremporal budge consrain faced by he household is [ ] E M,+τP $ +τ C +τ E M $,+τ+τ LI +τ + τ= τ= I {H,L} 1 D I,+τ j)dj, 4) where M,+τ $ > is he nominal pricing kernel ha discouns nominal cash flows a ime + τ o ime, is he price of he final good, LI is he real labor income from supplying labor o he producion secor, and D I, j) is he real profi from he producion of he differeniaed good j in indusry I. 8

10 The maximizaion of 1) subjec o 4) provides us wih he ineremporal marginal rae of subsiuion of consumpion for he economy. The marginal raes of subsiuion of consumpion beween period and period + 1 in real and nominal erms are and M,+1 = β C+1 C ) ) ψ V 1/1 ψ) ψ γ +1, 5) Q 1/1 γ) ) 1 M,+1 $ P+1 = M,+1, 6) respecively. From hese wo equaions we can compue he real and nominal gross) one-period risk-free raes as R f, = 1 E [M,+1 ], and R$ f, = 1 E [ M $,+1 ], 7) respecively. These raes are imporan o compue excess real and nominal reurns on socks. The one-period nominal risk-free rae is he insrumen of moneary policy. Wage Seing We follow Schmi-Grohé and Uribe 27) o model an imperfecly compeiive labor marke where he represenaive household monopolisically provides a coninuum of labor ypes indexed by k [, 1] 6. Specifically, he supply of labor ype k saisfies he demand equaion ) θw N s W k) k) = N d, 8) W 6 This approach is differen from he sandard heerogeneous households approach o model wage rigidiies as in Erceg, Henderson and Levin 2), where each household supplies a differeniaed ype of labor. In he presence of recursive preferences, his approach inroduces heerogeneiy in he marginal rae of subsiuion of consumpion across households since i depends on labor. We avoid his difficuly and obain a unique marginal rae of subsiuion by modeling a represenaive agen who provides all differen ypes of labor. 9

11 where N d is he aggregae labor demand of he producion secor, W k) is he wage for labor ype k, and W is he aggregae wage index given by [ 1 W = ] 1 1 θw W 1 θw k) dk. The labor demand equaion 8) is derived in he producion secor secion below. The household chooses opimal wages W k) for all labor ypes k under Calvo 1983) saggered wage seing. Specifically, each period he household is only able o adjus wages opimally for a fracion 1 α of labor ypes. A fracion α of labor ypes keeps heir previous period wages. The opimal wage maximizes 1) subjec o demand funcions 8) and he budge consrain 4), where real labor income is given by LI = 1 W k) N s k)dk. Because he demand curve and he cos of labor supply are idenical across differen labor ypes, he household chooses he same opimal wage W for all he labor ypes subjec o a wage change a ime. The appendix shows ha he opimal wage saisfies W = µ w κ N s ) ω C ψ G w, H w,, 9) where N s = 1 N s k)dk is he aggregae labor supply, µ w θw θ w 1, H w, = 1 + αe [M $,+1 and G w, = 1 + αe [M $,+1 ) N d +1 W N d P+1 W +1 ) θw H w,+1], ) ) ψ ) ) ) C+1 N d +1 κ+1 N s ω +1 W C N d κ N s W +1 ) θw G w,+1]. In he absence of wage rigidiies α = ), he opimal wage is given by he markup-adjused marginal rae of subsiuion beween labor and consumpion, wih opimal markup µ w. 1

12 2.2 Firms The producion of he final consumpion good uses wo inermediae goods from indusry H and L via he aggregaor [ Y = ϕ 1/η H η 1 Y η H, + ϕ 1/η L ] η η 1 Y η 1 η L,. Wihin each indusry, here is a coninuum of firms indexed by j [, 1]. The final oupu of indusry I {H, L} is given by he Dixi-Sigliz aggregaor [ 1 Y I, = ] θ Y θ 1 θ 1 θ I, j) dj. The producion echnology of firm j in indusry I is given by Y I, j) = A N d I,j), where A is labor produciviy and NI, d j) is firm j s labor demand. We assume ha labor produciviy conains permanen and ransiory componens. Specifically, A = A p Z, where he permanen and ransiory componens follow processes log A p +1 = φ a log A p + σ a ε a,+1, and log Z +1 = φ z log Z + σ z ε z,+1, respecively, wih as he difference operaor, and innovaions ε a, and ε z, IIDN, 1). The labor inpu in producion is a composie of a coninuum of differeniaed labor ypes indexed by 11

