Does Lumpy Investment Matter for Business Cycles?

Transcription

1 Does Lumpy Invesmen Maer for Business Cycles? Jianjun Miao Pengfei Wang January 8, 21 Absrac We presen an analyically racable general equilibrium business cycle model ha feaures micro-level invesmen lumpiness. We prove an exac irrelevance proposiion which provides sufficien condiions on preferences, echnology, and he fixed cos disribuion such ha any posiive upper suppor of he fixed cos disribuion yields idenical equilibrium dynamics of he aggregae quaniies normalized by heir deerminisic seady sae values. We also give wo condiions for he fixed cos disribuion, under which lumpy invesmen can be imporan o a firs-order approximaion: (i) The seady-sae elasiciy of he adjusmen rae is large so ha he exensive margin effec is large. (ii) More mass is on low fixed coss so ha he general equilibrium price feedback effec is small. JEL Classificaion: E22, E32 Keywords: generalized (S,s) rule, lumpy invesmen, general equilibrium, business cycles, marginal Q, exac irrelevance proposiion We hank Fernando Alvarez, Simon Gilchris, Bob King, and Yi Wen for helpful commens. Deparmen of Economics, Boson Universiy, 27 Bay Sae Road, Boson, MA Tel.: miaoj@bu.edu. Homepage: hp://people.bu.edu/miaoj. Deparmen of Economics, Hong Kong Universiy of Science and Technology, Clear Waer Bay, Hong Kong. Tel: (+852) pfwang@us.hk

2 1 Inroducion In his paper, we presen an analyically racable general equilibrium business cycle model ha incorporaes micro-level convex and nonconvex adjusmen coss. Recen empirical sudies have documened ha nonconvexiies of microeconomic capial adjusmen are widespread phenomena. Examining a 17-year sample of large, coninuing US manufacuring plans, Doms and Dunne (1998) find ha ypically more han half of a plan s cumulaive invesmen occurs in a single episode. In addiion, hey find ha long periods of relaively small changes are inerruped by invesmen spikes. Using he Longiudinal Research Daabase, Cooper and Haliwanger (26) find ha abou 8 percen of observaions enail an invesmen rae near zero. These observaions of inacion are complemened by periods of raher inensive adjusmen of he capial sock. Cooper and Haliwanger (26) also esimae srucural parameers of a rich specificaion of convex and nonconvex adjusmen coss. Given he above evidence, an imporan quesion is wheher micro-level nonconvexiies maer for aggregae macroeconomic dynamics. 1 This quesion is under significan debae in he lieraure. In a seminal sudy, Thomas (22) challenges he previous parial equilibrium analyses (e.g., Caballero e al. (1995), Cooper e al. (1999), and Caballero and Engel (1999)) by providing a general equilibrium model wih lumpy invesmen. 2 She applies he Dosey, King, and Wolman (1999) mehod and shows quaniaively ha lumpy invesmen is irrelevan for business cycles. Subsequenly, Khan and Thomas (23, 28) build more general models and use a differen numerical mehod (Krusell and Smih (1998)) o solve he models. 3 They sill obain a similar finding. Their key insigh is ha he general equilibrium price feedback effec offses changes in aggregae invesmen demand. 4 However, some researchers remain unconvinced by he Khan-Thomas finding. Bachmann e al. (28) and Gourio and Kashyap (27) argue ha boh fixed adjusmen coss and general equilibrium price movemens are imporan for business cycle analysis. The relaive imporance of hese wo effecs is sensiive o 1 Embedding a parial equilibrium model similar o Abel and Eberly (1998) in a coninuous-ime general equilibrium framework, Miao (28) sudies he effec of corporae ax policy on long-run equilibrium in he presence of fixed coss and irreversibiliy. 2 Veraciero (22) embeds he parial equilibrium cosly irreversibiliy model of Abel and Eberly (1996) in a general equilibrium business cycles model. Wang and Wen (29) presen a general equilibrium model wih irreversible invesmen o sudy aggregae and firm-level volailiy. 3 Miao (26) proves he exisence of sequenial compeiive equilibrium for he Krusell-Smih-syle incomplee markes economy wih heerogeneous agens. However, i is an open quesion wheher or no he Krusell-Smihype recursive equilibrium exiss. See Heahcoe e al. (29) for a survey of heerogeneous agens models. 4 House (28) finds an approximae irrelevance resul numerically in a differen seup. In his model, he source of he irrelevance resul is no he general equilibrium price movemens, bu is he nearly infinie ineremporal subsiuion for he iming of invesmen resuling from long-lived capial. 1

3 calibraion. Using he Dosey e al. (1999) mehod as in Thomas (22), Gourio and Kashyap (27) calibrae a larger size of fixed adjusmen coss, and argue ha for he exensive margin effec o be large, he fixed cos disribuion mus be compressed in he sense ha many firms face roughly he same sized fixed coss. One reason causing he debae is due o he complexiy of he general equilibrium models wih heerogeneous firms in his lieraure. Researchers have o apply complicaed numerical mehods o solve hese models. 5 There is no general heoreical resul for comparison. Wihou a heoreical resul under explicily saed assumpions, one may doub he accuracy and generaliy of numerical soluions. To he bes of our knowledge, here is no accuracy es of he Dosey, King, and Wolman (1999) mehod applied in he lieraure of equilibrium models wih lumpy invesmen. In addiion, den Haan (21b) poins ou ha he accuracy es of he Krusell-Smih mehod based on he R 2 and he sandard error is inadequae and has flaws. There is also no comparison of numerical soluions obained from hese wo mehods in he lumpy invesmen lieraure. Thus, i is imporan o develop a reference model ha can deliver an explici characerizaion of equilibrium. This characerizaion can be used no only for esablishing heoreical resuls, bu also for obaining accurae and efficien numerical soluions. In he presen paper, we propose such an analyically racable general equilibrium model o undersand he aggregae implicaions of various forms of adjusmen coss. Our model feaures boh convex and nonconvex adjusmen coss. Firms face aggregae labor-augmening echnology shocks and invesmen-specific echnology shocks. 6 In addiion, firms face idiosyncraic fixed cos shocks, resuling in a generalized (S,s) invesmen rule as in Caballero and Engel (1999). Our model is similar o he Khan and Thomas (23) model wih wo main differences. Firs, we assume ha he firm-level producion funcion has consan reurns o scale raher han decreasing reurns o scale. This assumpion makes he aggregae producion funcion have consan reurns o scale, consisen wih he real business cycle (RBC) lieraure. Second, we assume ha a firm s fixed coss are proporional o is exising capial sock raher han labor coss. These wo assumpions allow us o exploi he homogeneiy propery of firm value o derive a closed-form soluion for he generalized (S,s) invesmen rule. They also allow us o derive exac aggregaion so ha we can represen aggregae equilibrium dynamics by a sysem 5 Recenly, researchers have developed some new numerical mehods o solve models wih heerogeneous agens. These mehods are summarized in he January 21 issue of he Journal of Economic Dynamics and Conrol. den Haan (21a) provides a comparison of hese mehods when applied o he Krusell and Smih (1998) model. I is ineresing o see how hese mehods are applied o solve he lumpy invesmen model. 6 Greenwood e al. (2) and Fisher (26) emphasize ha invesmen-specific echnology shocks are imporan for business cycles. 2

