Putty-Clay and Investment: A Business Cycle Analysis

Putty-Clay and Investment: A Business Cycle Analysis1 Simon Gilchrist Boston University and NBER 270 Bay State Road Boston, MA 02215 sgilchri@bu.edu and John C. Williams Board of Governors of the Federal Reserve System Mail Stop 67 Washington, DC 20551 jwilliams@frb.gov June, 1998 Abstract This paper develops a dynamic stochastic general equilibrium model with putty-clay technology that incorporates embodied technology, investment irreversibility, and variable capacity utilization. Low short-run capital-labor substitutability nativeto theputty-clayframeworkinduces the putty-clay e(cid:11)ect of a tight link between changes in capacity and movements in employment and output. As a result, persistent shocks to technology or factor prices generate business cycle dynamics absent in standard neoclassical models, including a prolonged hump-shaped response of hours, persistence in output growth, and positive comovementin the forecastable components of output and hours. Capacity constraints result in a nonlinear aggregateproduction function that implies asymmetric responses to large shocks with recessions steeper and deeper thanexpansions. Minimumdistanceestimationofatwo-sectormodelthatnests putty-clay and neoclassical production technologies supports a signi(cid:12)cant role for putty-clay capital in explaining business-cycle and medium-run dynamics. Keywords: putty-clay, vintage capital, business cycle, irreversibility, capacity utilization. JEL Classi(cid:12)cation: D24, E22, E32 1We are indebted to Steven Sumnerfor exemplary research assistance. We wish to thank Flint Brayton, Je(cid:11) Cambell, Thomas Cooley, Russel Cooper, Sam Kortum, John Leahy, Scott Schuh, DanSichel,andparticipantsatpresentationsattheNBERImpulseandPropagationWorkshopand EconomicFluctuationsMeeting, ColumbiaUniversity,HarvardUniversity,NYU,theUniversityof Maryland, the Federal Reserve Bank of New York, and the Board of Governors of the Federal Reserveforhelpfulcomments. TheopinionsexpressedherearenotnecessarilysharedbytheBoard of Governors of theFederal Reserve Systemor its sta(cid:11)

1 Introduction In this paper we develop a dynamic stochastic general equilibrium model based on the putty-clay technology introduced by Johansen (1959). The putty-clay model possesses a number of attractive features typically absent from models based on neoclassical productionfunctions,includinganonlinearshort-runaggregate production function, irreversible investment, variable capacity utilization, and endogenous machine replacement. We investigate the implications of putty-clay technology for macroeconomic dynamics at business cycle and medium-run frequencies.2 A key (cid:12)ndingin the paper is that low short-runcapital-labor substitutability native to the putty-clay frameworkinducestheputty-clay e(cid:11)ect ofatight linkbetween changes in capacity and movements in employment and output. In the short-run, an expansion of employment quickly confronts sharp increases in marginal costs, owing to capacity constraints. Once new capacity is in place, a sustained boom in employment and output ensues, as (cid:12)rms fully employ new machines without reducing utilization rates on existing capacity. The putty-clay technology described above has two major implications for business-cycle dynamics. First, persistent shocks to technology or factor prices generate a prolonged hump-shaped response of hours, persistence in output growth, andpositivecomovement between theforecastable components ofoutputandhours. These features of the business cycle, documented by Cogley and Nason (1995) and Rotemberg and Woodford (1996), are absent in standard neoclassical models where the response of hours peaks upon the impact of the shock and the dynamic response of output closely follows that of the shock.3 Second, large shocks generate asymmetric responses of output and hours, with recessions steeper and deeper than expansions. This asymmetric response is consistent with the empirical evidence documented by Neftci (1984) and others, and reflects the fact that the short-run elasticity of output with respect to labor is decreasing in the quantity of labor employed. The empirical relevance of putty-clay technology is con(cid:12)rmed by minimum distance estimation of a two-sector model that formally nests both the putty-clay and neoclassical model within a common econometric framework. We (cid:12)nd that the dis- 2Inarecentpaper,Caballero andHammour(1998)studymedium-runissuesusingaputty-clay model. See also Malinvaud (1980) and Blanchard (1997). 3ClassicexamplesofneoclassicalmodelsarefoundintherealbusinesscycleliteratureofKydland and Prescott (1982) and Hansen (1985). 1

tance between model and data moments is minimized for an estimated putty-clay shareoftotal outputthatis ontheorderof 50-75%. Thedataoverwhelminglyreject the restriction of no role for putty-clay capital. The putty-clay model provides an intrinsically appealing description of capital accumulation. In this framework, capital goods embody the level of technology and the choice of capital intensity made at the time of their creation. Ex ante, the choice of capital intensity|the amount of capital to be used in conjunction with oneunitof labor|is basedon a standardneoclassical productionfunction. Expost, the production function is of the Leontief form with a zero-one utilization decision based on the output per hour of a given piece of capital relative to the prevailing wage rate. Over time, as the economy grows and real wages rise, older vintages of capital become too costly to operate given their current labor requirements and they are mothballed or scrapped.4 Putty-clay models have a long history in both the growth (Johansen (1959), Solow (1962), Phelps (1963), Cass and Stiglitz (1969), Sheshinski (1967) and Calvo (1976)) and investment literatures (Bischo(cid:11) (1971), Ando, Modigliani, Rasche and Turnovsky (1974)). Due in part to computational complexities, these past literatures mainly limited themselves to characterizing the long-run features of a puttyclay economy or to partial equilibrium analysis of the investment sector. More recently, interest in real business cycle models has spurred a revival in alternative speci(cid:12)cations of technology, including putty-clay.5 Atkeson and Kehoe (1994) develop a model where the energy-intensity of production is the putty-clay factor. Their model possesses the property of a cuto(cid:11) rule for utilizing capital|based on the price of energy as opposed to the wage|but their dynamic analysis focuses on the case where capital is always fully utilized. Cooley, Hansen and Prescott (1995) study a model where each period physical capital is assigned to plots of land, the supply of which is assumed to be (cid:12)xed for the dynamic analysis. Although this model features variable capacity utilization, the assumption that the land intensity of capital can be freely changed after one period e(cid:11)ectively cuts the dynamic link 4The e(cid:11)ects of technological lock-in motivate the machine replacement problem (cid:12)rst addressed by Johansen (1959) and Calvo (1976), and more recently formalized in a dynamic programming environment byCooper and Haltiwanger (1993) and Cooley, Greenwood and Yorukoglu (1994). 5Vintage models havealso experienced a resurgence of late as witnessed byBenhabib and Rustichini(1991), BenhabibandRustichini(1993), Caballero andHammour(1996), Campbell(1994), Cooper, Haltiwanger and Power (1995), Boucekkine, Germain and Licandro (1997), and Greenwood, Hercowitz and Krusell (1997). 2

between capital and labor that is key to the putty-clay e(cid:11)ect. Our model incorporates what we view as the essential features of the puttyclay framework, including variable capacity utilization and investment irreversibility. Although the assumption of ex post Leontief technology may at (cid:12)rst seem unrealistically stark, the resulting aggregate production function embeds, depending on the model’s parameterization, both the relatively flat short-run supply curve usually associated with a neoclassical model and a reverse L-shaped supply curve traditionally associated with the putty-clay framework. In addition, the model’s micro-foundationsarelargelyconsistent withmicroeconomicevidenceontheimportance of plant shutdowns as a short-run adjustment margin (Bresnahan and Ramey (1994)) andthelumpinessofinvestment attheplantlevel (Doms andDunne(1993), Cooper et al. (1995), Caballero, Engel and Haltiwanger (1995)). The distinguishing features of the model developed in this paper are nicely illustrated by the experiment of a reduction in the cost of producing new capital goods. This reduction in capital cost raises the return to new capital and causes a surge in investment, which over time leads to rising aggregate output and consumption. Initially, however, (cid:12)rms’ e(cid:11)orts to raise employment encounter capacity constraints owing to the putty-clay nature of capital. As a result of this low short-run substitutibility of labor for capital, the initial aggregate response of both output and hours is muted. To e(cid:14)ciently increase production, (cid:12)rms invest in new capacity, that, once in place, can be utilized by an expanded workforce. Ex post (cid:12)xity of the capital-labor ratio for existing capacity implies that labor can only be reallocated to new machines at the cost of mothballing existing capacity. Therefore, there is an incentive to simultaneously utilize both new and exisiting capital. This dynamic linkage between capital and labor causes hours and output to rise together for a sustained period of time following the initial burst of investment, generating the putty-clay e(cid:11)ect. The dynamicresponseto a large increase in the cost of new capital goods di(cid:11)ers insomerespectsfromthatdescribedabove. Inthiscase, alargefraction of theoveralladjustmentof outputandhoursis accomplished throughanimmediate reduction in capacity utilization. Thus, the putty-clay model naturally delivers asymmetric responses to positive and negative shocks, with the asymmetries increasing in the magnitude of the shock. 3

2 The Model In this section we describe the model and derive the equilibrium conditions. Each capital good possesses two de(cid:12)ning qualities: its level of embodied technology and its capital intensity. The underlying or ex ante production technology is assumed to be Cobb-Douglas with constant returns to scale, but for capital goods in place, production possibilities take the Leontief form: there is no ex post substitutability of capital and labor. In addition to aggregate technological change, we allow for the existence of idiosyncratic uncertainty regarding the productivity of investment projects. As in Campbell (1994), the introduction of heterogeneity within vintages smooths the aggregate allocation and simpli(cid:12)es computation of the equilibrium. More importantly, such idiosyncratic uncertainty implies the existence of a well-de(cid:12)ned aggregate production function despite the Leontief nature of the microeconomic utilization choice. Once in place, capital goods are irreversible, that is, they cannot be converted into consumption goods or capital goods with di(cid:11)erent embodied characteristics, and have zero scrap value. Firms can choose, however, whether or not to operate a given unit of capital depending on the pro(cid:12)tability of doing so in the current economic environment. We assume that there are no costs of taking machines or workers on- and o(cid:11)-line. As such, the utilization choice is purely atemporal. The optimal utilization choice for each unit of capital is determined by the di(cid:11)erence between the (labor) productivity of the capital and the cost of utilizing the capital, which in the absence of other costs equals the wage rate. If the productivity of a unit of capital exceeds the wage rate, the capital is used in production, otherwise, it is not. In equilibrium, the wage rate, capacity utilization rate, and levels of employment, production, consumption, and investment are determined jointly by the dynamic optimizing behavior of households and (cid:12)rms. To characterize the equilibrium allocation, we (cid:12)rst discuss the optimization problem at the project level and then describe aggregation from the project level to the aggregate allocation. 2.1 The Investment Decision Each periodasetof newinvestment \projects"becomes available. Constantreturns to scale implies an indeterminacy of scale at the level of projects, so without loss of generality, we normalize all projects to employ one unitof labor at fullcapacity. We 4

refer to these projects as \machines." Capital goods require one period for initial installation and then are productive for M (cid:21) 1 periods. The productive e(cid:14)ciency of machine i initiated at time t is a(cid:11)ected by two stochastic productivity terms, one idiosyncratic, one aggregate. In addition, we assume all machines, regardless of their relative e(cid:14)ciency, fail at an exogenously given rate that varies by the age of the machine. Insummary, capital goods are heterogeneous and are characterized by three attributes: vintage (age and level of aggregate embodied technology), capitalintensity, and the realized value of the idiosyncratic productivity term. The productivity of each machine, initiated at time t, di(cid:11)ers according to the log-normally distributed random variable, (cid:18) , where i;t 1 log(cid:18) (cid:24) N(log(cid:18) − (cid:27)2;(cid:27)2): i;t t 2 Theaggregate index (cid:18) measures the mean level of embodied technology of vintage t t capital goodsand (cid:27)2 is thevariance of theidiosyncratic shock. Themean correction term −1(cid:27)2 implies E((cid:18) j(cid:18) ) = (cid:18) . We assume (cid:18) follows a stochastic process with 2 i;t t t t meangrossgrowthrate(1+g)1−(cid:11). Forthesakeofnotationalclarity, inthefollowing discussion we abstract from disembodied aggregate technological change of the type typicalintherealbusinesscycleliterature. Theinclusionofastochasticdisembodied technology process is straightforward and used in section 3 when analyzing model dynamics. Before investment decisions are made, the economy-wide level of vintage technology (cid:18) is observed but the idiosyncratic shock to individual machines is not. We t also assume that after the revelation of the idiosyncratic shock, further investments inexistingmachinesarenotpossible. Subjecttotheconstraintthatlaboremployed, L , is nonnegative and less than or equal to unity (capacity), (cid:12)nal goods output i;t+j produced in period t+j by machine i of vintage t is Y = (cid:18) k(cid:11) L ; i;t+j i;t i;t i;t+j where k is the capital-labor ratio chosen at the time of installation. Denote the i;t labor productivity of a machine by X (cid:17) (cid:18) k(cid:11) : i;t i;t i;t The only variable cost to operating a machine is the wage rate, W . Idle mat chinesincurnovariablecostsandhavethesamecapitalcostsasoperatingmachines.6 6The model can be extended to allow for a (cid:12)xed labor cost per unit of capital. Under such a 5