13 k [, 1] via he aggregaor [ 1 NI,j) d = NI,j, d k) θw 1 θw ] θw θw 1 dj, 1) where θ w is he elasiciy of subsiuion across differeniaed labor ypes. Producers have marke power o se he price of heir differeniaed goods in a Calvo 1983) saggered price seing. Tha is, wih some posiive probabiliy a producer is unable o change he produc price a any poin of ime. We allow for differen probabiliies across he wo indusries o capure heerogeneous degrees of price rigidiies. The probabiliy of no changing he price of a differeniaed good a a paricular ime in indusry I is α I. When he producer is able o se a new price for he good, he price is se o maximize he expeced presen value of all fuure profis, aking ino accoun he probabiliy of no changing ha price in he fuure. The maximizaion problem is max {P I, j)} E [ αi τ M,+τ $ PI, j)y I,+τ j) W +τ j)ni,+τ j) )] d, 11) τ= subjec o he demand funcion see appendix B for is derivaion) ) YI,+τ j) 1/θ P I, j) = P I,+τ, 12) Y I,+τ and he producion funcion Y I,+τ j) = A +τ N d I,+τ j), 13) where Y I,+τ j) is he level of oupu of firm j in indusry I a ime + τ when he las ime he price was rese was a. A similar definiion applies o N d I,+τ j) and W +τ j). All firms wihin an indusry adjusing heir produc price opimally face he same opimizaion problem 12

14 and choose he same opimal price PI,. The appendix shows ha his price saisfies P ) ) I, PI, H I, = µ W G I,, 14) A P I, where µ = θ, θ 1 H I, = 1 + α I E [M $,+1 and G I, = 1 + α I E [M $,+1 YI,+1 Y I, YI,+1 ) ) θ PI,+1 H I,+1], P I, ) ) θ ) PI,+1 W+1 A Y I, P I, W A +1 ) G I,+1 ]. In he absence of price rigidiies α I = ), he opimal price is he markup adjused marginal cos, wih opimal markup µ. 2.3 Moneary Auhoriy We model a moneary auhoriy ha ses he level of a shor-erm nominal ineres rae. For simpliciy, we define he coninuously compounded one-period nominal rae, i logr f, $ ). Moneary policy is described by he policy rule i = ρ i ρ) ī + ı π π + ı x x ) + u, 15) where he ineres rae is se responding o he lagged ineres rae, aggregae inflaion π log log 1, he oupu gap x, and a policy shock u. The oupu gap is defined as he deviaion of oal oupu wih respec o he oupu ha would be obained under perfecly flexible prices and wages, Y f. Tha is, x log Y log Y f. 16) 13

15 I can be shown ha real oupu and real wages in a flexible price and wage economy are A Y f 1+ω ) 1/ω+ψ) =, and W real,f µµ w κ = A µ, 17) respecively. The policy shock follows he process u +1 = φ u u + σ u ε u,+1, 18) wih ε u IIDN, 1). 2.4 Asse Reurns We define socks as financial claims on all fuure profis. We are ineresed in analyzing sock reurns a aggregae and indusry levels. The real sock price and associaed one-period gross) reurn for indusry I are, respecively, [ ] S D,I, = E M,+n D I,+n, and R D,I,+1 = D I,+1 + S D,I,+1. 19) S D,I, n=1 Similarly, he aggregae sock is a claim on aggregae profis D = D H, + D L,. The price and reurn of his claim are [ ] S D, = E M,+n D +n, and R D,+1 = D +1 + S D,+1, 2) S D, n=1 respecively. I is also convenien o compare sock reurns wih reurns of claims on aggregae and indusry oupu. The price and reurn of a claim on all fuure real oupu of indusry I are, respecively, S Y,I, = E [ n=1 M,+n Y real I,+n ], and R Y,I,+1 = Y real I,+1 + S Y,I,+1 S Y,I,, 21) 14