4 of nonlinear difference equaions as in he RBC lieraure. In paricular, he disribuion of capial maers only o he exen of is mean. We hen prove ha he compeiive equilibrium is consrained efficien in he sense ha if a social planner decides allocaions aken firm-level convex and nonconvex adjusmen coss as given, hen he opimal allocaions are he same as hose in a compeiive equilibrium. This resul also implies ha a recursive equilibrium exiss and unique, which provides he heoreic foundaion for a recursive mehod o solve he model numerically. The benefi of our modelling is ha we do no need o use a complicaed numerical mehod (e.g., Krusell and Smih (1998)) o approximae he disribuion of capial. Because we characerize equilibrium dynamics by a sysem of nonlinear difference equaions, we can use a sandard numerical mehod o obain accurae soluions. In paricular, we apply he secondorder approximaion mehod (Schmi-Grohe and Uribe (24)), which proves quie efficien and accurae for analyzing business cycles (see Aruoba e al. (26)). In addiion, we can also use he sandard log-linear approximaion mehod o flesh ou inuiion ransparenly by pencil and paper. Boh mehods can be easily implemened numerically using he publicly available package Dynare. The cos of our modelling is ha our model canno address disribuional asymmery and nonlineariy emphasized by Caballero e al. (1995). Neverheless, our model is sill rich enough for us o analyze business cycles wih he essenial feaure of micro-level lumpiness, bu also is racable enough for us o analyze heoreically he effecs of inensive margin, exensive margin, and general equilibrium price movemens, which are he mos imporan elemens emphasized in he lieraure. We derive he following main resuls. Firs, we prove an exac irrelevance proposiion: If he producion funcion is Cobb-Douglas, preferences are represened by a ime-addiive expeced uiliy funcion consisen wih balanced-growh pah, and he idiosyncraic fixed cos shocks are drawn independenly and idenically from a power funcion disribuion, hen any posiive upper suppor of he fixed cos disribuion yields idenical equilibrium dynamics of he aggregae quaniies normalized by heir deerminisic seady-sae values. Second, we derive condiions under which lumpy invesmen is imporan for aggregae dynamics o a firs order approximaion. Essenially, we need he exensive margin effec o be large and he general equilibrium price feedback effec o be small. We show ha he exensive margin effec is deermined by he seady-sae elasiciy of he adjusmen rae wih respec o he invesmen rigger. The larger is his elasiciy, he larger is he exensive margin effec. The general equilibrium price feedback effec is deermined by preferences and he seady-sae raio of he opion value of waiing o he price of capial. When he elasiciy of ineremporal 3

5 subsiuion is large, he ineres rae feedback effec is small. When he fixed cos disribuion is more righ skewed (i.e. more firms have small fixed coss), he opion value of waiing is larger, leading o a weaker general equilibrium wage feedback effec. Third, we show numerically ha inroducing fixed coss o a model wih convex adjusmen coss raises business cycle volailiy, bu reduces persisence of oupu, consumpion, invesmen, and hours. In addiion, when lumpy invesmen becomes more imporan, i brings business cycle momens closer o hose in he sandard fricionless RBC model. Our heoreical resuls may reconcile some of he debae and some of he numerical findings in he lieraure. For example, Khan and Thomas (23, 28) find ha when hey increase he maximal fixed cos by 1 folds, he equilibrium dynamics nearly have no change. This is due o he fac ha hey assume a uniform disribuion of fixed coss and a nearly consanreurns-o-scale producion funcion (heir calibraed value of reurns o scale is.95 or.896). For he maximal size of fixed coss o maer, we need he producion funcion o have high curvaure as shown numerically by Gourio and Kashyap (27) and Bachmann e al. (28). Gourio and Kashyap (27) also argue ha he fixed cos disribuion mus be compressed. We show ha his feaure of he disribuion is no essenial. Wha is essenial is ha he fixed cos disribuion mus be righ skewed and mus have a high seady-sae elasiciy of he adjusmen rae. We emphasize ha he size of oal fixed coss is no essenial for he lumpy invesmen o be imporan. Gourio and Kashyap (27) argue ha Khan and Thomas calibraed fixed coss are oo small and ha raising he size of oal fixed coss will make lumpy invesmen more imporan. By conras, we use numerical examples o show ha even he size of fixed coss is smaller, lumpy invesmen can be more imporan for he reason discussed before. The remainder of he paper proceeds as follows. Secion 2 presens he model. Secion 3 analyzes equilibrium properies. Secion 4 provides numerical resuls. Secion 5 concludes. An appendix conains all proofs. 2 The Model We consider an infinie horizon economy. Time is discree and indexed by =, 1, 2,... There is a coninuum of heerogeneous producion unis, indexed by j and disribued uniformly over [, 1]. We idenify a producion uni wih a firm or a plan. There is a coninuum of idenical households, who rade all firms shares. Each firm is subjec o aggregae labor-augmening produciviy shocks and invesmen-specific echnology shocks. In addiion, each firm is subjec 4

6 o idiosyncraic shocks o fixed adjusmen coss of invesmens. To focus on he implicaions of fixed coss for business cycles, we absrac from long-run growh. I is sraighforward o incorporae growh because our model assumpions are consisen wih balanced growh. 2.1 Firms All firms have an idenical producion echnology ha combines labor and capial o produce oupu. Specifically, if firm j owns capial K j and hires labor N j according o he producion funcion: Y j = F, i produces oupu Y j ( ) K j, A N j, (1) where A represens aggregae labor-augmening echnology shocks and follows a Markov process given by: ln A +1 = ρ A ln A + 1 ρ 2 A σ Ae A,+1. Here, ρ A (, 1), σ A > and e A, is an idenically and independenly disribued (iid) sandard normal random variable. Assume ha F is sricly increasing, sricly concave, coninuously differeniable, and saisfies he usual Inada condiions. In addiion, i has consan reurns o scale. Each firm j may make invesmen I j o increase is exising capial sock Kj. Invesmen incurs boh nonconvex and convex adjusmen coss. As in Uzawa (1969), Baxer and Crucini (1993), and Jermann (1998), capial accumulaion follows he law of moion: ( ) K j +1 = (1 δ)kj + Kj Φ I j, K j given, (2) where δ is he depreciaion rae and Φ represens convex adjusmen coss. To faciliae analyical soluions, we follow Jermann (1998) and specify he adjusmen cos funcion as: Φ (x) = K j ψ 1 θ x1 θ + ς, (3) where ψ > and θ (, 1). Nonconvex adjusmen coss are fixed coss ha mus be paid if and only if he firm chooses o inves. As in Cooper and Haliwanger (26), we measure hese coss as a fracion of he firm s capial sock. 7 Tha is, if firm j makes new invesmen, hen i pays fixed coss ξ j Kj, which is independen of he amoun of invesmen. As will be clear laer, his 7 There are several ways o model fixed adjusmen coss in he lieraure. Fixed coss may be proporional o he demand shock (Abel and Eberly (1998)), profis (Caballero and Engel (1999) and Cooper and Haliwanger (26)), or labor coss (Thomas (22) and Khan and Thomas (23, 28)). 5

