Firm-Specific Capital, Nominal Rigidities and the Business Cycle

Board of Governors of the Federal Reserve System International Finance Discussion Papers Number 990 January 2010 Firm-Specific Capital, Nominal Rigidities, and the Business Cycle David E. Altig Lawrence J. Christiano Martin Eichenbaum and Jesper Lindé NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors. Recent IFDPs are available on the Web at www.federalreserve.gov/pubs/ifdp/.

Firm-Speci(cid:133)c Capital, Nominal Rigidities and the Business Cycle (cid:3) David Altig Lawrence J. Christiano Martin Eichenbaum y z x Jesper LindØ { January 25, 2010 Abstract This paper formulates and estimates a three-shock US business cycle model. The estimated model accounts for a substantial fraction of the cyclical variation in output and is consistent with the observed inertia in in(cid:135)ation. This is true even though (cid:133)rms in the model reoptimize prices on average once every 1.8 quarters. The key feature of our model underlying this result is that capital is (cid:133)rm-speci(cid:133)c. If we adopt the standard assumption that capital is homogeneous and traded in economy-wide rental markets, we (cid:133)nd that (cid:133)rms reoptimize their prices on average once every 9 quarters. We argue that the micro implications of the model strongly favor the (cid:133)rm-speci(cid:133)c capital speci(cid:133)cation. JEL: E3, E4, E5 WearegratefulforthecommentsofDavidAndolfatto, GadiBarlevy, JesœsFernÆndez-Villaverde, Andy (cid:3) Levin and Harald Uhlig. In addition we have bene(cid:133)tted from the reactions of the participants in the Third Banco de Portugal Conference in Monetary Economics, Lisbon, Portugal, June 10, 2004, the conference on Dynamic Models and Monetary Policymaking, September 22-24, 2004, held at the Federal Reserve Bank of Cleveland,andtheconference(cid:147)SDGEModelsandtheFinancialSector(cid:148)organizedbyDeutscheBundesbank inEltville,November26-27,2004. Wearealsogratefulforthecommentsoftheparticipantsinthe(cid:145)Impulses andPropagations(cid:146)workshopoftheNBERSummerInstituteintheweekofJuly19,2004andtheconference (cid:147)MacroeconomicsandReality,25YearsLater(cid:148)heldinBarcelona,April1-2,2005. Finally,weareparticularly grateful for the insights and for the research assistance of Riccardo Dicecio. The views expressed in this paper are solely the responsibility of the authors and should not be interpreted as re(cid:135)ecting the views of the Board of Governors of the Federal Reserve System, Federal Reserve Bank of Atlanta, or of any other person associated with the Federal Reserve System. Federal Reserve Bank of Atlanta. y NorthwesternUniversity,NationalBureauofEconomicResearch,andFederalReserveBanksofChicago, z Atlanta and Minneapolis. NorthwesternUniversity,NationalBureauofEconomicResearch,andFederalReserveBanksofChicago x and Atlanta. Board of Governors of the Federal Reserve System and CEPR. {

1. Introduction Macroeconomic data indicate that in(cid:135)ation is inertial. To account for this inertia, macro modeler embed assumptions that are either implausibe on a priori grounds or directly in con(cid:135)ictwithmicrodata. Forexample, inmanynew-Keynesianmacroeconomicmodels, (cid:133)rms index non-optimized prices to lagged in(cid:135)ation. These models account for in(cid:135)ation inertia by assuming that (cid:133)rms re-optimize their prices every six quarters or even less often.1 Other new-Keynesian models don(cid:146)t allow for indexing to lagged in(cid:135)ation. In estimated versions of thesemodels, (cid:133)rms changepricesonceeverytwoyearsorlessoften.2 Thispropertycontrasts sharply with (cid:133)ndings in Bils and Klenow (2004), Golosov and Lucas (2007) and Klenow and Kryvstov (2008) who argue that (cid:133)rms change prices more frequently than once every two quarters.3 In this paper we formulate and estimate a model which is consistent with the evidence of inertia in in(cid:135)ation, even though (cid:133)rms re-optimize prices on average once every 1.8 quarters.4 In addition our model accounts for the dynamic response of 10 key U.S. macro time series to monetary policy shocks, neutral technology shocks and capital embodied shocks.5 In our model aggregate in(cid:135)ation is inertial despite the fact that (cid:133)rms re-optimize prices frequently. The inertia re(cid:135)ects that when (cid:133)rms do re-optimize prices, they change prices by a small amount. Firms change prices by a small amount because each (cid:133)rm(cid:146)s short run marginal cost curve is increasing in its own output.6 This positive dependency re(cid:135)ects our assumption that in any given period, a (cid:133)rm(cid:146)s capital stock is pre-determined. In standard equilibrium business cycle models a (cid:133)rm(cid:146)s capital stock is not pre-determined and all factors of production, including capital, can be instantly and costlessly transferred across (cid:133)rms. Theseassumptionsareempiricallyunrealisticbutaredefendedonthegroundsoftractability. The hope is that these assumption are innocuous and do not a⁄ect major model properties. In fact these assumptions matter a lot. 1For example, Eichenbaum and Fisher (2007) (cid:133)nd that estimated versions of Calvo pricing models with indexing to lagged in(cid:135)ation imply that (cid:133)rms reoptimize prices roughly once every six quarters. Smets and Wouters(cid:146)(2003) estimated model implies that (cid:133)rms reoptimize prices on average once every nine quarters. 2SeeforexampletheestimatesinRabanalandRam(cid:237)rez(2005)forthepost1982eraortheNBERworking paper 10617 version of Eichenbaum and Fisher (2007). 3For example in calibrating their model to the micro data, Golosov and Lucas (2007) select parameters to ensure that (cid:133)rms change prices on average once every 1.5 quarters. 4TomaintaincontactwiththebulkofestimatednewKeynesianmodelsweassumethat(cid:133)rmsindexprices to lagged in(cid:135)ation. The NBER working paper 10617 version of Eichenbaum and Fisher (2007) establishes that the introduction of (cid:133)rm speci(cid:133)c capital has similar e⁄ects in models with and without indexation to lagged prices 5See also Dicecio (2009) for a multi-sectoral general equilibrium model which allows for the same shocks that we consider. Also Edge, Laubach and Williams (2003) consider a general equilibrium model with two types of technology shocks. Gal(cid:237), L(cid:243)pez-Salido and VallØs (2003) consider neutral technology shocks only. 6For early discussions of this idea, see Ball and Romer (1990) and Kimball (1995). 2

In our model, a (cid:133)rm(cid:146)s capital is pre-determined and can only be changed over time by varying the rate of investment. These properties follow from our assumption that capital is completely (cid:133)rm-speci(cid:133)c. Our assumptions about capital imply that a (cid:133)rm(cid:146)s marginal cost curvedependspositivelyonitsoutputlevel.7 Toseetheimpactofthisdependenceonpricing decisions, consider a (cid:133)rm that contemplates raising its price. The (cid:133)rm understands that a higher price implies less demand and less output. A lower level of output reduces marginal cost, which other things equal, induces a (cid:133)rm to post a lower price. Thus, the dependence of marginal cost on (cid:133)rm-level output acts as a countervailing in(cid:135)uence on a (cid:133)rm(cid:146)s incentives to raise price. This countervailing in(cid:135)uence is why aggregate in(cid:135)ation responds less to a given aggregate marginal cost shock when capital is (cid:133)rm-speci(cid:133)c. Anything, including (cid:133)rm-speci(cid:133)city of some other factor of production or adjustment costs in labor, which causes a (cid:133)rm(cid:146)s marginal cost to be an increasing function of its output works in the same direction as (cid:133)rm-speci(cid:133)city of capital. This fact is important because our assumption that the (cid:133)rm(cid:146)s entire stock of capital is predetermined probably goes too far from an empirical standpoint. We conduct our analysis using two versions of the model analyzed by Christiano, Eichenbaum, and Evans (CEE henceforth, 2005): in one, capital is homogeneous whereas in the other, it is (cid:133)rm speci(cid:133)c. We refer to these models as the homogeneous and (cid:133)rm-speci(cid:133)c capital models, respectively. We show that the only di⁄erence between the log-linearized equations characterizing equilibrium in the two models pertains to the equation relating in- (cid:135)ation to marginal costs. The form of this equation is identical in both models: the change in in(cid:135)ation at time t is equal to discounted expected change in in(cid:135)ation at time t+1 plus a reducedformcoe¢ cient, (cid:13), multiplyingtimeteconomy-wideaveragerealmarginalcost. The di⁄erencebetweenthetwomodelsliesinthemappingbetweenthestructuralparametersand (cid:13). In non-linear framework, however, it is not true that the solutions to the homogeneous and (cid:133)rm-speci(cid:133)c capital models are obserbationally equivalent with respect to macro data.8 In the homogeneous capital model, (cid:13) depends only on agents(cid:146)discount rates and on the fraction, 1 (cid:24) , of (cid:133)rms that re-optimize prices within the quarter. In the (cid:133)rm-speci(cid:133)c p (cid:0) capital model, (cid:13) is a function of a broader set of the structural parameters. For example, the more costly it is for a (cid:133)rm to vary capital utilization, the steeper is its marginal cost curve and hence the smaller is (cid:13). A di⁄erent example is that in the (cid:133)rm-speci(cid:133)c capital model, the parameter (cid:13) is smaller the more elastic is the (cid:133)rm(cid:146)s demand curve.9 This result re(cid:135)ects that the more elastic is a (cid:133)rm(cid:146)s demand, the greater is the reduction in demand and output 7A closely related assumption that generates an upward sloping marginal cost curve is that there are internal costs of adjusting capital. 8See Levin, Lopez-Salido, Nelson and Yun (2008) who emphasize this point and analyze the nature of optimal policy in the non-linear versions of the two models. 9See Ball and Romer (1990) and Kimball (1995) for an early discussion of this point. 3

in response to a given price increase. A bigger fall in output implies a bigger fall in marginal cost which reduces a (cid:133)rm(cid:146)s incentive to raise its price. The only way that (cid:24) enters into the reduced form of the two models is via its impact p on (cid:13). If we parameterize the two models in terms of (cid:13) rather than (cid:24) , they have identical p implications for all aggregate quantities and prices in a standard (log-)linearized framework. This observational equivalence result implies that we can estimate the model in terms of (cid:13) without taking a stand on whether capital is (cid:133)rm-speci(cid:133)c or homogeneous. The observational equivalence result also implies that we cannot assess the relative plausibility of the homogeneous and (cid:133)rm-speci(cid:133)c capital models using macro data. However, the two models have very di⁄erent implications for micro data. To assess the relative plausibility of the two models, we focus on the mean time between price re-optimization, and the dynamic response of the cross - (cid:133)rm distribution of production and prices to aggregate shocks. These implications depend on the parameters of the model, which we estimate. We follow CEE (2005) in choosing model parameter values to minimize the di⁄erences between the dynamic response to shocks in the model and the analog objects estimated using a vector autoregressive representation of 10 post-war quarterly U.S. time series.10 To computevectorautoregression(VAR)basedimpulseresponsefunctions, weuseidenti(cid:133)cation assumptions satis(cid:133)ed by our economic model: the only shocks that a⁄ect productivity in the long run are innovations to neutral and capital-embodied technology; the only shock that a⁄ects the price of investment goods in the long run is an innovation to capital-embodied technology;11 monetary policy shocks have a contemporaneous impact on the interest rate, but they do not have a contemporaneous impact on aggregate quantities or the price of investment goods. We estimate that together these three shocks account for almost 60 percent of cyclical (cid:135)uctuations in aggregate output and other aggregate quantities. We now discuss the key properties of our estimated model. First, the model does a good job of accounting for the estimated response of the economy to both monetary policy and technology shocks. Second, according to our point estimates, households re-optimize wages on average about once a year. Third, our point estimate of (cid:13) is 0:014. In the homogeneous capital version of the model, this value of (cid:13) implies that (cid:133)rms change prices on average once every 9:4 quarters. But in the (cid:133)rm-speci(cid:133)c capital model, this value of (cid:13) implies that (cid:133)rms change price on average once every 1:8 quarters. The reason why the models have such di⁄erent implications for (cid:133)rms(cid:146)pricing behavior is that according to our estimates, (cid:133)rms(cid:146) 10The quality of our estimation strategy depends on the ability of identi(cid:133)ed VARs to generate reliable estimates of the dynamic response of economic variables to shocks. In an appendix, available upon request, weevaluatethereliabilityofVARmethodsinourapplicationbymeansofMonteCarlosimulationmethods. We(cid:133)ndthattheMonteCarloperformanceofourVARbasedestimatesofimpulseresponsefunctionsisvery good. 11Our strategy for identifying technology shocks follows Fisher (2006). 4