16 ) where YI, real PI, = Y I, is he real oupu of indusry I. The price and reurn of a claim on all aggregae oupu are, respecively, [ ] S Y, = E M,+n Y +n, and R Y,+1 = Y +1 + S Y,+1. 22) S Y, n=1 Asse reurns also are useful o provide an alernaive specificaion of he pricing kernel in equaion 5). Appendix D shows ha i can be wrien in erms of consumpion and a porfolio reurn as M,+1 = [ β C+1 C ) ψ ] 1 γ 1 ψ 1 R Y L,+1 ) 1 1 γ 1 ψ, 23) where R Y L,+1 = 1 ν )R Y,+1 + ν R LI,+1, R LI,+1 = LI +1 + S LI,+1 and ν = S LI, νs LI, νs LI, S C,., LI W N s H w, G w,, for ν 1 ψ 1 1+ω µ w. Therefore, if ψ γ, he pricing kernel depends on he porfolio reurn R Y L,. This reurn is a weighed average of he reurns of claims on aggregae oupu and adjused labor income LI. Noice ha LI = LI in he absence of wage rigidiies. Also, if here is no disuiliy of labor ω ), he porfolio reurn converges o R Y L, = R Y,, as in he original specificaion in Epsein and Zin 1989). 2.5 Equilibrium The equilibrium of he economy requires produc, labor, and financial marke clearing. Produc marke clearing 15

17 The produc marke clearing condiions are C = Y, and C I, = Y I,, for I = {H, L}. Labor marke clearing In equilibrium, supply and demand of labor ype k employed by firm j in indusry I are equal. Tha is, Ni,j, s k) = NI, d j, k). From i, Appendix C shows ha equilibrium in he aggregae labor marke implies N s = N d F w,, where aggregae demand saisfies N d = Y A F. The disorion F w, = 1 [ W k) W ) 1 θw ] θw 1 θw dk, is he resul of wage dispersion across labor ypes from wage rigidiies. The disorion F = ϕ H PH, ) η ) η PL, F H, + ϕ L F L,, where F I, = 1 ) θ PI, j) dj, P I, is he resul of price dispersion across firms and indusries from price rigidiies. The appendix shows ha he inefficiencies F w, and F resuling from nominal rigidiies ranslae ino Y < A N s. Financial marke clearing In equilibrium, he nominal ineres rae from household maximizaion in equaion 7) maches he ineres rae se by he moneary auhoriy. Tha is, log [ M $,+1] = ρ i ρ) ī + ı π π + ı x x ) + u. Appendix E provides a summary of he sysem of equaions describing he equilibrium of he model. In order o obain balanced growh, we make κ = A p ) 1 ψ. This condiion ensures ha Y, Y I,, W, and W are growing a he same rae. We solve he model numerically, applying 16

18 a second-order approximaion of he opimaliy condiions. 7 A second-order approximaion is required o capure expeced excess reurns on financial claims. 3 Analysis We analyze he implicaions of nominal rigidiies and moneary policy on expeced asse reurns a aggregae and indusry level. We focus on expeced excess reurns of claims on all fuure oupu consumpion) and profis. The effecs of nominal rigidiies on expeced excess reurns can be undersood by heir impac on he pricing kernel, oupu, and producion markups. We calibrae he model o capure imporan dynamics of U.S. macroeconomic variables and sock reurns. We compare differen model specificaions o highligh he mos imporan channels driving he resuls. 3.1 Undersanding he Mechanism: Oupu and Markup Effecs Expeced excess asse reurns over he risk-free rae reflec a compensaion for macroeconomic risk. The compensaion is deermined by he join dynamics of he marginal rae of subsiuion of consumpion and asse reurns, which is affeced by nominal rigidiies and moneary policy. We are ineresed in analyzing he reurns on he oupu and profi claims in equaions 19) o 22). To illusrae he mechanism, we assume ha he log-pricing kernel, oupu growh, profi growh and asse log-reurns follow normal disribuions. Consider firs a claim on aggregae oupu. Appendix F shows ha he expeced excess reurn of his claim over he risk-free rae is log E [R Y,+1 ] log R f, = cov m,+1, log R Y,+1 ) = cov m,+1, y +1 ) cov m,+1, log 1 + P Y,+1 )), 24) 7 We use he Dynare code available in 17

