Interpreting Shocks to the Relative Price of Investment with a Two-Sector Model

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Interpreting Shocks to the Relative Price of Investment with a Two-Sector Model Luca Guerrieri, Dale Henderson, and Jinill Kim 2016-007 Please cite this paper as: Guerrieri, Luca, Dale Henderson, and Jinill Kim (2016). “Interpreting Shocks to the Relative Price of Investment with a Two-Sector Model,” Finance and Economics Discussion Series 2016-007. Washington: Board of Governors of the Federal Reserve System, http://dx.doi.org/10.17016/FEDS.2016.007. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Interpreting Shocks to the Relative Price of Investment with a Two-Sector Model∗ Luca Guerrieri† Dale Henderson‡ Jinill Kim§ Federal Reserve Board Center for Applied Korea University Macroeconomic Analysis February 2016 Abstract Consumption and investment comove over the business cycle in response to shocks that permanently move the price of investment. The interpretation of these shocks has relied on standard one-sector models or on models with two or more sectors that can be aggregated. However, the same interpretation continues to go through in models that cannot be aggregated into a standard one-sector model. Furthermore, such a two-sector model with distinct factor input shares across production sectors and commingling of sectoral outputs in the assembly of final consumption and investment goods, in line with the U.S. Input-Output Tables, has implications for aggregate variables. It yields a closer match to the empirical evidence of positive comovement for consumption and investment. JEL Classification: E13, E32. Key Words: DSGE models, multi-sector models, vector auto-regressions,long-run restrictions. ∗The views expressed in this paper are solely the responsibility of the authors and should not be interpreted as reflecting the views of the Board of Governors of the Federal Reserve System or of any other person associated with the Federal ReserveSystem. Jinill Kim acknowledges the financial support from theNational Research Foundation of Korea (NRF-2014-S1A3A2044238). †Luca Guerrieri, Office of Financial Stability Policy and Research, Federal Reserve Board. Email: luca.guerrieri@frb.gov. ‡Dale Henderson, Centre for Applied Macroeconomic Analysis. Email: dale.henderson@rcn.com. §Jinill Kim (corresponding author), Department of Economics, Korea University. Email: jinillkim@korea.ac.kr.

1. Introduction One of the striking features of post-WWII U.S. data is that the relative price of investment has a downward trend and displays notable cyclical variation. Exploring these features, Fisher (2006), Smets and Wouters (2007), Justiniano and Primiceri (2008), and Papanikolaou (2011) argued that shocks that affect the relative price of investment can explain a large part of business cycle fluctuations. In particular, building on the long-run identification scheme of Gali (1999), Fisher (2006) used a VAR to show that shocks to the relative price of investment can explain more than 70% of the fluctuations in hours worked over the business cycle. To interpret the permanent shock to the relative price of investment identified from the VAR, Fisher (2006) focused on a one-sector model with investment-specific technology (IST) shocks that increase the efficiency of investment in a capital accumulation equation. This aggregate approach is based on the results of Greenwood, Hercowitz, and Krusell (2000),who showedthat, under certain conditions, a two-sectormodel with a multi-factor productivity (MFP) shock in each sector can be recast as an aggregate model with IST shocks as well as neutral MFP shocks.1 These conditions include equal factor shares across production sectors, assembly of each final good using the output of a single production sector, and perfect mobility ofcapital acrossproduction sectors. Guerrieri,Henderson, andKim (2014)showed that the conditions for aggregation are inconsistent with key features of U.S. Input-Output Tables and other data.2 Nonetheless, much of the literature has proceeded with an aggregate approach. This paper makes three contributions: 1) We show analytically and numerically that a twosector model that cannot be aggregated to a one-sector model is still compatible with the long-run identification scheme proposed by Fisher; 2) Extending the VAR estimated by Fisher to include household consumption and investment, we find a positive correlation between consumption and investment—conditional on shocks that move the price of investment permanently; 3) Estimates from our two-sector and aggregate models indicate that the two-sector model is more likely to be consistent with the positive correlationuncovered from the VAR. Our results indicate that the sectoral sources of technology shocks have implications not just at the sectoral level, but also at the level of commonly scrutinized macroeconomic aggregate series, 1Guerrieri, Henderson,and Kim (2014) set out conditionsunderwhich this“aggregate equivalence”result holds and– since the conditions are quite restrictive–referred to shocks that influences a capital accumulation equation in a general two-sector model as marginal efficiency of investment (MEI) shocks. 2According to the U.S. Input-OutputTables, different production sectors display different intensities of factor inputs and assembly of each final good uses outputs from more than one production sector. Moreover, as shown, for example, by Ramey and Shapiro (1998) it is quitecostly to movecapital across sectors. 2

such as consumption and investment. Moreover, our results indicate that relative prices can be informativewithregardtosectoralproductivitydevelopments. Inthis respect,ourfindingsstandin contrastwith those ofBasu, Fernald,Fisher, andKimball(2013)who arguedthatthe identification scheme of Fisher (2006) does not apply when sectoral production functions display different factor intensities. We proceed by extending two alternative DSGE models from Guerrieri, Henderson, and Kim (2014): atwo-sectormodelandanaggregatemodel. Thesectoralmodelhastwoproductionsectors, amachinery-producingsectoranditscomplementthatisdubbedanon-machinery-producingsector. It also allows for the assembly of consumption and investment goods each of which uses sectoral outputs indifferentproportions. Thesetwofeaturesofthe modelallowus toreflectkeyinformation from the U.S. Input-Output Tables and other sectoral statistics. We estimate the extended models by matchingkeymomentsofU.S. dataextractedfromthe same variablesincludedinthe VAR. The extensions include a broader set of shocks, habit persistence in consumption, and an endogenous labor supply. Becausethesectoralproductionfunctionsdisplaydifferentfactorintensities,ourtwo-sectormodel cannot be aggregated into a one-sector model. Nonetheless, we prove that relative prices are still informativeaboutsectoralproductivitydevelopments. Weproceedintwostages. First,forasimpler version of our two-sector model in which each sectoral output is used in the assembly of one final good, we offer an analyticalproof. Second for a fuller, empirically-relevantversionof the model, we rely on numerical illustrations that the results in the analytical proof continue to apply. When the two extended models are estimated to match the same aggregate features, MFP increasesinthe machinery-producingsectorofthe two-sectormodelhaveeffects thatarequalitatively different from IST shocks in the aggregate model. One important difference is that, conditional on shocks that move the price of investment permanently, the correlation between consumption and investment is positive in the two-sector model with MFP shocks and negative for the aggregate model with IST shocks. The commingling of sectoral outputs in the assembly of both consumption and investment goods implies that an increase in productivity in one production sector lowers the cost of assembly of both final goods, creating an incentive to increase the assembly of both goods. Allowing for differences in factor intensities across production sectors and restricting capital stocks to be predetermined at the sectoral level both reduce the attractiveness of substituting between 3

consumption and investment.3 The imprecisionofestimatesfromlong-runidentificationstrategiesappliedto smallsamplescan make it difficult to discriminate between alternative hypotheses.4 To investigate the small sample propertiesoftheVARestimates,werelyonaMonteCarloexperiment. Were-estimatethesameVAR used on observed U.S. data on random samples of data generated from the two alternative DSGE models. Thecumulativedensityfunctionforthecorrelationbetweenconsumptionandinvestmentfor thetwo-sectormodelisuniformlyclosertothatfortheVARestimatedonobserveddata,confirming that the two-sectormodel is a more plausible candidate data-generatingprocess than the aggregate model. The rest of the paper proceeds as follows. Section 2 describes the VAR identified with longrun restrictions and documents the positive comovement between consumption and investment in response to shocks that move the price of investment permanently. Section 3 proves that sectoral shocksinatwo-sectormodelareconsistentwiththe identificationscheme. Section4describessome extensions of the model framework and Section 5 revisits the identification issues in line with these extensions. Section 6 shows that the two-sector model is more likely to be consistent with the positive comovement uncoveredby the VAR than the aggregatemodel. 2. New Empirical Evidence on the Correlation Between Consumption and Investment Akeyfeaturefordiscriminatingbetweenaone-sectormodelwithISTshocksandatwo-sectormodel with MFP shocks is the comovement of consumption and investment conditional on technology shocks. Fisher’sseminalworkonidentifyingISTshocksdidnotincludemeasuresofconsumptionor investment in the VAR, making it impossible to investigate this comovement. We update Fisher’s results and extend them to gauge this comovement by including measures of consumption and investment in the VAR. The VAR that we estimate includes five variables: 1. the growth rate of the relative price of investment, constructed as the log-differenced implicit 3Using a calibrated DSGE model, similar to the one considered here, Guerrieri, Henderson, and Kim (2014) showed that allowing for commingling in the assembly is sufficient, by itself, to change the consumption-investment correlation from negative to positive. Furthermore, they showed that incorporating each of these model features by itself makes the consumption-investmentcorrelation less negative. 4See,forinstance,FaustandLeeper(1997)andErceg,Guerrieri,andGust(2005)foranexaminationoftheeconometric issues related to long-run restriction schemes. 4