7 modelling of fixed coss is imporan o ensure ha firm value is linearly homogenous. Following Caballero and Engel (1999), we assume ha ξ j is idenically and independenly drawn from a disribuion wih densiy φ over [, ξ max ] across firms and across ime. These idiosyncraic coss cause firm heerogeneiy. Each firm j pays dividends o households who are shareholders of he firm. Dividends are given by: D j = Y j w N j Ij z ξ j Kj 1 I j (4) where w is he wage rae, and z represens aggregae invesmen-specific echnology shocks. Here 1 I j is an indicaor funcion aking value 1 if Ij, and value, oherwise. Assume z follows a Markov process given by: ln z +1 = ρ z ln z + 1 ρ 2 z σ z e z,+1, where ρ z (, 1), σ z >, and e z, is an iid sandard normal random variable. All random variables A, z and ξ j are muually independen. Firm j s objecive is o maximize cum-dividends marke value of equiy P j : [ ] max P j E β s Λ +s D j +s, (5) Λ s= subjec o (2) and (4). Here, β s Λ +s /Λ is he sochasic discoun facor beween period and + s. We will show laer ha Λ +s is a household s marginal uiliy in period + s. 2.2 Households All households are idenical and have he same uiliy funcion: [ ] E β U (C, 1 N ), (6) = where β (, 1) is he discoun facor, and U is a sricly increasing, sricly concave and coninuously differeniable funcion ha saisfies he usual Inada condiions. Each household chooses consumpion C, labor supply N, and share holdings α j +1 subjec o he budge consrain: C + α j +1 ( ) P j Dj dj = o maximize uiliy (6) α j P j dj + w N. (7) 6

8 The firs-order condiions are given by: Λ (P j Dj ) = E βλ +1 P j +1, (8) U 1 (C, 1 N ) = Λ, (9) U 2 (C, 1 N ) = Λ w. (1) Equaions (8)-(9) imply ha he sock price P j is given by he discouned presen value of dividends as in equaion (5). In addiion, Λ is equal o he marginal uiliy of consumpion. 2.3 Compeiive Equilibrium The sequences of quaniies {I j, N j, Kj }, {C, N }, and prices {w, P j } for j [, 1] consiue a compeiive equilibrium if he following condiions are saisfied: (i) Given prices {w }, {I j, N j } solves firm j s problem (5) subjec o he law of moion (2). (ii) Given prices { w, P j }, N, α j +1 } maximizes uiliy in (6) subjec o he budge consrain (7). (iii) Markes clear in ha: I j C + dj + z 3 Equilibrium Properies α j = 1, N = N j dj, ξ j Kj 1 I j dj = F ( ) K j, A N j dj. (11) We sar by analyzing a single firm s opimal invesmen policy, holding prices fixed. We hen conduc aggregaion and characerize equilibrium aggregae dynamics by a sysem of nonlinear difference equaions. We show ha he equilibrium is consrained efficien. Nex, we analyze seady sae and prove an exac irrelevance resul. Finally, we log-linearize he equilibrium dynamic sysem and examine he condiions under which lumpy invesmen can be imporan. 3.1 Opimal Invesmen Policy To simplify problem (5), we firs solve a firm s saic labor choice decision. Le n j = N j /Kj. The firs-order condiion wih respec o labor yields: f ( A n j ) A = w, (12) 7

9 where we define f ( ) = F (1, ). This equaion reveals ha all firms choose he same laborcapial raio in ha n j = n = n(w, A ) for all j. We can hen derive firm j s operaing profis: ( ) max F K j N j, A N j w N j = R K j, where R = f(a n ) w n is independen of j. Noe ha R also represens he marginal produc of capial because F has consan reurns o scale. Le i j = Ij /Kj invesmen rae. We can hen express dividends in (4) as: [ ] D j = R ij ξ j z 1 i j K j, denoe firm j s and rewrie (2) as K j +1 = [ (1 δ) + Φ(i j ) ] K j. (13) The above wo equaions imply ha equiy value or firm value are linear in capial K j. We can hen wrie firm value as V j Kj [ V j Kj = max i j and rewrie problem (5) by dynamic programming: ] R ij z ξ j 1 i j K j + E [ βλ+1 Λ ] V j +1 Kj +1, (14) subjec o (13). Subsiuing equaion (13) ino equaion (14), we rewrie problem (14) as: V j = max i j [ ] R ij ξ j z 1 i j + g(ij )E βλ+1 V j +1, (15) Λ where we define: g(i j ) = 1 δ + Φ(ij ). (16) Noe ha R and Λ depend on he curren aggregae sae (K, A, z ) only. equilibrium law of moion for aggregae capial is given by: Suppose he Given his law of moion, he sae variable for V j K +1 = G (K, A, z ). (17) V j = V ( K, A, z, ξ j is (K, A, z, ξ j ). We can wrie i as: for some funcion V. We aggregae each firm s price of capial V j and define he aggregae value of he firm per uni of capial condiioned on aggregae sae (K, A, z ) as: V = V (K, A, z ) = ξmax 8 ), V (K, A, z, ξ) φ(ξ)dξ, (18)