19 where m,+1 log M,+1, y log Y, and P Y, S Y, Y is he wealh-consumpion raio. The covariance of he pricing kernel wih oupu growh and he wealh-consumpion raio capures he expeced excess reurn of he claim on producion over he risk-free rae. Similarly, he expeced excess reurn of he claim on aggregae profis over he risk-free rae is log E [R D,+1 ] log R f, = cov m,+1, d +1 ) cov m,+1, log 1 + P D,+1 )), 25) where d log D, and P D, S D, D is he price-dividend raio. Nominal rigidiies affec he dynamics of consumpion and profis, and herefore, he responses o shocks of he marginal rae of subsiuion of consumpion, oupu and profi growh, and he wealh-consumpion and pricedividend raios. We use he model calibraion o analyze and compare hese responses in he presence and absence of nominal rigidiies. We also explore he difference in expeced reurns beween oupu and profi claims implied by he rigidiies. Consider a poenially ime-varying markup µ such ha profis are D = 1 1µ ) Y. From equaions 24) and 25), i follows ha log ) E [R D,+1 ] = cov m,+1, log 1 1 )) cov m,+1, log E [R Y,+1 ] µ PD, P Y,+1 )). 26) Two erms capure he difference in expeced excess reurns beween oupu and profi claims. The firs erm is he difference in he covariance of oupu and profi growh wih he pricing kernel. This difference is driven by he markup dynamics. The expeced reurn of a claim on profis is lower higher) han he expeced reurn of a claim on oupu if he markup and he discoun facor are posiively negaively) correlaed. Tha is, claims on profis are less more) risky han claims on oupu if profis represen a larger smaller) fracion of producion when he marginal rae of subsiuion of consumpion is high. The second componen capures he difference in he dynamics of wealh-consumpion and price-dividend raios wih he pricing kernel. This difference 18

20 is also driven by markup dynamics. Nominal rigidiies generae ineresing markup dynamics. In he absence of nominal rigidiies, producers choose a level of producion o charge a consan opimal markup µ = µ over marginal coss. Since he markup is consan, claims on oupu and profis have he same expeced reurns. On he oher hand, if he economy is affeced by price and wage rigidiies, he markup is where W real µ = µ ) W real 1, F W real,f = W is he real wage. The markup is driven by he aggregae price dispersion across firms, F, and he wage gap, or he value of he real wage index relaive o is value in a reference economy wih flexible prices and wages. Since F > 1, a direc effec of price dispersion is a reducion in he markup relaive o µ. In addiion, he wage gap has an inuiive effec on he markup. If real wages are higher han hose under flexible prices and wages, he markup is lower han µ. Therefore, analyzing he effecs of he ime variaion in markups induced by he nominal rigidiies amouns o analyzing he covariance beween he marginal rae of subsiuion and he wage gap. Equaions 24), 25), and 26) also apply o indusry claims. Differences in expeced sock reurns across indusries resul from differences in he covariance of heir respecive oupus and markups wih he discoun facor. We analyze he responses o shocks of indusry oupus and markups o undersand he effecs of nominal rigidiies on he cross secion of asse reurns. 3.2 Calibraion We use quarerly U.S. daa from 1987:1 o 21:4 for consumpion, inflaion, he shor-erm nominal ineres rae, and sock reurns. We focus on he Greenspan-Bernanke period o avoid changes in he moneary policy regime, as suggesed by Clarida, Galí and Gerler 2). The consumpion series was consruced using daa on real consumpion of nondurable and services 19