demand curves are highly elastic and their marginal cost curves are very steep. Finally, we show that the two versions of the model di⁄er sharply in terms of their implications for the cross-sectional distribution of production. In the homogeneous capital model, a very small fraction of (cid:133)rms produce the bulk of the economy(cid:146)s output in the periods after a monetary policy shock. The implications of the (cid:133)rm-speci(cid:133)c model are much less extreme. We conclude that both the homogeneous and (cid:133)rm-speci(cid:133)c capital models can account for in(cid:135)ation inertia and the response of the economy to monetary policy and technology shocks. But only the (cid:133)rm-speci(cid:133)c model can reconcile the micro-macro pricing con(cid:135)ict without obviously unpalatable micro implications. It is useful to place this paper in the context of the literature. That (cid:133)rm-speci(cid:133)c capital can rationalize a lower estimate of (cid:24) (more frequent price re-optimization) was (cid:133)rst demonp strated by Sbordone (1998, 2002) and further discussed by Gali, Gertler and Lopez-Salido (2001) and Woodford (2003). In these papers, the stock of capital owned by the (cid:133)rm is (cid:133)xed. Woodford (2005) analyzes the impact of (cid:133)rm speci(cid:133)c capital allowing for investment. As our discussion above indicates, whether (cid:133)rm speci(cid:133)c capital actually does rationalize a lower estimate of (cid:24) depends critically on the other parameters characterizing (cid:133)rms(cid:146)environp ments. To the extent that capital utilization rates can easily be varied, the assumption of (cid:133)rm speci(cid:133)c capital loses its ability to rationalize low values of (cid:24) : A maintained assumption p of the papers just cited is that (cid:133)rms cannot vary capital utilization rates. So these papers leave open the question of how important (cid:133)rm speci(cid:133)c capital is once (cid:133)rms can vary capital utilization rates. Similarly, the smaller is the elasticity of demand for a (cid:133)rm(cid:146)s output, the smaller is the impact of (cid:133)rm speci(cid:133)c capital on inference about (cid:24) : The above cited papers p condition their inference on particular assumed values for this elasticity. The key contribution of this paper is to estimate the key parameters governing the operational importance of (cid:133)rm speci(cid:133)c capital and to assess the importance for (cid:133)rm speci(cid:133)c capital in an estimated dynamic stochastic general equilibrium model. Our key result is that the assumption of (cid:133)rm speci(cid:133)c capital does infact rationalize relativelylowvalues of (cid:24) ; therebyhelpingtoreconcile p the apparent con(cid:135)icting pictures of pricing behavior painted by micro and macro data.12 From a broader perspective, this paper belongs to a larger literature that tries to explain the mechanisms by which nominal shocks have e⁄ects on real economic activity that last longer than the frequency with which (cid:133)rms re-optimize prices. Perhaps the most closely related mechanisms are (cid:133)rm speci(cid:133)c labor (Woodford (2005)), sector speci(cid:133)c labor (Gertler and Leahy (2008)) and strategic complementarities arising from an elasticity of (cid:133)rm demand that is increasing in the (cid:133)rm(cid:146)s price (see for example Kimble (1995) and Eichenbaum and 12Since our paper was written, an interesting literature has arisen incorporating (cid:133)rm-speci(cid:133)c capital into analyzes of price setting. See for example Christo⁄el, Coenen, and Levin (2007), de Walque, Smets and Wouters (2006) Eichenbaum and Fisher (2007), and Sveen and Weinke (2007a, 2007b). 5

Fisher(2007)). Adi⁄erentpropagationmechanismstemsfromheterogeneityacrosssectorsin the frequency of price changes (see for example Bils and Klenow(2004), Carvalho (2006) and Steinsson and Nakamura (2008)) and the presence of intermediate inputs (see for example Basu (1995) and Huang (2006)). An alternative and promising propagation mechanism arises from the assumption that (cid:133)rms cannot attend perfectly to all available information (see Sims (1998, 2003)). Mackowiak and Wiederholdt (2008) present a model in which (cid:133)rms decide what variables to pay attention to, subject to a constraint on information (cid:135)ow. When idiosyncratic conditions are more variable or more important than aggregate conditions, (cid:133)rms in their model pay more attention to idiosyncratic conditions than to aggregate conditions. Their model has the important property that (cid:133)rms react fast and by large amounts to idiosyncratic shocks, but only slowly and by small amounts to nominal shocks. As a result nominal shocks have strongandpersistentreale⁄ects. Woodford(2008)developsageneralizationofthestandard, full-information model of state-dependent pricing in which decisions about when to review a (cid:133)rm(cid:146)s existing price must be made on the basis of imprecise awareness of current market conditions. He endogenizes imperfect information using a variant of the theory of (cid:147)rational inattention(cid:148)proposed by Sims (1998, 2003)). In related work, Mankiw and Reis (2002) and Reis (2006) stress the potential importance of slow dissemination of information to (cid:133)rms for generating persistent e⁄ects of nominal shocks. The merits of these alternative propagation mechansims is a subject of an ongoing, vigorous debate. A detailed assessment of their empirical strengths is beyond the scope of this paper. It is clear however that the debate will be settled on the (cid:133)eld of microeconomic data. For recent reviews of how (cid:133)rm level data on prices bears on alternative approaches we refer the reader to Mackowiak and Smets (2008), Eichenbaum, Jaimovich and Rebelo (2009), and Klenow and Malin (2009). While we emphasize the importance of (cid:133)rm speci(cid:133)c capital in this paper, we leave open the possibility that the other propagation mechanisms discussed above may be at least as important. The remainder of this paper is organized as follows. In Section 2 we describe our basic modeleconomy. Section3describesourVAR-basedestimationprocedure. Section4presents our VAR-based impulse response functions and their properties. Sections 5 and 6 present and analyze the results of estimating our model. Section 7 discusses the implications of the homogeneous and (cid:133)rm-speci(cid:133)c capital models for the cross-(cid:133)rm distribution of prices and production in the wake of a monetary policy shock. Section 8 concludes. 2. The Model Economy In this section we describe the homogeneous and (cid:133)rm-speci(cid:133)c capital models. 6

2.1. The homogeneous capital model The model economy is populated by goods-producing (cid:133)rms, households and the government. 2.1.1. Final Good Firms Attimet,a(cid:133)nalconsumptiongood,Y ;isproducedbyaperfectlycompetitive,representative t (cid:133)rm. The (cid:133)rm produces the (cid:133)nal good by combining a continuum of intermediate goods, indexed by i [0;1]; using the technology 2 1 1 (cid:21)f Y t = y t (i)(cid:21)fdi ; (2.1) (cid:20)Z0 (cid:21) where 1 (cid:21) < and y (i) denotes the time t input of intermediate good i: The (cid:133)rm takes f t (cid:20) 1 its output price, P ; and its input prices, P (i); as given and beyond its control. The (cid:133)rst t t order necessary condition for pro(cid:133)t maximization is: (cid:21)f P t (cid:21)f(cid:0) 1 = y t (i) : (2.2) P (i) Y (cid:18) t (cid:19) t Integrating (2.2) and imposing (2.1), we obtain the following relationship between the price of the (cid:133)nal good and the price of the intermediate good: 1 1 (1 (cid:0) (cid:21)f ) P t = P t (i)1 (cid:0) (cid:21)fdi : (2.3) (cid:20)Z0 (cid:21) 2.1.2. Intermediate Good Firms Intermediate good i (0;1) is produced by a monopolist using the following technology: 2 K (i)(cid:11)(z h (i))1 (cid:11) (cid:30)z if K (i)(cid:11)(z h (i))1 (cid:11) (cid:30)z y t (i) = 0 t t t (cid:0) (cid:0) t(cid:3) othe t rwise t t (cid:0) (cid:21) t(cid:3) (2.4) (cid:26) where 0 < (cid:11) < 1: Here, h (i) and K (i) denote time t labor and capital services used t t to produce the ith intermediate good. The variable, z ; represents a time t shock to the t technology for producing intermediate output. We refer to z as a neutral technology shock t and denote its growth rate, z =z ; by (cid:22) : The non-negative scalar, (cid:30); parameterizes (cid:133)xed t t 1 zt (cid:0) costs of production. The variable, z ; is given by: t(cid:3) (cid:11) z = (cid:7)1 (cid:11)z ; (2.5) t(cid:3) t(cid:0) t where (cid:7) represents a time t shock to capital-embodied technology. We choose the structure t of the (cid:133)rm(cid:146)s (cid:133)xed cost in (2.5) to ensure that the non-stochastic steady state of the economy 7

exhibits a balanced growth path. We denote the growth rate of z and (cid:7) by (cid:22) and (cid:22) t(cid:3) t z t(cid:3) (cid:7)t respectively, so that: (cid:11) (cid:22) = (cid:22) 1 (cid:11) (cid:22) : (2.6) z ;t (cid:7);t (cid:0) z;t (cid:3) Throughout, we rule out entry into and ex(cid:0)it fro(cid:1)m the production of intermediate good i: Let (cid:22)^ denote ((cid:22) (cid:22) )=(cid:22) ; where (cid:22) is the growth rate of (cid:22) in non-stochastic steady z;t z;t z z z z;t (cid:0) state. We de(cid:133)ne all variables with a hat in an analogous manner. The variables (cid:22)^ evolves z;t according to: (cid:22)^ = (cid:26) (cid:22)^ +" (2.7) z;t (cid:22) z z;t (cid:0) 1 (cid:22) z ;t where (cid:26) < 1 and " is uncorrelated over time and with all other shocks in the model. j (cid:22) zj (cid:22) z ;t We denote the standard deviation of " by (cid:27) : Similarly, we assume: (cid:22) ;t (cid:22) z z (cid:22)^ = (cid:26) (cid:22)^ +" ; (2.8) (cid:7);t (cid:22) (cid:7) (cid:7);t (cid:0) 1 (cid:22) (cid:7) ;t where " has the same properties as " : We denote the standard deviation of " by (cid:22) ;t (cid:22) ;t (cid:22) ;t (cid:7) z (cid:7) (cid:27) : (cid:22) (cid:7) Intermediate good (cid:133)rms rent capital and labor in perfectly competitive factor markets. Pro(cid:133)tsaredistributedtohouseholdsattheendofeachtimeperiod. LetP rk andP w denote t t t t the nominal rental rate on capital services and the wage rate, respectively. We assume that the (cid:133)rm must borrow the wage bill in advance at the gross interest rate, R : t Firms set prices according to a variant of the mechanism spelled out in Calvo (1983). In each period, an intermediate goods (cid:133)rm faces a constant probability, 1 (cid:24) ; of being able to p (cid:0) re-optimize its nominal price. The ability to re-optimize prices is independent across (cid:133)rms and time. As in CEE (2005), we assume that a (cid:133)rm which cannot re-optimize its price sets P (i) according to: t P (i) = (cid:25) P (i): (2.9) t t 1 t 1 (cid:0) (cid:0) Here, (cid:25) denotes aggregate in(cid:135)ation, P =P : t t t 1 (cid:0) An intermediate goods (cid:133)rm(cid:146)s objective function is: 1 E (cid:12)j(cid:29) P (i)y (i) P w R h (i)+rk K (i) ; (2.10) t t+j t+j t+j (cid:0) t+j t+j t+j t+j t+j t+j j=0 X (cid:2) (cid:0) (cid:1)(cid:3) where E is the expectation operator conditioned on time t information. The term, (cid:12)t(cid:29) ; is t t+j proportional to the state-contingent marginal value of a dollar to a household.13 Also, (cid:12) is a scalar between zero and unity. The timing of events for a (cid:133)rm is as follows. At the beginning of period t; the (cid:133)rm observes the technology shocks and sets its price, P (i). Then, a shock t to monetary policy is realized, as is the demand for the (cid:133)rm(cid:146)s product, (2.2). The (cid:133)rm then 13The constant of proportionality is the probability of the relevant state of the world. 8