Figure 1: Steady-state Distribution of Labor Productivity 44 0.5 Aggregate (left axis) Newest Vintage(right axis) 0.4 33 0.3 22 0.2 11 0.1 00 0.0 00 111 222 333 444 555 666 Project level Productivity (x) tnemtsevni yb dethgiew noitubirtsiD Equilibrium Wage Given the Leontief structure of production, these assumptions imply a cuto(cid:11) value for the minimum e(cid:14)ciency level of machines used in production: those with productivity X (cid:21) W are run at capacity, while those less productive are left idle. i;t t To illustrate these ideas, Figure 1 shows the steady-state distribution of labor productivity for the model calibrated to parameters speci(cid:12)ed below. The height of thedistributionreflectsthenumberofmachinesatanygivenproductivitylevel. The cuto(cid:11) valueforthewage isshown as avertical line. Capital goodswith productivity lying to the right of the cuto(cid:11) are used in production, those to the left are idle. Capital utilization is given by the area in the shaded region divided by the total area under the distribution. Figure 1 also shows the distribution of labor productivity for the most recent vintage (rightscale). Itspositiononthehorizonalaxisreflects boththecurrentlevel of technology and the capital intensity of new machines. Owing to trend growth speci(cid:12)cation, it is optimal topermanently scrap machineswhose e(cid:14)ciency falls below some cuto(cid:11). This modi(cid:12)cation substantially complicates the investmentdecision and is left for future research. 6

and relatively long-lived capital, the average labor productivity of the most recent vintage is substantially higher than the average labor productivity of existing machines. Obsolescence through embodied technical change implies that old vintages have lower average utilization rates than new vintages. Note that trend growth in investment|due to population growth and technological change|causes the aggregate distribution to be skewed. To derive the equilibrium allocation of labor, capital intensity, and investment, we begin by analyzing the investment and utilization decision for a single machine. Q De(cid:12)ne the time t discount rate for time t+j income by R~ (cid:17) j R −1 , where t;t+j s=1 t+s R is the one period gross interest rate at time t + s. At the machine level, t+s capital intensity is chosen to maximize the present discounted value of pro(cid:12)ts to the machine: (cid:26) (cid:27) XM max E −k + R~ (1−(cid:14) )(X −W )L ; (1) t i;t t;t+j j i;t t+j i;t+j ki;t;fLi;t+j gM j=1 j=1 s:t: 0 (cid:20) L (cid:20) 1; j = 1;:::;M; i;t+j 0 < k < 1; i;t where (cid:14) is the probability a machine has exogenously failed by j periods and exj pectations are taken over labor productivity, whose realization depends on the time t idiosyncratic shock, and future values of wages and interest rates. Because investment projects are identical ex ante, the optimal choice of the capital-labor ratio is equal across all machines in a vintage; that is, k = k ;8i. i;t t Denote the average productivity of the entire stock of vintage t capital by X = t (cid:18) k(cid:11). Capital utilization for vintage s at time t is the ratio of labor employed to t t employment capacity of the vintage, given by Pr(X > W jW ;(cid:18) ). Given the i;s t t t log-normal distribution for (cid:18) we obtain: i;t Pr(X > W jW ;(cid:18) ) = 1−(cid:8)(zs); i;s t t t t where (cid:8)((cid:1)) is the c.d.f. of the standard normal and (cid:18) (cid:19) 1 1 zs (cid:17) logW −logX + (cid:27)2 ; t (cid:27) t s 2 Similarly, capacity utilization for vintage s at time t is the ratio of actual output produced from the capital of a given vintage to the level of output that could be 7

produced at full capital utilization. Letting F((cid:1)) denote the cdf of X , capacity i;t utilization is formally de(cid:12)ned as R 1 X dF(X ) XRis 1 >W X t d i; F s (X ) i;s = (1−(cid:8)(z t s−(cid:27))) 0 i;s i;s where the equality follows from the log-normality of X .7 i;t If all machines were fully utilized, labor productivity would simply equal X . t With partial utilization, labor productivity also depends on capital and capacity utilization. The average product of labor for vintage s capital at time t, is 1−(cid:8)(zs −(cid:27)) APLs = t X : t 1−(cid:8)(zs) s t Letting (cid:30)((cid:1)) denote the p.d.f. of the standard normal, the marginal productof labor for vintage s capital at time t is (cid:30)(zs−(cid:27)) MPLs = t X : t (cid:30)(zs) s t For any given vintage, the marginal product of labor is equal to the e(cid:14)ciency of the least productive machine of the vintage in operation.8 Expected net income in period t from a vintage s machine, (cid:25)s, conditional on t W , is given by t (cid:18)(cid:16) (cid:17) (cid:16) (cid:17) (cid:19) (cid:25) t s = (1−(cid:14) t−s ) 1−(cid:8)(z t s−(cid:27)) X s − 1−(cid:8)(z t s) W t : Substituting this expression for net income into equation 1 eliminates the future choices of labor from the investment problem. The remaining choice variable is k . t The (cid:12)rst order condition for an interior solution for k is given by9 t ( (cid:18) (cid:19) (cid:27) XM k = (cid:11)E R~ (1−(cid:14) ) 1−(cid:8)(zt −(cid:27)) X : (2) t t t;t+j j t+j t j=1 7Capacity utilization may be expressed as [E(X jX > W )=E(X )]Pr(X > W ): We i;s i;s t i;t i;s t then use the formula for the expectation of a truncated log-normal: If log((cid:22)) (cid:24) N((cid:16);(cid:27)2), then E((cid:22)j(cid:22)>(cid:31))= (1−(cid:8)(γ−(cid:27)))E((cid:22)) where γ =(log((cid:31))−(cid:16))=(cid:27)(Johnson, Kotzand Balakrishnan 1994). (1−(cid:8)(γ)) 8Normalizing the quantity of machines at unity,marginal product equals the increment to machineoutputobtainedbyreducingthee(cid:14)ciencycuto(cid:11)W ,dividedbytheincrementtolaborinput t obtainedbyreducingthecuto(cid:11)W . Theincrementtooutputequals @(1−(cid:8)(zt s−(cid:27))Xs) @zt s ,whilethe t @z t s @Wt increment to labor equals @(1−(cid:8)(zt s)) @zt s . @z t s @Wt 9This(cid:12)rstorderconditionisobtainedbytakingthederivativeofthepro(cid:12)tfunctionwithrespect to k , recognizing that in equilibrium, the marginal machine earns zero quasi-rents so that @(cid:25)t s (cid:17) t @zs t 1(cid:30)(zs−(cid:27))X − 1(cid:30)(zs)W =0: (cid:27) t s (cid:27) t t 8

New machines are put into place until the value of a new machine (the present discounted value of net income) is equal to the cost of a machine (k ) t (cid:26) XM (cid:16) (cid:17) k = E R~ (1−(cid:14) ) 1−(cid:8)(zt −(cid:27)) X (3) t t t;t+j j t+j t j=1 XM (cid:16) (cid:17) (cid:27) − R~ (1−(cid:14) ) 1−(cid:8)(zt ) W t;t+j j t+j t+j j=1 This is the free-entry or zero-pro(cid:12)t condition. The (cid:12)rst term on the right hand side of equation 3 reflects the expected present discounted value of output adjusted for the probability that the machine’s idiosyncratic productivity draw is too low to pro(cid:12)tably operate the machine in period t+j. The second term likewise reflects the expected present value of the wage bill, adjusted for the probability of such a shutdown. This condition must hold as long as there is gross investment in period t.10 Equations 2and 3jointly implythat, in equilibrium,the expected presentvalue of the wage bill equals (1−(cid:11)) times the expected present value of revenues. 2.2 Aggregation Aggregation of machine-level labor inputs and output is a two-step process. First, labor input and output of machines in each vintage are aggregated into vintage totals. Second, inputs and outputs from the M productive vintages are summed to yield aggregate values. Total labor employment, L , is t XM (cid:16) (cid:17) L t = 1−(cid:8)(z t t−j) (1−(cid:14) j )Q t−j : (4) j=1 whereQ t−j isthequantityofnewmachinesstartedinperiodt−j,(cid:8)(z t t−j)istheidle rate of those machines in period t, and (cid:14) reflects the fact that a subset of machines j has failed completely. Aggregate (cid:12)nal output, Y is t XM (cid:16) (cid:17) Y t = 1−(cid:8)(z t t−j −(cid:27)) (1−(cid:14) j )Q t−j X t−j : (5) j=1 In the absence of government spending or other uses of output, aggregate consumption, C , satis(cid:12)es t C = Y −k Q ; (6) t t t t where k Q is gross investment in new capital machines. t t 10If the cost of a machine exceeds the value of a machine for all admissible values of k no t investment is undertaken. 9