pricedeflatorforequipmentandsoftwarefromNIPATable1.1.9minuslog-differencenon-farm business output prices (net of equipment and software using the Laspeyres formula);5 2. laborproductivitygrowth,measuredaslog-differencedlaborproductivityinthenonfarmbusiness sector from the Bureau of Labor Statistics; 3. hours per capita, constructedas the log of hours workedin the nonfarmbusiness sector minus the log of civilian non-institutional population 16 years and overfrom the CurrentPopulation Survey; 4. thegrowthrateofrealequipmentandsoftwarepercapita,definedasthelog-differencedequipment and software (nominal equipment and software divided by its implicit deflator) minus the log-differenced civilian non-institutional population 16 years and over from the Current Population Survey; 5. the growth rate of real consumption per capita, constructed as the log-differenced real personal consumption expenditures from NIPA Table 1.1.6, minus the log-differenced civilian non-institutional population 16 years and over from the Current Population Survey. Several recent papers have replaced or augmented labor productivity growth in the VAR with the growthof total factor productivity (TFP) measures obtained from growthaccounting exercises. See,forinstance,BeaudryandLucke(2010),Schmitt-GroheandUribe(2011),andSims(2011). All those exercises rely, in one form or another, on aggregation of production function across sectors. We continue to use labor productivity growth since the conditions for aggregationunderlying those TFP measures do not hold in our model. We estimate a VAR of order 4. The start date for the estimation sample is 1982:Q3, avoiding the adjustment from the Volcker disinflation. We end the sample in 2008:Q3 to avoid a possible break associated with the zero lower bound on nominal interest rates. In robustness analysis, we also consider a longer sample, spanning all available data. We follow the long-run identification scheme of Fisher (2006). Building on the idea of Greenwood,Hercowitz,andKrusell(2000)thatrelativepricesareinformativeaboutsectoraltechnological developments, Fisher also focused on relative prices. However, to resolve the problem that, in the shortrun,inthepresenceofrealrigiditiesrelativepricescanbeinfluencedbynon-technologyshocks, he considered long-run movements in relative prices. Following Fisher’s scheme, the identification 5Throughout thebody of this paper, we take“investment” to mean investment in equipmentand software. 5

Table 1: Historical Variance Decomposition Implied by the VAR Shock Growth of Growth of Labor Hours Growth of Growth of Price of Investment Productivity Consumption Investment Price of Investment 0.60 0.10 0.71 0.40 0.45 Neutral MFP 0.10 0.56 0.03 0.04 0.19 Variable definitionscan be found in Section 2. scheme we use imposes that only a shock to the relative price of investment can move that price permanently. Moreover, only shocks to the relative price of investment and to labor productivity can move the level of labor productivity permanently. All other shocks are left unidentified. The thick dashed lines in Figure 1 show the effects of a one-standard-deviationshock estimated by our VAR to reduce the price of investment permanently. The point estimate for the decline in the relative price is close to 3 percent. The areas shaded with vertical dashed lines show 90% confidence intervals following Runkle (1987), and based on 1000 bootstrap replications of the data. While the confidence intervals are strikingly large, they exclude a positive response for the relative priceofinvestment,andnegativeresponsesforoutput,consumption(inallbutthefirstperiod),and investment. From the point estimates for the impulse responses, it can be correctly inferred that there is conditional comovement between consumption and investment. Table 1 offers a decomposition of the variance of the variables included in the VAR on average over the estimation sample. Shocks to the price of investment account for 60% of the variation in the growth rate of the relative price of investment and they also account for more than 70% of the variation in hours worked, in line with the results presented by Fisher (2006) and confirmed with estimates from a DSGE model by Justiniano, Primiceri, and Tambalotti (2010). In addition, the same shocks are important for the variation in the growth of consumption and investment, accounting for 40% and 45% of this variation, respectively. The top panel of Figure 2 shows the cumulative density function (CDF) for the correlation between consumption and investment at business cycle frequencies, conditional on a shock that changes the relative price of investment permanently, as estimated from the VAR on our baseline sample from 1982q3 to 2008q3. The cumulative density function captures the sampling uncertainty for the estimate of the VAR coefficients and is traced from a bootstrap exercise. First, we sample with replacement from the VAR residuals to construct 1000 new synthetic samples of the same length as the original historical sample. Second, we re-estimate the VAR on each synthetic sample. Third, by another bootstrapon the residuals from the VAR estimated onthe synthetic samples, we 6

obtain a population estimate for the correlation between consumption and investment at business cyclefrequencies,conditionalonashockthatchangesthe relativepriceofinvestmentpermanently.6 The median correlationis 0.95. The CDF indicates that negative values for the correlationbetween consumption and investment are an unlikely occurrence. The lowerpanelofFigure 2showsthe sameCDF basedona longersample,spanningthe period from 1948q2 to 2015q1, which includes all the publicly available data at the point of writing. The results from the smaller sample appear robust. The median estimate of the conditional correlation between consumption and investment at business cycle frequencies is still a high 0.8, and the CDF still indicates that negative values are unlikely, with probability lower than 2%. Insum,ourextensionsproduceestimatesofthecorrelationbetweenconsumptionandinvestment thatpointtosignificantcomovementoverthebusinesscycleconditionalonshocksthatpermanently vary the price of investment. This comovement is robust to alternative sample choices. Moreover, we verified that our extensions do not overturn previously emphasized results on the importance of shocks to the relative price of investment in explaining business cycle fluctuations. 3. The Identification of Technology Shocks in Two-Sector Models: Part I To interpret his identification scheme, Fisher (2006) wrote down a one-sector model with neutral MFPshocksandIST shocksthatenterthe capitalaccumulationequation. TheworkofGreenwood, Hercowitz, and Krusell (2000) implies that Fisher’s identification scheme is consistent with a twosector model under some restrictive assumptions, including equal factor shares across sectors and complete specialization in the assembly of consumption and investment. These assumptions are at odds with the U.S. Input-Output Tables. We show that Fisher’s identification scheme is consistent withourextendedtwo-sectormodelinwhichtheseassumptionsarerelaxed. Ourdemonstrationhas two components. First, in this this section, we present a baseline version of our two-sector model with factor shares that differ across sectors, with which we can prove analytically that Fisher’s identification scheme continues to apply. Specifically, the proof shows that relative prices respond permanently only to sector-specific shocks while labor productivity (aggregated at constant prices or in units of consumption) responds permanently both to equiproportionate sectoral shocks and to sector- 6Thepopulationestimateofthecorrelationbetweenconsumptionandinvestmentisobtainedonabootstrappedsample of 1050 observations, ten times as many as in the original sample. We used a bandpass filter to isolate the oscillations with frequencies between 6 and 32 quarters, typically used to definethebusiness cycle. 7