chooses productive inputs to satisfy this demand. The problem of the ith intermediate good (cid:133)rm is to choose prices, employment and capital services, subject to the timing and other constraints described above, to maximize (2.10). 2.1.3. Households There is a continuum of households, indexed by j (0;1): The sequence of events in a 2 period for a household is as follows. First, the technology shocks are realized. Second, the household makes its consumption and investment decisions, decides how many units of capital services to supply to rental markets, and purchases securities whose payo⁄s are contingent upon whether it can re-optimize its wage decision. Third, the household sets its wage rate. Fourth, the monetary policy shock is realized. Finally, the household allocates its beginning of period cash between deposits at the (cid:133)nancial intermediary and cash to be used in consumption transactions. Each household is a monopoly supplier of a di⁄erentiated labor service, and sets its wage subject to Calvo-style wage frictions. In general, households earn di⁄erent wage rates and work di⁄erent amounts. A straightforward extension of arguments in Erceg, Henderson, and Levin (2000) and Woodford (1996) establishes that in the presence of state contingent securities, householdsarehomogeneouswithrespecttoconsumptionandassetholdings. Our notation re(cid:135)ects this result. The preferences of the jth household are given by: h2 Ej 1 (cid:12)l t log(C bC ) j;t+l ; (2.11) t (cid:0) t+l (cid:0) t+l (cid:0) 1 (cid:0) L 2 l=0 (cid:20) (cid:21) X where 0 and Ej is the time t expectation operator, conditional on household j(cid:146)s time t L t (cid:21) information set. The variable, C ; denotes time t consumption, and h denotes time t hours t j;t worked. When b > 0; (2.11) exhibits habit formation in consumption preferences. The household(cid:146)s asset evolution equation is given by: M = R [M Q +(x 1)Ma]+A +Q +W h (2.12) t+1 t t (cid:0) t t (cid:0) t j;t t j;t j;t +P rku K (cid:22) +D (1+(cid:17)( ))P C P (cid:7) 1 I +a(u )K (cid:22) : t t t t t (cid:0) V t t t (cid:0) t (cid:0)t t t t (cid:0) (cid:1) Here, M ; Q and W denote the household(cid:146)s beginning of period t stock of money, cash t t j;t (cid:22) balances and time t nominal wage rate, respectively. In addition, K ; u ;D and A denote t t t j;t the household(cid:146)s physical stock of capital, the capital utilization rate, (cid:133)rm pro(cid:133)ts and the net cash in(cid:135)ow from participating in state-contingent securities at time t, respectively. The variable x represents the gross growth rate of the economy-wide per capita stock of money, t Ma: The quantity (x 1)Ma is a lump-sum payment made to households by the monetary t t (cid:0) t 9

authority. The household deposits M Q +(x 1)Ma with a (cid:133)nancial intermediary. The t (cid:0) t t (cid:0) t variable, R ; denotes the gross interest rate. t In (2.12), the price of investment goods relative to consumption goods is given by (cid:7) 1 (cid:0)t which we assume is an exogenous stochastic process. One way to rationalize this assumption isthatagentstransform(cid:133)nalgoodsintoinvestmentgoodsusingalineartechnologywithslope (cid:7) : This rationalization also underlies why we refer to (cid:7) as capital-embodied technological t t progress. The variable, ; denotes the time t velocity of the household(cid:146)s cash balances: t V P C t t = ; (2.13) t V Q t where (cid:17)( ) is increasing and convex. The function (cid:17)( ) captures the role of cash balances t t V V in facilitating transactions. Similar speci(cid:133)cations have been used by a variety of authors including Sims (1994) and Schmitt-Grohe and Uribe (2004). For the quantitative analysis of our model, we require the level and the (cid:133)rst two derivatives of the transactions function, (cid:17)( ); evaluated in steady state. We denote these by (cid:17); (cid:17) ; and (cid:17) ; respectively. We chose 0 00 V values for these objects as follows. The (cid:133)rst order condition for Q is: t P C P C 2 t t t t R = 1+(cid:17) : t 0 Q Q (cid:18) t (cid:19)(cid:18) t (cid:19) Let (cid:15) denote the interest semi-elasticity of money demand: t 100 dlog(Qt) (cid:15) (cid:2) Pt : t (cid:17) (cid:0) 400 dR t (cid:2) Denote the curvature of (cid:17) by ’: (cid:17) 00 ’ = V: (cid:17) 0 Then, the (cid:133)rst order condition for Q implies that the interest semi-elasticity of money t demand in steady state is: 1 1 1 (cid:15) = ; 4 R 1 2+’ (cid:18) (cid:0) (cid:19)(cid:18) (cid:19) where the steady state value of R is (cid:25)(cid:22) =(cid:12): We parameterize (cid:17)( ) indirectly using values z (cid:3) (cid:1) for (cid:15); and (cid:17): V The remaining terms in (2.12) pertain to the household(cid:146)s capital-related income. The (cid:22) services of capital, K ; are related to stock of physical capital, K ; by t t (cid:22) K = u K : t t t The term P rku K (cid:22) represents the household(cid:146)s earnings from supplying capital services. The t t t t (cid:22) function a(u )K denotes the cost, in investment goods, of setting the utilization rate to u : t t t 10

We assume a(u ) is increasing and convex. These assumptions capture the idea that the t more intensely the stock of capital is utilized, the higher are maintenance costs in terms of investment goods. Our log-linear approximation solution strategy requires the level and (cid:133)rst two derivatives of a( ) in steady state. We treat (cid:27) = a (1)=a(1) 0 as a parameter to a 00 0 (cid:1) (cid:21) be estimated and impose that u = 1 and a(1) = 0 in steady state. Although the steady t state of the model does not depend on the value of (cid:27) ; the dynamics do. Given our solution a procedure, we do not need to specify any other features of the function a: The household(cid:146)s stock of physical capital evolves according to: I (cid:22) (cid:22) t K = (1 (cid:14))K +(1 S )I ; (2.14) t+1 t t (cid:0) (cid:0) I (cid:18) t 1(cid:19) (cid:0) where (cid:14) denotes the physical rate of depreciation, and I denotes time t investment goods. t The adjustment cost function, S; is assumed to be increasing, convex and to satisfy S = S = 0 in steady state. We treat the second derivative of S in steady state, S > 0; as a 0 00 parameter to be estimated. Although the steady state of the model does not depend on the value of S ; the dynamics do. Given our solution procedure, we do not need to specify any 00 other features of the function S: 2.1.4. The Wage Decision As in Erceg, Henderson, and Levin (2000), we assume that the jth household is a monopoly supplier of a di⁄erentiated labor service, h . It sells this service to a representative, comj;t petitive (cid:133)rm that transforms it into an aggregate labor input, H ; using the technology: t 1 1 (cid:21)w H = h(cid:21)wdj ; 1 (cid:21) < : t j;t (cid:20) w 1 (cid:20)Z0 (cid:21) The demand curve for h is given by: j;t (cid:21)w h = W t (cid:21)w (cid:0) 1 H : (2.15) j;t t W (cid:18) j;t(cid:19) Here, W is the aggregate wage rate, i.e., the nominal price of H : It is straightforward to t t show that W is related to W via the relationship: t j;t 1 1 (cid:21)w 1 (cid:0) W t = (W j;t )1 (cid:0) (cid:21)w dj : (2.16) (cid:20)Z0 (cid:21) The household takes H and W as given. t t Households set their nominal wage according to a variant of the mechanism by which intermediate good (cid:133)rms set prices. In each period, a household faces a constant probability, 1 (cid:24) ;ofbeingabletore-optimizeitsnominalwage. Theabilitytore-optimizeisindependent w (cid:0) 11

across households and time. If a household cannot re-optimize its wage at time t; it sets W j;t according to: W = (cid:25) (cid:22) W : (2.17) j;t t 1 z j;t 1 (cid:0) (cid:3) (cid:0) The presence of (cid:22) in (2.17) implies that there are no distortions fromwage dispersion along z (cid:3) the steady state growth path. 2.1.5. Monetary and Fiscal Policy We adopt the following speci(cid:133)cation of monetary policy: x^ = x^ +x^ +x^ : t z;t (cid:7);t M;t Here x represents the gross growth rate of money, M =M : We assume that t t+1 t x^ = (cid:26) x^ +" (2.18) M;t xM M;t 1 M;t (cid:0) x^ = (cid:26) x^ +c " +cp" z;t xz z;t 1 z z;t z z;t 1 (cid:0) (cid:0) x^ = (cid:26) x^ +c " +cp" (cid:7);t x(cid:7) (cid:7);t 1 (cid:7) (cid:7);t (cid:7) (cid:7);t 1 (cid:0) (cid:0) Here, " represents a shock to monetary policy. We denote the standard deviation of " M;t M;t by (cid:27) . The dynamic response of x^ to " is characterized by a (cid:133)rst order autoregression, M M;t M;t so that (cid:26)j is the response of E x^ to a one-unit time t monetary policy shock. The term t t+j xM x^ captures the response of monetary policy to an innovation in neutral technology, " : z;t z;t We assume that x^ is characterized by an ARMA(1,1) process. The term, x^ ; captures z;t (cid:7);t the response of monetary policy to an innovation in capital-embodied technology, " : We (cid:7);t assume that x^ is also characterized by an ARMA(1,1) process. (cid:7);t In models with nominal rigidities, it is generally the case that the dynamic response functions to shocks depend heavily on the nature of monetary policy. CEE (2005) show that for a very closely related model the impulse response function to a monetary policy shock as parameterized above are very similar to the response obtained when the central bank is instead assumed to follow an explicit Taylor rule. Finally, we assume that the government adjusts lump sum taxes to ensure that its intertemporal budget constraint holds. 2.1.6. Loan Market Clearing, Final Goods Market Clearing and Equilibrium Financial intermediaries receive M Q + (x 1)M from the household. Our notation t t t t (cid:0) (cid:0) re(cid:135)ects the equilibrium condition, Ma = M : Financial intermediaries lend all of their money t t to intermediate good (cid:133)rms, which use the funds to pay labor wages. Loan market clearing requires that: W H = x M Q : (2.19) t t t t t (cid:0) 12