2.3 Preferences To close the model, we specify the economic relationships that determine labor supplyandsavingsdecisions. Weassumethattheeconomyismadeupofrepresentative households whose preferences are given by (cid:18) (cid:19) 1 X1 C (N −L ) 1−γ E (cid:12)s t+s t+s t+s ; (7) 1−γ t N s=0 t+s where (cid:12) 2 (0;1);γ > 0; > 0, and N = N (1+n)t is the household’s growing time t 0 endowment.11 Households optimize over these preferences subject to the standard intertemporal budget constraint. We assume that claims on the pro(cid:12)t streams of individualmachinesaretraded;inequilibrium,householdsownadiversi(cid:12)edportfolio of all such claims. The (cid:12)rst-order condition with respect to consumption is given by (cid:12) U = E R U ; (8) c;t t t;t+1 c;t+1 1+n where U denotes the marginal utility of consumption. The (cid:12)rst-order condition c;t+s with respect to leisure and work is given by U W +U = 0; (9) c;t t L;t where U denotes the marginal utility associated with an incremental increase in L;t work (decrease in leisure). This completes our description of the economy. The rational expectations equilibrium is de(cid:12)ned to be the set of sequences of prices and quantities such that each household and (cid:12)rm solves its respective maximization problem as described above, taking prices as given, and all markets clear. The derivation of the deterministic steady state and its properties are found in Gilchrist andWilliams (1998b). In conductingthe dynamicanalysis we focus on deviations from the balanced growth path. The solution methodology for the dynamic analysis is described in the appendix. 3 Model Dynamics In this section we describe the model’s implications for business cycle and mediumrun dynamics in response to persistent shocks to factor prices; in the next section 11WealsoconsideredthecaseofindivisiblelaborasinHansen(1985)andRogerson(1988). This speci(cid:12)cation altersthemagnitudeofthedynamicresponsesreportedbelowbutnotthequalitative properties of themodel. 10

we turn our attention to permanenttechnology shocks. Thepurposeof this analysis is not to argue for a speci(cid:12)c theory of business cycles based on particular shocks, butinstead to document the putty-clay model’s dynamic properties and their correspondenceto those evident in the data for a wide variety of shocks. For the purpose of comparison, we construct a neoclassical model of vintage capital, initially introduced by Solow (1962). Details of this model are provided in the appendix. In the vintage model, the restriction that ex post capital-labor ratios are (cid:12)xed is removed. Thus, this model takes the standard Cobb-Douglas putty-putty formulation. The two models are otherwise identical. We focus primarily on three key business cycle properties: comovement, persistence, and asymmetries. First, as emphasized by Lucas (1977), a fundamental featureofthebusinesscycleisthatoutputmovements acrosssectorsexhibitpositive comovement. The real business cycle literature tends to focus on a particular interpretation of comovement that is based almost exclusively on a model’s implications for unconditional second moments of (cid:12)ltered data; see, for example, Kydland and Prescott (1991). RotembergandWoodford(1996) extend thenotion of comovement to the forecastable components of output, hours, and consumption. Using a VAR model, they document that the forecastable components of these variables also exhibit strongly positive comovement. Second, Cogley and Nason (1995) document that output growth displays signi(cid:12)cant positive serial correlation; that is, output growth is persistent. Finally, there is a wide range of evidence that asymmetries exist with respect to the business cycle and the dynamic response to particular shocks. As documented below, the putty-clay model possesses a powerful internal propagation mechanism that yields dramatic improvements over the neoclassical model in all three dimensions. 3.1 The Short-Run Aggregate Supply Curve Insight into the dynamicresponses of the putty-clay modelis provided by the shortrun aggregate supply curve (the inverse of the marginal product of labor). Here, short-run refers to the time period during which the capital stock is (cid:12)xed. Figure 2 shows the short-run aggregate supply curve, computed as the markup 1=W; in log deviations from the deterministic steady state, for the neoclassical vintage model and three versions of the putty-clay model with di(cid:11)erent degrees of idiosyncratic uncertainty (measured by (cid:27)). For the neoclassical vintage model, the production 11

Figure 2: Short-run Aggregate Supply Curve 0000....8888 s = .05 s = .15 0000....6666 s = .25 Cobb-Douglas 0000....4444 0000....2222 0000....0000 ----0000....2222 ----0000....4444 ---000...333000 ----0000....22225555 ----0000....22220000 ----0000....11115555 ----0000....11110000 ----0000....00005555 0000....00000000 0000....00005555 00..1100 Output (log deviation from steady state) )etats ydaets morf noitaived gol( pukraM function is Cobb-Douglas implying that the SRAS is linear in logs. The SRAS of the putty-clay model, however, is distinctly nonlinear. For very low values of (cid:27), the putty-clay SRAS curve becomes vertical for levels of output a few percent above steady state. As (cid:27) increases, the SRAS curve of the putty-clay model becomes less sharply curved and approaches that of the neoclassical vintage model as (cid:27) approaches in(cid:12)nity. Thus, the model developed in this paper embeds both the reverse-L shaped aggregate supply curve traditionally associated with putty-clay technology and the log-linear aggregate supply curve of the neoclassical production function. Thedegreeofidiosyncraticuncertaintydeterminestheextenttowhichthe model’s short-run aggregate production function and dynamic responses are more putty-clay or neoclassical in flavor. In the putty-clay model, the variable slope of the SRAS curve results from varying utilization rates of existing machines. Variable utilization is often suggested as anexplanationforthefactthatempiricalestimates oftheshort-runelasticity ofproduction with respect to labor inputs, dlnY; are much closer to unity than to labor’s dlnL 12

share.12 The intuition here is that a 1% increase in labor e(cid:11)ectively causes a 1% increaseincapital, throughincreasedutilization, andhencea1%increaseinoutput. This simple calculation relies on the assumption that capital goods are homogenous however. It also ignores equilibrium determinants of utilization and capacity. In the putty-clay model, dY = w where w is the e(cid:14)ciency of the marginal machine. dL As labor inputs increase, the quality of the marginal machine falls, guaranteeing dln(Y) <1: dln(L) dln(Y) By considering optimal capacity choice, we can explicitly quantify . In dln(L) steady state, the short-run elasticity of output with respect to labor for the puttyclay model equals 1 − (cid:11); the long-run labor share. If utilization rates rise above steady-state, costs increase rapidly and dlnY < 1 − (cid:11). The only way to justify dlnL dlnY > 1−(cid:11) is to argue that (cid:12)rms frequently hold costly excess capacity. This is dlnL sub-optimal from the (cid:12)rm’s point of view however. Hence, except in response to large negative shocks, (cid:12)rms typically operate in a region where dlnY ’ 1−(cid:11); and dlnL variable utilization does not provide measured short-run increasing returns to labor in the putty-clay model. Besides having important implications for utilization rates and their influence on labor productivity, the variable slope of the putty-clay SRAS also implies that dynamic responses to positive shocks di(cid:11)ers from those to negative shocks, as discussed below. Although not examined here, the nonlinear aggregate supply curve also implies asymmetries in price adjustment in models with nominal rigidities. 3.2 Calibration ThemodelsarecalibratedusingparametervaluestakenfromChristianoandEichenbaum (1992) and Kydland and Prescott (1991), except for the trend growth rates, which are averages over 1954{96.13 We assume a period is one quarter of one year. In annual basis terms, the calibrated parameters are (cid:12) = 0:97; (cid:26) = 1; = 3; g = 0:018; n = 0:015; (cid:14) = 0:084; (cid:11) = 0:36; M = 1. The results reported in this paper are not sensitive to reasonable variations in these parameters. When calibrating the model, the only parameter for which we do not have a prior estimate is the variance of the idiosyncratic component of a machine’s productivity, (cid:27)2. As 12Basu and Fernald (1997) provide a recent discussion. 13These estimates are obtained from long-run restrictions, and, with one caveat, are therefore valid for both the putty-clay model and the neoclassical model. The caveat is that variation in (cid:27) has a small impact on steady-state properties through endogenous depreciation. Endogenous depreciation alters theestimated (cid:14) by 1-2% and has only a veryminor e(cid:11)ect on model properties. 13

discussed above, for large (cid:27), the short-run aggregate supply curve is very close to Cobb-Douglas. By lowering (cid:27) we increase the curvature of the short-run aggregate supply curve and increase the degree to which the model displays dynamics unique to the putty-clay structure. To make clear distinctions between the neoclassical and putty-clay models we set (cid:27) = 0:15. Lowering (cid:27) to 0.1 does not alter model results in any substantial manner; lowering (cid:27) much further causes numerical problems for the dynamic solution methods. On the other hand, raising (cid:27) to 0.5 for almost all essential purposes replicates the neoclassical model dynamics, while intermediate values ((cid:27) = 0:2 −0:25) provide results that are a combination of the neoclassical and putty-clay model with low (cid:27). 3.3 Capital Cost Shocks We start by characterizing the e(cid:11)ect of a temporary but persistent shock to the cost of producing capital goods relative to consumption goods. This is identical to an increase in technology embodied in capital goods; henceforth, we describe it as such.14 We assume that embodied technology follows the process (1−(cid:26) L)ln(cid:18) = (cid:18) t (1−(cid:26) L)tln(1+g)+u , where u is an i.i.d. innovation and L is the lag operator. (cid:18) t t We set the autocorrelation coe(cid:14)cient of the shock process at (cid:26) = 0:95 implying a (cid:18) half life for the shock of just under 14 quarters. Figure 3 shows the impulse response function to output for both the putty-clay model (upper panel) and the neoclassical model (lower panel) to a one percentage point positive shock to embodied technology. The di(cid:11)erence in output dynamics between the two models is striking. For the putty-clay model, output rises very little initially, steadily increases for a period of (cid:12)ve years, and eventually returns to steady state. For the neoclassical model, the peak response occurs at the onset of the shock, after which output exhibits something close to exponential decay as it returns to steady state. In the neoclassical model, output dynamics simply mirror shock dynamics with no evidence of any interesting cyclical pattern. In the puttyclay model, output exhibits a clear \hump-shaped" response that creates a slowly unfolding and long-lasting business cycle. Figure 3 also plots the impulse response of the Solow residual, conventionally measured,fortheputty-clayandneoclassicalmodels. Inbothmodels,movements in 14Itis,not,however,thesameasadistortionaryshocktothecostofcapitalgoodsfromachange intaxesorthatmightresultfromamonetarydisturbanceinamodelwithnominalrigidities. Such shocks a(cid:11)ect theequilibrium allocation but not thefeasible allocation for the economy. 14

Figure 3: Persistent Cost of Capital Shock Figure 3: Persistent Cost of Capital Shock Putty-Clay Model 0.8 Output Solow Residual Adjusted Solow Residual 0.6 0.4 0.2 0.0 0 5 10 15 20 years Neoclassical Model 1.2 Output 1.0 Solow Residual 0.8 0.6 0.4 0.2 0.0 0 5 10 15 20 years theSolowresidualaremuchsmaller(byafactorof3to5)thanmovementsinoutput. Hence, in both models, embodied technological change provides \magni(cid:12)cation" as usually measured by movements in output vis-a-vis the Solow residual. In addition to magni(cid:12)cation, the putty-clay model provides signi(cid:12)cant positive comovement, in levels and growth rates, between output and the Solow residual over the cycle. This positive comovement stems from the fact that both output and the Solow residual exhibit similar hump-shaped responses to embodied technological change. In the neoclassical model, embodied shocks generate a negative correlation between growth in output and the Solow residual over the (cid:12)rst (cid:12)ve years. The third line in the top panel of (cid:12)gure 3 shows the Solow residual, after correcting for capital utilization. Initially, consistent with the arguments made above, 15

