Putty-Clay and Investment: A Business Cycle Analysispeople.bu.edu/sgilchri/research/puttyclay_jpe00.pdf · Putty-Clay and Investment: A ... assistance. We wish to thank Flint Brayton,

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[Journal of Political Economy, 2000, vol. 108, no. 5]q 2000 by The University of Chicago. All rights reserved. 0022-3808/2000/10805-0006$02.50

Putty-Clay and Investment: A Business CycleAnalysis

Simon GilchristBoston University

John C. WilliamsBoard of Governors of the Federal Reserve System

This paper develops a general equilibrium model with putty-clay tech-nology, investment irreversibility, and variable capacity utilization. Lowshort-run capital-labor substitutability induces the putty-clay effect ofa tight link between changes in capacity and movements in employ-ment and output. Permanent shocks to technology or factor pricesgenerate a hump-shaped response of hours, persistence in outputgrowth, and positive comovement in the forecastable components ofoutput and hours. Capacity constraints result in asymmetric responsesto large shocks with recessions deeper than expansions. Estimation ofa two-sector model supports a significant role for putty-clay capital inexplaining business cycle and medium-run dynamics.

I. Introduction

In this paper we develop a dynamic stochastic general equilibrium modelbased on the putty-clay technology introduced by Johansen (1959). Theputty-clay model possesses a number of attractive features typically ab-sent from business cycle models, including irreversible investment, var-

We are indebted to John Shin, Steven Sumner, and Joanna Wares for exemplary researchassistance. We wish to thank Flint Brayton, Jeff Campbell, Thomas Cooley, Russell Cooper,Sam Kortum, John Leahy, Scott Schuh, Dan Sichel, John Willis, and two anonymousreferees for helpful comments. We also thank the National Science Foundation for fi-nancial support (SES-9876548). The opinions expressed here are not necessarily sharedby the Board of Governors of the Federal Reserve System or its staff.

putty-clay and investment 929

iable capacity utilization, and endogenous machine replacement.1 Weinvestigate the implications of putty-clay technology for macroeconomicdynamics at business cycle frequencies. A key finding is that low short-run capital-labor substitutability native to the putty-clay framework in-duces the putty-clay effect of a tight link between changes in capacity andmovements in employment and output. In the short run, an expansionof employment quickly confronts sharp increases in marginal costs, ow-ing to capacity constraints. Once new capacity is in place, a sustainedboom in employment and output can occur, as firms fully employ newmachines while maintaining high rates of utilization on preexistingcapacity.

Putty-clay technology has important implications for business cycledynamics. In particular, permanent shocks to technology or factor pricesgenerate a prolonged hump-shaped response of hours, persistence inoutput growth, and positive comovement between the forecastable com-ponents of output and hours. These features of the business cycle, doc-umented by Cogley and Nason (1995) and Rotemberg and Woodford(1996), are absent in standard business cycle models, where the responseof hours peaks on the impact of the shock and the dynamic responseof output closely follows that of the shock.2 In addition, capacity con-straints inherent to the putty-clay framework imply asymmetric responsesto large shocks, with recessions steeper and deeper than expansions,consistent with the empirical evidence documented by Neftci (1984)and Sichel (1993).

The empirical relevance of putty-clay technology is confirmed by min-imum distance estimation of a two-sector model that formally nests boththe putty-clay model and Solow’s (1960) vintage capital model within acommon econometric framework. We find that the distance betweenmodel and data moments is minimized for an estimated putty-clay shareof total output that is on the order of 50–70 percent. The data over-whelmingly reject the restriction of no role for putty-clay capital.

In addition to its empirical relevance, the putty-clay model providesan intrinsically appealing description of investment, capacity utilization,and machine replacement. In this framework, capital goods embody thelevel of technology and the choice of capital intensity made at the timeof their creation. Ex ante, the choice of capital intensity—the amountof capital to be used in conjunction with one unit of labor—is basedon a standard Cobb-Douglas production function. Ex post, the produc-tion function has the Leontief form with a zero-one utilization decision

1 Putty-clay models have a long history in both the growth (Johansen 1959; Solow 1962;Phelps 1963; Sheshinski 1967; Cass and Stiglitz 1969) and investment literatures (Bischoff1971; Ando et al. 1974).