The aggregate resource constraint is: (1+(cid:17)( ))C +(cid:7) 1 I +a(u )K (cid:22) Y : (2.20) V t t (cid:0)t t t t (cid:20) t (cid:2) (cid:3) We adopt a standard sequence-of-markets equilibrium concept. In the technical appendix to this paper, Altig et al. (ACEL henceforth, 2004), we discuss our computational strategy for approximating that equilibrium. This strategy involves taking a (log-)linear approximation about the non-stochastic steady state of the economy and using the solution methods discussed in Anderson and Moore (1985) and Christiano (2002). 2.2. The Firm-Speci(cid:133)c Capital Model Inthismodel,(cid:133)rmsowntheirowncapital. The(cid:133)rmcannotadjustitscapitalstockwithinthe period. It can only change its stock of capital over time by varying the rate of investment. In all otherrespects theproblemof intermediategood(cid:133)rms is thesameas before. Inparticular, they face the same demand curve, (2.2), production technology, (2.4)-(2.8), and Calvo-style pricing frictions, including the updating rule given by (2.9). The technology for accumulating physical capital by intermediate good (cid:133)rm i is given by I (i) (cid:22) (cid:22) t K (i) = (1 (cid:14))K (i)+(1 S )I (i): t+1 t t (cid:0) (cid:0) I (i) (cid:18) t 1 (cid:19) (cid:0) The present discounted value of the ith intermediate good(cid:146)s net cash (cid:135)ow is given by: E 1 (cid:12)j(cid:29) P (i)y (i) P R w (i)h (i) P (cid:7) 1 I (i)+a(u (i))K (cid:22) (i) : t t+j t+j t+j (cid:0) t+j t+j t+j t (cid:0) t+j (cid:0)t+j t+j t+j t+j j=0 X (cid:8) (cid:2) (cid:3)(cid:9) (2.21) Time t net cash (cid:135)ow equals sales, less labor costs (inclusive of interest charges) less the costs associated with capital utilization and capital accumulation. The sequence of events as it pertains to the ith (cid:133)rm is as follows. At the beginning of (cid:22) period t; the (cid:133)rm has a given stock of physical capital, K (i). After observing the technology t shocks, the (cid:133)rmsets its price, P (i); subject to the Calvo-style frictions described above. The t (cid:133)rm also makes its investment and capital utilization decisions, I (i) and u (i); respectively. t t The time t monetary policy shock then occurs and the demand for the (cid:133)rm(cid:146)s product is realized. The (cid:133)rm then purchases labor to satisfy the demand for its output. Subject to these timing and other constraints, the problem of the (cid:133)rm is to choose prices, employment, the level of investment and utilization to maximize net discounted cash (cid:135)ow. 2.3. Implications for In(cid:135)ation The equations which characterize equilibrium for the homogenous and (cid:133)rm-speci(cid:133)c capital model are identical except forthe equationwhichcharacterizes aggregate in(cid:135)ationdynamics. 13

This equation is of the form: (cid:1)(cid:25)^ = E[(cid:12)(cid:1)(cid:25)^ +(cid:13)s^ (cid:10) ]; (2.22) t t+1 t t j where 1 (cid:24) 1 (cid:12)(cid:24) p p (cid:13) = (cid:0) (cid:0) (cid:31); (cid:24) (cid:0) (cid:1)(cid:0)p (cid:1) and (cid:1) is the (cid:133)rst di⁄erence operator. The information set (cid:10) includes the current realization t ofthetechnologyshocks,butnotthecurrentrealizationoftheinnovationtomonetarypolicy. The variable s denotes the economy-wide average marginal cost of production, in units of t the (cid:133)nal good. In ACEL (2004) we establish the following14: Proposition 1 (i) In the homogeneous capital model, (cid:31) = 1; (ii) In the (cid:133)rm-speci(cid:133)c capital model, (cid:31) is a particular non-linear function of the parameters of the model. We parameterize the (cid:133)rm-speci(cid:133)c and homogeneous capital model in terms of (cid:13); rather than (cid:24) : Consequently, the list of parameters for the two models remains identical. Given p values for these parameters, the two models are observationally equivalent with respect to aggregate prices and quantities. This means that we do not need to take a stand on which version of the model we are working with at the estimation stage of our analysis. 3. Econometric Methodology WeemployavariantofthelimitedinformationstrategyusedinCEE(2005)(seealsoRotemberg and Woodford (1997)). De(cid:133)ne the ten dimensional vector, Y : t (cid:1)ln(relative price of investment ) (cid:1)p t It (cid:1)ln(GDP =Hours ) t t 1 1 0 (cid:1)ln(GDP de(cid:135)ator ) 1 0 (cid:1) (cid:2) a 1 t t |{z} B Capacity Utilization C B C B t C B 1 1 C B ln(Hours ) C B Y (cid:2) C Y t = B B ln(GDP =Hours ) t ln(W =P ) C C = B B |{ 1 z t } C C (3.1) 10 (cid:2) 1 B B t ln(C t =G t D (cid:0) P t ) t t C C B B 6 R (cid:2) t 1 C C B C B |{z} C |{z} B ln(I t =GDP t ) C B 1 1 C B C B (cid:2) C B Federal Funds Rate C B Y C t 2t B C B |{z} C B ln(GDP de(cid:135)ator )+ln(GDP ) ln(MZM ) C B C t t t 1 1 B (cid:0) C B (cid:2) C @ A @ A We embed our identifying assumptions as restrictions on the parameter|s{ozf}the following reduced form VAR: 14See Christiano (2004) for a discussion of the solution to (cid:133)rm-speci(cid:133)c capital models in simpler settings. 14

Y = (cid:11)+B(L)Y +u ; (3.2) t t 1 t (cid:0) Eu u = V; t 0t where B(L) is a pth-ordered polynomial in the lag operator, L: The (cid:147)fundamental(cid:148)economic shocks, " ; are related to u as follows: t t u = C" ; E" " = I, (3.3) t t t 0t where C is a square matrix and I is the identity matrix. We assume that " is a martingale t di⁄erence stochastic process, so that we allow for the presence of conditional heteroscedasticity.15 We require B(L) and the ith column of C; C ; to calculate the dynamic response of i Y to a disturbance in the ith fundamental shock, " : t i;t According to our economic model, the variables in Y ; de(cid:133)ned in (3.1), are stationary t stochastic processes. We partition " conformably with the partitioning of Y : t t " = " (cid:7);t " z;t " 01t " M;t " 2t 0 : (3.4) t (cid:18) 1 (cid:2) 1 1 (cid:2) 1 1 (cid:2) 6 1 (cid:2) 1 1 (cid:2) 1 (cid:19) Here, " is the innovation to a n|e{uz}tral|{tze}chn|o{lzo}gy|s{hzo}ck|,{"z} is the innovation in capitalz;t (cid:7);t embodied technology, and " is the monetary policy shock: M;t 3.1. Identi(cid:133)cation of Impulse Responses We assume that policy makers set the interest rate so that the following rule is satis(cid:133)ed: R = f((cid:10) )+#" ; (3.5) t t M;t where " is the monetary policy shock and # > 0 is a constant. We interpret (3.5) as a M;t reduced form Taylor rule. To ensure identi(cid:133)cation of the monetary policy shock, we assume f is linear, (cid:10) contains Y ; :::;Y and the only date t variables in (cid:10) are {(cid:1)a ;(cid:1)p ;Y }. t t 1 t q t t It 1t (cid:0) (cid:0) Finally, we assume that " is orthogonal to (cid:10) : M;t t As in Fisher (2006), we assume that innovations to technology (both neutral and capitalembodied) are the only shocks which a⁄ect the level of labor productivity in the long run. In addition, we assume that capital embodied technology shocks are the only shocks that a⁄ect the price of investment goods relative to consumption goods in the long run. These assumptions are satis(cid:133)ed in our model. 15Justiniano and Primiceri (2008) argue that conditional heteroscedasticity in fundamental shocks is importantforexplainingthe(cid:145)GreatModeration(cid:146). SimilarargumentshavebeenmadebyCEE(1999)andSmets and Wouters (2007). 15

To compute the responses of Y to " ; " ; and " ; we require estimates of the paramet (cid:7);t z;t M;t ters in B(L); as well as the 1st; 2nd and 9th columns of C: We obtain these estimates using a suitably modi(cid:133)ed variant of the instrumental variables strategy proposed by Shapiro and Watson (1988). See ACEL (2004) for further details. 4. Estimation Results Based on a Structural Vector Autoregression In this section we describe the dynamic response of the economy to monetary policy shocks, neutral technology shocks and capital embodied shocks. In addition, we discuss the quantitative contributionof these shocks tothe cyclical (cid:135)uctuations inaggregate economic activity. In the (cid:133)rst subsection we describe the data used in the analysis. In the second and third subsections we discuss the impulse response functions and the importance of the shocks to aggregate (cid:135)uctuations. 4.1. Data With the exception of the price of investment and of monetary transactions balances, all data were taken from the FRED Database available through the Federal Reserve Bank of St. Louis.16 The price of investment corresponds to the (cid:145)total investment(cid:146)series constructed and used in Fisher (2006).17 Our measure of transactions balances, MZM; was obtained from the Federal Reserve Bank of St. Louis(cid:146)s online database. Our data are quarterly, and the sample period is 1982:1-2008:3.18 We work with the monetary aggregate, MZM; for the following reasons. First, MZM is constructed to be a measure of transactions balances, so it is a natural empirical counterpart to our model variable, Q : Second, our statistical procedure requires that the velocity of t 16Nominal gross output is measured by GDP, real gross output is measured by GDPC96 (real, chainweightedGDP).NominalinvestmentisPCDG(householdconsumptionofdurables)plusGPDI(grossprivate domestic investment). Nominal consumption is measured by PCND (nondurables) plus PCESV (services) plus GCE (government expenditures). Real private domestic investment is given by GPDIC96. Real private consumption expendutures are given by PCEC96. Our MZM measure of money is MZMSL. Variables were converted into per capita terms by CNP16OV, a measure of the US civilian non-institutional population over age 16. A measure of the aggregate price index was obtained from the ratio of nominal to real output, GDP/GDPC96. Capacity utilization is measured by CUMFN, the manufacturing industry(cid:146)s capacity index. Theinterestrateismeasuredbythefederalfundsrate,FEDFUNDS.HoursworkedismeasuredbyHOANBS (Non-farmbusinesshours). Hourswereconvertedtopercapitatermsusingourpopulationmeasure. Nominal wages are measured by COMPNFB, (nominal hourly non-farm business compensation). This was converted to real terms by dividing by the aggregate price index. 17Wealsore-estimatedtheVARandthestructuralmodelusingasourmeasuresofhoursandproductivity, private business hours and business sector productivity, respectively. In these estimation runs, we measure consumption and output as private sector consumption and private sector output, respectively. Taking sampling uncertainty, we (cid:133)nd that our results are robust to these alternatives data measures. 18The estimation period for the vector autoregression drops the (cid:133)rst p quarters, to accommodate the p lags. 16

money is stationary. The velocity of MZM is reasonably characterized as being stationary. The stationarity assumption is more problematic for the velocity of aggregates like the base, M1 and M2: 4.2. Estimated Impulse Response Functions In this subsection we discuss our estimates of the dynamic response of Y to monetary policy t and technology shocks. To obtain these estimates we set p; the number of lags in the VAR, to 4: Various indicators suggest that this value of p is large enough to adequately capture the dynamics in the data. For example, the Akaike, Hannan-Quinn and Schwartz criteria supportachoiceofq = 2;2;1;respectively.19 WealsocomputethemultivariatePortmanteau (Q) statistic to test the null hypothesis of zero serial correlation up to lag n in the VAR disturbances. We consider n = 4; 6; 8; 10: The test statistics are, respectively, Q = 262; 475, 680; 880. Using conventional asymptotic sampling theory, these Q statistics all have a p-value very close to zero, indicating a rejection of the null hypothesis. However, we (cid:133)nd evidence that the asymptotic sampling theory rejects the null hypothesis too often. When we simulate the Q statistic using repeated arti(cid:133)cial data sets generated from our estimated VAR, we (cid:133)nd that the p-values of our Q statistics are 97; 87; 90 and 95 percent, respectively. For these calculations, each arti(cid:133)cial data set is of length equal to that of our actual sample, and is generated by bootstrap sampling from the (cid:133)tted disturbances in our estimated VAR. On this basis we do not strongly reject the null hypothesis that the disturbance terms in a VAR with p = 4 are serially uncorrelated. Figure 1 displays the response of the variables in our analysis to a one standard deviation monetary policy shock (roughly 30 basis points). In each case, there is a solid line in the centerof a grayarea. The grayarea represents a95 percent con(cid:133)dence interval, and the solid line represents the point estimates.20 Except for in(cid:135)ation and the interest rate, all variables are expressed in percent terms. So, for example, the peak response of output is about 0:15 percent. The Federal Funds rate is expressed in units of percentage points, at an annual rate. In(cid:135)ation is expressed in units of percentage points, at a quarterly rate. Six features of Figure 1 are worth noting. First, the e⁄ect of a policy shock on the money growth rate and the interest rate is completed within roughly one year. Other quantity variables respond over a longer period of time. Second, there is a signi(cid:133)cant liquidity e⁄ect, i.e. the interest rate and money growth move in opposite directions after a policy shock. Third, in(cid:135)ation responds very weakly to the policy shock. Fourth, output, consumption, 19See Bierens (2004) for the formulas used and for a discussion of the asymptotic properties of the lag length selection criteria. 20The con(cid:133)dence intervals are symmetric about our point estimates. They are obtained by adding and subtracting1.96timesourestimateofthestandarderrorsofthecoe¢ cientsintheimpulseresponsefunctions. These standard errors were computed by bootstrap simulation of the estimated model. 17