Figure 4: Persistent Cost of Capital Shock Putty-Clay Model Investment Labor 3.5 0.5 3.0 0.4 2.5 0.3 2.0 0.2 1.5 1.0 0.1 0.5 0.0 0.0 0 5 10 15 20 0 5 10 15 20 years years Q k 5 1.0 4 0.5 3 2 0.0 1 -0.5 0 0 5 10 15 20 0 5 10 15 20 years years the correction has only a trivial e(cid:11)ect on the Solow residual. Over time, we see a somewhat larger adjustment, but this adjustment reflects expanded capacity that lowers the utilization rate of existing machines in later periods. The upper panels of Figure 4 show the responses of investment and labor hours in the putty-clay model to the same shock. Investment rises immediately as the economy seeks to build new capital goods that embodythelatest technology. Hours increase and consumption falls in response to high real interest rates. The initial expansion in hours is muted, however, owing to the sharply increasing short-run aggregate supplycurve embedded in the putty-clay model. As more capital is brought 16

online, theshort-runsupplycurveshiftsoutandlaborexpandsfurther. Asaresult, the peak labor response occurs three years after the onset of the shock. Theslow butsustained risein hoursabove steady-state levels occursbecausethe bene(cid:12)ts to building new capital goods are much greater if existing, e(cid:14)cient capital is not scrapped in the process. With ex post Leontief technology, labor cannot be reallocated across machines to equate marginal products. To bene(cid:12)t from new machines without losing the productive services of existing capital, the economy must hire new workers to operate these machines. Hence, as new machines become operative, morelaborishired. Eventually, astheproductivevalueofthesemachines falls, labor returns to steady state. We refer to this dynamic linkage between labor and machines as the putty-clay e(cid:11)ect. Decomposing the investment dynamics into the quantity of new machines, Q, and the capital intensity of each new machine, k, adds further insight into the model’s dynamics. As shown in the lower panels of Figure 4, at the onset of the shock, Q rises and k falls as a large number of low capital-intensity machines are produced. This rapid expansion of inexpensive (in terms of foregone consumption) machines shifts the short-run aggregate supply curve to the right and facilitates the increase in labor input. This reliance on low e(cid:14)ciency capital does not persist, however. As the real interest rate falls and the real wage rises, (cid:12)rms substitute into high capital-intensity capital goods. The capital-intensity of new machines remains at elevated levels for a number of years as households store the bene(cid:12)ts of the temporary shock through increased saving implying capital deepening. Rising capital intensities dramatically o(cid:11)set the exponential rate of decay in technology and consequently provide a sustained increase in the e(cid:14)ciency levels of new machines for a number of years after the shock has occurred.15 Because machines are long-lived, this sustained increase in e(cid:14)ciency levels of new machines translates into a sustained increase in total labor productivity over a long horizon. 15In the initial period, (cid:18) is 1% above steady state while k is 1.3% below steady state, implying t thatx=(cid:18)k(cid:11), themean e(cid:14)ciency of new machines, is 0.75% percent abovesteady state. Six years later (cid:18) is 0.3% abovesteady state while k is 1.1% above steady state, implying that x is still 0.7% above steady state. 17

3.4 Labor Cost Shocks We now consider the e(cid:11)ect of a temporary but highly persistent shock to the marginal cost of labor. This shock may be interpreted as either a reduction in the tax on wage income or a shock to preferences that reduces the marginal utility of leisure relative to the marginal utility of consumption. Formally, we specify the labor cost shock ln(cid:17) t = (cid:26) (cid:17) ln(cid:17) t−1 + e t , where e t is an i.i.d. innovation, and set (cid:26) = 0:98. We embed (cid:17) in the labor-leisure (cid:12)rst-order condition given by equa- (cid:17) t tion 9. Figure 5 shows the responses of output, hours, and consumption to a one percentage point reduction in the marginal cost of labor for the two models. The hump-shaped response of output and hours in the putty-clay model observed in response to the capital cost shock carries over to the labor cost shock. The labor cost shock illustrates the ability of the two models to generate comovement between output, hours, and consumption. As seen in the (cid:12)gure, this comovement is especially strong in the putty-clay model after the initial shock period and is therefore present in the forecastable components. That is, starting from the (cid:12)rst period, output, hours and consumption are rising for a number of years, after which time they decline in unision. The comovement is much weaker in the neoclassical model, especially during the (cid:12)rst (cid:12)ve years following the onset of the shock. Moreover, what little positive comovement does occur during this period is mostly unforecastable. While consumption is rising over several years, hours and output are falling, making the forecastable comovement between consumption and the other two series negative rather than positive. 3.5 Nonlinear Dynamics The degree of curvature embedded in the short-run aggregate supplycurve plays an important role in conditioning the putty-clay model’s response to shocks. For small shocks such as those discussed above, the magnitude of the response to positive and negative shocks is roughly the same. As we consider larger disturbances, the curvature of the short-run aggregate supply curve away from steady-state becomes important, and the model’s dynamic responses display interesting asymmetries. In particular, the model delivers the result that a response to negative shocks is more rapidandlargerthanthattopositiveshocks. Thisisconsistentwiththeevidenceon the response to monetary shocks documented by Cover (1992) and that to oil price shocks studied by Tatom (1988) and Mork (1989). In a business cycle context, this 18

Figure 5: Persistent Cost of Labor Shock Putty-Clay Model 0.8 Output Hours Consumption 0.6 0.4 0.2 0.0 0 5 10 15 20 25 30 years Neoclassical Model 1.2 Output Hours 1.0 Consumption 0.8 0.6 0.4 0.2 0.0 0 5 10 15 20 25 30 years pattern of asymmetric responses can imply that recessions are deeper and steeper than expansions, a result consistent with the time series evidence documented by Neftci (1984), Sichel (1993), and Potter (1995). For the type of shocks considered here, the most likely source of large shocks in an economy such as the U.S. come through changes in tax policy that a(cid:11)ect factor prices.16 An investment tax credit or a revision in personal income tax rates are plausiblesources of persistentmovements in factor costs of 10% or more. To display the model’s ability to generate asymmetries, we therefore consider the di(cid:11)erential 16Otherpotential sources of large shocksare energy prices and monetary disturbances. 19

(cid:3) Figure 6: Asymmetries in Response to Labor Cost Shocks Labor Output 5 5 4 4 3 3 2 2 1 1 0 0 0 2 4 6 8 10 0 2 4 6 8 10 years years Consumption Investment 5 16 10% Increase 4 14 10% Decrease 12 3 10 2 8 1 6 0 4 -1 2 -2 0 0 2 4 6 8 10 0 2 4 6 8 10 years years * Impulse Responses to 10% Increase in Cost of Labor Displayed with Reverse Sign e(cid:11)ect of a 10% increase versus decrease in the labor cost shock considered above. The results reported in Figure 6 reveal the basic source of the asymmetry. A labor cost increase causes an immediate shutdown of machines as the economy moves down the relatively flat portion of the short-run aggregate supply curve. This immediate shutdown produces a sharp contraction in output and hours in both the initial and subsequent periods. Owing to the shutdown, the economy has excess machine capacity, and investment drops sharply in response to the shock. In contrast,alaborcostdecreasehaslittleimmediatee(cid:11)ectoneitheroutputorhoursas theeconomy is pushedupthesteep portion of theshort-runaggregate supplycurve. 20

Evidence of non-linearity only disappears after 6-8 years as capacity eventually adjusts. Asaresult,intheputty-claymodel,largecontractionaryshockscausesteep immediate declines in output and hours while large expansionary shocks generate a hump-shaped dynamic response even more pronounced than in the case of small expansionary shocks. Of particular interest here is the fact that the asymmetries on labor are the most pronounced, a result supported by Neftci (1984)’s non-linear time series analysis. 4 Permanent Technology Shocks An important unresolved issue in macroeconomics is the extent to which permanent innovations in technology can explain output fluctuations at the business cycle frequency. While past research has claimed varying degrees of success, more recent work has tempered enthusiasm for business cycle theories based on permanent technology shocks. Cogley and Nason (1995) and Rotemberg and Woodford (1996) demonstratethatthestandardneoclassicalmodelwithpermanentdisembodiedproductivity shocks is unable to match the persistence and comovement properties of key aggregate variables. Christiano and Eichenbaum (1992) show that such models predict excessive contemporaneous correlation between output growth and laborproductivity growth. In this section, we extend this literature in two directions by analyzing the e(cid:11)ects of permanent embodied, as well as disembodied, technology shocks and allowing for putty-clay technology. Greenwood et al. (1997) argue that the evidence supports embodied technology as the primary source of technological change, making the analysis of such shocks of particular interest. As shown below, the putty-clay model generates dynamic responses to permanent technology shocks that accord well with key properties of the data. We begin with an analysis of the persistence properties of the two models. The (cid:12)rstrow of table 1 shows the unconditional autocorrelation of outputgrowth for the two models for each type of technology shock.17 For comparison, this statistic is estimated to be0.3 in thedata. An alternative measure of persistence, emphasized by Rotemberg and Woodford (1996), is the ratio of standard deviations of forecastable output growth to total output movements. The second and third rows of the table report this statistic at the four- and eight-quarter horizons. For comparison, these 17All model moments reported in thispaper are computed using thelinearized model. 21

Table 1: Output Persistence with Permanent Technology Shocks Neoclassical Model Putty-clay Model Disembodied Embodied Disembodied Embodied cor((cid:1)y t ;(cid:1)y t−1 ) 0.01 0.07 0.03 0.80 (cid:27) =(cid:27) 0.06 0.25 0.14 0.83 (cid:1)y^t;4 (cid:1)yt;4 (cid:27) =(cid:27) 0.08 0.3 0.17 0.77 (cid:1)y^t;8 (cid:1)yt;8 Notes: Shocks are permanent. y denotes the log of output. (cid:1)y^ = t;j E (y − y ), where expectations are based on date t information. All t t+j t reported moments are asymptotic means. are estimated to be about 0.6{0.7 in the data. Two results stand out clearly in table 1. First, the putty-clay model delivers signi(cid:12)cantly more persistence in output growth (by either measure and for either type of permanent technology shock) than the neoclassical model. Second, embodied technology shocks generate much more persistence in output growth than disembodied shocks. This is especially true for the putty-clay model, which, when driven solely by permanent embodied technology shocks, actually overpredicts the persistence in output growth observed in U.S. post war data. Theintuitionbehindtheseresultsisprovidedbytheimpulseresponsestopermanent technology shocks, plotted in Figure 7. In both the putty-clay and neoclassical models, the long run e(cid:11)ect of a 1−(cid:11) percentage increase in technology is to raise output, consumption and investment by 1% whileleaving the long-run level of labor unchanged. In the case of disembodied shocks (shown in the panels on the right) total factor productivity increases immediately, causing an immediate expansion of output. This initial increase in output represents a large fraction of the permanent increase. As a result, for both the neoclassical and putty-clay models, nearly all of the output dynamics are unforecastable, and output growth displays very little persistence in response to disembodied shocks to technology. If technology shocks are embodied in capital (shown in the panels on the left) total factor productivity does not increase until the economy invests in new capital goods. In the neoclassical model, a positive shock to embodied technology still causes a large immediate expansion in outputas hours surge in responseto the high 22

Figure 7: Permanent Productivity Shocks Embodied Technology Shock Disembodied Technology Shock Output Output 1.2 1.2 1.0 1.0 0.8 0.8 0.6 0.6 0.4 0.4 Putty-Clay 0.2 0.2 Neoclassical 0.0 0.0 0 5 10 15 20 0 5 10 15 20 years years Consumption Consumption 1.5 1.2 1.0 1.0 0.8 0.5 0.6 0.4 0.0 0.2 -0.5 0.0 0 5 10 15 20 0 5 10 15 20 years years Labor Labor 1.0 0.30 0.8 0.25 0.20 0.6 0.15 0.4 0.10 0.2 0.05 0.0 0.00 0 5 10 15 20 0 5 10 15 20 years years rate of return to investment. Because of the rapid expansion in production, the initial output response represents a large fraction of the permanent response and output movements are mostly unpredictable. In the putty-clay model, a positive shock to embodied technology causes only a small initial expansion despite the increaseddesirefornewinvestment, owingtothehighcostsofexpandingproduction in the short-run. Both output and hours expand slowly as new capital is brought on-line, with labor reaching its peak response a number of years after the shock occurs. As a result, mostof the outputdynamics are predictable andoutputgrowth displays a high degree of persistence in response to embodied shocks to technology. 23