21 Table 1: Baseline Parameer Values Parameer Descripion Value β Subjecive discoun facor.9778 ψ Inverse of elasiciy of ineremporal subsiuion 5.3 γ Risk aversion parameer ω Inverse of Frisch labor elasiciy.35 ϕ H Weigh of indusry H good in he baske.5 θ Elasiciy of subsiuion of differeniaed goods 6 η Elasiciy of subsiuion of indusry goods 6 θ w Elasiciy of subsiuion of labor ypes 21 α H Price rigidiy parameer for indusry H.8 α L Price rigidiy parameer for indusry L α Wage rigidiy parameer.78 ρ Ineres-rae smoohing coefficien in policy rule.76 ī Consan in he policy rule.224 ı π Response o inflaion in he policy rule 1.5 ı x Response o oupu gap in he policy rule.125 φ u Auocorrelaion of policy shock.15 σ u 1 2 Condiional volailiy of policy shock.175 φ a Auocorrelaion of permanen produciviy shock.275 σ a 1 2 Condiional volailiy of permanen produciviy shock.246 φ z Auocorrelaion of ransiory produciviy shock.957 σ z 1 2 Condiional volailiy of permanen produciviy shock from he Bureau of Economic Analysis. The series is de-rended using a Hodrick-Presco filer. The inflaion series was consruced o capure inflaion relaed only o consumpion of nondurables and services, following he mehodology in Piazzesi and Schneider 27). The shor-erm nominal rae is he 3-monh T-bill rae from he Fama risk-free raes daabase. The sock marke daa are he quarerly reurns of he marke porfolio obained from he Cener for Research in Securiy Prices CRSP). Our calibraion sraegy has wo seps. Firs, we choose values for all parameers bu γ o mach second momens of de-rended consumpion, consumpion growh, inflaion, and he nominal ineres rae. Second, we choose γ o mach he Sharpe raio of he aggregae sock marke. Table 1 presens he parameer values for he baseline calibraion. 2

22 For he firs sep, we mosly rely on Alig e al. 211) hereafer ACEL). All parameers values are sandard in he macroeconomic lieraure. The value of θ is chosen o provide a markup of 2%, which is he value for he high markup specificaion in ACEL. We se he elasiciy of subsiuion across indusry goods, η, equal o θ. In he analysis of he cross secion of sock reurns, we presen resuls for specificaions for η differen han θ, since his difference has imporan cross secional implicaions. The price rigidiy parameer values for α H and α L are chosen such ha he average price duraion dur and dispersion of price duraion across indusries σdur) are, respecively, dur = ϕ H 1 log α H ϕ L 1 log α L = 2.2 quarers, and σdur) = [ϕ H 1 ) 2 dur + ϕ L 1 ) ] 2 1/2 dur = 2.13 quarers. log α H log α L These values are consisen wih he empirical evidence in Bils and Klenow 24). For simpliciy, we assume same weighs for indusries H and L, ϕ H = ϕ L =.5. The value of θ w is chosen o have an average markup of wages over he marginal rae of subsiuion of leisure and consumpion of 5%. The parameer α implies a duraion of wages of four quarers, as esimaed in ACEL. The parameer β and ī = logβ)) is chosen o mach he average level of he nominal risk-free rae. The ineres rae rule parameers ρ, ı π, and ı x are chosen o be consisen wih he evidence for he Greenspan era according o Clarida, Galí and Gerler 2). The parameer values for he elasiciies ψ, ω, and he auocorrelaions and condiional volailiies of produciviy and policy shocks are chosen o mach some empirical resuls presened in ACEL. ACEL use a VAR o idenify produciviy and policy shocks and obain a variance decomposiion for differen macroeconomic variables. Table 2 presens heir variance decomposiion for inflaion, consumpion and he shor-erm ineres rae. 8 Produciviy and policy shocks ex- 8 ACEL refers o hese produciviy shocks as neural echnology shocks. The variance decomposiion in 21