specific shocks. Specifically, we derive steady-state relationships for a versionof our model with the followingfeatures: themodelincludesonlyonestockofcapitalusedinbothsectors;bothcapitaland labor are perfectly mobile across sectors; there is complete sectoral specialization in the assembly of consumption and investment. Second, in Section 4 we present numerical simulations of a more general version of our model in which important implications of the model for the identification of technology shocks carry through. Accordingly based on the analytical and numerical results, our extended two-sector model is consistent with the identification scheme used in Section 2, despite different factor input shares across sectors and despite the commingling of sectoral outputs in the assembly of final goods. 3.1. The Baseline Model In period t, the representative household supplies a fixed amount of labor L, and maximizes the intertemporal utility function ∞ max βs−tlogC , (1) s Cs,Is,KNs,KMs,Bs s=t X by choosing paths for consumption (C), investment (I), capital for M goods (K ), capital for M N goods (K ), and for bonds (B) that pay the rate of return ρ after one period. The utility N maximization problem is subject to a budget constraint given by W s L+R Ms K Ms +R Ns K Ns +ρ s−1 B s−1 =P Cs C s +P Is I s +B s , (2) where W is the wage rate, R and R are the rental rates for K and K , respectively, P is M N M N C the price of N goods but also of consumption (P =P ), and P is the price of M goods but also C N I of investment (P = P ). Furthermore, the utility maximization problem is also subject to the I M following law of motion for the accumulation of capital K +K =(1−δ)(K +K )+I , (3) M,s+1 N,s+1 Ms Ns s with capital predetermined at the aggregate level and with δ denoting the depreciation rate for capital. There is complete specialization in the assembly of consumption and investment goods. Investment exhausts the output of the M sector (I = Y ), and consumption exhausts the output M 8

of the N sector (C =Y ). N Ineachsector,perfectly competitivefirmsminimize productioncoststomeetdemandsubjectto the technology constraint as reflected in the following Lagrangianproblems: min R K +W L +P (Y −KαM (A L )1−αM), (4) Ms Ms s Ms Ms Ms Ms Ms Ms KMs,LMs,PMs min R K +W L +P (Y −KαN (A L )1−αN), (5) Ns Ns s Ns Ns Ns Ns Ns Ns KNs,LNs,PNs where α and α determine capital intensities of the production of M and N goods respectively. M N In addition to satisfying the first-order conditions for the optimization problems of households and firms given above, an equilibrium of the model also requires that all factor and product markets clear. For the purposes of analyzing the implications of the model in the long run, we focus on the steady-state conditions for an equilibrium, which are summarized in Table 2. 3.2. Proving that the Baseline Model is Consistent with Fisher’s Long Run Identification Scheme In this section we prove analytically that the baseline two-sector model described in Section 3.1 satisfies the restrictions imposed by the identification scheme in Fisher (2006) despite its multisector structure with different factor intensities across sectors. Theorem 1. Inthelongrun,equiproportionateshockstotechnologyinthetwoproductionsectors M and N affect aggregatelabor productivity but do not affect relative prices. Furthermore, shocks to technology in one production sector affect both aggregate labor productivity and relative prices. The proof to this theorem is given in two parts below and relies on the steady-state conditions in Table 2. A corollary of this theorem is that the two-sector model of Section 3.1 can be used to interpret the permanent shocks to the relative price of investment and to labor productivity identified in Section 2. 9

Table 2: Steady State Restrictions I) RN − PM +β PM (1−δ)= 0 II) R = R PNC PNC PNC Nt Mt III) R = P α YM IV) W = P (1−α )YM M M MKM M M LM V) R = P α YN VI) W = P (1−α )YN N N NKN N N LN VII) Y = K αM(A L ) 1−αM VIII) Y = K αN(A L ) 1−αN M M M M N N N N IX) Y = I X) Y = C M N 1 XI) L +L = L XII) K +K = Y M N M N δ M 3.2.1. The Long-Run Response of Relative Prices Some quick preliminary manipulations are in order. Notice that the rental rates for the two types of capital will be equalized in steady state, as shown in II) in Table 2, so I) implies R =R =P (1−β(1−δ)). (6) M N M Next, from III) and VII), and from V) and VIII) in Table 2, one can relate labor productivity at the sectoral level to the ratio of the sectoral price and the sectoral rate of return for capital: αM αN Y M P M 1−αM Y N P N 1−αN =A α , (7) =A α . (8) M M N N L R L R M (cid:18) M(cid:19) N (cid:18) N(cid:19) The final preliminary manipulation involves using IV) and VI) in Table 2 to relate the relative price of goods in the two-sectors to the sectoral labor productivities: P (1−α ) Y L M N N M = . (9) P (1−α )L Y N M N M Substituting equations 6, 7, and 8 into equation9, one can solvefor PM in terms of parametersand PN the ratio of sector-specific technologies AN: AM αN 1−αN P M A N 1−αN (1−α N ) α N(1−β( 1 1−δ)) 1−αN =ψ , where ψ = . (10) P N 1 (cid:18) A M(cid:19) 1 (1−α M ) (cid:16) α 1 (cid:17) 1− α α M M   M(1−β(1−δ))   (cid:16) (cid:17)  Thus, changesin technology in a single production sector will affect relative prices, but equiproportionate changes in technology in the two production sectors, dubbed neutral MFP shocks for the VAR of Section 2, will not affect relative prices. Looking beyond the model at hand with complete specialization, variation in relative prices at the sectoral level is a precondition for variation in rel- 10

ative prices at the level of final goods even in models with incomplete specialization. Accordingly, one can grasp how the result derived here also extends to richer models with incomplete sectoral specializationintheassemblyofconsumptionandinvestmentgoodsandisreflectedinthenumerical simulations offered below. 3.2.2. The Long-Run Response of Labor Productivity Define aggregate labor productivity (at constant prices) as: Y +Y Y L Y L Mt Nt Mt Mt Nt Nt = + . (11) L L L L L Mt Nt First work on relating LMt and LNt to the conditions for an equilibrium in Table 2. Using V, VIII, L L 6 and III, VII, 6 one can obtain, respectively: K α K α P M M N N N = , (12) = . (13) Y (1−β(1−δ)) Y (1−β(1−δ))P M N M KN can be related to technology levels through (10). From XII, one has that KN YN + KM = 1, YN YN YM YM δ which can be used with to (12) and (13) to solve for YN: YM Y A 1−αN (1−β(1−δ)) α N N M =ψ , where ψ =ψ − . (14) Y 2 A 2 1 δα α M (cid:18) M(cid:19) (cid:18) N N (cid:19) Combining IV, VI, and XI, one obtains: L (1−α )P Y M M Mt Mt = , (15) L (1−α )P Y +(1−α )P Y N Nt Nt M Mt Mt which can be expressed as a function of parameters and technology levels as in Equation 16 below, and since L +L =L, Equation 17 also follows: N M L (1−α )ψ L (1−α )ψ M = M 1 , (16) N = N 2 . (17) L (1−α )ψ +(1−α )ψ L (1−α )ψ +(1−α )ψ M 1 N 2 M 1 N 2 Next, work on YMt and on YNt. Combining equations 7 and 8 with equation 6 yields: LMt LNt 11

αM αN Y M α M 1−αM Y N α N P N 1−αN =A , (18) =A . (19) M N L (1−β(1−δ)) L (1−β(1−δ))P M (cid:18) (cid:19) N (cid:18) M(cid:19) 1−αN Summing up, remembering that PM =ψ AN , one can see that at constant prices: PN 1 AM (cid:16) (cid:17) Y +Y Y L Y L M N M M N N = + = L L L L L M N αM A α M 1−αM (1−α M )ψ 1 (20) M (1−β(1−δ)) (1−α )ψ +(1−α )ψ (cid:18) (cid:19) M 1 N 2 αN +AαNA(1−αN) α N 1−αN (1−α N )ψ 2 . M N ψ (1−β(1−δ)) (1−α )ψ +(1−α )ψ (cid:18) 1 (cid:19) M 1 N 2 According to Equation 20, in the long run, aggregate labor productivity is a function of constant parameters and of the levels of multi-factor productivity in sectors M and N. Accordingly, labor productivity will vary permanently both in response to sectoral MFP shocks that vary the relative level of A and A , and in response to neutral MFP shocks that vary the levels of A and A M N M N equiproportionately. In sum, based on equations 10 and 20, our baseline model is consistent with the scheme in Fisher (2006).7 4. A Richer Model To arrivemore speedily at ournovelresults regardingthe use of empiricalestimates to discriminate between the aggregate and sectoral models, we give here an overview of the salient features of the richer model and relegate a full description to the appendix. In order to incorporate empirically relevant features, we extend the baseline model along the lines of Guerrieri, Henderson, and Kim (2014). We augment the utility function in Equation 1 to allow for habit persistence in consumption and for endogenous labor supply, using an additively separable function between consumption and leisure. We modify Equation 3 so that the capital stocks are distinct and predetermined across sectors, rather than being predetermined only at the aggregate level, and we introduce investment adjustment costs. We allow for the investment and consumptionaggregatestobeconstant-elasticityfunctionsofmachineryandnon-machineryoutputs. 7Notice that Fisher (2006) defined aggregate labor productivity in terms of consumption units, i.e., YMt LMtPM + LMt L PN YN LN usingournotation, ratherthanat constantprices. Evenunderthatalternative aggregation, labor productivityis LN L affected both by equiproportionate shocks across production sectors and byshocks to a single production sector. 12