Table 2: Forecastable Comovement Between Output and Hours Neoclassical Model Putty-clay Model Disembodied Embodied Disembodied Embodied cor((cid:1)h^ ;(cid:1)y^ ) -1.00 -1.00 0.44 0.44 t;4 t;4 cor((cid:1)h^ ;(cid:1)y^ ) -1.00 -1.00 0.28 0.28 t;8 t;8 Notes: h denotes the log of hours. See also Table 1. In addition to the limited degree of internal propagation, Rotemberg and Woodford(1996)criticizethehighnegativecorrelationsbetweenpredictablemovementsin outputand hoursimplied by the standardneoclassical model in responseto random walk technology shocks. Table 2 formalizes this point by reporting the correlation between forecastable growth of output and hours four and eight quarters ahead. For comparison, these correlations are estimated to be about 0.86 in the data. For the neoclassical model, the negative correlation follows from the fact that the peak responsein hoursoccurs at the onset of the shock while outputcontinues to grow as capital accumulation proceeds. Thus, while hours are falling in a predictable fashion, outputis risingin apredictable fashion. Theputty-clay model’s slow expansion of output and hours reverses this correlation and provides a closer match with the data on this dimension. Finally, we consider the models’ predictions regarding the contemporaneous correlation between output and productivity growth. Table 3 reports this correlation for the two models for each of the two sources of permanent productivity shocks. For comparison, this correlation is estimated to be 0.76 in the data. If hours were held constant, both models would predict a perfectly positive correlation between growth in output and productivity. With disembodied shocks, this correlation is nearly unity as the e(cid:11)ect of the movements in hours on productivity are dwarfed by the e(cid:11)ects of the shock itself. Inthecaseof embodiedtechnology shocks, themodels’predictionsdi(cid:11)ergreatly. The neoclassical model predicts a highly negative correlation between growth in output and productivity. This negative correlation results from the immediate expansion of hours which produces a large decline in productivity at the same time as the largest increase in output. The putty-clay model, on the other hand, predicts 24

Table 3: Output and Productivity Growth Comovement Neoclassical Model Putty-clay Model Disembodied Embodied Disembodied Embodied cor((cid:1)y ;(cid:1)p )) 0.99 -0.71 1.00 0.50 t t Notes: p denotes the log of output per hour. See also Table 1. a positive correlation between output and productivity growth. This di(cid:11)erence lies in the muted expansion of hours, which does less to o(cid:11)set the positive comovement in productivity and output directly resulting from the shock. The putty-clay model thus provides an explanation for why the correlation between growth in output and productivity may be positive but less than unity, even in the absence of shocks to demand. Although embodied shocks help explain a numberof empirical regularities, some results are not consistent with the business cycle. In particular, both the neoclassical and putty-clay models create excess volatility of consumption and investment relative to the data (as seen in the set of moments reported in the appendix). This excess volatility occurs because factor cost movements create investment patterns that overwhelm the usual desire to smooth consumption. 5 Estimation Theresultsintheprevioussection highlighttheputty-claymodel’sabilitytoexplain keybusinesscyclefactssuchaspersistenceinoutputgrowthandpositivepredictable comovement between output and hours. In this section we provide a more formal evaluation of the empirical relevance of a putty-clay production process in matching key moments of U.S. aggregate data. To perform this evaluation, we construct a two-sector model that nests both the putty-clay model and the neoclassical model within the same econometric framework. Letting (cid:18) denote the level of technology embodied in capital, we assume that t sector 1 output is derived from the putty-clay production process describe above, while sector 2 output is derived from the Solow vintage-capital model described in the appendix. Letting A denote the level of disembodied technology, we assume t 25

that (cid:12)nal-goods output is a Cobb-Douglas function of sectoral output: Y = A Y(cid:21)Y1−(cid:21) t t 1t 2t We then estimate (cid:21), the share of output obtained from the putty-clay sector. If our estimate of (cid:21) is close to unity, the data e(cid:11)ectively put a large weight on puttyclay production in order to match the vector of moments that we consider. If our estimate of (cid:21) is close to zero, the data suggest little if any role for putty-clay in explaining the moments we choose to match. ToaddressrecentcriticismsregardingstandardRBC-stylemoment-matchingexercises, ourmethodologyreliesonbothunconditionalmomentstraditionallyemphasizedintheRBCliteratureandconditionalmoments emphasizedbyRotembergand Woodford (1996).18 The unconditional moments include the standard deviations of (the growth rates of) investment and hours relative to the standard deviation of output, the correlations of investment and hours with output, and the (cid:12)rst-order autocorrelations of these variables. The conditional moments include correlations and regression coe(cid:14)cients among predictable changes in output, investment and hours over a four-quarter horizon.19 These moments also include the ratio of the standard deviation of the predictable change in output relative to the standard deviation of the total change in output at this horizon. To compute moments constructed from predictable components we specify a VAR process for (y ;h ;i ), the logs of output, hours, and investment in the data.20 t t t As shown in the appendix, the statistical properties and hence all relevant moments ofthisVARaresummarizedbyanunknownparametervector(cid:24). Weestimate(cid:24) using standard time-series techniques and then use the resulting parameter estimate to compute a set of moments g((cid:24)^), along with the variance of these moments V . g Our estimation strategy is to choose (cid:21), the share of output accounted for by the 18Christiano and Eichenbaum (1992) provide a GMM procedure for estimating and evaluating business-cycle models based on unconditional moments. A contribution of this paperis to provide aGMM procedurethat allows for bothtypesof momentswithin auni(cid:12)edeconometric framework. 19Resultsusingacombination ofmomentscomputedfrom predictablechangesat the4,8and16 quarterhorizon donot alter our conclusions. 20Ourinvestment series is business(cid:12)xed investment(non-residential equipment and structures). In the model, investment is a linear combination of output and consumption so that consumption andinvestmentcontainthesameinformationwhencombinedwithoutput. Analternativeapproach to the data is to de(cid:12)ne investment as a linear combination of output and non-durables consumption. Although less desirable for a model explicitly designed to capture short-run capital/labor complementarities wehaveconsideredthisapproach,aswellasmatchingmomentscomputedfrom total consumption ratherthan investment. Neitherof thesealternatives alter ourempirical results in any substantial way. 26

putty-clay sector, along with other relevant parameters, to minimize the distance betweenmodelmomentsanddatamoments. Foragiven vector ofunknownmodel parameters,weuseourmodelsolutiontocomputeg ( ),themodel’sanalogofg((cid:24)). M By minimizing L( ) = (g ( )−g((cid:24)^)) 0 V −1(g ( )−g((cid:24)^)) M g M with respect to we obtain the minimum distance estimator ^. For a time-series sample of size T, T (cid:3)L( ^) provides a (cid:31)2 test for equality between g ( ^)and g((cid:24)^): M For our moment matching exercise, we consider two independantsources of fluctuations: disembodiedtechnological changeandembodiedtechnological change. We assumethatshocksfollowanAR1processandthenfreelyestimate[(cid:26) ;(cid:26) ],theauto- A (cid:18) correlations of the shock processes, and (cid:27) (cid:18) , the relative importance of embodied (cid:27)A+(cid:27) (cid:18) shocks. Togeneralizeourresultsbeyondtechnologyshocks,wealsoconsideramodel that includes labor cost shocks. Under this speci(cid:12)cation, we also estimate the autocorrelation of labor cost shocks, and the percentage of fluctuations attributable to labor cost shocks. Estimation results based on unconditional moments reported in table 4 place a large weight on the putty-clay production technology { on the order of 50% for the modelthatdoesnotincludelaborcostshocks.21 Theestimate of (cid:27) (cid:18) isabout0.2, (cid:27)A+(cid:27) (cid:18) suggestingasubstantialroleforembodiedtechnology shocksinexplainingaggregate dynamics.22 For comparison purposes, table 4 also reports h ( ) and T (cid:3)L( ) for M the standard RBC model with disembodied shocks and (cid:26) = 0:95. Relative to this A baseline, allowing forembodiedshocks andanon-zero weight onputty-clay provides a substantial gain in terms of (cid:12)t, reducing T (cid:3)L( ) by 50%.23. In this speci(cid:12)cation, the estimated values of (cid:26) and (cid:26) reach their upper bound of unity, emphasizing A (cid:18) the importance of persistent shocks when matching unconditional moments. Introducinglaborcostshocksprovidesfurthergainsin(cid:12)tandplacesevengreater emphasis on putty-clay technology. The gain in (cid:12)t comes from a relatively large fraction of fluctuations being accounted for by labor cost shocks. Our estimation results also set (cid:26) , the autocorrelation of labor cost shocks, at an imposed upper L 21The putty-clayshare is estimated precisely with a standard error of about 0.02. 22As an alternative to estimating (cid:27)(cid:18) , we considered (cid:12)xing this ratio at 0.6, Greenwood et (cid:27)A+(cid:27)(cid:18) al. (1997)’s estimate of the share of post-war technological change embodied in capital. Assuming randomwalkshocksandsetting(cid:27) =0,weestimateaputty-clayshareof0.69,indicatingthatour L estimate of theputty-clayshare is robust to changes in themix of technology shocks. 23Nonetheless, we still reject a test of equality between model and data moments. This may partly reflect thepoor small-sample properties of such tests (Burnside and Eichenbaum1996). 27

Table 4: Unconditional Moments: Estimation of Two-Sector Model Neoclassical Two-Sector Data Benchmark Model Est. S. E. Model Parameters: Putty-Clay Share 0 0.47 0.67 (cid:27) (cid:18) 1 0 0.23 0.21 (cid:27) (cid:18) +(cid:27)A (cid:27)L 2 0 0 0.55 (cid:27) (cid:18) +(cid:27)A+(cid:27)L (cid:26) 1.00 1.00 (cid:18) (cid:26) 0.95 1.00 1.00 A (cid:26) 0.993 L Moments: (cid:27) =(cid:27) 0.57 0.39 0.66 0.65 0.04 (cid:1)h (cid:1)y (cid:27) =(cid:27) 2.85 2.49 2.39 1.80 0.11 (cid:1)i (cid:1)y cor((cid:1)h ;(cid:1)y ) 0.99 0.69 0.61 0.74 0.04 t t cor((cid:1)i ;(cid:1)y ) 1.00 0.87 0.89 0.61 0.05 t t cor((cid:1)y t ;(cid:1)y t−1 ) -0.02 0.08 0.12 0.32 0.08 cor((cid:1)h t ;(cid:1)h t−1 ) -0.04 0.21 0.26 0.61 0.06 cor((cid:1)i t ;(cid:1)i t−1 ) -0.03 0.07 0.12 0.50 0.07 Minimized Objective: 708 361 246 1. (cid:27)(cid:18) measures the share of technology shocks embodied in capital. (cid:27)(cid:18)+(cid:27)A 2. (cid:27)L measures the relative magnitude of labor market shocks. (cid:27)(cid:18)+(cid:27)A+(cid:27)L 3. Estimate constrained by upper bound of 0.99. bound of 0.99. Thus, even with the introduction of labor cost shocks, our moment matching exercise still places a strong emphasis on highly persistent shocks. Given the importance of all three shocks in the estimation procedure, it is interesting to ask what fraction of output variance is accounted for by each shock. Variance decompositionsusingestimatedparametervaluesimplythatdisembodied,embodied and labor cost shocks account for 76%, 4% and 20% of output fluctuations at the one-year horizon and 63%, 11% and 26% at the (cid:12)ve-year horizon. Estimation results based on the full set of moments reported in Table 5 also imply a large putty-clay share { on the order of two-thirds, regardless of the shock processes. Introducing putty-clay technology provides substantial gain in (cid:12)t, which isfurtherimprovedthroughtheintroductionoflaborcostshocks. Whenconsidering the full set of moments, the estimate of (cid:27) (cid:18) drops from about 0.2 to zero, and (cid:27)A+(cid:27) (cid:18) the persistence of disembodied shocks falls to 0.94. The drop in (cid:27) (cid:18) is o(cid:11)set (cid:27)A+(cid:27) (cid:18) 28