10 for some funcion V. Because ξ j is iid across boh ime and firms and is independen of aggregae shocks, we obain: [ ] [ ξmax ] [ ] Λ+1 E V j Λ+1 Λ+1 +1 = E V (K +1, A +1, z +1, ξ) φ(ξ)dξ = E V+1, (19) Λ Λ Λ Now, we rewrie problem (15) as: V ( ) K, A, z, ξ j = max i j [ ] R ij ξ j z 1 i j + g(ij )E βλ+1 V+1. (2) Λ Using his equaion, we can characerize a firm s opimal invesmen policy by a generalized (S,s) rule (Caballero and Engel (1999)). In so doing, we firs define marginal Q as he (riskadjused) presen value of a marginal uni of invesmen: Q = E [ βλ+1 Λ V+1 ]. (21) Since invesmen becomes producive wih a one period delay, marginal Q is equal o he discouned expeced value of he firm of an addiional uni of capial in he nex period. In coninuous ime, he difference beween marginal Q and he aggregae price of capial V+1 disappears. Because firm value is linearly homogeneous in capial, Tobin s average Q is equal o he marginal Q (Hayashi (1982)). Equaion (2) reveals ha g or Φ mus be concave o deermine opimal i j. In addiion, opimal i j is relaed o marginal Q (Abel and Eberly (1994)). Wihou convex adjusmen coss, Φ (x) = x and i j is indeerminae unless one assumes decreasing-reurns-o-scale echnology. Proposiion 1 Firm j s opimal invesmen policy is characerized by he (S, s) policy in ha here is a unique rigger value ξ > such ha he firm invess if and only if ξ j min{ξ, ξ max }. The rigger value ξ saisfies he equaion: The opimal arge invesmen level is given by: When ξ ξ max, marginal Q saisfies: [ { βλ +1 Q = E R +1 + (1 δ + ς) Q +1 + Λ θ 1 θ z 1 θ θ (ψq ) 1 θ = ξ. (22) i j = (ψz Q ) 1 θ. (23) +1 [ ξ +1 ξ ] φ(ξ)dξ }]. (24) 9

11 Equaion (22) says ha, a he value ξ, he benefi from invesmen is equal o he fixed cos of invesmen. The benefi from invesmen increases wih Q and z. Thus, he invesmen rigger ξ also increases wih Q and z. If ξ ξ max, hen he firm always invess. In he aggregae wih a cross secion of firms, his means ha all firms decide o inves. analysis below, we will focus on an inerior soluion for which ξ < ξ max. Noe ha he invesmen rigger ξ In he depends on he aggregae sae (K, A, z ) only. I does no depend on he firm-specific sae (K j, ξj ). This observaion implies ha condiioned on he aggregae sae, he adjusmen hazard, ξ φ (ξ) dξ, is a consan. This resul is due o our assumpions of compeiive markes, consan-reurns-o-scale producion funcion, and he iid disribuion of ξ j. When he producion funcion has decreasing reurns o scale or here is monopoly power, he invesmen rigger ξ and he adjusmen hazard will depend on he firm-specific capial sock, as discussed in Caballero e al. (1995), Caballero and Engel (1999), and Khan and Thomas (23, 28). Equaion (23) implies ha all firms choose an idenical arge invesmen level, which is inconsisen wih empirical evidence on invesmen spikes (Cooper and Haliwanger (26)). One way o make invesmen arges depend on firm-specific characerisics is o inroduce a persisen idiosyncraic produciviy shock (Khan and Thomas (28)). This exension will complicae our analysis significanly because in his case he invesmen rigger ξ depends on he idiosyncraic produciviy shock, which makes aggregaion complicaed. 8 Equaion (23) shows ha he opimal invesmen level is posiively relaed o marginal Q if and only if he firm s idiosyncraic shock ξ j is lower han he rigger value ξ, condiioned on he aggregae sae (K, A, z ). When ξ j > ξ, firm j chooses no o inves. This zero invesmen is unrelaed o marginal Q. As a resul, invesmen may no be relaed o marginal Q in he presence of fixed adjusmen coss, a poin made by Caballero and Leahy (1996). Equaion (24) is a ype of asse-pricing equaion. Ignoring he inegraion erm inside he condiional expecaion operaor in equaion (24), his equaion saes ha he expeced price of capial or marginal Q is equal o he risk-adjused presen value of he marginal produc of capial. The inegraion erm in (24) reflecs he opion value of waiing because of he fixed adjusmen coss. When he shock ξ j > ξ, i is no opimal o pay he fixed coss o make invesmen. Firms will wai o inves unil ξ j ξ and here is an opion value of waiing. 8 An alernaive way is o inroduce idiosyncraic invesmen-specific echnology shocks. We have worked ou his exension and proved an exac irrelevance proposiion. The analysis is available upon reques. 1

12 3.2 Aggregaion and Equilibrium Characerizaion Given he linear homogeneiy feaure of firm value, we can conduc aggregaion racably. We define aggregae capial K = K j dj, aggregae labor demand N = N j dj, aggregae oupu Y = Y j dj, and aggregae invesmen expendiure in consumpion unis I = I j /z dj. Proposiion 2 The aggregae equilibrium sequences {Y, N, C, I, K, Q, ξ } are characerized by he following sysem of difference equaions: 9 K +1 = (1 δ + ς)k + ξ = θ 1 θ z 1 θ θ (ψq ) 1 θ, (25) [ I = (ψq ) 1 1 θ ] ξ θ z θ φ(ξ)dξ K, (26) [ ψ ξ 1 θ K (z I /K ) 1 θ φ(ξ)dξ] θ, (27) Y = F (K, A N ) = I + C + K ξφ(ξ)dξ, (28) U 2 (C, 1 N ) U 1 (C, 1 N ) = A F 2 (K, A N ), (29) { βu1 (C +1, 1 N +1 ) Q = E [F 1 (K +1, A +1 N +1 ) + (1 δ + ς) Q +1 U 1 (C, 1 N ) +1 ( + ξ +1 ξ ) ]} φ(ξ)dξ. (3) Equaion (25) is idenical o (22). We derive equaions (26) and (27) by aggregaing equaions (2) and (23). Equaion (26) shows ha aggregae invesmen rae I /K is posiively relaed o marginal Q as prediced by he sandard Q-heory. However, unlike his heory, marginal Q is no a sufficien saisic for he invesmen rae. In paricular, he aggregae sae (K, A, z ) also helps explain he aggregae invesmen rae besides marginal Q, via is effec on ξ. Equaion (28) is he resource consrain. The las erm in he equaion represens he aggregae fixed adjusmen coss. The firs equaliy of equaion (28) gives aggregae oupu using a single firm s producion funcion F. This resul is primarily due o he consan reurns o scale propery of F. The represenaive household s consumpion/leisure choice gives equaion 9 We omi he sandard ransversaliy condiions here. 11