Inthe productionfunctions embeddedinEquation4andinEquation5,wedistinguishbetweentwo types of capital: equipment and structures. This greater degree of flexibility permits differences in factor intensities across sectors and the commingling of sectoral outputs consistent with the U.S. Input-Output Tables. Finally, we augment the stochastic structure of the model with nontechnology shocks, namely government spending shocks, consumption preference shocks, and labor supply shocks, which help match key moments of U.S. data. We estimate two variants of this richer model: 1. SectoralModel withMFPshocks Withalltheextensionsjustdescribedthatincreasetheempiricalrelevanceofthe model,the resultingmodel cannotbe aggregatedto a standardone-sector model. We estimate this richer model capturing the variation in sectoral MFP levels with a neutral shock that varies the levels of MFP in equal ways across sectors and with an MFP shock specific to the machinery sector. 2. Aggregate Model with IST shocks. Under special parametric restrictions that impose complete sectoral specialization in the assembly of final goods, equal factor shares across sectors, capital stocks that are predetermined only at the aggregate level, our richer model can still be aggregatedto a one-sectormodel. Moreover,under the same restrictions,sectoralvariation in multi-factor productivity canbe capturedwith a neutralMFP shock in the aggregateproductionfunctionandwithISTshocksthatvarythe efficiencyofinvestmentinproducinginstalled capital right in the aggregate capital accumulation equation. We estimate the aggregate variant of the model with IST shocks that are in line with Fisher’s original interpretation of the shocks that yield a permanent movement in the relative price of investment. For each variant, the estimated parameters include the autoregressive coefficients and the standard deviations for all the shock processes. In addition, we estimate the elasticity of substitution betweensectoraloutputsintheassemblyfunctionsforfinalgoods,includingconsumptionandinvestmentinbothmachineryandstructures(forthesectoralmodelonly),thedegreeofhabitpersistence in consumption, and the investment adjustment costs. We focus on matching the variances, the covariances, and the first autocorrelations of the same five variables used in the VAR: the growth rateofthe relativepriceofinvestment,laborproductivitygrowth,hourspercapita,thegrowthrate of equipment and software per capita, and the growth rate of consumption per capita. To weight the various moments we use the diagonal of the simulated method of moments weighting matrix. 13

5. The Identification of Technology Shocks in Two-Sector Models: Part II Theempiricalextensionoftheaggregatemodeldonotinfluenceitslong-runproperties. Accordingly, ouraggregatemodelremainsinline with the identificationschemedescribedinSection2. While we donotprovideananalyticalproofthattheempiricalextensionsconsideredinthesectoralmodelare consistentwithFisher’sidentificationscheme,Figure3offersanumericalsubstantiationbyshowing the response of the relative price of investment and of labor productivity to all the shocks included in the model. Among the shocks included in the model, the only shock that affects the price of investment permanently is an MFP shock in the machinery sector. Moreover, the only two shocks that affect the level of labor productivity permanently are the MFP shock in the machinery sector and the neutral MFP shock (constructed as MFP shocks in both sectors). 6. Discriminating Across Models Based on the VAR Results Having established that the identification scheme for the VAR estimates is consistent with both variants of our richer model, we proceed by comparing model and VAR estimates. One approach typically used to discriminate across models based on VAR evidence is to check whether the model response to a certain shock is consistent or not with the empirical evidence from the VAR.8 For our purposes, the problem with this approach is that the VAR confidence intervals for standard significance levels are so wide, as noted above in the description of Figure 1, that we would not be able to tell the models apart. AsnotedinErceg,Guerrieri,andGust(2005),evenimprecisetoolssuchasourVARcanstillbe usefulindiscriminatingacrossmodels. Forinstance,takingoneofthemodelsasthedata-generating process, one could check if the VAR implies a bias in the point estimates of the impulse response functions in a certain direction. If that bias is reversed under the alternative model, then even an imprecise tool can offer sharp discriminating evidence. To investigate this possibility, we estimated the same VAR and used the same identification scheme to constructthe impulse response functions in Figure 1 based on data generated from the two alternative DSGE models. For this experiment, weused1000randomlydrawnsamplesofthesamelengthasthebaselinesample. Wefoundthatthe differential implications of the two alternative models are swamped by the uncertainty associated with our empirical tool and still do not allow us to tell the models apart.9 8See, for instance, Gali (1999) and Gali and Rabanal (2004). 9The results for this experiment are reported in the appendix. 14

While the estimated impulse response functions do not offer discriminating evidence, a key difference between the two models is the correlationbetween consumption and investment at business cycle frequencies,conditionalonshocksto the price ofinvestment. The populationestimate for this correlationis negative andequals -0.74for the aggregatemodel with IST shocks andis positive and equal to 0.97 for the two-sector model with MFP shocks. The vertical lines in Figure 4 show these twocorrelations. Recallthatthemedianpopulationestimatefortheconditionalcorrelationbetween consumption and investment from the VAR is 0.95. For convenience, the red shaded area reproduces the CDF of the same correlation produced from the VAR. The CDF from the VAR indicates that the negative correlationfrom the aggregatemodel would be extremely unlikely pointing to the two-sector model as the more plausible candidate to explain the comovement properties extracted from the observed U.S. data. In addition to the CDF from the VAR, Figure 4 also reports CDFs for the correlation between consumption and investment, obtained through the same Monte Carlo experiment described above for the impulse responsefunctions. These CDFs allowus to gaugehow samplinguncertainty affects the estimates for the correlationbetween consumption and investment when eachof the alternative models is taken to be the data-generatingprocess. The solid line shows the CDF for the two-sector model. The dashed line shows the CDF for the aggregate model. As for the case of the impulse response functions, the CDFs indicate that the VAR is an imprecise tool with substantial mass for the density function away from the pseudo-true values for each of the two models. Nonetheless, the CDF for the two-sector model is uniformly closer to the CDF for the VAR estimated on observed U.S.data,indicatingthatthetwo-sectormodelisamoreplausiblecandidatedata-generatingprocess even when sampling uncertainty is considered. 7. Conclusion Consumptionandinvestmentcomoveoverthebusinesscycle. Ourestimatesshowthatconsumption andinvestmentalsocomoveconditionalonshocksthat changethe price ofinvestmentpermanently. Our finding obtains in our baseline sample, from 1982:Q3 to 2008:Q3, broadly coinciding with the GreatModeration,aswellasinourfullsampleencompassingallpubliclyavailabledataandspanning the period from 1948:Q2 through 2015:Q1. We show that this comovement can be used to discriminate between alternative models of the business cycle. Heretofore,the set ofmodels used to interpret permanentmovements in the relative 15

priceofinvestmentincludedone-sectormodelswithISTshocks,ormulti-sectormodelsthatcouldbe aggregatedtoaone-sectormodel. Weshowedthat,infact,thesetofadmissiblemodelsalsoincludes a two-sector model that cannot be aggregated. We found that this two-sector model matches more closely the evidence of a positive correlation between consumption and investment, conditional on shocks that move the price of investment permanently. In this paper we have examined the connection between empirical evidence from movements in the relative price of investment with sectoral and aggregate treatments of multi-factor productivity changes using DSGE models. A fruitful avenue for further research would be to explore the relationship between sectoralMFP shocks inferredfromidentified VARs and sectoralmeasuresof MFP levels obtained from growth accounting exercises in the tradition of Solow (1957)and Griliches and Jorgenson(1966). A relateddirectionfor further researchwouldbe to characterizethe generalclass of DSGE models that is consistent with the restrictions implied by growth accounting exercises. 16