23 Table 2: Daa and Model Volailiy. The able conains he oal volailiy of macroeconomic variables and he volailiy explained by he model shocks in he daa and he model. The variance decomposiion is obained from Alig e al. 211). Columns labeled All refer o he volailiy explained by policy and produciviy shocks. Columns labeled Prod. refer o produciviy shocks permanen and ransiory). The column labeled Perm. refers o permanen produciviy shocks. The column labeled Trans. refers o ransiory produciviy shocks. The row labeled ĉ refers o de-rended log consumpion. The volailiy figures are presened in annualized percenage erms. Toal Variance Volailiy explained by he shocks volailiy decomp. %) Daa Model Daa ) Policy Prod. All Policy Prod. All Policy Prod. Perm. Trans. i π ĉ c plain a small fracion of he oal volailiy of he hree macroeconomic variables. Based on his decomposiion, we choose parameer values o mach he conribuion of hese shocks o he oal variabiliy of he macroeconomic series. Since our model has boh permanen and ransiory produciviy shocks, we require addiional resricions o idenify how much of he variabiliy explained by produciviy shocks is he resul of permanen and ransiory shocks. We choose a mix of shocks ha maches he volailiy of consumpion growh. Specifically, a calibraion in which all produciviy shocks are permanen implies a volailiy of consumpion growh significanly higher han in he daa. On he oher hand, a calibraion where all produciviy shocks are ransiory implies a very low volailiy in consumpion growh. The combinaion of permanen and produciviy shocks wih policy shocks maches he volailiy of consumpion growh in he daa. 9 A significan fracion of his volailiy is aribued o permanen shocks. The able shows ACEL for he shor-erm rae refers o he Federal Funds rae. We assume ha he same variance decomposiion applies o he hree-monh T-bill rae. ACEL esimae heir VAR using daa for We assume ha heir variance decomposiion also applies o our sample period. 9 Ideally, we would like o mach only he volailiy of consumpion growh explained by produciviy and policy shocks. However, his informaion is no available. Also, maching he oal volailiy of consumpion growh is helpful o make more direc comparisons wih oher asse pricing models. 22

24 ha he model is able o mach he conribuions of produciviy and policy shocks o he oal variabiliy of consumpion, inflaion, and he nominal ineres rae. The calibraion implies a low elasiciy of ineremporal subsiuion of consumpion of 1/5.3.19, and a Frisch elasiciy of labor supply of 1/ In he second calibraion sep, we choose γ o mach he sock marke quarerly annualized) Sharpe raio of.4 for he period. Since he empirical Sharpe raio is compued using nominal sock reurns and he nominal risk-free rae, we use he nominal expeced asse reurns and riskfree rae of he model o mach he daa. 11 Alernaively, we could have chosen γ o mach he equiy premium. However, profi claims in he model are no direcly comparable o dividend claims of he aggregae U.S. sock marke for several reasons: The model absracs from capial accumulaion and firm leverage, and profis are only he resul of monopolisic compeiion, among ohers. Insead, we mach he Sharpe raio and quanify he conribuion of nominal rigidiies o he equiy premium. The recursive uiliy specificaion allows us o increase risk aversion wihou affecing he elasiciy of ineremporal subsiuion. By doing so, he macroeconomic properies of he model are no significanly affeced by he degree of risk aversion, as shown by Tallarini 2). In he presence of leisure preferences, he coefficien of consan relaive risk aversion is no only deermined by γ. The household s aiude oward risk is also affeced by heir willingness o supply labor in differen saes of he world. As shown by Swanson 211), he average) coefficien 1 Macroeconomic models usually rely on elasiciies of subsiuion beween and 1. The Bansal and Yaron 24) model requires an elasiciy of subsiuion greaer han 1 in order o capure he observed equiy premium. Empirical esimaes range beween and 1. For insance, Hall 1988) provides an esimae very close o zero, and Vissing-Jorgensen 22) finds an elasiciy for sockholders around.3 o.4., and very close o zero for non-sockholders. 11 The difference beween nominal and real expeced excess reurns depends on he join dynamics of inflaion, reurns, and he pricing kernel. To see his, consider he real and nominal asse reurns R and R $, respecively. Under log-normaliy assumpions, i can be shown ha nominal and real expeced excess reurns are linked by E [R +1 $ log ] ) ) E [R +1 ] = log cov π +1, m,+1 log R +1 ) + var π +1 ). R $ f, R f, 23