Table 5: All Moments: Estimation of Two-Sector Model Neoclassical Two-Sector Data Benchmark Model Est. S. E. Model Parameters: Putty-Clay Share 0 0.68 0.64 (cid:27) (cid:18) 0 0.01 0.00 (cid:27) (cid:18) +(cid:27)A (cid:27)L 0 0 0.64 (cid:27) (cid:18) +(cid:27)A+(cid:27)L (cid:26) 0.98 (cid:18) (cid:26) 0.95 0.95 0.94 A (cid:26) 0.991 L Moments: (cid:27) =(cid:27) 0.57 0.30 0.67 0.65 0.04 (cid:1)h (cid:1)y (cid:27) =(cid:27) 2.85 2.60 2.67 1.80 0.11 (cid:1)i (cid:1)y cor((cid:1)h ;(cid:1)y ) 0.99 0.96 0.75 0.74 0.04 t t cor((cid:1)i ;(cid:1)y ) 1.00 0.99 0.99 0.61 0.05 t t cor((cid:1)y t ;(cid:1)y t−1 ) -0.02 0.06 0.09 0.32 0.08 cor((cid:1)h t ;(cid:1)h t−1 ) -0.04 0.24 0.25 0.61 0.06 cor((cid:1)i t ;(cid:1)i t−1 ) -0.03 0.04 0.08 0.50 0.07 cor((cid:1)h^ ;(cid:1)y^ ) 0.89 0.90 0.82 0.89 0.04 t;4 t;4 cor((cid:1)^i ;(cid:1)y^ ) 0.94 0.84 0.80 0.86 0.05 t;4 t;4 (cid:11) 0.80 0.43 0.53 0.64 0.10 ((cid:1)h^ t;4;(cid:1)y^t;4) (cid:11) 3.60 2.99 2.80 1.83 0.27 ((cid:1)^it;4;(cid:1)y^t;4) (cid:27) =(cid:27) 0.27 0.25 0.26 0.62 0.07 (cid:1)y^t;4 (cid:1)yt;4 Minimized Objective: 1148 786 444 1. Estimate constrained by upper bound of 0.99. by an increase in the percentage of fluctuations obtained from labor cost shocks. Computing variance decompositions using estimates in Table 5, we (cid:12)nd that labor cost shocks account for 15% of outputfluctuations at theone-year horizon, and47% at the (cid:12)ve-year horizon. These results imply that disembodied shocks to technology do well at explaining output movements at high frequencies, while persistent shocks to factor costs do better at lower frequencies.24 24Asrobustnesscheckstoourestimationresults,wehaveconsideredanumberofissues,including alternativeparameterizationsoftheutilityfunctionandthepresenceofconvexadjustmentcostsfor investment. Forestimation based on unconditional moments, lowering theintertemporal elasticity of substitution or adding adjustment costs increases both the putty-clay share and (cid:27)(cid:18) . By (cid:27)(cid:18)+(cid:27)A raising γ or introducing adjustment costs, we reducethe volatility of consumption and investment 29

6 Conclusion Bycombininginvestmentirreversibilities,capacityconstraints,andvariablecapacity utilization, the putty-clay model developed in this paper provides a rich framework for analyzing a number of issues regarding investment, labor, capacity utilization, andproductivity. Inthispaperwehighlightsomeimplications foremployment, output, and investment at business cycle and medium-run frequencies. Compared to standard neoclassical models, the putty-clay model displays a substantial degree of persistence and propagation for both output and hours in response to shocks to factor costs and technology. And, unlike standard neoclassical models, the putty-clay modelgenerates forecastable comovements between labor, output, andconsumption consistent with the data. Finally, owing to the existence of a nonlinear aggregate supply curve, the putty-clay model generates interesting asymmetries with recessions steeper and deeper than expansions. Beyond its descriptive appeal, the putty-clay production process is also found to be empirically relevant for explaining business-cycle and medium-run dynamics. Estimates obtained from a two-sector model that minimize the distance between moments generated by the model and those obtained from the data place a sizable weight on putty-clay production { on the order of one-half to three-fourths. This (cid:12)nding is robust to the speci(cid:12)cation of the shock processes and the choice of moments used in the estimation. In addition to supporting a major role for putty-clay technology, our results suggest that factor price shocks may be key to explaining fluctuations, especially at the medium-run frequencies of two to eight years. This paper focuses on the e(cid:11)ects of factor-cost shocks and technological change for business-cycle dynamics. The putty-clay model developed here has broader applications for(cid:12)scal,monetaryandtradepolicy, andthestudyoftransitionaldynamics for growing economies. In particular, this paper highlights the notion that the short-run e(cid:11)ects of policy may be substantially di(cid:11)erent from their medium-term consequences, owing to the linkage between the capital accumulation process and labor market dynamics. in response to movements in factor costs. When considering the full set of moments we (cid:12)nd little sensitivity to such speci(cid:12)cations. 30

References Anderson, Gary and George Moore,\ALinearAlgebraicProcedureforSolving Linear Perfect Foresight Models," Economics Letters, 1985, 17, 247{252. Ando, Albert, Franco Modigliani, Robert Rasche, and Stephen Turnovsky, \On the Role of Expectations of Price and Technological Change inanInvestmentFunction,"International Economic Review,1974,15,384{414. Atkeson, Andrew and Patrick J. Kehoe, \Putty-Clay Capital and Energy," August 1994. NBER Working Paper No. 4833. Basu, Susanto and John G. Fernald, \Returns to Scale in U.S. Production: Estimates and Implications," Journal of Political Economy, 1997, 105, 249{83. Benhabib, Jess and Aldo Rustichini, \Vintage Capital, Investment and Growth," Journal of Economic Theory, 1991, 55, 323{339. and , \A Vintage Capital Model of Investment and Growth: Theory and Evidence," in Becker R. et al., ed., General Equilibrium, Growth and Trade. II The Legacy of Lionel W. McKenzie,NewYork: AcademicPress,1993, pp.248{ 301. Bischo(cid:11), Charles W., \The E(cid:11)ect of Alternative Lag Distributions," in G. Fromm, ed., Tax Incentives and Capital Spending, Washington, D.C.: The Brookings Institution, 1971, chapter 3, pp. 61{130. Blanchard, Olivier J., \The Medium Run," Brookings Papers on Economic Activity, 1997, pp. 89{158. Boucekkine, Raouf, Marc Germain, and Omar Licandro, \Replacement Echoes in the Vintage Capital Growth Model," Journal of Economic Theory, 1997, 74, 333{48. Bresnahan, Timothy F. and Valerie A. Ramey, \Output Fluctuations at the Plant Level," Quarterly Journal of Economics, 1994, 109, 593{624. Burnside, Craig and Martin Eichenbaum, \Small Sample Properties of GMM Based Wald Tests," Journal of Business Economics and Statistics, 1996. 31

Caballero, Ricardo J. and Mohamad L. Hammour, \On the Timing and E(cid:14)ciency of Creative Destruction," Quarterly Journal of Economics, 1996, 111, 805{52. and , \Incomplete Contracts, Factor Proportions, and Unemployment," Carnegie Rochester Conference Series on Public Policy, 1998, 48. , Eduardo Engel, and John Haltiwanger, \Plant-Level Adjustment and Aggregate Investment Dynamics," Brookings Papers on Economic Activity, 1995, pp. 1{54. Calvo, Guillermo A., \Optimal Growth in a Putty-Clay Model," Econometrica, 1976, 44, 867{878. Campbell, Je(cid:11)rey R., \Technical Change, Di(cid:11)usion and Productivity," 1994. Mimeo, Rochester University. Cass, DavidandJosephE.Stiglitz,\TheImplicationsofAlternativeSavingand Expectations Hypotheses for Choices of Technique and Patterns of Growth," Journal of Political Economy, 1969, 77, 586{627. Christiano, Lawrence J. and Martin Eichenbaum, \Current Real-Business- Cycle Theories and Aggregate Labor-Market Fluctuations," American Economic Review, 1992, 82, 430{450. Cogley, Timothy and James M. Nason, \Output Dynamics in Real Business Cycle Models," American Economic Review, 1995, 85, 492{511. Cooley, Thomas F., Gary D. Hansen, and Edward C. Prescott, \Equilibrium business cycles with idle resources and variable capacity utilization," Economic Theory, 1995, 6, 35{49. , Jeremy Greenwood, and Mehmet Yorukoglu, \TheReplacement Problem," 1994. Institute for Empirical Macroeconomics Discussion Paper 95. Cooper, Russel and John Haltiwanger, \The Macroeconomic Implications of Machine Replacement: Theory and Evidence," American Economic Review, 1993, 83, 360{382. 32

, , and Laura Power, \Machine Replacement and the Business Cycle: Lumps and Bumps," 1995. NBER Working Paper #5260. Cover, JamesPerry,\AsymmetricE(cid:11)ectsofPositiveandNegativeMoney-Supply Shocks," Quarterly Journal of Economics, 1992, 107, 1261{1282. Doms, Mark and Timothy Dunne, \An Investigation into Capital and Labor Adjustment at the Plant Level," 1993. Mimeo, Center for Economic Studies, Bureau of the Census. Fair, Ray C. and John B. Taylor, \Solution and Maximum Likelihood Estimation of Dynamic Nonlinear Rational Expectations Models," Econometrica, July 1983, 51 (4), 1169{1185. Gilchrist, Simon and John C. Williams, \Cutting Models Down to Size: E(cid:14)cient Methods for Solving Large Rational Expectations Models," 1998. Mimeo, Board of Governors of the Federal Reserve. and , \Putty-Clay and Investment: A Steady State Analysis," 1998. Mimeo, Board of Governors of the Federal Reserve. Greenwood, Jeremy, Zvi Hercowitz, and Per Krusell, \Long-Run Implications of Investment-Speci(cid:12)c Technological Change," American Economic Review, 1997, pp. 342{362. Hamilton, James D., Time Series Analysis, Princeton University Press, 1994. Hansen, Gary, \Indivisible Labor and the Business Cycle," Journal of Monetary Economics, 1985, 16, 309{327. Johansen, Leif, \Substitution versus Fixed Production Coe(cid:14)cients in the Theory of Economic Growth: A Synthesis," Econometrica, 1959, 27, 157{176. Johnson, Norman L., Samuel Kotz, and N. Balakrishnan, Continuous Univariate Distributions, Volume 1, Second Edition, New York: John Wiley & Sons, 1994. Judd, Kenneth L., \Projection Methods for Solving Aggregate Growth Models," Journal of Economic Theory, December 1992, 58, 410{452. , Numerical Methods in Economics, Cambridge, MA.: MIT Press, 1998. 33