13 (29). Equaion (3) is an asse pricing for he price of capial Q. I is obained from equaion (24). Noe ha by equaions (26) and (25), we can show ha he opion value of waiing in he second line of (3) is equal o 3.3 Consrained Efficiency θ I +1 1 θ K +1 ξ ξφ(ξ)dξ. Is he compeiive equilibrium we sudied efficien? To answer his quesion, we consider a social planner s problem in which he faces he same invesmen fricions as individual firms. Suppose he planner selecs an invesmen rigger ξ such ha all firms make invesmens when he idiosyncraic fixed adjusmen cos shock ξ ξ. We can hen aggregae individual firms capial and invesmens o obain he resource consrain (28) and he capial accumulaion equaion (27). The social planner s problem is o maximize he represenaive agen s uiliy (6) subjec o hese wo equaions. Proposiion 3 If φ, hen he compeiive equilibrium allocaion and he invesmen rigger characerized in Proposiion 2 are consrained efficien in he sense ha hey are idenical o hose obained by solving a social planner s problem. In addiion, he soluion is unique. The condiion φ and he assumpions for he preferences and echnology given before ensure ha he social planner s problem is a concave problem and hence i has a unique soluion. Proposiion 3 is imporan because we can use i o esablish he exisence of a recursive equilibrium for our economy by a sandard argumen as in Sokey and Lucas (1989). As a resul, i provides he heoreic foundaion for applying a recursive mehod o solve our model numerically. To he bes of our knowledge, no similar resul is proven in oher models, e.g., he models in Khan and Thomas (23, 28) or Bachmann e al. (28). In addiion, i is also an open quesion wheher or no a recursive equilibrium exiss for hese models. 3.4 Seady Sae We consider a deerminisic seady sae in which here is no aggregae shock o labor augmening echnology and no aggregae shock o invesmen-specific echnology, bu here is sill idiosyncraic fixed coss shock. In his case, seady-sae aggregae variables (Y, C, N, K, I, Q, ξ ) are deerminisic consans by a law of large numbers. Proposiion 4 Consider he lumpy invesmen model. Suppose δ > ς and δ ς < ξmaxψ 1 θ ξmax (1 θ) θ φ(ξ)dξ. θ (1 θ) 12

14 Then he seady-sae invesmen rigger ξ (, ξ max ) is he unique soluion o he equaion: ξ 1 θ ψ δ ς = (1 θ) θ θ (1 θ) φ(ξ)dξ. (31) Given his value ξ, he seady-sae value of Q is given by: ( ξ ) (1 θ) θ. (32) Q = 1 ψ θ The oher seady-sae values (I, K, C, N) saisfy: I K = (δ ς) (1 θ) Q, (33) F (K, N) = I + C + K ξφ (ξ) dξ, (34) U 2 (C, 1 N) U 1 (C, 1 N) = F 2 (K, N), (35) { } β ξ Q = F 1 (K, N) + (ξ ξ) φ(ξ)dξ. (36) 1 β (1 δ + ς) The invesmen rigger ξ is uniquely deermined by equaion (31), which saes ha, for he aggregae capial sock o be consan over ime, new invesmen mus offse capial depreciaion. The seady-sae aggregae price of capial is deermined by equaion (32), which follows from equaion (25). A his price, a firm is jus willing o pay he fixed cos o inves if he shock o is new invesmen jus his he rigger value ξ. The oher seady-sae values (I, K, C, N) are deermined by a sysem of four equaions (33)-(36). In paricular, equaion (33) implies ha he seady-sae invesmen rae increases wih he aggregae price of capial Q. Equaion (36) shows ha Q mus saisfy a seady-sae version of an asse-pricing equaion, which saes ha i is equal o he presen value of he marginal produc of capial plus he opion value of waiing. We are unable o derive analyical comparaive saics resuls for he seady sae values of (I, K, C, N) under general condiions because hey are deermined by a sysem of four nonlinear equaions. If we make some specific assumpions on preferences and echnology, we have he following sharp comparaive saics resuls: Corollary 1 Consider he power funcion disribuion wih densiy φ (ξ) = ηξη 1 (ξ max ) η, η >. Assume ha he parameer values are such ha he inequaliy in (38) holds, i.e., ξ max > [ψ 1 (δ ς) (1 θ) θ θ (1 θ)] 1 1 θ >. (37) 13

15 Then he seady-sae rigger value is given by: [ ξ = ψ 1 (δ ς) (1 θ) θ θ (1 θ) ξmax η ] 1 η+1 θ (, ξ max ). (38) In addiion, consider he following specificaions: F (K, AN) = K α (AN) 1 α, α (, 1), { (39) U (C, 1 N) = C 1 γ 1 γ v(1 N) if γ >, 1, log (C) + v(1 N) if γ = 1 (4) where v is sricly increasing, sricly concave, coninuously differeniable, and saisfies he Inada condiions. ξ max. In addiion, (I/K) >, ξ max Then he seady-sae values R/Q, I/Y, C/Y and N are independen of K ξ max <, Q >, ξ max Y <, ξ max R >, ξ max C <, and ξ max As ξ max increases, he power funcion disribuion is more spread ou. w <, ξ max (41) I <. ξ max (42) Thus, less firms will adjus capial for a given invesmen rigger. To raise he aggregae invesmen rae o compensae capial depreciaion, he seady-sae invesmen rigger ξ mus rise, as shown in equaion (38). As a resul, he seady-sae invesmen rae I/K and Q increase wih ξ max. Under he addiional assumpions on preferences and echnology, boh I and K decrease wih ξ max, bu K decreases faser han I. This in urn implies ha he seady-sae oupu Y and consumpion C decrease wih ξ max. In addiion, he seady-sae renal rae of capial R increases wih ξ max, bu he seady-sae wage rae w decreases wih ξ max because capial becomes relaively scarce. The surprising resul is ha he seady-sae values of R/Q, I/Y, C/Y and N are independen of ξ max. An imporan assumpion for his resul is ha he disribuion of he fixed coss is a power funcion, which has a homogeneiy propery. Our assumed funcions for preferences and echnology also have a homogeneiy propery. These wo homogeneiy properies are key o he independence resul. We emphasize ha he assumpions on preferences and echnology in Corollary 1 are sandard in macroeconomics and are consisen wih balanced growh (e.g., King e al. (22)). We nex use Corollary 1 o sudy aggregae dynamics. 3.5 An Exac Irrelevance Resul We normalize an aggregae variable by is seady-sae value characerized in Proposiion 4. We le X = X /X denoe his normalized value of X when is deerminisic seady-sae value is X. We have he following irrelevance resul: 14