investment, hours worked and capacity utilization all display hump-shaped responses. With the exception of hours worked, the peak response in these aggregates occurs roughly one year after the shock. The hump shaped response in hours worked is more drawn out, with the peak occurring after approximately two years. Fifth, velocity co-moves with the interest rate, with both initially falling in response to a monetary policy shock, and then rising. Sixth, the real wage does not respond signi(cid:133)cantly to a monetary policy shock, but after a delay the price of investment does. Figure 2 displays the response of the variables in our analysis to a positive, one standard deviation shock in neutral technology, e : By construction, the impact of this technology z;t shock on output, labor productivity, consumption, investment and the real wage can be permanent. Because the roots of our estimated VAR are stable, the impact of a neutral technology shock on the variables whose levels appear in Y must be temporary. These varit ables are the Federal Funds rate, capacity utilization, hours worked, velocity and in(cid:135)ation. According to Figure 2 a positive, neutral technology shock leads to a persistent rise in output with a peak rise of roughly 0:35 percent over the period displayed. In addition, hours worked, investment and consumption rise in response to the technology shock. These rises are only marginally statistically signi(cid:133)cant. Finally notice that a neutral technology shock leads to an initial sharp fall in the in(cid:135)ation rate.21. Overall, these e⁄ects are broadly consistent with what a student of real business cycle models might expect. Figure 3 displays the response of the variables in our analysis to a one standard deviation positive capital-embodied technology shock, " : This shock leads to statistically signi(cid:133)cant (cid:7);t rises in output, hours worked, capacity utilization, investment and the federal funds rate. At the same time, it leads to an initial fall in the price of investment of roughly 0:2 percent, followed by an ongoing signi(cid:133)cant decline. Finally, the shock also leads a marginally signi(cid:133)cant decline in real wages. 4.3. The Contribution of Monetary Policy and Technology Shocks to Aggregate Fluctuations We now brie(cid:135)y discuss the contribution of monetary policy and technology shocks to cyclical (cid:135)uctuations in economic activity. Table 1 summarizes the contribution of the three shocks to the variables in our analysis. We de(cid:133)ne business cycle frequencies as the components of a timeserieswithperiodsof8to32quarters. ThecolumnsinTable1reportthefractionofthe variance in the cyclical frequencies accounted for by our three shocks. Each row corresponds to a di⁄erent variable. Using the techniques described in Christiano and Fitzgerald (2003), we calculate the fractions as follows. Let fi(!) denote the spectral density at frequency ! 21Alves (2004) also (cid:133)nds that in(cid:135)ation drops after a positive neutral technology shocks using data for non-U.S. G7 countries 18

of a given variable, when only shock i is active. That is, the variance of all shocks in " ; t apart from the ith; are set to zero and the variance of the ith shock in " is set to unity. Let t f(!) denote the corresponding spectral density when the variance of each element of " is t set to unity. The contribution of shock i to variance in the business cycle frequencies is then de(cid:133)ned as: !2fi(!)d! 2(cid:25) 2(cid:25) !1 ; ! = ; ! = : !2f(!)d! 1 32 2 8 R!1 Our estimate of the spectral dRensity is the one implied by our estimated VAR.22 Numbers in parentheses are the standard errors, which we estimate by bootstrap methods. Finally, the fraction of the variance accounted for by all three shocks is just the sum of the individual fractions of the variance. Table1showsthatthethreeshockstogetheraccountforasubstantialportionofthecyclical variance in the aggregate quantities. For example, they account for roughly 60 percent of the variation in aggregate output, with the capital-embodied technology shock playing the largest role. Indeed the capital-embodied technology shock is the largest contributor to the cyclical variation in all of the variables included in the VAR. Intriguingly, the capital embodiedtechnologyshockaccounts fornearly30% of the cyclical variationinthe real wage, a variable whose cyclical variation is typically di¢ cult to account for empirically. 5. Estimation Results for the Equilibrium Model In this section we discuss the estimated parameter values. In addition, we assess the ability of the estimated model to account for the impulse response functions discussed in Section 4. 5.1. Benchmark Model Parameter Estimates We partition the parameters of the model into three groups. The (cid:133)rst group of parameters, (cid:16) ; is: 1 (cid:16) = [(cid:12);(cid:11);(cid:14);(cid:30); ;(cid:21) ;(cid:22) ;(cid:22) ;x; ;(cid:17)]: 1 L w (cid:7) z V The second group of parameters, (cid:16) ; pertain to the (cid:145)non-stochastic part(cid:146)of the model: 2 (cid:16) = [(cid:21) ;(cid:24) ;(cid:13);(cid:27) ;b;S ;(cid:15)]: 2 f w a 00 22We found that the analog statistics computed using the Hodrick-Prescott (cid:133)lter yielded essentially the same results. We computed this as follows: (cid:25) g(!)fi(!)d! 0 ; (cid:25) g(!)f(!)d! R0 where g(!) is the frequency-domain representaRtion of the Hodrick-Prescott (cid:133)lter with (cid:21)=1600. 19

The third set of parameters, (cid:16) ; pertain to the stochastic part of the model: 3 (cid:16) = (cid:26) ;(cid:27) ;(cid:26) ;(cid:27) ;(cid:26) ;c ;cp;(cid:26) ;(cid:27) ;(cid:26) ;c ;cp : 3 xM M (cid:22) z (cid:22) z xz z z (cid:22) (cid:7) (cid:22)(cid:7) x(cid:7) (cid:7) (cid:7) (cid:2) (cid:3) We estimate the values of (cid:16) and (cid:16) and set the values of (cid:16) a priori. We assume 2 3 1 (cid:12) = 1:03 0:25, which implies a steady state annualized real interest rate of 3 percent. We set (cid:0) (cid:11) = 0:36; which corresponds to a steady state share of capital income equal to roughly 36 percent.23 We set (cid:14) = 0:025, which implies an annual rate of depreciation on capital equal to 10 percent. This value of (cid:14) is roughly equal to the estimate reported in Christiano and Eichenbaum (1992). The parameter, (cid:30); is set to guarantee that pro(cid:133)ts are zero in steady state. As in CEE (2005), we set the parameter, (cid:21) ; to 1:05. We set the parameter to w L one. The steady state growth of real per capita GDP, (cid:22) ; is given by y (cid:11) (cid:22) = (cid:22) 1 (cid:11) (cid:22) : y (cid:7) (cid:0) z (cid:0) (cid:1) Given an estimate of (cid:22) and (cid:22) ; we use this equation to estimate (cid:22) : We use data over y (cid:7) z the sample period 1959II - 2001IV, the sample period in ACEL (2005), to estimate the parameters (cid:22) and (cid:22) : If we use the sample period 1982:1-2008:3, then the implied point (cid:7) x estimate of (cid:22) is less than one, a value that seems implausible to us. It seems reasonable to z extend the sample back in time because the value of (cid:22) should not be a⁄ected by any change z in the monetary policy regime that occurred in the early 1980(cid:146)s. For comparability with ACEL (2005) we stopped the sample period at 2001IV.24 With these considerations in mind, we set the parameter (cid:22) to 1:0042. At an annualized rate, this value is equal to the negative (cid:7) of the average growth rate of the price of investment relative to the GDP de(cid:135)ator which fell at an annual average rate of 1:68 percent over the ACEL (2005) sample period. The average growth rate of per capita GDP in the ACEL (2005) sample period is (cid:22) = 1:0045: y Solving the previous equation for (cid:22) yields (cid:22) = 1:00013; which is the value of (cid:22) we use in z z z our analysis.25 We set the average growth rate of money, (cid:22) ; equal to 1:017:26 This value x corresponds to the average quarterly growth rate of money (MZM) over the ACEL (2005) sample period. Wesettheparameters and(cid:17) to0:45and0:036;respectively. Thevalueof corresponds V V to the average value of P C =Q in the ACEL (2005) sample period, where Q is measured by t t t t 23In our model, the steady state share of labor income in total output is 1 (cid:11): This result re(cid:135)ects our (cid:0) assumption that pro(cid:133)ts are zero in steady state. 24Adding the years 2002 - 2008 makes little di⁄erence to our estimate of (cid:22) and (cid:22) : (cid:7) x 25For the 1982-2008 period, the investment de(cid:135)ator drops on average by 2:3 percent. Over the same (cid:0) sample period GDP per capita growth was 1:75 percent at an annualized rate, implying that (cid:22) is less than z one. 26The average annual growth rate of MZM over the sample period 1982-2008 and 1959-2008 is 2 and 1.7 percent, respectively. 20

MZM: We chose (cid:17) so that in conjunction with the other parameter values of our model, the steady state value of (cid:17)C=Y is 0:025. This corresponds to the percent of value-added in the (cid:133)nance, insurance and real estate industry (see Christiano, Motto, and Rostagno (2004)). Therowlabeled(cid:145)benchmark(cid:146)inTable2summarizesourpointestimatesoftheparameters in the vector (cid:16) : Standard errors are reported in parentheses. The lower bound of unity is 2 binding on (cid:21) : So we simply set (cid:21) to 1:01 when we estimate the model. However, the f f estimation criterion displays very little curvature with respect to (cid:21) . When we individually f test the hypotheses that (cid:21) is 1:05 or 1:20 against the null that (cid:21) = 1; we obtain a chif f square statistic equal to 0:01 and 0:2, respectively, with associated probability values 0.0002 and 0:024, respectively. So we cannot clearly reject either hypothesis. Tables 2 and 3 report point estimates for (cid:16) and (cid:16) when we re-estimate the model setting (cid:21) to 1:05 and 1:20. 2 3 f Our point estimate of (cid:24) implies that wage contracts are re-optimized, on average, once w every 4:5 quarters. To interpret our point estimate of (cid:13); recall that in the homogeneous capital model, (cid:13) = (1 (cid:24) )(1 (cid:12)(cid:24) )=(cid:24) : So our point estimate of (cid:13) implies a value of (cid:24) p p p p (cid:0) (cid:0) equal to 0:896: This implies that (cid:133)rms re-optimize prices roughly every 9:36 quarters (see Table 4). This value is much larger than the value used by Golosov and Lucas (2007) who calibrate their model to micro data to ensure that the (cid:133)rms re-optimize prices on average once every 1:5 quarters. Table 4 shows that if we adopt the assumption that capital is (cid:133)rm-speci(cid:133)c, then our estimates imply that (cid:133)rms re-optimize prices on average once every 1:8 quarters.27 So the assumption that capital is (cid:133)rm-speci(cid:133)c has a very large impact on inference about the frequency at which (cid:133)rms re-optimize price. To interpret the estimated value of (cid:27) , we consider the homogeneous capital model. a Linearizing the household(cid:146)s (cid:133)rst order condition for capital utilization about steady state yields: 1 E ( r^k u^ ) (cid:10) = 0: (cid:27) t (cid:0) t j t (cid:26) a (cid:27) According to this expression, 1=(cid:27) , equal to 0:08 of a percent, is the elasticity of capital a utilization with respect to the rental rate of capital. Our estimate of (cid:27) is larger than the a value estimated by CEE (2005) and indicates that it is relatively costly for (cid:133)rms to vary the utilization of capital. Our point estimate of the habit parameter b is 0:76. This value is reasonably close to the point estimate of 0:66; reported in CEE (2005) and the value of 0:7 reported in Boldrin, Christiano, and Fisher (2001). The latter authors argue that the ability of standard general equilibrium models to account for the equity premium and other asset market statistics is considerably enhanced by the presence of habit formation in preferences. 27This number was obtained using the algorithm discussed in ACEL (2004). 21