2 Kydland and Prescott (1982) provide a classic example of a real business cycle (RBC)model with putty-putty technology.

930 journal of political economy

based on the output per hour of a given piece of capital relative to theprevailing wage rate. Over time, as the economy grows and real wagesrise, older vintages of capital become too costly to operate given theirlabor requirements, and they are mothballed or scrapped.3

Our model incorporates what we view as the essential features of theputty-clay framework, including variable capacity utilization and invest-ment irreversibility.4 Although the assumption of ex post Leontief tech-nology may at first seem unrealistically stark, the resulting aggregateproduction function embeds, depending on the model’s parameteri-zation, both the relatively flat short-run marginal cost curve usually as-sociated with a putty-putty production function and a reverse L-shapedmarginal cost curve traditionally associated with the putty-clay frame-work. Moreover, the model’s microfoundations are largely consistentwith empirical evidence on the importance of plant shutdowns as a short-run adjustment margin (Bresnahan and Ramey 1994) and the lumpinessof investment at the plant level (Cooper, Haltiwanger, and Power 1999).

II. The Model

In this section we describe the model and derive the equilibrium con-ditions. As noted above, each capital good possesses two defining qual-ities: its level of embodied technology and its capital intensity. In ad-dition to aggregate technological change, we allow for the existence ofidiosyncratic uncertainty regarding the productivity of investment proj-ects. As in Campbell (1998), the introduction of heterogeneity withinvintages smooths the aggregate allocation and simplifies computationof the equilibrium. More important, such idiosyncratic uncertainty im-plies the existence of a well-defined aggregate production function de-spite the Leontief nature of the microeconomic utilization choice.

Once in place, capital goods are irreversible; that is, they cannot beconverted into consumption goods or capital goods with different em-bodied characteristics, and they have zero scrap value. Firms can choose,however, whether or not to operate a given unit of capital dependingon the profitability of doing so in the current economic environment.We assume that there are no costs of taking machines or workers on-and off-line. As such, the utilization choice is purely atemporal. The

3 The effects of technological lock-in motivate the machine replacement problem firstaddressed by Johansen (1959) and Calvo (1976) and more recently formalized in a dynamicprogramming environment by Cooper and Haltiwanger (1993) and Cooley, Greenwood,and Yorukoglu (1997).

4 In Atkeson and Kehoe (1999), the energy intensity of production is the putty-clayfactor, but their dynamic analysis focuses on the case in which capital is always fully utilized.See also Struckmeyer (1986). Cooley, Hansen, and Prescott (1995) model variable capitalutilization but do not impose the irreversibility of factor proportions central to the putty-clay effect described in this paper.


optimal utilization choice for each unit of capital is determined by thedifference between the (labor) productivity of the capital and the costof utilizing the capital, which in the absence of other costs equals thewage 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, pro-duction, consumption, and investment are jointly determined by thedynamic optimizing behavior of households and firms. To characterizethe equilibrium allocation, we first discuss the production technologyand then describe utilization and investment decisions.

A. The Production Technology

Each period a set of new investment “projects” becomes available. Con-stant returns to scale implies an indeterminacy of scale at the level ofprojects, so without loss of generality, we normalize all projects to employone unit of labor at full capacity. We refer to these projects as “ma-chines.” The productive efficiency of a machine initiated at time t isaffected by three terms: the amount of capital invested per machine, kt,the economywide level of vintage technology, vt, and an idiosyncraticshock to productivity, mt. We assume that capital goods require oneperiod for initial installation and fail at the exogenously given rate d.5

The economywide level of vintage technology, vt, follows a stochasticprocess with mean gross growth rate The idiosyncratic shock12a(1 1 g) .

is drawn from a continuous distribution with unit mean and prob-m 1 0t

ability density function (pdf) denoted by fm(7). The realized level ofmachine productivity is assumed to be permanent and thus embodiedin the machine. For the sake of notational clarity, in the following dis-cussion we abstract from disembodied aggregate technologicalchange—that is, economywide shifts in total factor productivity—of thetype typical in the RBC literature. Such a process is incorporated at theend of this section.