16 Proposiion 5 Suppose he assumpions in Corollary 1 are saisfied. Then any maximal fixed cos ξ max > does no affec he equilibrium sysem of nonlinear difference equaions ha characerizes aggregae dynamics of he normalized variables {Ỹ, Ñ, C, Ĩ, K, Q, ξ }. Proposiion 5 demonsraes ha he maximal fixed cos ξ max > maers for aggregae dynamics only o he exen ha i affecs he seady sae. The sysem of difference equaions ha characerizes he normalized variables relaive o heir seady-sae values do no depend on ξ max. As a resul, ξ max does no affec he second momen and impulse response properies of he normalized aggregae variables or he logarihms of hese variables. The key o his proposiion is ha he sysem of nonlinear difference equaions for he normalized equilibrium variables has a cerain homogeneiy propery such ha i is fully deermined by he seady-sae values R/Q, I/Y, C/Y and N, and he model parameers oher han ξ max. By Corollary 1, hese seady sae values are also independen of ξ max. Thus, he dynamic sysem is independen of ξ max. The key condiion for his resul is ha he disribuion of he idiosyncraic fixed cos shock is a power funcion. Oher condiions are sandard in he RBC lieraure. Noe ha he irrelevance proposiion is sill valid when we inroduce mainenance invesmen as in Bachmann e al. (28). The reason is ha mainenance invesmen is proporional o he capial sock and hus sill preserves he homogeneiy propery of he equilibrium sysem discussed above. We emphasize ha his resul does no imply ha aggregae dynamics wih fixed adjusmen coss (ξ max > ) are he same as hose in a model wihou fixed adjusmen coss (ξ max = ), because he dynamic sysems of he (normalized) aggregae variables in he wo models are differen. Tha is, here is disconinuiy when ξ max moves from o a posiive number. Formally, when ξ max goes o zero, he economy converges o he model wihou fixed adjusmen coss. Is equilibrium sysem is given by: K +1 = (1 δ + ς)k + Q = E { βu1 (C +1, 1 N +1 ) U 1 (C, 1 N ) I = (ψq ) 1 1 θ θ z θ K, (43) ψ 1 θ K (z I /K ) 1 θ, (44) Y = F (K, A N ) = I + C, (45) U 2 (C, 1 N ) U 1 (C, 1 N ) = A F 2 (K, A N ), (46) } [F 1 (K +1, A +1 N +1 ) + (1 δ + ς) Q +1 ]. (47) 15

17 Clearly, he equilibrium sysem in Proposiion 2 does no converge o he above sysem as ξ max : For (28) o converge o (45), we need ξ =. Bu his implies ha I = by (26), which conradics wih (43). Imporanly, he shape of he fixed cos disribuion plays an imporan role in he lumpy invesmen model. To analyze his issue more ransparenly, we nex consider a log-linearized equilibrium sysem. 3.6 Log-Linearized Sysem We firs noe ha he equilibrium wage rae w = A F 2 (K, A N ) and he equilibrium gross ineres rae r +1 saisfies U 1 (C, 1 N ) = E [βu 1 (C +1, 1 N +1 ) r +1 ]. Using hese wo equaions, we log-linearize he dynamic sysem given in Proposiion 2 around he deerminisic seady sae and obain he following proposiion afer some edious algebra. We use ˆX = (X X) /X o denoe he deviaion of a variable X from is seady sae value X. Proposiion 6 The log-linearized equilibrium dynamic sysem is given by: ˆξ = 1 θ ẑ + 1 θ θ ˆQ, (48) ( Î ˆK 1 = θ ˆQ + 1 θ ) ẑ + ξ φ(ξ ) θ ξ ˆξ, (49) φ(ξ)dξ ˆK +1 = (1 δ + ς) ˆK + θ ˆK + (1 θ)(î + ẑ ) + θξ φ(ξ ) ˆξ, (5) φ(ξ)dξ Ŷ = F ( 1K ˆK + 1 F ) 1K [Â + ˆN ], (51) Y Ŷ = I Y Î + C Y Ĉ + ( 1 I Y C Y Y ) [ ˆK + (ξ ) 2 φ(ξ ) ˆξ ξφ(ξ)dξ ], (52) ˆQ = βe ˆQ+1 E ˆr +1 + β(δ ς)(1 θ)e ẑ +1 (53) + F 11F 2 βr ] F 21 F 1 Q E [Â+1 ŵ +1, ( ) E ˆr +1 = u C,C Ĉ u C,N ˆN E u C,C Ĉ +1 u C,N ˆN+1, (54) where we denoe u N,C = CU 21(C,1 N) NU 12 (C,1 N) U 1 (C,1 N). ŵ = (u N,C u C,C )Ĉ + (u C,N u N,N ) ˆN = KF 21 ˆK + NF ( 22 ˆN NF ) 22 Â, (55) F 2 F 2 F 2 U 2 (C,1 N), u N,N = NU 22(C,1 N) U 2 (C,1 N), u C,C = CU 11(C,1 N) U 1 (C,1 N), and u C,N = 16

18 This proposiion demonsraes explicily how parameers for preferences, echnology, and he fixed cos disribuion deermine he log-linearized equilibrium sysem. Equaion (48) shows ha changes in he invesmen-specific echnology shock or in he price of capial deermine changes in he invesmen rigger, and hus changes in he likelihood of capial adjusmen and in he number of adjusors. This effec is ofen referred o as he exensive margin effec. Equaion (49) shows ha changes in he aggregae invesmen rae are deermined by an inensive margin effec and an exensive margin effec. The inensive margin effec represened by he expression in he bracke on he righ hand side of (49) deermines he size of he aggregae invesmen rae. The magniude of he exensive margin effec on he aggregae invesmen rae is deermined by he seady-sae elasiciy of he adjusmen rae wih respec o he invesmen rigger, ξ φ(ξ )/ φ(ξ)dξ. In order for lumpy invesmen o maer for business cycles, he exensive margin effec mus be large. This requires he elasiciy of he adjusmen rae wih respec o he invesmen rigger o be large. 1 We will give some examples in he nex secion o illusrae his poin. Boh he inensive margin and exensive margin effecs are affeced by he general equilibrium price movemens because changes in he wage rae and in he ineres rae affec he changes in he price of capial, as revealed by equaion (53). The change in he ineres rae is deermined by preferences. As equaion (54) shows, when he elasiciy of ineremporal subsiuion is larger, he consumpion smoohing incenive is weaker, leading o a smaller ineres rae movemen. The magniude of he wage movemens is deermined by he preferences and echnology parameers as revealed by equaion (55). Equaion (53) shows ha he wage feedback effec is magnified by he seady-sae raio of he marginal produc of capial o Q or R/Q. Because he seady-sae Q is equal o he presen value of R and he opion value of waiing as revealed by (36), he larger is he opion value of waiing, he smaller is R/Q. We can show ha holding he adjusmen rae and he invesmen rigger fixed, if more low fixed coss have high probabiliies or he fixed cos disribuion is more righ skewed, he opion value of waiing is higher. In his case, R/Q is smaller and hus he wage feedback effec is smaller. In summary, boh he micro-level invesmen lumpiness and he general equilibrium price movemens are imporan o deermine aggregae dynamics. The relaive imporance of hese wo effecs is deermined by he preference and echnology parameers and he disribuion of he idiosyncraic fixed cos shock. In paricular, holding preferences and echnology fixed, if 1 Even hough his elasiciy is an endogenous concep, we can compue i ex pos in equilibrium. For he power funcion disribuion, he elasiciy is equal o he shape parameer, which is exogenous. 17