Kydland, Finn E. and Edward C. Prescott, \Time to Build and Aggregate Fluctuations," Econometrica, 1982, 50, 1345{1370. and , \Hours and Employment Variation in Business Cycle Theory," Economic Theory, 1991, 1 (1), 63{81. Lucas, Robert E., \Understanding Business Cycles," Carnegie Rochester Conference Series on Public Policy, 1977, 5, 7{29. Malinvaud, Edmond, Pro(cid:12)tability and Unemployment, Cambridge, England.: Cambridge Press, 1980. Mork, Knut A., \Oil and the Macroeconomy when Prices go Up and Down," Journal of Political Economy, 1989, 79, 173{189. Neftci, Salih N., \Are Economic Time Series Asymmetric over the Business Cycle?," Journal of Political Economy, 1984, 92, 307{28. Phelps, EdmundS.,\Substitution,FixedProportions,GrowthandDistribution," International Economic Review, 1963, 4, 265{288. Potter, Simon M., \A Nonlinear Approach to U.S. GNP," Journal of Applied Econometrics, 1995, 10, 109{125. Rogerson, Richard, \Indivisible Labor, Lotteries and Equilibrium," Journal of Monetary Economics, 1988, 21, 3{16. Rotemberg, Julio J.andMichaelWoodford,\RealBusiness-CycleModelsand the Forecastable Movements in Output, Hours, and Consumption," American Economic Review, 1996, 86, 71{89. Sheshinski, E., \Balanced Growth and Stability in the Johansen Vintage Model," Review of Economic Studies, 1967, 34, 239{248. Sichel, Daniel E., \Business Cycle Asymmetry: A Deeper Look," Economic Inquiry, 1993, 31, 224{236. Solow, R.M., \Substitution and Fixed Proportions in the Theory of Capital," Review of Economic Studies, 1962, 29, 207{218. Tatom, John A., \Are the Macroeconomic E(cid:11)ects of Oil Changes Symmetric?," Carnegie Rochester Conference Series on Public Policy, 1988, 28, 325{368. 34

Appendix In this appendix we describe the derivation of the neoclassical vintage model, the solution methodsfor the dynamicmodel, details of the econometric procedurealong with a complete description of the two-sector model used in section 5. At the end of this appendix we also provide complete tables for the moments of the neoclassical and putty-clay models with permanent technology shocks. Derivation of the Neoclassical Vintage Model: For the purposeof comparison, we construct a neoclassical modelof vintage capital, initially introduced by Solow (1962). In this model, the restriction that ex post capital-labor ratios are (cid:12)xed is removed. The two models are otherwise identical. LetI t−j denote aggregate investment inperiodt−j, andde(cid:12)nethefollowing capital aggregator XM K t = (cid:18) t 1 − =(cid:11) j (1−(cid:14) j )I t−j j=1 Period t labor and output of machine i created in period t−j satis(cid:12)es: L t;t−j (i) =((1−(cid:11))(cid:18) it−j =W t )1=(cid:11)(1−(cid:14) j )I t−j and Y t;t−j (i) = (cid:18) i;t−j 1=(cid:11)((1−(cid:11))=W t )(1−(cid:11))=(cid:11)(1−(cid:14) j )I t−j where (1−(cid:14) j )I t−j represents the capital remaining at time t that was put in place at time t−j. Integrating over machines with respect to the distribution of (cid:18) i;t−j , and using the result that (cid:27)2 1−(cid:11) E((cid:18) i;t−j 1=(cid:11))= (cid:18) t 1 − =(cid:11) j exp( 2(cid:11) (cid:11) ) provides the relationship between total vintage t−j output and labor inputs: (1−(cid:11))Y t;t−j = W t L t;t−j where (cid:16) (cid:17) Y t;t−j = A1−(cid:11) (cid:18) t 1 − =(cid:11) j (1−(cid:14) j )I t−j (cid:11) L1 t; − t− (cid:11) j and A= exp((cid:27)2 ). Summing across vintages we obtain 2(cid:11) Y = ((1−(cid:11))A=W )(1−(cid:11))=(cid:11)K t t t 35

L =((1−(cid:11))A=W )1=(cid:11)K : t t t Combining these two expressions gives the aggregate production function Y = A1−(cid:11)K(cid:11)L1−(cid:11) t t t where L is aggregate labor input and A= exp((cid:27)2 ) is a scale correction that results t 2(cid:11) from aggregating across machines at the idiosyncratic level. If we assume that (cid:14) = 1 − (1 − (cid:14))j−1 and M = 1; we obtain the following j capital accumulation equation K t = (1−(cid:14))K t−1 +(cid:18) t 1 − =(cid:11) 1 I t−1 Thus, shocks to embodied technological change are identical to shocks to the true economic cost of new capital goods. Solution Methods: The model consists of 2M +N state variables, including average machine e(cid:14)ciencies, X t−j ;j = 1;:::;M, and the quantity of new machines per vintage, Q t−j ;j = 1;:::;M, for each of the M vintages in existence, and N shocks. Owing to the log-normal distribution of X , these state variables completely summarize the exit isting distribution of machines depicted in Figure 1. By choosing M su(cid:14)ciently large we provide an arbitrarily good approximation to the case M = 1. For values of M large enough to analyze business-cycle frequency properties of the model, the state-space is too large for the type of nonlinear state-space methods discussed in Judd (1998). Instead, depending on the purpose, we apply one of two methods that yield approximate solutions at relatively low computational cost. The(cid:12)rstmethodusesalog-linearization aroundthedeterministicsteadystateof theequations describingtheequilibrium. Theresultinglinear model,whichincludes M leads and lags of ln(X) and ln(Q), is then solved using the AIM implementation of the Blanchard-Kahn method due to Anderson and Moore (1985). This algorithm yields accurate solutions for values at relatively low computational cost for M up to 40, which is su(cid:14)ciently large for an annual version of the model. For the quarterly version of the model, M = 40 implies capital goods completely depreciate within 10 years, which seems to be an unrealistically short lifespan. Thus, following Gilchrist and Williams (1998a), we use polynomial distributed leads and lags to approximate the M = 1 leads and lags of variables. For example, for some variable u and 36

P coe(cid:14)cient sequence fa j g1 1 , we approximate the sum 1 j=1 a j u t−j bythe polynomial distributedlag(orlead) ut−1,whereB(L) = b −b L−b L2−:::−b Lp andpis(cid:12)nite. B(L) 0 1 2 p For the putty-clay model we use p = 1 and chose values corresponding to b and 0 b that minimize the weighted squared deviations between the PDL representation 1 and the original lag or lead structure; this approximations yields virtually no loss in accuracy for model simulations. This adds 8 equations to our model and reduces the maximum lead and lag from M to 1, thus drastically reducing the size of the companion form of the model. The solution time for this approximate model is trivial. The log-linearized model, approximated in the manner described above, is used for the simulations reported in the paper except for those illustrating the asymmetrical response to large shocks shown in (cid:12)gure 6, which requires a method that preserves the nonlinearities of the model. For those simulations an extended path algorithm based on Fair and Taylor (1983) is used for a model with M = 160. This method requires far more memory and CPU time than the linearization method described above. For small shocks, the two methods yield nearly identical answers. Both of these methods compute \certainty-equivalent" solutions; that is, expectations are computed assuming all futureshocks equal their mean values of zero. To address this issue, we used projection methods to solve a version of the model with small M. This approach, described in Judd (1992), uses polynomial approximations to the decision rules and multi-point quadrature to approximate expectation integrals and therefore does not impose certainty equivalence on the solutions. We found that the solutions were very close to those generated using the extended path algorithm, implying that the certainty equivalent solution provides a good approximation even in the presence of substantial non-linearities and reasonably large aggregate shocks. Econometric Methodology for Moment Matching Exercise Inthissection ofthepaperwepresentourmomentmatchingmethodology. Westart by specifying a stochastic process for the log of output, hours and investment in the data which depends on an unknown parameter vector (cid:18) which is to be estimated. Our approach follows Rotemberg and Woodford’s method of specifying a tightly parameterized low-order VAR system to characterize the data. Our data consists of private non-farm output, total private hours and non-residential business (cid:12)xed 37

investment (equipment and structures) for the period 1960-1997. Relative to GDP, the output and hours series exclude government and farm output as well as the imputed output obtained from owner-occupied housing. The output and hours series are thus de(cid:12)ned in a mutually consistent manner. With the exception of agricultural investment in structures and machinery, the investment series is also consistent with the output and hours series. Both the output and the investment series are deflated using 1997 chain weighted-deflators. After (cid:12)rst removing a linear time trend from the hours series, we assume that the (cid:12)rst di(cid:11)erence of the log of output, the log of the investment/output ratio and the log of hours, [(cid:1)y ;i −y ;h ] can be represented using a two-lag stationary VAR t t t t representation. De(cid:12)ning u 0 t = [(cid:1)y t ;i t −y t ;h t ;(cid:1)y t−1 ;i t−1 −y t−1 ;h t−1 ]; E(u t u 0 t )=(cid:6) u e 0 = [ey;ei;eh]; E(e e 0 ) = (cid:6) ; E(e e 0 ) = 0for s 6= 0 t t t t t t e t t+s we express the stochastic process for u in companion form as: t " # " # u t = Au t−1 +v t ; A = I (cid:5) 0 ; v t = e 0 t ; E(v t v t 0 )=(cid:6) v where(cid:5)is amatrixof VARcoe(cid:14)cients. De(cid:12)ningD as theduplication matrixsuch n that D vech((cid:6) ) = vec((cid:6) ); the parameter vector (cid:24) and its associated variance is n e e then: " # " # vec((cid:5)) (cid:6) ⊗(cid:6) −1 0 (cid:24) = ; V = e u vech((cid:6) ) (cid:24) 0 (cid:6) e 22 where (cid:6) = 2D+((cid:6) ⊗ (cid:6) )(D+) 0 for D+ = (D 0 D ) −1D 0 . (Hamilton (1994) pp 22 n e e n n n n n 301-302 provides details.) Given this speci(cid:12)cation for the stochastic process for output, hours and investment, we are interested in computing second moments of both the kth di(cid:11)erences of these variables as well as second moments of the forecastable components of the kth di(cid:11)erences, where the forecast is made conditional on time t−k information. To obtain expressions for these second moments as functions of the underlyingVAR parameters, we (cid:12)rst express the kth di(cid:11)erences in y ;i and h as functions t+k t+k t+k of data known at time t and shocks that occur between t and t+k: For x = y;i;h we have: Xk (cid:1)x (cid:17) x −x = bxu + dxv t;k t+k t k t j t+j j=1 38