Final-goods output produced at time t by a machine built in periodwith embodied technology and capital ist 2 j m v kt2j t2j t2j

a a aY(m v k ) p 1{L (m v k ) p 1}m v k , (1)t t2j t2j t2j t t2j t2j t2j t2j t2j t2j

5 Although our model allows for endogenous depreciation owing to machine obsoles-cence, the long-run rate of economic growth is too low to fully explain capital depreciationpurely through this mechanism. Introducing exogenous machine failure thus captureswear and tear, which constitutes a major portion of measured rates of depreciation. Froma qualitative point of view, setting has no consequence as long as economic growthd p 0is positive and some form of machine retirement occurs along the balanced growth path.Quantitatively, rapid obsolescence, either endogenous or exogenous, reduces the eco-nomic consequences of irreversible capital-labor ratios embedded in the putty-clayframework.


where is labor employed at the machine, and the zero-aL (m v k )t t2j t2j t2j

one indicator function reflects the Leontief naturea1{L (m v k ) p 1}t t2j t2j t2j

of machine production with unit labor capacity.6 LetaX { m vk (2)t t t t

denote the efficiency level (labor productivity) of a particular machine.To characterize the production possibilities of this economy, we notethat, once produced, machines are distinguished only by their efficiencylevel X.7 Let Ht(X) denote the quantity of machines of efficiency levelX that are available for production at time t. Integrating over X yieldsexpressions for aggregate output,

`

Y p 1{L (X ) p 1}XH(X )dX, (3)t E t t0

and aggregate labor,`

L p L (X )H(X )dX, (4)t E t t0

where if machines of efficiency X are utilized.L (X ) p 1t

The quantity of machines with efficiency X available for productionat time equals the quantity of machines that survive from last periodt 1 1plus the quantity of new machines with efficiency X that are put intoplace at time t. Owing to the idiosyncratic variation in project outcomes,the realized efficiency of a new machine, Xt, is randomly distributedaround the mean efficiency of new machines. The quantityaX p vk ,t t t

of machines of a given efficiency X evolves according to

H (X ) p (1 2 d)H(X ) 1 f (X; X )Q , (5)t11 t x t t

where

X21f (X; X ) p (X ) fx t t m ( )Xt

is the pdf of Xt, and Qt is the aggregate quantity of new machines thatare put in place at time t.

In the absence of government spending or other uses of output,aggregate consumption, Ct, satisfies

6 In equilibrium, machines are used at full capacity or not at all; hence, we couldequivalently express eq. (1) using the more traditional Leontief notation:

a a aY (m v k ) p m v k min [L (m v k ), 1].t t2j t2j t2j t2j t2j t2j t t2j t2j t2j

7 Because all existing machines have a constant probability of exogenous failure d, re-gardless of machine age, we may eliminate the time subscript associated with the realizedefficiency of a given machine.


C p Y 2 I , (6)t t t

where denotes total investment expenditures. Along with laborI p k Qt t t

supply, which is assumed to grow at a constant rate n, equations (3)–(6)define the production possibilities for this economy.

B. The Utilization and Investment Decision

The only variable cost to operating a machine is the cost associated withemploying labor, the wage rate Wt. Idle machines incur no loss in currentresources.8 Given the Leontief structure of production, these assump-tions imply a cutoff value for the minimum efficiency level of machinesused in production: those with productivity are run at capacity,X ≥ Wt

whereas those less productive are left idle.Conditional on the current wage rate, equation (3) implies that ag-

gregate output may be expressed as

Y p XH(X )dX, (7)t E t1X Wt

and equation (4) implies that aggregate labor may be similarly expressedas

L p H(X )dX. (8)t E t1X Wt

Define the time t discount rate for time income by Before˜t 1 j R .t,t1j

making investment decisions, project managers observe the economy-wide level of vintage technology vt. The idiosyncratic shock to individualmachines mt is revealed only after kt is chosen, however. These timingassumptions imply that capital intensity is chosen to maximize the pres-ent discounted value of profits to the machine

`

j˜max E 2k 1 R (1 2 d) (X 2 W )1{L (X ) p 1} , (9)O{ }t t t,t1j t t1j t1j t` jp1k ,{L (X )}t t1j t jp1

where expectations are taken over the time t idiosyncratic shock andfuture values of wages and interest rates.