19 he seady-sae elasiciy of he adjusmen rae is larger, hen he exensive margin effec is sronger. If he fixed cos disribuion is more righ skewed, hen he general equilibrium wage feedback effec is weaker. 4 Numerical Resuls We evaluae our lumpy invesmen model quaniaively and compare his model wih wo benchmark models. The firs one is obained by removing fixed adjusmen coss only (ξ j = ). We call his model he parial adjusmen model. Is equilibrium sysem is given by (43)-(47). The second one is a fricionless RBC model, obained by removing all adjusmen coss in he model presened in Secion 2. Is equilibrium sysem is obained by seing θ = ς = and ψ = 1 in (44)-(47) and Q = 1/z. In boh benchmark models, all firms make idenical decisions, and hus hese models are equivalen o sandard represenaive-firm RBC models (e.g., Fisher (26) and Greenwood e al. (2)). Because we have characerized he equilibria for all hree models by sysems of nonlinear difference equaions as shown in he previous secion, we can use he sandard second-order approximaion mehod o solve he models numerically. 11 To do so, we need firs o calibrae he models. 4.1 Baseline Paramerizaion For all model economies, we ake he Cobb-Douglas producion funcion, F (K, AN) = K α (AN) 1 α, and he period uiliy funcion, U (C, 1 N) = log(c) an, where a > is a parameer. We fix he lengh of period o correspond o one year, as in Thomas (22), and Khan and Thomas (23, 28). Annual frequency allows us o use empirical evidence on esablishmen-level invesmen in selecing parameers for he fixed adjusmen coss and he disribuion of idiosyncraic invesmen-specific shocks. We firs choose parameer values for preferences and echnology o ensure ha he seadysae of he fricionless RBC model is consisen wih he long-run values of key poswar U.S. aggregaes. Specifically, we se he subjecive discoun facor o β =.96, so ha he implied annual real ineres rae is 4 percen (Presco (1986)). We choose he value of a so ha he seady-sae hours are abou 1/3 of available ime spen in marke work. We se he capial share α =.36, implying a labor share of.64, which is close o he labor income share in he NIPA. We ake he depreciae rae δ =.1, as in he lieraure on business cycles (e.g., Presco (1986)). 11 The Dynare code is available upon reques. 18

20 I is ofen argued ha convex adjusmen coss are no observable direcly and hence canno be calibraed based on average daa over he long run (e.g., Greenwood e al. (2)). Thus, we impose he wo resricions: ψ = δ θ and ς = θ δ, (56) 1 θ so ha he parial adjusmen model and he fricionless RBC model give idenical seady-sae allocaions. 12 As in our paper, Baxer and Crucini (1993), Jermann (1998), and Greenwood e al. (2) make similar assumpions for he parameers in he adjusmen cos funcion. We assume condiion (56) hroughou our numerical experimens below. We nex follow Khan and Thomas (23) o selec parameers for he aggregae shocks. They use Sock and Wason (1999) daa se o esimae he persisence and volailiy of he Solow residuals equal o.9225 and.134, respecively. Transforming he oal facor produciviy shocks o our labor-augmening echnology shocks, we se ρ A =.9225 and σ A =.134/.64 =.2 1. As in Khan and Thomas (23), we se ρ z =.76 and σ z =.17 in he invesmen-specific echnology shock process. Following Kiyoaki and Wes (1996), Thomas (22), and Khan and Thomas (23), we se θ = 1/5.98, implying ha he Q-elasiciy of he invesmen rae is We adop he power funcion disribuion for he idiosyncraic fixed cos shock. We need o calibrae wo parameers ξ max and η. We ry o mach micro-level evidence on he invesmen lumpiness repored by Cooper and Haliwanger (26). Cooper and Haliwanger (26) find ha he inacion rae is.81 and he posiive spike rae is abou.186. A posiive invesmen spike is defined as he invesmen rae exceeding.2. For he power funcion disribuion, he seady-sae inacion rae is given by 1 (ξ /ξ max ) η and he seady-sae invesmen rae is given by equaion (33). Because our model implies ha he arge invesmen rae I/K is idenical for all firms, our model canno mach he spike rae. Therefore, here are many combinaions of η and ξ max o mach he inacion rae. As baseline values, we follow Khan and Thomas (23, 28) and ake a uniform disribuion (η = 1). This implies ha ξ max =.242. In his case, oal fixed adjusmen coss accoun for 2.4 percen of oupu, 1 percen of oal invesmen spending and 1. percen of capial sock, which are reasonable according o he esimaion by Cooper and Haliwanger (26). We summarize he baseline parameer values in Table Under he log-linear approximaion mehod, only he curvaure parameer θ in he convex adjusmen cos funcion maers for he approximaed equilibrium dynamics. 19

21 Table 1. Baseline Parameer Vales β a α δ ρ A σ A ρ z σ z θ ξ max η / Parial Equilibrium Dynamics In order o undersand he general equilibrium effecs of fixed coss on business cycles, we sar wih a parial equilibrium analysis by fixing he wage rae and he ineres rae a heir seady sae values. For he power funcion disribuion, we can show ha he elasiciy of he adjusmen rae is equal o η. Using assumpion (56), he specificaion of he uiliy funcion and he producion funcion, and seing ŵ = ˆr =, we can rewrie equaions (49) and (53) as: Î ˆK = 1 θ ˆQ + 1 θ ẑ }{{ θ } inensive [ ( ˆQ = βe ˆQ+1 + δβe ẑ δ 1 θ + η ˆξ, (57) }{{} exensive ) β βθδ 1 θ η ] 1 α α E Â+1. (58) The las erm in he square bracke in equaion (58) represens he opion value of waiing in he presence of fixed coss. The log-linearized sysem for he parial adjusmen model wih fixed prices is obained by seing η = and ignoring equaion (48). We now analyze he impulse response properies based on he above log-linearized sysem. Figure 1 plos he impulse responses o a posiive one sandard deviaion shock o he laboraugmening echnology (N-shock). Following his shock, he marginal produc capial rises. Thus, he price of capial or he marginal Q rises. Because here is an opion value of waiing, he increase in Q is higher in he lumpy invesmen model han in he parial adjusmen model. The increase in Q has boh inensive and exensive margin effecs in he lumpy invesmen model as revealed by equaion (49). In paricular, i raises he adjusmen rae by 11 percen in he lumpy invesmen model. Due o his exensive margin effec, he increase in he invesmen rae in he lumpy invesmen model is higher han ha in he parial adjusmen model (22 percen versus 1 percen). [Inser Figures 1-2 Here.] Figure 2 plos he impulse responses o a posiive one sandard deviaion shock o he invesmen-specific echnology (I-shock). Following his shock, he marginal Q rises by he same magniude in boh he lumpy invesmen and in he parial adjusmen model because 2