References Basu, S., J. G. Fernald, J. Fisher, and M. Kimball (2013). Sector-Specific Technical Change. Mimeo, Federal Reserve Bank of San Francisco. Beaudry, P. and B. Lucke (2010). Letting Different Views about Business Cycles Compete. In NBER Macroeconomics Annual 2009, Volume 24, NBER Chapters, pp. 413–455. National Bureau of Economic Research, Inc. Erceg, C. J., L. Guerrieri, and C. Gust (2005). Can Long-Run Restrictions Identify Technology Shocks? Journal of the European Economic Association 3(6), 1237–1278. Faust, J. and E. M. Leeper (1997). When Do Long-Run Identifying Restrictions Give Reliable Results? Journal of Business & Economic Statistics 15(3), 345–53. Fisher, J. (2006). The Dynamic Effects of Neutral and Investment-Specific Technology Shocks. Journal of Political Economy 114(3), 413–452. Gali,J.(1999).Technology,Employment,andtheBusinessCycle: DoTechnologyShocksExplain Aggregate Fluctuations? American Economic Review 89, 249–271. Gali, J. and P. Rabanal (2004). Technology Shocks and Aggregate Fluctuations: How Well Does the RBC Model Fit Postwar U.S. Data? NBER Working Papers 10636, National Bureau of Economic Research, Inc. Greenwood,J.,Z.Hercowitz,andP.Krusell(1997).Long-RunImplicationsofInvestment-Specific Technological Change. American Economic Review 87, 342–362. Greenwood,J.,Z.Hercowitz,andP.Krusell(2000).TheRoleofInvestment-SpecificTechnological Change in the Business Cycle. European Economic Review 44, 91–115. Griliches, Z. and D. W. Jorgenson (1966). Sources of Measured Productivity Change: Capital Input. The American Economic Review 56(1/2), 50–61. Guerrieri, L., D. Henderson, and J. Kim (2014). Modeling Investment-Sector Efficiency Shocks: When Does DisaggregationMatter? International Economic Review 55, 891–917. Justiniano, A. and G. Primiceri (2008). The Time Varying Volatility of Macroeconomic Fluctuations. American Economic Review 98(3), 604–641. Justiniano,A.,G.E.Primiceri,andA.Tambalotti(2010).Investmentshocksandbusinesscycles. Journal of Monetary Economics 57(2), 132–145. Katayama,M.andK.H.Kim(2012).CostlyLaborReallocation,Non-SeparablePreferences,and 17

Expectations Driven Business Cycles. Mimeo, Louisiana State University. Papanikolaou, D. (2011). Investment Shocks and Asset Prices. Journal of Political Economy 119(4), 639–684. Ramey, V. A. and M. D. Shapiro (1998, October). Displaced Capital. NBER Working Papers 6775, National Bureau of Economic Research, Inc. Runkle, D. E. (1987). Vector Autoregressions and Reality. Journal of Business and Economic Statistics 5, 437–442. Schmitt-Grohe, S. and M. Uribe (2011). Business Cycles With A Common Trend in Neutral and Investment-Specific Productivity. Review of Economic Dynamics 14(1), 122–135. Sims,E.(2011).PermanentandTransitoryTechnologyShocksandtheBehaviorofHours.Mimeo, University of Notre Dame. Smets,F.andR.Wouters(2007).ShocksandFrictionsinUSBusinessCycles: ABayesianDSGE Approach. American Economic Review 97(3), 586–606. Solow, R. (1957). Technical change and the aggregate production function. Review of Economics and Statistics 39(3), 312–320. 18

Figure 1: VAR Estimates of the Response to a One-Standard Deviation Shock that Lowers the Level of the Relative Price of Investment Permanently 0 −1 −2 −3 −4 −5 20 40 60 80 100 tnecreP Price of investment 2 1.5 1 0.5 0 −0.5 20 40 60 80 100 VAR VAR. 90% Conf. Interval tnecreP Labor productivity Hours per capita 1.5 1 0.5 0 −0.5 20 40 60 80 100 tnecreP Output per capita 2 1.5 1 0.5 0 20 40 60 80 100 tnecreP Consumption per capita 2 1.5 1 0.5 0 −0.5 20 40 60 80 100 Quarters tnecreP Investment per capita 6 5 4 3 2 1 0 20 40 60 80 100 Quarters tnecreP 19

Figure 2: Cumulative Distribution Function for the Estimate of the Long-Run Correlation between Investment and Consumption at Business Cycle Frequencies Baseline Sample from 1982:Q3 to 2008:Q3 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Full Sample from 1948:Q2 to 2015:Q1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 20

Figure 3: Properties of the Sectoral Model: The Responses of the Relative Price of Investment and of Labor Productivity to Various Shocks 0 −0.5 −1 −1.5 −2 −2.5 −3 −3.5 10 20 30 40 50 60 70 80 90 100 Quarters .S.S morF .veD tnecreP 1. Relative Price of Investment Machinery MFP shock Neutral MFP shock Labor Supply Shock Consumption Shock Government Spending Shock 2. Labor Productivity 1 0.8 0.6 0.4 0.2 0 10 20 30 40 50 60 70 80 90 100 Quarters .S.S morF .veD tnecreP 21

Figure 4: Cumulative Distribution Function for the Estimate of the Correlation Between Consumption and Investment at Business Cycle Frequencies, Conditional on Shocks that Lower the Price of Investment Permanently: VAR and DSGE Model Results 1 0.9 0.8 0.7 Aggregate IST model 0.6 0.5 Sectoral MFP model 0.4 0.3 0.2 0.1 0 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 For convenience,the shaded area reports again theCDF for estimates the correlation between consumption and investment conditional on shocks that move theprice of investmentpermanently from a VAR for thebaseline sample 1982:q3-2008:Q3. The vertical lines denote estimates conditional on shocks that movetherelative price of investment permanently in theaggregate model with ISTshocks and in the sectoral model with MFP shocks. The CDF denoted by a dashed line pertains to a MonteCarlo experiment,in which the VAR is estimated on data generated from the aggregate modeldescribed in Section 4.The CDFdenotedbyasolid line pertainstoaMonteCarlo experiment,in which theVAR is estimated on data generated from thesectoral model also described in Section 4. 22

A. Appendix: Additional Results from the VAR Section 2 provides a description of our VAR, identification strategy, and estimated responses to a shock that moves permanently the relative price of investment. For completeness, Figure 5 shows theestimatesoftheresponsefromtoaonestandarddeviationshockthatincreasespermanentlythe level of labor productivity but that does not have a long-run effect on the level of the relative price of investment. Again, for the variables that overlap,our results are close to those in Fisher (2006). 23

Figure 5: VAR Estimates of the Response to a One-Standard Deviation Shock that Increases the Level of Labor Productivity Permanently 0.5 0.4 0.3 0.2 0.1 0 −0.1 −0.2 20 40 60 80 100 tnecreP Price of investment 0.5 VAR VAR. 90% Conf. Interval 0.4 0.3 0.2 0.1 0 20 40 60 80 100 tnecreP Labor productivity Hours per capita 0.6 0.4 0.2 0 −0.2 −0.4 20 40 60 80 100 tnecreP Output per capita 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 20 40 60 80 100 tnecreP Consumption per capita 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 20 40 60 80 100 Quarters tnecreP Investment per capita 2 1.5 1 0.5 0 −0.5 −1 20 40 60 80 100 Quarters tnecreP 24