We now discuss our point estimate of S : Suppose we denote by P the shadow price of 00 k;t 0 (cid:22) one unit of k ; in terms of output. The variable P is what the price of installed capital t+1 k;t 0 (cid:22) would be in the homogeneous capital model if there were a market for k at the beginning t+1 of period t: Proceeding as in CEE (2005), it is straightforward to show that the household(cid:146)s (cid:133)rst order condition for investment implies: ^{ =^{ + 1 1 (cid:12)jE[P ^ (cid:10) ]: t t 1 k;t+j t (cid:0) S 00 0 j j=0 X Accordingtothisexpression,1=S istheelasticityofinvestmentwithrespecttoaonepercent 00 temporary increase in the current price of installed capital. Our point estimate implies that this elasticity is equal to 0:66. The more persistent is the change in the price of capital, the larger is the percentage change in investment. This property holds because adjustment costs induce agents to be forward looking. Table 3 reports the estimated values of the parameters pertaining to the stochastic part of the model. With these values, the laws of motion for the neutral and capital-embodied technology shocks are: (cid:22)^ = 0:55(cid:22)^ +" ; 100 (cid:27) = 0:21 (cid:7);t (0:12) (cid:7);t (cid:0) 1 (cid:22) (cid:7) ;t (cid:2) (cid:22) (cid:7) (0:04) (cid:22)^ = 0:42(cid:22)^ +" ; 100 (cid:27) = 0:17 z;t (0:27) z;t (cid:0) 1 (cid:22) z ;t (cid:2) (cid:22) z (0:08) Numbers in parentheses are standard errors. Our estimates imply that a one-standard deviation neutral technology shock drives z up by 0:17 percent in the period of the shock t and by 0:29 (= 0:17=(1 0:42)) percent in the long run. A one-standard-deviation shock to (cid:0) embodied technology drives (cid:7) up by 0:21 percent immediately and by 0:47 percent in the t long run. Our estimates imply that shocks to neutral technology exhibit a high degree of serial correlation, while shocks to capital-embodied technology do not. It is interesting to compare our results for (cid:22)^ with the ones reported in Prescott (1986), z;t who estimates the properties of the technology shock process using the Solow residual. He (cid:133)nds that the shock is roughly a random walk, and its growth rate has a standard deviation of roughly 1 percent.28 By contrast, our estimates imply that the unconditional standard deviation of the growth rate of neutral technology is roughly 0:19 (= 0:17/ (1 0:422)) (cid:0) percent. So we (cid:133)nd that technology shocks are substantially less volatile but more persistent p than those estimated by Prescott. In principle, these di⁄erences re(cid:135)ect two factors. First, 28Prescott (1986) actually reports a standard deviation of 0:763 percent. However, he adopts a di⁄erent normalization for the technology shock than we do, by placing it in front of the production function. By assumption, the technology shock multiplies labor directly in the production and is taken to a power of labor(cid:146)s share. The value of labor(cid:146)s share that Prescott uses is 0:70. When we translate Prescott(cid:146)s estimate into the one relevant for our normalization, we obtain 0:763=:7 1. (cid:25) 22

from the perspective of our model, Prescott(cid:146)s estimate of technology confounds technology with variable capital utilization. Second, our analysis is based on di⁄erent data sets and di⁄erent identifying assumptions than Prescott(cid:146)s. 5.2. Impulse Responses The dotted lines in Figures 1 through 3 display the impulse response functions of the estimated model to monetary policy, neutral technology shocks and capital-embodied shocks, respectively. Recall that the solid lines and the associated con(cid:133)dence intervals (the gray areas) pertain to the impulse response functions from the estimated, identi(cid:133)ed VARs. 5.2.1. Response to a Monetary Policy Shock Webeginbydiscussingthemodel(cid:146)sperformancewithrespecttoamonetarypolicyshock(see Figure1). First, consistentwithresultsinCEE(2005), themodel doeswell ataccountingfor the dynamic response of the U.S. economy to a monetary policy shock. Most (but not all) of the model responses lie within the two-standard deviation con(cid:133)dence interval computed fromthe data. This is true even though (cid:133)rms in the (cid:133)rm-speci(cid:133)c capital version of the model change prices on average once every 1:8 quarters. Second, the model generates a very persistent response in output. The peak e⁄ect occurs roughlyone yearafterthe shock. The output response is positive foroverthree years. Third, the model accounts for the dynamic response of the interest rate to a monetary policy shock. Consistent with the data, an expansionary monetary policy shock induces a sharp decline in the interest rate which then returns to its pre-shock level within a year. The model does not account for the overshooting pattern of the interest rate in the data. The growth rate of transactions balances rises for a brief period of time after the policy shock, but then quickly reverts to its pre-shock level. But the model does not account for the overshooting pattern seen in transaction balances. Figure 1 shows that the e⁄ects of a policy shock on aggregate economic activity persist beyond the e⁄ects on the policy variable itself, regardless of whether the policy variable is measured as the interest rate or the money supply. This property re(cid:135)ects the strong internal propagation mechanisms in the model. Fourth, as in the data, the real wage remains essentially una⁄ected by the policy shock. Fifth,consumption,investment,andhoursworkedexhibitpersistent,hump-shapedrisesthat are consistent with our VAR-based estimates. Sixth, consistent with the data, velocity falls after the expansionary policy shock. This fall re(cid:135)ects the rise in money demand associated with the initial fall in the interest rate. However this fall is nearly as strong as the VAR based response of velocity to a monetary policy shock. Seventh, by construction, the relative price of investment is not a⁄ected by a policy shock in the model. At least for the (cid:133)rst 23

two years after the policy shock, this lack of response is consistent with the response of the relative price of investment to a policy shock in the identi(cid:133)ed VAR. It is not consistent with the rise in that price in the third year after the shock. Finally, capacity utilization in the model rises by only a very small amount, and understates the estimated rise in the data. Overall the response of the model to a monetary policy shock is quite similar to the response of the estimated model in CEE (2005). This result holds even though the models are estimated over very di⁄erent sample periods. The main di⁄erence is that the size of the monetary policy shock is almost twice as large as in ACEL (2005). But conditional on a given monetary policy shock, the transmission mechanism in the two estimated models is very similar. 5.2.2. Response to a Neutral Technology Shock We now discuss the model(cid:146)s performance with respect to a neutral technology shock (see Figure 2). First, the model does well at accounting for the dynamic response of the U.S. economy to a neutral technology shock. Speci(cid:133)cally, the model accounts for the rise in aggregate output, hours worked, investment, consumption and the real wage. However, the model does not capture the extent of the fall in in(cid:135)ation that occurs immediately after the shock. 5.2.3. Response to a Capital Embodied Technology Shock We now discuss the model(cid:146)s performance with respect to a capital-embodied technology shock (see Figure 3). The model does very well in accounting for the response of the U.S. economytothisshock, exceptthatitdoesnotaccountfortherisesincapacityutilizationand the federal funds rate that occur after the capital-embodied technology shock. In addition moneygrowthishighrelativetotheestimatedresponsefromtheVAR.Toseetheimportance of monetary policy in the transmission of capital embodied technology shocks, we compute the response of the model economy to a positive, capital embodied technology shock under the assumption that money growth remains unchanged from its steady state level. We (cid:133)nd that output and hours worked rise by much less, while in(cid:135)ation falls compared to what happens when monetary policy is accommodative. We conclude that the model requires accommodative monetary policy to match the expansionary e⁄ects of a positive capital embodied technology shock. 6. The Key Features of the Model In this section we discuss the features of the data driving our estimates of the parameters determining the implications of the (cid:133)rm-speci(cid:133)c and homogeneous capital models for the 24

frequency at which (cid:133)rms re-optimize prices. Our point estimate of (cid:13) (0:014) implies that a temporary one percent change in marginal cost results in only a 0:02 percent change in the aggregate price level.29 The small value of (cid:13) liesattheheartofthetensionbetweenthemicroandmacroimplicationsofthehomogeneous capital model. We now argue that any reasonable estimate of (cid:13) must be low. In Figure 4a we plot (cid:1)(cid:25)^ (cid:12)(cid:1)(cid:25)^ against our measure of the log of marginal cost, s^.30 The distribution of t t+1 t (cid:0) (cid:1)(cid:25)^ (cid:12)(cid:1)(cid:25)^ is at best weakly related to the magnitude of s^:31 The relatively (cid:135)at curve t t+1 t (cid:0) in Figure 4a has a slope equal to our point estimate of (cid:13) (0:014). Signi(cid:133)cantly, this curve passes through the central tendency of the data. The steeper curve in Figure 4a is drawn for a value of (cid:13) equal to 0:68; a value which implies that in the homogeneous capital model (cid:133)rms change prices roughly once every 1:8 quarters. Figure 4a shows that raising (cid:13) to 0:68 leads to a drastic deterioration in (cid:133)t. Equation (2.22) implies that the magnitude of the residuals from the lines in Figure 4a cannotbeusedasaformalmeasureofmodel(cid:133)t. Weshouldfocusonthesizeofresidualswhen thedataare replacedbytheirprojectionontodatet information, because then(2.22) implies that least squares consistently recovers the true value of (cid:13). Figure 4b is the analog to Figure 4a, with variables replaced by their projection onto zt (cid:1)(cid:25) t s (cid:12)(cid:1)(cid:25) t+1 s ;s^ t s ;s = 1;2 : (cid:17) f (cid:0) (cid:0) (cid:0) (cid:0) g Figures 4a and 4b are very similar so that our conclusions are unchanged: the data on in(cid:135)ation and marginal cost suggest that (cid:13) is small.32 The low estimated value of (cid:13) provides a di⁄erent perspective on the in(cid:135)ation inertia puzzle, particularly the weak response of in(cid:135)ation to monetary policy shocks. Solving (2.22) forward we obtain 1 (cid:1)(cid:25)^ = (cid:13) (cid:12)jE s^ : (6.1) t t t+j j=0 X This relation makes clear why many authors incorporate features like variable capital utilization and sticky wages into their models. These features can reduce the response of expected marginal cost to shocks.33 Relation (6.1) reveals another way to account for in(cid:135)ation inertia: assign a small value to (cid:13): The evidence in Figure 4a and 4b indicates that a small value of (cid:13) must be part of any successful resolution of the in(cid:135)ation inertia puzzle. 29This estimate is consistent with results in the literature. See Eichenbaum and Fisher (2007) and the references therein. 30Weset(cid:12) =1:03 :25:Also,wemeasuremarginalproductivitybylabor(cid:146)sshareinGDP.Inourmodelthis (cid:0) is the correct measure if (cid:133)xed costs are zero. This measure is approximately correct here, since our estimate of (cid:30) is close to zero. 31Eichenbaum and Fisher (2007) argue that their estimates of (cid:13) are robust to alternative measures of marginal cost. 32We obtain the same results whether we work with (cid:1)(cid:25)^ or with (cid:25)^ : t t 33See,forexample,BallandRomer(1990),CEE(2005),DotseyandKing(2009),Gal(cid:237)andGertler(1999), LindØ (2005) and Smets and Wouters (2003). 25