25 for he recursive preferences in equaion 1) is ψ 1 + ψ ωµ + γ ψ 1 1 ψ 1+ω 26. This value is sill high according o empirical and experimenal evidence, 12 bu significanly lower han he values required by sandard real business cycle models o mach Sharpe raios. This reducion is he resul of permanen produciviy shocks and leisure preferences in he model. Table 3 presens summary saisics for our benchmark calibraion along wih hose from alernaive model specificaions. The alernaive specificaions help us undersand he main channels driving he resuls. In he benchmark calibraion, he volailiy of consumpion growh is equal o he volailiy in he daa by consrucion. The volailiy of profi growh is more han wice as large as he volailiy of consumpion growh. This is a direc resul of nominal rigidiies inducing variabiliy in producion markups. Maching a nominal Sharpe raio for profi claims of.4 implies a real Sharpe Raio of.48 and a real expeced excess reurn of.76%. The expeced excess reurn of a claim on oupu is slighly lower a.71%, indicaing ha he markup volailiy increases he riskiness of profis wih respec o oupu. Columns 2) o 4) allow us o quanify he conribuion of he hree differen shocks o he resuls. Each column corresponds o he baseline calibraion wih only one shock affecing he economy he volailiy of he wo oher shocks is se o zero). I is clear from his able ha mos of he premium is a compensaion for permanen produciviy shocks. These shocks conribue wih almos 73 bps o he premium, while he individual conribuion of ransiory produciviy and policy shocks is less han 2 bps each. The differences are also refleced in he implied Sharpe raios. The Sharpe raio for permanen shocks is significanly higher han he Sharpe raios for he wo oher shocks. Columns 5) o 7) presen model specificaions wih flexible prices, or flexible wages, or boh. The economy relaed o column 5) can be seen as a fricionless only-labor real business cycle 12 See, for insance, Barsky e al. 1997). 24

26 Table 3: Model Summary Saisics for Differen Specificaions. The baseline parameer values are presened in Table 1. Benchmark indicaes an economy wih boh price and wage rigidiies. Only A p indicaes only permanen produciviy shocks σ z = σ u = ). Only Z indicaes only ransiory produciviy shocks σ a = σ u = ). Only u indicaes only policy shocks σ a = σ z = ). No Rig. indicaes no price and wage rigidiies α H = α L = α = ). Only WR indicaes no price rigidiies α H = α L = ). Only PR indicaes no wage rigidiies α=). The expeced excess reurns and he Sharpe raio for asse b are XR b, = R b,+1 R f,, and SR b = E[XR b,] σxr b,, respecively. Volailiies and expeced excess ) reurns are presened in annualized percenage erms. Sharpe raios are annualized. 1) 2) 3) 4) 5) 6) 7) Benchmark Only A p Only Z Only u No Rig. Only WR Only PR σπ) σx) σi) σr) σlog µ) σ c) σ d) σ w) ρπ, c) ρ c, d) E[R f, $ ] E[XR Y,+1 ] E[XR D,+1 ] σr Y ) σr D ) SR Y SR D economy. In he absence of nominal rigidiies, he calibraion parameers imply negaive expeced excess reurns and Sharpe raios. This is he resul of permanen produciviy shocks as explained in he nex secion. Column 6) presens resuls for he benchmark calibraion assuming ha prices are perfecly flexible α H = α L = ). I becomes clear ha inroducing wage rigidiies significanly increases he premium in oupu and profi claims from -2 bps o 85 bps. Since wage rigidiies do no generae markup volailiy, claims on aggregae oupu and profis have he same expeced reurns. Column 7) shows implicaions for a model wih price rigidiies only α = ). 25

27 Price rigidiies induce posiive expeced excess reurns. Profi claims have lower expeced reurns han oupu claims, since he ime variaion in markups induced by price rigidiies decreases he riskiness of profis relaive o oupu. Quaniaively, he effec of wage rigidiies on he premium is significanly larger han he effec of price rigidiies. 3.3 Risk Compensaions in Expeced Asse Reurns under Nominal Rigidiies Our model calibraion shows ha nominal rigidiies conribue o increase expeced excess reurns on oupu and profi claims. The increased premium is mainly a compensaion for exposure o permanen produciviy shocks and wage rigidiies. To undersand hese resuls, we compare he conribuion of each shock o expeced asse reurns in economies wih and wihou nominal rigidiies. Compensaion for Permanen Produciviy Shocks Panel A of Table 4 shows ha nominal rigidiies increase he premium for permanen produciviy shocks in oupu and profi claims. To see why, consider firs an economy wih perfecly flexible prices and wages. Oupu and profi claims have he same expeced reurns in his economy since he markup is consan. Figure 1 shows ha a negaive permanen shock increases he marginal uiliy of consumpion and he reurn on he oupu claim. This claim hen ac as hedging insrumen and has a negaive expeced excess reurn. Two opposie effecs drive his resul. The firs erm in equaion 24) generaes a posiive premium since he negaive shock leads o lower consumpion growh. The second erm generaes a negaive premium since he negaive shock leads o a higher wealh-consumpion raio, P Y,+1. The raio increases because he permanen increase in he pricing kernel raises he price of he claim. 13 The negaive conribuion of his 13 There are wo channels affecing P Y,+1 : he effec of he shock on all fuure pricing kernels and he effec of he shock on all fuure oupu. Afer a negaive shock, he higher pricing kernel increases P Y,+1, and he lower oupu decreases P Y,+1. Bansal and Yaron 24) refer o hese effecs as wealh and subsiuion effecs, respecively. 26