B. Appendix: Full Description of the Extended Models This appendix describes in detail our extended two-sector model with MFP shocks. Under some parametric restrictions the two-sector model collapses to an aggregate model. Section C reports the estimates of key parameters for both the two-sector model and the aggregate model and the parametric restrictions that allow the two-sector model to nest the aggregate model. B.1. Production Sectors Our two production sectors, the M (for Machinery) and N (for Non-machinery) sectors, comprise perfectlycompetitivefirms. Considertherepresentativefirminsectori(wherei∈{M,N})inperiod s. It hires labor (L ) from households at a wage (W ) that is same for both sectors because labor is s is perfectly mobile between sectors. It also rents two types of capital from households: equipment capital (KE) and structures capital KS at rentals (RE and RS) that are sector-specific when is is is is it is costly to reallocate capital. The(cid:0) firm(cid:1) minimizes the unit cost of producing a given number of physical units of its sector’s output (Y ) subject to a sector-specific Cobb-Douglas production is function: Y =(L )1−αE i −αS i KE αE i KS αS i . (21) is is is is (cid:0) (cid:1) (cid:0) (cid:1) The factor shares for the two types of capital are αE and αS. There is a multi-factor productivity i i (MFP) process A s which determines the efficiency units generated by physical machinery output i (i.e., YA =A Y ). Ms Ms Ms Since it is competitive andthere areconstantreturns to scale,the firmends upselling ata price equal to unit cost. Let P represent the factor cost of a unit of physical output i. The factor cost is of a physicalunit ofmachinery is P andthe costof anefficiency unit ofmachineryis PA = PMs Ms Ms As so that P P Y = Ms A Y =PA YA . (22) Ms Ms A Ms Ms Ms Ms (cid:18) Ms(cid:19) . Similarly, P P Y = Ns A Y =PA YA . (23) Ns Ns A Ns Ms Ns Ms (cid:18) Ns(cid:19) . 25

B.2. Final Goods There are three final goods: a consumptiongood(C ) andtwo investmentgoods,one (JE) usedfor s s gross investment in E (for Equipment) capital stocks and the other (JS) used for gross investment s in S (for Structures)capitalstocks. These goodsareassembledby perfectly competitive finalgoods firmsthatuseasinputstheoutputsofthetwoproductionsectors,andthesefinalgoodsaremeasured in efficiency units. When we find it expedient for the exposition, we us an upper bar to denote final goods measured in physical units. The assembly function for consumption C and exogenousgovernmentspending G are a Cobbs s Douglas function of two inputs, efficiency units of M goods along with N goods: σC σC−1 σC−1 σC−1 C = φC A Ms C Ms σC +φC A Ns C Ns σC , (24) s  M φC N φC  (cid:18) M (cid:19) (cid:18) N (cid:19)   σC σC−1 σC−1 σC−1 G = φC A Ms G Ms σC +φC A Ns G Ns σC , (25) s  M φC N φC  (cid:18) M (cid:19) (cid:18) N (cid:19)   where φC and φC are the weights for M and N goods and σ is the elasticity of substitution M N C between M and N goods in the assembly of C and of G . s s TheassemblyfunctionsforJE andJS areCobb-Douglasfunctionsofthetwoinvestmentinputs, s s efficiency units of M goods along with N goods: σE JE = φE A Ms I M E s σE σE −1 +φE A Ns I N E s σE σE −1 σE−1 , (26) s  M φE N φE  (cid:18) M (cid:19) (cid:18) N (cid:19)   σS JS = φS A Ms I M S s σS σS −1 +φS A Ns I N S s σS σS −1 σS−1 , (27) s  M φS N φS  (cid:18) M (cid:19) (cid:18) N (cid:19)   where φE ,φE,φS and φS are the weights given to M and N goods, and σ and σ are the M N M N S E elasticities of substitution between M and N goods. The assembly firms minimize the unit cost of producing efficiency units of consumption, equipment, and structures. Because they are perfectly competitive, firms end up selling final goods at prices that are equal to these costs and that are indicated by PC, PJE , and PJS . We assume that s s s the assembly functions for both C and JS are intensive in N goods relative to the function for JE. s s s There is an investment specific technology (IST) shock Z which further enhances the efficiency s 26

of JE, the efficiency unit ofequipment assembledusing M and N inputs. The final total amountof s equipment efficiency units is given by Z JE and the all-in unit cost is Ps JE so that s s Zs PJE JE = P s JE Z JE. (28) s s Z s s s ! B.3. Tastes and Constraints In period t, the representative household supplies a fixed amount of labor L and maximizes 10 the following intertemporal utility function 1−γ ∞ Cs−ηCs−1−Us −1 1−η βs−t (cid:16) 1−γ(cid:17) −σ0V s L s , (29) s=t X     where U and V represent aggregate demand shocks and labor supply shocks. The houses s hold also chooses holdings of a single bond (B ) denominated in the N good (the nus meraire good for the model). In addition, for each of the four inherited capital stocks (DE ,DE ,DS , and DS ), the household decides how much to adapt to obtain the four Ms Ns Ms Ns capital stocks rented out for use in production (KE ,KE ,KS , and KS ) as well as the Ms Ns Ms Ns fractions (jE ,jE ,jS , and jS ) of investment of the two types (JE or JS) to be added to Ms Ns Ms Ns s s the four capital stocks. The distinction between capital inherited from the previous period, j j the D stocks, and capital used in production, the K stocks, allows us to nest in the same is is model the case in which capital is predetermined only at the aggregate level and the case in which capital is essentially predetermined also at the sectoral level. Thehouseholdissubjecttoperiodbudgetconstraints. Ineachperiod,factorincomeplusincome from bonds held in the previous period must be at least enough to cover purchases of final goods (consumption goods and the two types of investment goods), as well as bonds: W s L+R M E s K M E s +R M S s K M S s +R N E s K N E s +R N S s K N S s +ρ s−1 B s−1 =PCC +PJE JE +PJS JS +B +T , (30) s s s s s s s s 10Theassumptionsoffixedaggregatelaborsupplyandperfectmobilityoflaboracrosssectorsweremadeforsimplicity, given our already involved structure with many sectors. Relaxing either of these assumptions matters for the issue of comovement. Katayama and Kim (2012) relax both assumptions. 27

whereRE , RS ,RE ,RS aretherentalratesforthecapitalstocksusedinproduction. Theterm Ms Ms Ns Ns ρ s−1 is the gross return on bonds, and T s represent lump-sum tax. The household is subject to technological constraints when allocating capital. It inherits four capital stocks from the previous period. Inherited capital suited for one sector can be adapted for use in the other sectorbefore being rentedout, but onlyby incurring increasingmarginalcosts. For example, inherited equipment capital (DE ) suited for the M sector can be adapted for use in the Ms N sector (KE ). Therefore, the capital of type h actually available for production in sector i in Ns period s depends on how much has been adapted for production in that sector: ωh Kh 2 Kh +Kh = Dh 1− Ms −1 Ms Ns Ms " 2 (cid:18) D M h s (cid:19) # ωh Kh 2 + Dh 1− Ns −1 , h∈{E,S}. (31) Ns " 2 (cid:18) D N h s (cid:19) # We consider two special cases: the case in which capital can be adapted at no cost (ωh = 0), so that capital is predetermined only at the aggregate level, and the case in which the marginal cost of adapting capital becomes prohibitive (ωh →∞), so that capital is predetermined at the sectoral level as well. The household is also subject to technological constraints when accumulating capital. The accumulation equations for structures capital are more straightforward and we consider them first. Let DS represent the amount of S capital available for production in sector i in period s without is incurring any costs of adaptation: DS = 1−δS KS +jS JS − νS 0ijS JS ji S s−1Js S −1 −1 2 , i∈{M,N}, (32) is i is−1 is−1 s−1 2 is−1 s−1 ji S s−2Js S −2 (cid:16) (cid:17) (cid:16) (cid:17) period s−1 that is added to the structures capital suitable for sector i in that period. DS has is three components represented by the three terms on the right hand side of equation (32). The first is the amount of S capital actually used in production in sector i in period s−1 remaining after depreciation. The second is the amount of S investment added to structures capital suitable for sector i in period s−1. The third represents the adjustment costs incurred if the S investment in a given type of capital in period s−1 differs from that in period s−2. It is important to note that while the IST shock Z does not enter the accumulation equations for structures capital by s assumption, the MFP shock A and A do enter through JS. Ms Ns s 28