A low value of (cid:13) is clearly a problem for the homogeneous capital model. This is because the model then implies that (cid:133)rms re-optimize prices very infrequently, e.g., at intervals of roughly 9:5 quarters.34 So to get the macro data right (i.e., a low (cid:13)) we must make assumptions about the frequency at which (cid:133)rms re-optimize prices that seem implausible in light of the micro data. In contrast, suppose we adopt the more plausible assumption that (cid:133)rms re-optimize prices on average once every 1:8 quarters. Then the homogeneous capital model implies (cid:13) = 0:68: But this means that the model gets the macro data wrong. In the (cid:133)rm-speci(cid:133)c capital model it is possible to reconcile the low value of (cid:13) with a low value of (cid:24) : This re(cid:135)ects two features of that model. The (cid:133)rst is that not all (cid:133)rms set prices p at the same time (i.e., (cid:145)staggered pricing(cid:146)). The second is that capital is (cid:133)rm-speci(cid:133)c so that the only way a (cid:133)rm can adjust its capital stock is by varying its investment over time. To understand the role of these features, suppose there is an increase in the quantity of money. Flexible price (cid:133)rms respond by increasing their prices. Depending on how elastic demand is, these price increases cause demand to shift away from (cid:135)exible price (cid:133)rms and towards the sticky price (cid:133)rms. Consequently, (cid:135)exible price (cid:133)rms need less capital services. If (cid:133)rms could trade physical capital, capital would (cid:135)ow from (cid:135)exible price (cid:133)rms to sticky price (cid:133)rms. With (cid:133)rm-speci(cid:133)c capital this (cid:135)ow cannot occur. To the extent that (cid:135)exible price (cid:133)rms cannot easily reduce capital utilization rates, the shadow value of capital to (cid:135)exible price (cid:133)rms plummets. This fall acts like a decline in marginal cost and reduces the incentive of (cid:135)exible price (cid:133)rms to raise prices in the (cid:133)rst place. This mechanism is enhanced the more elastic is demand and the less (cid:135)exible is capital utilization. No doubt, the assumption that capital is completely immobile between (cid:133)rms is unrealistic. At the same time, anything which causes a (cid:133)rm(cid:146)s marginal cost to be an increasing function of its output works in the same direction as (cid:133)rm-speci(cid:133)city of capital.35 The key parameter which governs the (cid:135)exibility of capital utilization is (cid:27) : The logic in a the previous paragraph suggests that for a (cid:133)xed value of (cid:24) , the larger is (cid:27) ; the lower is (cid:13): p a But other things equal, a lower (cid:24) implies a higher (cid:13): These observation suggest that for a p given value of (cid:13); (cid:24) is a decreasing function of (cid:27) : In fact, our point estimate of (cid:27) is large p a a which helps explain why the value of (cid:24) implied by the (cid:133)rm-speci(cid:133)c model is low. p What is it about the data that leads to a large (albeit imprecise) point estimate for (cid:27) ? a We recompute the impulse responses implied by our model, holding all but one of the model parametersattheirestimatedvalues. Theexceptionis(cid:27) whichwesetto0:01. Thenewvalue a of (cid:27) has two major e⁄ects on the model impulse response functions. First, the responses of a capital utilization to both technology shocks are stronger. The responses are so strong that, 34This is a straightforward implication of the homogeneous capital model discussed above, according to which (cid:13) =(1 (cid:24) )(1 (cid:12)(cid:24) )=(cid:24) : (cid:0) p (cid:0) p p 35Marginal costs could be increasing because of (cid:133)rm-speci(cid:133)city of other factors of production or costs of adjusting production inputs. 26

at several horizons, they lie substantially outside the corresponding empirical con(cid:133)dence intervals. This e⁄ect is particularly strong for a capital-embodied technology shock. Also the model has di¢ culty in matching the rise in output, measured net of capacity utilization costs, after a capital-embodied technology shock. Basically capacity utilization rises by such a large amount that it leads to a drag in output net of capacity utilization costs. These two e⁄ects explain why our estimation criterion settles on a high value of (cid:27) : a In Table 2 and 3 we report the results of estimating the model subject to the constraint that (cid:27) is a small number, 0:01:(see the row labelled (cid:145)Low Cost of Varying Capacity Utilizaa tion). Note that the point estimate rises from0:014 to0:065, a value that is inconsistent with the estimated value of (cid:13) discussed in the context of Figure 4. Table 4 shows that consistent with our intuition, the homogeneous and (cid:133)rm-speci(cid:133)c capital model now yield very similar implicationsforthefrequencywithwhich(cid:133)rmsre-optimizeprices, namelyaboutonceayear. To verify our intuition about why our benchmark estimate of (cid:27) is high, we re-estimate a themodelincludingonlytheresponsestoamonetarypolicyshockinthecriterion. Wereport our results in Tables 2, 3 and 4. Our point estimate of (cid:27) falls from 11:42 to 3:93. The lower a value of (cid:27) allows the model to better capture the estimated rise in capital utilization that a occurs after a monetary policy shock, without paying a penalty for a counterfactually large rise in capacity utilization and a fall in output after a capital-embodied technology shock. This result reconciles our (cid:133)ndings with those reported in CEE (2005) who report a low estimated value of (cid:27) based on an estimation criterion that includes only the responses to a a monetary policy shock. Topursueourintuitionaboutthebenchmarkestimateof(cid:27) wealsore-estimatethemodel a including only a capital-embodied technology shock in the estimation criterion. Tables 2, 3 and 4 show that our results are similar to the benchmark results except that our estimate of (cid:27) is higher. The higher value of (cid:27) dampens the response of capital utilization to a a a capital-embodied technology shock, bringing the model response closer to the VAR-based response. Figure 5 suggests that a high elasticity of demand also works to reduce a (cid:133)rm(cid:146)s incentive to raise price after an exogenous increase in marginal cost, i.e. a low value of (cid:21) reduces (cid:13): f While our estimation criterion is very insensitive to (cid:21) ; it weakly prefers a very low value f for this variable. To examine the role played by (cid:21) ; we re-estimate the model imposing f (cid:21) = 1:05 and 1:20: The (cid:133)rst of these values of (cid:21) is close to Bowman(cid:146)s (2003) estimate f f of the markup for the economy as a whole. The second value of (cid:21) is equal to the point f estimate in CEE (2005). Table 2 shows that imposing di⁄erent values of (cid:21) has very little f impact on the estimated values of the key structural parameters of the model. Table 4 shows that the main qualitative e⁄ect of a higher value of (cid:21) is to reduce the frequency f with which (cid:133)rms re-optimize prices in the (cid:133)rm-speci(cid:133)c capital model. For (cid:21) equal to 1.05 f 27

and 1.20 respectively, the frequency with which (cid:133)rms re-optimize prices rises to once every 3:15 and 4:90 quarters respectively. We conclude that to resolve the micro - macro pricing puzzle in our framework we are compelled to take the view that (cid:21) is close to one. This last f result may re(cid:135)ect our assumption that intermediate good (cid:133)rms face a constant elasticity of demand. Other speci(cid:133)cations of demand, like the one proposed in Kimball (1995), break the link between the steady state markup and the elasticity of demand away from steady state. Incorporating changes like these may make it possible to rationalize a low (cid:13) with a low value of (cid:24) and a higher value of (cid:21) : p f 7. Examining the Microeconomic Implications of the Homogeneous Versus Firm-Speci(cid:133)c Capital Models The homogeneous and (cid:133)rm-speci(cid:133)c capital models imply that (cid:133)rms re-optimize prices on average once every1:8 and9:4 quarters, respectively. These results point infavorof the (cid:133)rmspeci(cid:133)c capital model. We now document an even more powerful reason for preferring that model: the estimated homogeneous capital model predicts, implausibly, that a small subset of (cid:133)rms produce the bulk of total output after a monetary policy shock. The (cid:133)rm-speci(cid:133)c capital model does not su⁄er from this shortcoming. However, the benchmark (cid:133)rm-speci(cid:133)c capital model does have an important shortcoming. It implies that (cid:133)rm-level output is too volatile, relative to (cid:133)rm-level prices. We display a variant of the benchmark model which does not su⁄er from this shortcoming. To document these (cid:133)ndings we begin by considering the impact of a monetary policy shock on the cross-(cid:133)rm distribution of prices and output. We suppose that the economy is in a steady state up until period 0. In the steady state, each (cid:133)rm(cid:146)s price and quantity is the same. An expansionary monetary policy shock occurs in period 1: Given the timing convention in our model, prices and output levels are the same across (cid:133)rms at the end of period 1. In period 2, a fraction, 1 (cid:24) ; of (cid:133)rms re-optimize their price. The other (cid:133)rms p (cid:0) update their price according to (2.9). In period 3 there are four types of (cid:133)rms: (i) a fraction, 1 (cid:24) 2 ; of (cid:133)rms that re-optimize in periods 2 and 3; (ii) a fraction, (cid:24)2; of (cid:133)rms that do (cid:0) p p not re-optimize in periods 2 or 3; (iii) a fraction, 1 (cid:24) (cid:24) , which re-optimize in period (cid:0) (cid:1) p p (cid:0) 2 and not in period 3; and (iv) a fraction, (cid:24) 1 (cid:24) , of (cid:133)rms that do not re-optimize in p (cid:0)p (cid:1) (cid:0) period 2, but do re-optimize in period 3. In period s there are 2s 1 di⁄erent types of (cid:133)rms. (cid:0) (cid:1) (cid:0) We calculate the distribution of output and relative prices across (cid:133)rms in period s = 4: Figures 6a and 6b summarize our (cid:133)ndings for the homogeneous capital version of the model. The integers 1, 2, 3, and 4 on the horizontal axes of these (cid:133)gures refer to di⁄erent groups of (cid:133)rms. The integer, 1; pertains to (cid:133)rms that did not re-optimize their price in periods 2, 3 and 4. The integers j = 2;3 and 4, pertain to (cid:133)rms which last re-optimized in period 28

j. Figure 6a shows the share of output (black bars) and the fraction of (cid:133)rms (grey bars) corresponding to the di⁄erent groups of (cid:133)rms. For the (cid:133)rms in each group, Figure 6b shows the log deviation of their price from the aggregate price. We note several features of Figure 6a and 6b. First, a small fraction of the (cid:133)rms are producing a disproportionate share of the output. Indeed, roughly 70% of the (cid:133)rms who did not re-optimize their prices in periods 2;3 and 4 produce 100 percent of output. The remaining (cid:133)rms e⁄ectively shut down. A key factor driving this result is the high elasticity of demand for a (cid:133)rm(cid:146)s output ((cid:21) is small) in the estimated benchmark model. As we show f below, we can overturn this implication by imposing a higher value of (cid:21) : f We now turn to the (cid:133)rm-speci(cid:133)c capital model. Figures 6c and 6d are the analogs to Figures 6a and 6b. Figure 6c shows that the dramatic degree of inequality of production associated with the homogeneous capital model no longer obtains. Still, there is some inequality in the level of production at individual (cid:133)rms. The average level of production by (cid:133)rms in a particular category corresponds to the ratio of the black bar (total production in that group) to the grey bar (number of (cid:133)rms in that group). In period 4, these averages are 1.8, 1.3, 1.0, and 0.8 for (cid:133)rms that last optimized in periods 1, 2, 3 and 4, respectively. So, the typical (cid:133)rm that has not been able to re-optimize its price since the monetary policy shock produces over twice as much as a (cid:133)rm that has not been able to re-optimize since the shock occurred. In later periods, the extent of the inequality in production is substantially mitigated.36 Figure 7 is the analog to Figure 6 for the estimated model when we set (cid:21) to to 1:05. f Comparing (cid:133)gures 6a and 7a we that the implications of the homogeneous capital model becomelessextremeforthehighervalueof(cid:21) : Itisstilltruethatasmallfractionof(cid:133)rmsare f producing a disproportionate share of the output. But now the fraction of output produced by the roughly 70 percent of the (cid:133)rms who did not re-optimize their prices in periods 2;3 and 4 is only a bit larger than 70 percent.37 However as Table 4 indicates, with (cid:21) equal to f 1:05; in the homogeneous capital model (cid:133)rms re-optimize prices once every 9 quarters. In the (cid:133)rm speci(cid:133)c capital model (cid:133)rms re-optimize prices roughly once every 3 quarters. On this basis we would still prefer the (cid:133)rm speci(cid:133)c capital model. 8. Conclusion We construct a dynamic general equilibrium model of cyclical (cid:135)uctuations that accounts for in(cid:135)ation inertia even though (cid:133)rms re-optimize prices on average once every 1:8 quarters. 36One measure of the degree of inequality in production is provided by the Gini coe¢ cient. In periods 4, 8 and 16, these are 0.12, 0.15 and 0.26 for the (cid:133)rm-speci(cid:133)c capital version of the model. 37Thebehaviorof(cid:133)rmswhencapitalis(cid:133)rmspeci(cid:133)cisnotsubstantiallya⁄ectedbythehighervalueof(cid:21) : f 29