where Xk by = e 0 Ai; bi = by +e 0 (Ak −I); bh = e 0 (Ak −I) k 1 k k 2 k 3 i=1 Xj dy = e 0 As−1; di = dy +e 0 Aj−1; dh = e 0 Aj−1: j 1 j j 2 j 3 s=1 Taking expectations as of time t, the predictable components of the kth di(cid:11)erence of x is: (cid:1)x^ (cid:17) E fx −x g= bxu t;k t t+k t k t Computing second moments, we obtain an expressions for the variance of (cid:1)x t;k Xk (cid:27)2 (cid:17) E((cid:1)x )2 = (bx)(cid:6) (bx) 0 + (dx)(cid:6) (dx) 0 x;k t;k k u k j v j j=1 The covariance between (cid:1)x ; and (cid:1)y is obtained from t;k t;k Xk (cid:27) (cid:17) E((cid:1)x (cid:1)y )=(bx)(cid:6) (b y ) 0 + (dx)(cid:6) (d y ) 0 : xy;k t;k t;k k u k j v j j=1 Similarly, the variance of the predictable component of the kth di(cid:11)erence in x is simply (cid:27)2 (cid:17) E((cid:1)x^ )2 = (bx)(cid:6) (bx) 0 x^;k t;k k u k while the covariance between the predictable components of the kth di(cid:11)erences of x with y is obtained from: (cid:27) (cid:17) E((cid:1)x^ (cid:1)y^ )=(bx)(cid:6) (by) 0 x^y;k t;k t;k k u k Tocomputeautocorrelationsamongkthdi(cid:11)erences,notethatE((cid:1)x (cid:1)x ) = t+k;k t;k E((cid:1)x (cid:1)x ) − E((cid:1)x )2. From the expressions obtained above we then have t;2k t;k t;k E((cid:1)x (cid:1)x )−E((cid:1)x )2 = bx (cid:6) (bx) 0−bx(cid:6) (bx) 0 implying that t;2k t;k t;k 2k u k k u k E((cid:1)x ;(cid:1)x ) (bx −bx)(cid:6) (bx) 0 (cid:26) (cid:17) t+k;k t;k = 2k k u k x;k E((cid:1)x )2 (bx)(cid:6) (bx)0 t;k k u k Usingtheseexpressions,wede(cid:12)netwosets ofmoments. The(cid:12)rstsetofmoments are the unconditional standard deviations of (cid:1)i ;(cid:1)h relative to the uncondit;k t;k tional standard deviation of (cid:1)y , the unconditional correlations of (cid:1)i ;(cid:1)h t;k t;k t;k with (cid:1)y , and the autocorrelations of (cid:1)y ;(cid:1)i and (cid:1)h : t;k t;k t;k t;k " # 0 (cid:27) (cid:27) (cid:27) (cid:27) i;k h;k iy;k hy;k h = ; ; ; ;(cid:26) ;(cid:26) ;(cid:26) : 1;k y;k i;k h;k (cid:27) (cid:27) ((cid:27) (cid:27) ) ((cid:27) (cid:27) ) y;k y;k i;k y;k h;k y;k 39

For k = 1 these moments are the unconditional moments considered in section 6. The second set of moments computes regression coe(cid:14)cients and correlations between the predictable components of (cid:1)i ;(cid:1)h and the predictable component t;k t;k of (cid:1)y . According to Rotemberg and Woodford, these measures of the underlying t;k dynamic response of the system when away from steady state. Intuitively, these statistics capture the strength and magnitude of the expected comovements among output, investment and hours. By varying k; we vary the horizon over which the comovements between forecastable components are computed. The second set of moments also includes the ratio of the variance of the predictable component of (cid:1)y relative to the total variance of (cid:1)y . Again, Rotemberg and Woodford view t;k t;k thisstatistic as providingagoodindicator ofthedegreetowhichthemodelcontains a propagation mechanism. Thus the second set of moments may be written as: " # 0 (cid:27) i^y;k (cid:27) h^y;k (cid:27) i^y;k (cid:27) h^y;k (cid:27) y^;k g = ; ; ; ; : 2;k (cid:27)2 (cid:27)2 (cid:27) (cid:27) (cid:27) (cid:27) (cid:27) y^;k y^;k ^{;k y^;k h^;k y^;k y;k where the (cid:12)rst two elements are regression coe(cid:14)cients, the second two elements are correlations, and the (cid:12)nal element is the variance ratio. Recognizing that both g and g are functions of A;(cid:6) ;(cid:6) and hence ultimately functions of the VAR 1;1 2;k u v 0 0 0 parameter vector (cid:24); we obtain our moment vector of interest: g((cid:24)) = [g g ]: To 1;1 2;k obtain an estimate of the variance of this moment vector, we use a Taylor series expansion of g (the delta method) to obtain V = @gV @g: g @(cid:24)0 (cid:24)@(cid:24) We estimate the model parameters = [a;(cid:26) ;(cid:26) ;(cid:26) ; (cid:27) (cid:18) ; (cid:27)L ] by min- A (cid:18) L (cid:27) (cid:18) +(cid:27)A (cid:27) (cid:18) +(cid:27)A+(cid:27)L imizing the distance between g((cid:24)) and g ( ) where g is the model’s analog of M M g((cid:24)). To obtain g as a function of we rely on the fact that our model solution M is linear and may be expressed in the (cid:12)rst order companion form: X t = AX t−1 +BU t 0 0 0 where E(U U ) = I and X is a vector of model variables with y = e X ;c = e X t t t t y t t y t 0 and h = e X : The moment vector g may then be computed as a function of t h t M A;B. Because the matrices A;B are (nonlinear) functions of the underlying model parameters ; the moment vector g is also a function of : Standard errors for M may then be obtained from V = f@gMV −1@gMg−1: @ 0 g @ Two Sector Model: In this section we provide a full description of the system of equations that characterize the equilibrium of the two-sector model. We set the (cid:12)nal-goods price as 40

the numeraire and de(cid:12)ne P and P as the relative output prices for each sector. 1;t 2;t Disembodiedtechnology, A ,a(cid:11)ects bothsectorsequally.25 Wede(cid:12)ne(cid:17) asthelabor t t market shock. Sectoral Production: XM (cid:16) (cid:17) Y 1;t = 1−(cid:8)(z t t−j −(cid:27)) (1−(cid:14) j )Q t−j X t−j j=1 XM (cid:16) (cid:17) L 1;t = 1−(cid:8)(z t t−j ) (1−(cid:14) j )Q t−j j=1 Y = K(cid:11)L1−(cid:11) 2;t t 2;t K t = (1−(cid:14))K t−1 +(cid:18) t 1 − =(cid:11) 1 I t−1 : Optimality conditions for sectoral labor and capital accumulation: ( (cid:18) (cid:19) (cid:27) XM k = (cid:11)E R~ (1−(cid:14) ) 1−(cid:8)(zt −(cid:27)) P X t t t;t+j j t+j 1;t+j t j=1 (cid:26) XM (cid:16) (cid:17) k = E R~ (1−(cid:14) ) 1−(cid:8)(zt −(cid:27)) P X t t t;t+j j t+j 1;t+j t j=1 XM (cid:16) (cid:17) (cid:27) − R~ (1−(cid:14) ) 1−(cid:8)(zt ) W t;t+j j t+j t+j j=1 (cid:18) (cid:19) (cid:18) −1=(cid:11) = E R (cid:11)P 2;t+1 Y 2;t+1 +(1−(cid:14))(cid:18) −1=(cid:11) t t t;t+1 K t+1 t+1 (1−(cid:11))P Y 2;t 2;t = W t L 2;t (cid:30)(z t t−j −(cid:27))P 1;t X t−j = W ; j = 1;:::M: t−j t (cid:30)(z ) t This last expression equates marginal(cid:18)products across all vintages(cid:19). It could have alternatively been written as z t t−j (cid:17) (cid:27) 1 logW t −logP 1;t X t−j + 1 2 (cid:27)2 . Household (cid:12)rst order conditions: (cid:12) U = E R U c;t t t;t+1 c;t+1 1+n 25Notethatthee(cid:11)ectofthelevelofdisembodiedtechnologyonthe(cid:12)rst-orderconditionsof(cid:12)rms is captured by theoutput price terms. 41

U (cid:17) W +U = 0: c;t t t L;t Aggregate output and resource constraints: Y = A Y(cid:21)Y1−(cid:21) t t 1;t 2;t C = Y −k Q −I t t t t t L = L +L : t 1;t 2;t Shocks: ln(A t ) = (cid:26) A ln(A t−1 )+e A;t ; with E(e2 A;t )= (cid:27) A 2 ln((cid:18) t ) = (cid:26) (cid:18) ln((cid:18) t−1 )+e (cid:18);t ; with E(e2 (cid:18);t ) = (cid:27) (cid:18) 2 ln((cid:17) t ) = (cid:26) L ln((cid:17) t−1 )+e (cid:17);t ; with E(e2 (cid:17);t ) = (cid:27) L 2 Permanent Technology Shocks: Tables 6 and 7 report the unconditional and forecastable moments, respectively, resultingfrompermanenttechnologyshocksforthetwomodelsaswellasthestatistics estimated in the data. 42

Table 6: Unconditional Moments { Permanent Technology Shocks Neoclassical Model Putty-Clay Model Data Disemb. Embod. Disemb. Embod. Est. S. E. (cid:27) =(cid:27) 0.55 1.13 0.70 2.87 0.36 0.02 (cid:1)c (cid:1)y (cid:27) =(cid:27) 0.36 1.57 0.09 0.91 0.66 0.04 (cid:1)h (cid:1)y (cid:27) =(cid:27) 2.15 6.03 1.76 8.14 1.82 0.11 (cid:1)i (cid:1)y (cid:27) =(cid:27) 0.65 0.69 0.91 0.77 0.68 0.04 (cid:1)p (cid:1)y cor((cid:1)c ;(cid:1)y ) 0.99 -0.83 0.99 -0.12 0.52 0.06 t t cor((cid:1)h ;(cid:1)y ) 0.98 0.95 0.96 0.68 0.74 0.04 t t cor((cid:1)i ;(cid:1)y ) 0.99 0.97 0.99 0.53 0.60 0.05 t t cor((cid:1)p ;(cid:1)y ) 0.99 -0.71 1.00 0.50 0.76 0.03 t t cor((cid:1)y t ;(cid:1)y t−1 ) 0.01 0.07 0.04 0.80 0.31 0.08 cor((cid:1)c t ;(cid:1)c t−1 ) 0.10 0.05 0.10 0.06 0.34 0.08 cor((cid:1)h t ;(cid:1)h t−1 ) -0.03 -0.03 0.20 0.20 0.60 0.06 cor((cid:1)i t ;(cid:1)i t−1 ) -0.02 -0.02 -0.01 -0.02 0.50 0.07 cor((cid:1)p t ;(cid:1)p t−1 ) 0.06 0.18 0.03 0.73 0.60 0.06 Table 7: Forecastable Moments { Permanent Technology Shocks Neoclassical Model Putty-Clay Model Data Disemb. Embod. Disemb. Embod. Est. S. E. cor((cid:1)c^ ;(cid:1)y^ ) 1.00 1.00 1.00 1.00 0.95 0.07 t;4 t;4 cor((cid:1)h^ ;(cid:1)y^ ) -1.00 -1.00 0.44 0.44 0.86 0.06 t;4 t;4 cor((cid:1)^i ;(cid:1)y^ ) -1.00 -1.00 -0.98 -0.98 0.76 0.07 t;4 t;4 cor((cid:1)p^ ;(cid:1)y^ ) 1.00 1.00 0.95 0.95 0.76 0.10 t;4 t;4 (cid:11) 3.18 3.18 1.61 1.61 0.22 0.07 ((cid:1)c^t;4;(cid:1)y^t;4) (cid:11) -1.68 -1.68 0.14 0.14 0.59 0.09 ((cid:1)h^ t;4;(cid:1)y^t;4) (cid:11) -4.44 -4.44 -0.49 -0.49 1.70 0.24 ((cid:1)^it;4;(cid:1)y^t;4) (cid:11) 2.68 2.68 0.86 0.86 0.41 0.09 ((cid:1)p^t;4;(cid:1)y^t;4) (cid:27) =(cid:27) 0.06 0.25 0.14 0.83 0.64 0.06 (cid:1)y^t;4 (cid:1)yt;4 (cid:27) =(cid:27) 0.08 0.30 0.17 0.77 0.71 0.07 (cid:1)y^t;8 (cid:1)yt;8 (cid:27) =(cid:27) 0.09 0.31 0.18 0.67 0.65 0.06 (cid:1)y^t;16 (cid:1)yt;16 43