Theaccumulationequationsforequipmentcapitalarelessstraightforwardbecauseofthedistinctionbetweenphysicalunitsandefficiencyunits. LetDE representtheamountofE capitalavailable is for production in sector i in period s without incurring any costs of adaptation: D i E s = 1−δE i K i E s−1 +Z s−1 j i E s−1 J s E −1 + (cid:16) ν 2 E 0iZ s−1 (cid:17) j i E s−1 J s E −1 (cid:18) Z Z s s − − 1 2 j j i i E E s s − − 1 2 J J s s E E − − 1 2 −1 (cid:19) 2 , i∈{M,N}, (33) where jE is the proportion of total equipment investment that is devoted to accumulation of is−1 structures capital suited for sector i in period s−1. Like DS,DE has three components. The first is is components of DS and DE are completely analogous. The second component of DE is the amount is is is of investment in equipment capital suited for sector i measured in efficiency units. It reflects the increase in the efficiency of the machinery input resulting from the MFP shocks A or A which Ms Ns are embedded in JE and the increase in efficiency resulting from the IST shock Z . The third s s component represents investment adjustment costs. The final household constraint is that for each type of investment good the proportions of the total amount added to the two capital stocks of the same type must sum to one: 1 = jE +jE , 1 = jS +jS . Ms Ns Ms Ns B.4. Market Clearing and Stochastic Structure Market clearing requires that the outputs of the production sectors must be used up in the assembly of final goods: Y = C +IE +IS +G , Y = C +IE +IS +G , Ms Ms Ms Ms Ms Ns Ns Ns Ns Ns that labor demand equal labor supply, L +L = L , (34) Ms Ns s 29

and that the bond be in zero net supply, B = 0, (35) s and that lump sum taxes are levied to finance all government spending, T = G . (36) s s Theconditionsthatfirms’demandsforKE ,KE ,KS ,andKS equalhouseholds’supplies Ms Ns Ms Ns are imposed implicitly by using the same symbol for both. We consider five sources of shocks: 1. The MFP shocks for the M and N sectors are integrated of order 1, A Ms =A Ms−1 +ǫ AM +ǫ A , (37) A Ns =A Ns−1 +ǫ A , (38) with the innovations ǫ M, and ǫ each normally and independently distributed with mean 0 A A and standard deviation equal to σ M, σ , respectively. Notice that the innovation ǫ M is A A A sector-specific, while the innovation ǫ is sector-neutral. A 2. The IST shock is integrated of order 1, Z s =Z s−1 +ǫ Z , (39) with the innovation ǫ normally and independently distributed with mean 0 and standard Z deviation equal to σ . Z 3. The shock to consumption U follows an AR(1) process, s U s =ρ U U s−1 +ǫ U , (40) with the innovation ǫ normally and independently distributed with mean 0 and standard U 30

deviation equal to σ . U 4. The shock to labor supply V follows an AR(1) process, s V s =ρ V V s−1 +ǫ V , (41) with the innovation ǫ normally and independently distributed with mean 0 and standard V deviation equal to σ . V 5. And, finally, government spending G is governed by an AR(1) process, s G s =ρ G G s−1 +ǫ G , (42) with the innovation ǫ normally and independently distributed with mean 0 and standard V deviation equal to σ . V C. Appendix: Parameter Choices We fix the model parameters with a mix of calibration and estimation. The calibration pertains to steady-state ratios and features that allow the general model described in Section to nest both the aggregate model and the model with sectoral MFP shocks. C.1. Calibrated Parameters for the Aggregate Model All calibratedparametersfor the aggregatemodel arereportedin Table 3. To facilitate comparisons with previous work on shocks that move the price of investment permanently in an aggregate model, we adhere to the parameter choices of Greenwood et al. (1997) whenever possible.11 Accordingly, the output share of equipment in both the M and N sectors is 17% and the share of structures is 13%. The parameters governing the assembly functions are set so that there is complete specialization: consumptionand structuresinvestment areassembledusing inputs fromthe N sector only, while equipment investment is assembled using inputs from the M sector only.12 The depreciation rates for equipment and structures capital are 3.1% per quarter and 1.4% per quarter, respectively. The adaptation costs for capital are chosen so that capital is predetermined at the aggregate level and completely flexible in every period at the sectoral level. The discount factor is 11For simplicity, we abstract from trend growth as well as capital and labor taxes, while Greenwood et al. (1997) incorporate them in theirmodel. 12The substitution elasticities between inputsin assembly become irrelevant undercomplete specialization. 31

set at 0.99, consistent with an annualized real interest rate of 4%. The intertemporal substitution elasticity for consumption is set at 1. C.2. Calibrated Parameters for the Model with Sectoral MFP shocks All calibrated parameters for the sectoral model are reported in Table 4. We focus here on the parameters that vary relative to the aggregatemodel. Sector-specific production functions To differentiate the intensities of factor inputs across sectors, we used the following restrictions: (a) while allowing variation across sectors, we kept the aggregate factor input intensities the same as in Greenwood et al. (1997); (b) factor payments are equalized across sectors, making the factors’ shares of sectoral output proportional to the sectoral stocks 13 of capital (since production functions are Cobb-Douglas) ; (c) factor input intensities are equal regardless of where the output of a sector is used. We combined data for the net capital stock of private nonresidential fixed assets from the U.S. Bureau of Economic Analysis, with data from the Input-Output Bridge Table for Private Equipment and Software. The first data set contains data on the size of equipment and non-equipment capital stocks by sector. The second data set allowed us to ascertain the commodity composition of private equipment and software. Finally, we used BEA data to establish a sector’s value added output. We focused on the year 2004, but similar sector-specific production functions would be implied by different vintages of data. Ourcalculations show that the machinery-producingsector is less intensive in structures and labor than the aggregate economy, but more intensive in equipment capital. For the machinery sector, the share of structures is 11 percent, the labor share 46 percent, and the share of equipment capital the remaining 43 percent (thus, αS = 0.11,αN = 0.46, M M αE = 0.43). For the non-machinery sector the share of structures is 13 percent, the share M oflabor72percent,andtheshareofequipmentcapital15percent. Theadaptationscostsfor capital are fixed at number sufficiently high to imply that capital stocks are predetermined at the sectoral level. 13If capital stocks are predetermined at thesectoral level, rentals are equalized only in the long run. 32

Incomplete specialization The baseline calibration assumes complete specialization in the assembly of investment and consumption goods. Equipment investment is assembled using output from the M sector only. In contrast, structures investment and consumption goods are assembled using outputfromtheN sectoronly. Thiscompletespecializationdoesnotreflectthecomposition of finalgoodsrevealed intheInput-OutputBridgeTables thatlinkfinalusesintheNIPAto sectors (industries)intheU.S.Input-OutputTables. Forexample, accordingtothedatafor 2004, wholesale and retail services (part of our non-machinery sector) are important inputs not only for consumption but also for equipment investment, accounting for 15 percent of 14 the total output of private equipment and software. Furthermore, electric and electronic 15 products are used in the assembly of consumption, accounting for 4 percent of the total. The model captures the commingling implied by the bridge tables through assembly functions that specify how inputs from the M and N sectors are combined to obtain consumption, structures investment, and equipment investment. The share parameters for the assembly functions are set as follows: the shares for equipment investment are φE = 0.85,φE = 0.15 and the shares for consumption and structures investment are M N φC = φS = 0.04,φC = φS = 0.96. M M N N 14There are bridge tables for consumption as well as equipment and software investment but not for structures investment. Weassume that thesectoral composition of structures investment is thesame as that of consumption. 15The machinery sector of our model has two components. The first component is the NIPA definition of “Equipment and Software” Investment, after excluding the Transportation, Wholesale, and Retail Margins from the Input-Output Tables. Most of the industries whose output is used in “Equipment and Software” produce exclusively for “Equipment and Software.” The second component of our machinery sector comprises those inputs for consumption assembly from all the industries that produce inputs used in both the NIPA definition of “Equipment and Software” Investment and of “Consumption.” These IO Table industries are: (334) Computer and Electronic Products; (335) Electrical Equipment,Appliances,andComponents;(513)BroadcastingandTelecommunications;(514)InformationandDataProcessing Services; and (5412OP) Miscellaneous Professional, Scientificand Technical Services. 33