22 hese wo models deliver an idenical coefficien of ẑ in (58). Even hough he increase in marginal Q is idenical, he invesmen rae increases much more in he lumpy invesmen model han in he parial adjusmen model (15 percen versus 8 percen). The reason is ha he invesmen-specific echnology shock has a direc exensive margin effec by raising he adjusmen rae (see equaion (48)). In paricular, he adjusmen rae rises by abou 8 percen. 4.3 General Equilibrium Dynamics We now urn o general equilibrium dynamics by endogenizing he prices. In his case, he general equilibrium price movemens play an imporan role in shaping aggregae dynamics. To see his, we wrie he log-linearized equaion for he marginal Q as: ˆQ = βe ˆQ+1 + βδe ẑ +1 E [ˆr +1 ] (59) [ ( δ ) β βθδ ] 1 1 α 1 θ 1 θ 1 + η α E [A +1 ŵ +1 ], where he equilibrium ineres rae and wage rae saisfy E [ˆr +1 ] = E [Ĉ+1 ] Ĉ, ( ŵ = Ĉ = (1 α) Â + α ˆK ˆN ). In general equilibrium, a posiive N-shock or I-shock raises he ineres rae and he wage rae, and hus dampens he increases in marginal Q or he price of capial, as revealed by equaion (59). As a resul, boh he exensive and inensive margin effecs are weakened in general equilibrium. Thomas (22) and Khan and Thomas (23, 28) emphasize his general equilibrium effec. They also find ha movemens in ineres raes and wages yield quaniy dynamics ha are virually indisinguishable from a sandard RBC model wihou fixed adjusmen coss. However, we do no obain his finding because hey use differen numerical mehods han ours. Their models are also differen han ours in ha hey assume long-run growh and a decreasing-reurns-o-scale echnology. In addiion, fixed coss in heir models are measured in erms of labor coss raher han capial. [Inser Figures 3-4 Here.] Figures 3-4 plo impulse responses o a posiive one sandard deviaion N-shock. Compared o Figure 1, he increase in he invesmen rae is abou 1 imes smaller in general equilibrium for he lumpy invesmen and parial adjusmen models han ha in parial equilibrium. In 21

23 addiion, he responses in he lumpy invesmen and parial adjusmen models are similar, bu he lumpy invesmen model brings predicions closer o hose of he fricionless RBC model. The inuiion is ha he parial adjusmen model implies oo sluggish responses of invesmen due o convex adjusmen coss. The exensive margin effec in he lumpy invesmen model raises he responses of invesmen o shocks. Bu he price feedback effec parially offses his exensive margin effec. Figure 4 shows ha boh he ineres rae and he wage rae rise. As a resul, he increase in marginal Q in he lumpy invesmen model is much smaller in general equilibrium han in parial equilibrium (.1 percen versus 1.8 percen). This in urn causes he adjusmen rae o rise by less han 1 percen as revealed in Figure 3, compared o 11 percen in parial equilibrium. [Inser Figures 5-6 Here.] Figures 5-6 plo he impulse responses o a posiive one sandard deviaion I-shock. Comparing wih Figure 2, we find ha he effecs on he invesmen rae is much smaller in general equilibrium han in parial equilibrium. In addiion, he impulse responses in he lumpy invesmen and he parial adjusmen model are similar. In conras o he parial equilibrium case, a posiive I-shock lowers marginal Q in boh he lumpy invesmen and parial adjusmen models. The inuiion follows from equaion (59) and Figure 6. The increase in he ineres rae and he wage rae lowers he profiabiliy of he firm and hence raises he cos of invesmen. This effec dominaes he posiive effec of invesmen-specific echnology shock on Q. Why do he invesmen rae and he adjusmen rae sill rise? The reason is ha he increase in he I-shock decreases he price of new invesmen. Thus, i has a direc posiive effec on he invesmen rigger and he invesmen rae as revealed by equaions (48) and (49), respecively. 13 However, he effec is smaller han ha in parial equilibrium, due o he powerful general equilibrium price feedback effec. Figure 5 shows ha he adjusmen rae rises by 1.5 percen only, which is much smaller han 8 percen in parial equilibrium. Nex, we urn o he business cycle momens properies. Table 2 presens sandard deviaions, auocorrelaions, and conemporaneous correlaions for several model economies. We firs consider he resul for he fricionless RBC and parial adjusmen models. I is well known ha he parial adjusmen model delivers less volaile and more persisen equilibrium quaniies and prices han he fricionless RBC model because of he smoohing role of he convex adjusmen coss. We hen inroduce fixed coss ino he parial adjusmen model. Rows 13 In conras o he N-shock, he iniial response of consumpion is negaive because invesmen crowds ou consumpion as ypical in models wih invesmen-specific echnology shocks. 22

24 labelled Lumpy1 in Table 2 presen he resul for his lumpy invesmen model wih he baseline parameer values. They reveal ha alhough impulse responses in he parial adjusmen model and he lumpy invesmen model are similar, he difference in he model prediced second momens is non-negligible. The lumpy invesmen model delivers higher volailiy in all quaniies and prices han he parial adjusmen model as revealed in Panel A. In paricular, aggregae invesmen, he invesmen rae, and hours are 16, 13, and 28 percen, respecively, more volaile in he lumpy invesmen model han in he parial adjusmen model. Panel B of Table 2 shows ha he lumpy invesmen model predics less persisen equilibrium quaniies and prices, which are closer o he predicions of he fricionless RBC model. Panel C of Table 2 presens conemporaneous correlaions wih oupu. Marginal Q is negaively correlaed wih oupu for all models because a posiive invesmen-specific echnology shock lowers he price of capial direcly. All oher quaniies and prices move posiively wih oupu. In summary, Table 2 demonsraes ha he predicions of he lumpy invesmen model are closer o hose of he sandard fricionless RBC model. Thus, i also suffers from a number of difficulies in maching he US business cycle facs, as in he sandard fricionless RBC model. Thomas (22) repors a similar finding. So far, we have shown ha under he baseline calibraion, he general equilibrium price movemens dampen he exensive margin effec significanly, making predicions of he lumpy invesmen model and he parial adjusmen model similar. We now illusrae ha he shape parameer of he disribuion funcion of he idiosyncraic shock is imporan for he exensive margin effec. We se η = 2 and re-calibrae ξ max =.2232 such ha he inacion rae is equal o.81. In his case, he elasiciy of he adjusmen rae is 2 imes of ha in he baseline calibraion so ha he exensive margin effec is much larger. Of course, his calibraion is unreasonable because oal fixed coss are oo large, accouning for 4.3 percen of oupu, 19.1 percen of oal invesmen spending, and 1.9 percen of capial sock. Rows labelled Lumpy2 in Table 2 presen he resul for his calibraion. The resul reveals ha he difference beween he lumpy invesmen model and he parial adjusmen model becomes larger. In paricular, aggregae invesmen in he lumpy invesmen model is 4 percen more volaile han in he parial adjusmen model. The invesmen rae in he lumpy invesmen model is 46 percen more volaile han in he parial adjusmen model. However, he differences in he auocorrelaions and conemporaneous correlaions across hese wo models are small. Gourio and Kashyap (27) argue ha for he exensive margin effec o be large in he Thomas (22) model, he fixed cos disribuion mus be sufficienly compressed in he sense 23