28 erm o he premium offses he posiive conribuion of he firs erm, and resuls in a negaive expeced excess reurn. Consider now an economy wih wage rigidiies and flexible prices. The expeced excess reurns on oupu and profi claims increase o 8 bps in his economy. Afer a negaive shock, Figure 1 shows ha oupu declines by more han under flexible wages. Since some nominal wages are no adjused downwards, producers increase heir produc prices o resore he opimal markup µ. Higher prices imply a lower oupu demand, lower profis, and a higher marginal uiliy han under flexible wages. The permanen decline in oupu and profis also decreases he wealhconsumpion and price-dividend raios. As a resul, wage rigidiies induce a posiive compensaion for permanen shocks in expeced reurns on oupu and profi claims. Table 4 also shows resuls for an economy wih price rigidiies and flexible wages. In his case, Figure 1 shows ha a negaive permanen shock decreases oupu by more han under flexible prices and wages. Producers ha canno adjus heir produc prices downwards face a reducion in produc demand. Real wages decrease and producion markups increase. As a resul, profis iniially increase reducing he riskiness of profi claims relaive o oupu claims. However, here is a permanen decline in oupu and profis ha induces negaive reurns on boh oupu and profi claims. Tha is, price rigidiies generae a posiive premium in hese reurns for exposure o permanen shocks. Quaniaively, he premium generaed by price rigidiies is lower han he premium generaed by wage rigidiies. While he premium in profi claims for wage rigidiies is 8 bps, i is only 2 bps for price rigidiies. Finally, in an economy characerized by boh price and wage rigidiies, he premium for permanen shocks in profi claims is around 73 bps. The premium is lower han in an economy wih only wage rigidiies because he price rigidiy parially offses he negaive effec of wage rigidiies on oupu. Since prices are sicky, some producers canno increase heir prices o offse he high nominal wages given he negaive shock. Therefore, real wages do no decline as much as in he For a low elasiciy of ineremporal subsiuion ψ > 1), and posiively auocorrelaed consumpion growh, he effec of a higher pricing kernel dominaes he effec of a lower oupu sream. 27

29 Table 4: Conribuion of Individual Shocks o Expeced Excess Reurns. The baseline parameer values are presened in Table 1. Benchmark indicaes an economy wih boh price and wage rigidiies. No Rig. indicaes no price and wage rigidiies α H = α L = α = ). Only WR indicaes no price rigidiies α H = α L = ). Only PR indicaes no wage rigidiies α=). Expeced excess reurns and Sharpe raios for asse b are XR b, = R b,+1 R f,, and SR b = E[XR b,] σxr b,, respecively. Volailiies and expeced excess reurns are presened in annualized ) percenage erms. Sharpe raios are annualized. Benchmark No Rig. Only WR Only PR Panel A: Only A p E[XR Y,+1 ] E[XR D,+1 ] σr Y ) σr D ) SR Y SR D Panel B: Only Z E[XR Y,+1 ] E[XR D,+1 ] σr Y ) σr D ) SR Y SR D Panel C: Only u E[XR Y,+1 ] E[XR D,+1 ] σr Y ) σr D ) SR Y SR D economy wih only wage rigidiies, inflaion is lower, and oupu demand is less negaively affeced by he shock. The relaive increase in real wages reduces he markup, which ranslaes ino higher expeced reurns on profi claims han on oupu claims. Compensaion for Transiory Produciviy Shocks Panel B of Table 4 shows posiive expeced excess reurns in boh oupu and profi claims as a compensaion for ransiory shocks. This compensaion is significanly lower han he compen- 28