To obtain this result we assume that capital is (cid:133)rm-speci(cid:133)c. If we assume that capital is homogenous we can account for in(cid:135)ation inertia. However, this version of the model has micro implications that are implausible: (cid:133)rms re-optimize their prices on average once every 9:4 quarters and a monetary policy shock induces extreme dispersion in prices and output across (cid:133)rms. These considerations lead us to strongly prefer the (cid:133)rm-speci(cid:133)c capital model. We conclude by noting that, in this paper, we have take as given that (cid:133)rms re-optimize prices roughly once every two quarters. If we take the position that (cid:133)rms re-optimize prices on average roughly once a year, then we can reconcile the micro - macro pricing puzzle with a value of (cid:21) around roughly 1:10: As Table 4 indicates, for these values of (cid:21) ; (cid:133)rm speci(cid:133)c f f capital still plays a critical role in generating plausible implications for the frequency with which (cid:133)rms re-optimize prices. At the same time higher values of (cid:21) are associated with less f elastic demand curves than lower values of (cid:21) : This fact has important implications for the f micro implications of the model, like the volatility of (cid:133)rm level output. We will pursue these implications in future research. 30

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Table 1: Decomposition of Variance, Business Cycle Frequencies Variable Monetary Policy Shocks Neutral Technology Shocks Embodied Technology Shocks Output 9 11 41 [4] [11] [15] MZM Growth 20 1 30 [5] [7] [12] Ination 5 12 25 [3] [10] [13] Fed Funds 14 2 38 [5] [9] [16] Capacity Util. 8 2 43 [4] [9] [16] Avg. Hours 9 6 45 [5] [10] [17] Real Wage 4 3 29 [3] [10] [14] Consumption 5 8 19 [3] [12] [13] Investment 9 9 33 [4] [10] [14] Velocity 17 2 26 [5] [8] [13] Price of Inv. 11 2 41 [4] [8] [14] Notes: Numbers are the fraction of variance in the business cycle frequencies accounted for by the indicated shock; number in square brackets is an estimated of the standard error (see text). All variables, exceptMZM growth, ination and Fed Funds, are measured in log-levels.

TABLE 2: ESTIMATED PARAMETER VALUES ! 1 Model " # $ % & ’ ( ! " # 00 Benchmark 1)01 0)78 0)014 11)42 0)76 1)50 0)61 ($%#%) (0%08) (0%007) (6%86) (0%08) (0%83) (0%23) Monetary Shocks Only 1)01 0)71 0)018 3)93 0)81 5)32 0)94 ($%#%) (0%16) (0%021) (3%45) (0%05) (3%30) (0%23) Neutral Technology Shocks Only 1)01 0)76 1)367*+04 13)39 0)95 0)01 2)42*+05 ($%#%) (0%14) (3%7&+05) (50%22) (0%05) (0%20) (9%09&+06) Embodied Technology Shocks Only 1)35 0)96 0)017 16)70 0)00 0)40 3)17 (0%31) (0%09) (0%008) (49%54) ($%#%) (0%15) (2%41) Low Cost of Varying Capital Util. 1)84 0)56 0)065 0)01 0)79 0)49 0)26 (0%22) (0%05) (0%073) (0%19) (0%65) (0%23) Intermediate Markup 1)05 0)78 0)014 9)46 0)75 1)44 0)61 (0%08) (0%007) (5%48) (0%08) (0%80) (0%24) High Markup 1)20 0)77 0)018 5)67 0)74 1)26 0)59 (0%09) (0%008) (3%36) (0%08) (0%72) (0%24)

TABLE 3: ESTIMATED PARAMETER VALUES ! 2 + % + % + , ,+ + % + , ,+ ’ ’ ( ! ( ! )* * * ( ! (! )! ! ! Benchmark 0)09 0)13 0)42 0)17 0)58 0)14 0)46 0)55 0)21 0)61 0)33 0)39 !(0%07) (0%05) (0%27) (0%08) (0%26) (0%21) (0%29) (0%12) (0%04) (0%07) (0%22) (0%15) Monetary Shocks Only 0)13 0)20 n.a. n.a. n.a. n.a. n.a. n.a. n.a. n.a. n.a. n.a. ! (0%08) (0%05) Neutral Technology Shocks Only n.a. n.a. 0)65 0)11 0)65 0)36 1)37 n.a. n.a. n.a. n.a. n.a. (0%21) (0%06) (0%83) ! (1%04) (2%42) Embodied Technology Shocks Only n.a. n.a. n.a. n.a. n.a. n.a. n.a. 0)67 0)17 0)84 0)43 0)11 (0%12) (0%04) (0%11) (0%83) ! (0%75) Low Cost of Varying Capital Utilization 0)01 0)06 0)71 0)07 0)51 0)92 0)65 0)21 0)20 0)61 0)42 0)26 !(0%10) (0%05) (0%11) (0%04) (0%12) (0%68) (0%55) (0%26) (0%06) (0%11) (0%29) (0%16) Intermediate Markup " = 1)05 ! 0)09 0)13 0)40 0)17 0)58 0)15 0)45 0)54 0)21 0)61 0)34 0)38 !(0%08) (0%05) (0%30) (0%08) (0%25) (0%21) (0%27) (0%12) (0%04) (0%07) (0%22) (0%15) High Markup " = 1)20 ! 0)09 0)13 0)30 0)19 0)56 0)15 0)41 0)52 0)21 0)62 0)38 0)35 !(0%08) (0%05) (0%45) (0%11) (0%23) (0%21) (0%24) (0%12) (0%04) (0%07) (0%22) (0%14)

TABLE 4: IMPLIED AVERAGE TIME (Quarters) BETWEEN REOPTIMIZATION 1 1 , ! " Model Firm-Specic Capital Model Homogeneous Capital Model Benchmark 1.81 9.36 Monetary Shocks Only 1.76 8.10 Neutral Technology Shocks Only 1)00# 1)00# " " Embodied Technology Shocks Only 5.78 8.40 Low Cost of Varying Capital Util. 4.50 4.52 Intermediate Markup " = 1)05 3.15 9.12 ! High Markup " = 1)20 4.90 8.26 ! Note: # 1)00 implies that prices are reoptimized each period in our quarterly model. "

Figure 1: Response to a monetary policy shock (o − Model, − VAR, grey area − 95 % Confidence Interval) Output MZM Growth (Q) Inflation 0.2 2 0.1 0.1 0 0 0 −0.1 −2 −0.1 0 5 10 15 0 5 10 15 0 5 10 15 Federal Funds Rate Capacity Utilization Average Hours 0.3 0.4 0.2 0.2 0.2 0 0.1 0 0 −0.2 −0.2 −0.1 0 5 10 15 0 5 10 15 0 5 10 15 Real Wage Consumption Investment 0.1 0.1 0.5 0 0 0 −0.1 −0.5 −0.1 0 5 10 15 0 5 10 15 0 5 10 15 Velocity Investment Good Price Total money growth (M) 1 1 0.2 0.5 0.1 0 0 0 −1 −0.5 0 5 10 15 0 5 10 15 0 5 10 15 20 Quarters Quarters Quarters

Figure 2: Response to a neutral technology shock (o − Model, − VAR, grey area − 95 % Confidence Interval) Output MZM Growth (Q) Inflation 2 0.6 0 1 0.4 −0.2 0 0.2 −1 −0.4 0 0 5 10 15 0 5 10 15 0 5 10 15 Federal Funds Rate Capacity Utilization Average Hours 0.4 0.2 0.4 0.2 0 0 0.2 −0.2 0 −0.2 −0.4 −0.2 0 5 10 15 0 5 10 15 0 5 10 15 Real Wage Consumption Investment 2 0.6 0.4 0.4 1 0.2 0.2 0 0 0 0 5 10 15 0 5 10 15 0 5 10 15 Velocity Investment Good Price Total money growth (M) 1 0.2 0.4 0 0 0.2 −1 −0.2 0 0 5 10 15 0 5 10 15 0 5 10 15 20 Quarters Quarters Quarters

Figure 3: Response to an embodied technology shock (o − Model, − VAR, grey area − 95 % Confidence Interval) Output MZM Growth (Q) Inflation 0.6 0.2 1 0.4 0 0.1 0.2 −1 0 0 −2 −0.1 0 5 10 15 0 5 10 15 0 5 10 15 Federal Funds Rate Capacity Utilization Average Hours 0.6 1 0.6 0.4 0.5 0.4 0.2 0.2 0 0 0 −0.2 −0.5 −0.2 −0.4 −0.4 0 5 10 15 0 5 10 15 0 5 10 15 Real Wage Consumption Investment 0.4 0.4 2 0.2 0 0.2 1 −0.2 0 0 −0.4 0 5 10 15 0 5 10 15 0 5 10 15 Velocity Investment Good Price Total money growth (M) 1 2 −0.2 −0.4 1 −0.6 0.5 0 −0.8 −1 −1 0 0 5 10 15 0 5 10 15 0 5 10 15 20 Quarters Quarters Quarters

0.04 0.03 0.02 0.01 0 !0.01 !0.02 !0.03 !0.04 !0.05 !0.04 !0.03 !0.02 !0.01 0 0.01 0.02 0.03 0.04 0.05 s!hat t π∆β ! π∆ 1+t t Figure 4a: Quasi First Difference of Change in Inflation versus Log, Marginal Cost 0.04 0.03 0.02 0.01 0 !0.01 !0.02 !0.03 !0.04 !0.05 !0.04 !0.03 !0.02 !0.01 0 0.01 0.02 0.03 0.04 0.05 P[s!hat | Ω] t t ]Ω | π∆β ! π∆[P t 1+t t Figure 4b: Projection of Quasi First Difference of the Change in Inflation versus Projection of Log, Marginal Cost

Figure 5: Firm-Specific Capital and the Response of Price to Marginal Cost Shocks MC 1,f MC P 0,f 1 P 2 MC 1,h B P 0 B! MC 0,h A Q Q 0

100 80 60 40 20 0 1 2 3 4 Period of most recent optimization tnecreP Figure 6a: Share of output and firms in Period 4 Figure 6b: Average relative price in Period 4 Homogeneous Capital Model Homogeneous Capital Model 0.03 0.02 0.01 0 !0.01 !0.02 1 2 3 4 Period of most recent optimization 100 80 60 40 20 0 1 2 3 4 Period of most recent optimization tnecreP Figure 6: Features of the Distribution of Output and Prices Across Firms Figure 6c: Share of output and firms in Period 4 Figure 6d: Average relative price in Period 4 Firm!specific Capital Model Firm!specific Capital Model 0.03 0.02 0.01 0 !0.01 !0.02 1 2 3 4 Period of most recent optimization

100 80 60 40 20 0 1 2 3 4 Period of most recent optimization tnecreP Figure 7a: Share of output and firms in Period 4 Figure 7b: Average relative price in Period 4 Homogeneous Capital Model Homogeneous Capital Model 0.03 0.02 0.01 0 −0.01 −0.02 1 2 3 4 Period of most recent optimization 100 80 60 40 20 0 1 2 3 4 Period of most recent optimization tnecreP Figure 7: Features of the Distribution of Output and Prices Across Firms: Lower Demand Elasticity Figure 7c: Share of output and firms in Period 4 Figure 7d: Average relative price in Period 4 Firm−specific Capital Model Firm−specific Capital Model 0.03 0.02 0.01 0 −0.01 −0.02 1 2 3 4 Period of most recent optimization