Table 3: Calibration for Aggregate Model Parameter Determines Parameter Determines Utility Function γ =1 Intertemporal consumption elast. =1/γ β =0.99 Discount factor Depreciation Rates δE =0.031 Equipment capital δS =0.014 Structures capital Adaptation Costs ωE =0 M, N Equipment Capital ωS =0 M, N Structures Capital M Goods Production αN =0.7 Labor share αE =0.17 Equipment share M M αS =.13 Structures share M N Goods Production αN =0.7 Labor share αE =0.17 Equipment share N N αS =0.13 Structures share N Consumption Assembly φC =0 M goods intensity φC =1 N goods intensity M N Assembly of Equipment Investment φE =1 M goods intensity φE =0 N goods intensity M N Assembly of Structures Investment φS =0 M goods intensity φS =1 N goods intensity M N Table 4: Calibration for the Model with Sectoral MFP Shocks Parameter Determines Parameter Determines Utility Function γ =1 Intertemporal consumption elast. =1/γ β =0.99 Discount factor Depreciation Rates δE =0.031 Equipment capital δS =0.014 Structures capital Adaptation Costs ωE =100 M, N Equipment Capital ωS =100 M, N Structures Capital M Goods Production αN =0.46 Labor share αE =0.43 Equipment share M M αS =.11 Structures share M N Goods Production αN =0.72 Labor share αE =0.15 Equipment share N N αS =0.13 Structures share N Consumption Assembly φC =0.04 M goods intensity φC =0.96 N goods intensity M N Assembly of Equipment Investment φE =0.85 M goods intensity φE =0.15 N goods intensity M N Assembly of Structures Investment φS =0.04 M goods intensity φS =0.96 N goods intensity M N 34

C.3. Estimated Parameters For the estimation, we focus on matching the variance, the covariance, and the first autocorrelation of the same five variables used in the VAR: the growth rate of the relative price of investment, labor productivity growth, hours per capita, the growth rate of equipment and software per capita, and the growth rate of consumption per capita. To weigh the various moments we use the diagonal of the simulated method of moments weighting matrix. We estimate the parameters governing the shock processes (labor supply, consumption, and government spending shocks). We estimate the parameter η, governing consumption habits, and the parameters ν0M and ν0N , determining the investment adjustment costs. In line with our focus on aggregate data, we restrict the investment adjustment costs to be equal across sectors. Finally, for the sectoral model with MFP shocks, we estimate the elasticity of substitution between factor inputs in the assembly of final goods, governed by the parameters σ , σ , and σ , which are also imposed to equal each other. C E S WereadoutthestandarddeviationsfortheinnovationsfortheneutralMFPandsectoral MFP or IST shocks from the VAR estimates. The standard deviation of the neutral MFP shockischosentomatchtheVARlong-runresponseoflaborproductivitytoaone-standarddeviation MFP shock. The standard deviation of the sectoral MFP or IST shocks is chosen to match the VAR long-run response of the relative price of investment to a one-standarddeviation shock to the relative price of investment. Under the calibration for the aggregate model, sectoral MFP shocks and IST shocks are equivalent and we drop the sectoral MFP shocks. Under the calibration that maintains the sectoral detail, we drop the IST shocks. The estimation results are summarized in Tables 5 and 6. 35

Table 5: Estimated Parameters For the Aggregate Model Parameter Determines Parameter Determines Standard Deviations of Shocks σ = 0.0036 Neutral MFP σ = 0.030 IST A Z σ = 0.022 Consumption σ = 0.036 Labor supply U V σ = 0.11 Government spending G Autoregressive Coefficient of Shocks ρ = 0.71 Consumption ρ = 0.97 Labor supply U V ρ =0.94 Government spending G Other Structural Parameters η = 0.40 Habits ν0 = 0.25 Investment adj. costs Table 6: Estimated Parameters For the Model with Sectoral MFP Shocks Parameter Determines Parameter Determines Standard Deviations of Shocks σ = 0.0037 Neutral MFP σ M = 0.0576 Sectoral MFP A A σ =0.0055 Consumption σ = 0.012 Labor supply U V σ = 0.062 Government spending G Autoregressive Coefficient of Shocks ρ = 0.001 Consumption ρ = 0.99 Labor supply U V ρ = 0.94 Government spending G Other Structural Parameters η = 0.77 Habits ν0 = 0.14 Investment adj. costs σ = σ = σ = 10.77 Sub. Elast. between M and N goods C E S 36

D. Appendix: Additional Results of Monte Carlo Experiment The red lines in Figure 6 show the responses to an MFP shock in the machinery sector of our two-sector model. By construction, the long-run response of the relative price of investment matches the response estimated from the VAR, but the short-run response is left unconstrained. The responses of consumption, investment, and hours per capita fall within the 90% confidence intervals estimated from the VAR both in the short and the long run. The most glaring departures from the results of the VAR occur for the relative price of investment and for labor productivity in the short run. However, if we were to match with the model the response of the price of investment from the VAR in every period, the resulting path for labor productivity, as well as all the other variables shown, would fall 16 within the confidence interval of the VAR even in the short run. The areas shaded in solid red show the results of a Monte Carlo experiment in which 1000 samples of the same length as the observed data were drawn using our two-sector model. For each sample we re-estimated the same VAR as for the observed data. The shaded areas are 90% confidence intervalsfortheresponsetoashockthatlowerstherelativepriceofinvestmentpermanently. Thereis substantialoverlap between theareas shadedinsolid red andthosein dashedblack indicating that the VAR results could have been generated from a random sample from our two-sector model. Figure 7 reports results for the IST model analogous to those described above. For convenience, the VAR results from the observed data are repeated again, as thick dashed and vertical dashed lines. The responses of consumption, investment, and hours per capita to an IST shock in our one-sector model fall within the confidence interval from estimation of the VAR most of the time horizon, except in the short run. Again the most glaring departure concerns the response of the price of investment in the short run—the long-run response for this variable being matched by construction. However, if we were to match with the model the response of the price of investment from the VAR in every period, the resulting paths for all the variables shown would fall within the confidence interval of the VAR even in the short run, in this case, too. Accordingly, based only on the impulse response functions reported in the figure, we would fail to reject the aggregate model with IST shocks. 16We confirmed this result by feeding a path of unforeseen shocks for the MFP process of the machinery sector that was devised to replicate thepath from theVAR. 37

Figure 6: The VAR Response to a One-Standard Deviation Shock that Lowers the Relative Price of Investment Permanently, Compared Against the Response to an MFP shock in the Machinery Sector of the Two-Sector Model and Against VAR Estimates Based on a Monte Carlo Experiment 0 −1 −2 −3 −4 −5 20 40 60 80 100 tnecreP Price of investment 5 0 −5 20 40 60 80 100 Sectoral Model, 90% Conf. Interval Sectoral Model, MFP shock VAR VAR. 90% Conf. Interval tnecreP Labor productivity Hours per capita 5 0 −5 20 40 60 80 100 tnecreP Output per capita 2 1.5 1 0.5 0 −0.5 −1 20 40 60 80 100 tnecreP Consumption per capita 2 1.5 1 0.5 0 −0.5 −1 20 40 60 80 100 Quarters tnecreP Investment per capita 8 6 4 2 0 −2 20 40 60 80 100 Quarters tnecreP 38

Figure 7: The VAR Response to a One-Standard Deviation Shock that Lowers the Relative Price of Investment Permanently, Compared Against the Response to an IST shock in the Aggregate Model and Against VAR Estimates Based on a Monte Carlo Experiment 0 −1 −2 −3 −4 −5 20 40 60 80 100 tnecreP Price of investment 6 4 2 0 −2 −4 −6 20 40 60 80 100 Aggregate Model, 90% Conf. Interval Aggregate Model VAR VAR. 90% Conf. Interval tnecreP Labor productivity Hours per capita 8 6 4 2 0 −2 −4 −6 20 40 60 80 100 tnecreP Output per capita 4 3 2 1 0 −1 −2 20 40 60 80 100 tnecreP Consumption per capita 2 1 0 −1 −2 20 40 60 80 100 Quarters tnecreP Investment per capita 15 10 5 0 −5 20 40 60 80 100 Quarters tnecreP 39