Expected net income in period from a vintage t machine, tt 1 j p ,t1j

conditional on Wt1j, is given by

8 The model can be extended to allow for a fixed labor cost per unit of capital. Undersuch a specification, it is optimal to permanently scrap inefficient machines. Because suchmachines nearly always lie idle, such a modification would not alter the putty-clay effector the main results of this paper.


t jp p (1 2 d) (X 2 W )f (X; X )dX.t1j E t1j x t1X Wt1j

Substituting this expression for net income into equation (9) eliminatesthe future choices of labor from the investment problem. The remainingchoice variable is kt. An increase in kt increases expected net income byraising the mean value of machine efficiency. Taking the derivativeX ,tof the profit function with respect to kt yields the first-order conditionfor an interior solution for kt:

` tpt1j˜E R p 0. (10)Ot t,t1j{ }kjp1 t

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 amachine kt:

`

t˜E R p p k . (11)O{ }t t,t1j t1j tjp1

This is the free-entry or zero-profit condition. This condition must holdas long as there is gross investment in period t; if the cost of a machineexceeds the value of a machine for all admissible values of kt, no in-vestment is undertaken.

C. Preferences

To close the model, we specify the economic relationships that deter-mine labor supply and savings decisions and define the rational expec-tations equilibrium. We assume that the economy is made up of rep-resentative households whose preferences are given by

`

sE N b U(C , L , N ), (12)Ot t1s t1s t1s t1ssp0

where12g

z1 C LU(C, L, N ) p 1 2( )[ ]1 2 g N N

is utility per period, and is thetb P (0, 1), g 1 0, z 1 0, N p N (1 1 n)t 0

labor endowment of the representative household. Households optimizeover these preferences subject to the standard intertemporal budgetconstraint. We assume that claims on the profit streams of individualmachines are traded; in equilibrium, households own a diversified port-folio of all such claims, implying that the discount factor, used byR ,t,t1j


firms to discount future profits, is consistent with the household’s in-tertemporal decisions

Uc,t1jjR p b , (13)t,t1j Uc,t

where denotes the marginal utility of consumption. The first-orderUc,t1s

condition with respect to leisure and work is given by

U W 1 U p 0, (14)c,t t L,t

where denotes the marginal utility associated with an incrementalUL,t

increase in work (decrease in leisure).The rational expectations equilibrium is defined to be the set of

sequences of prices and quantities such that each household and firmsolves its respective maximization problem as described above, takingprices as given, and all markets clear. The derivation of the deterministicsteady state and its properties are found in Gilchrist and Williams (1998).In conducting the dynamic analysis, we focus on deviations from thebalanced growth path. The solution methodology for the dynamic anal-ysis is described in the Appendix.

D. The Lognormal Distribution

To parameterize the model, it is necessary to choose a distributionalform for mt, the idiosyncratic shock to machine productivity. A naturalchoice is to assume that mt is lognormally distributed:

1 2 2log m ∼ N(2 j , j ),t 2

where the mean correction term implies The lognormal1 22 j E(m ) p 1.t2distribution for mt implies that Xt is also lognormally distributed, with

1 2E(log X FX ) p log X 2 j .t t t 2

Hence, the distribution of machines expressed in equation (5) evolvesaccording to

H (X ) p (1 2 d)H(X )t11 t

1 121 21 (jX ) f (log X 2 log X 1 j ) Q , (15)( )t t2j

where f(7) denotes the pdf of the standard normal.9

9 As emphasized in Gilchrist and Williams (1998), the lognormal distribution facilitatesthe analysis of aggregate quantities while preserving the putty-clay characteristics of themicroeconomic structure.


Given the lognormal distribution, the expressions for aggregate out-put and aggregate labor developed in subsection C may be considerablysimplified by developing an explicit notion of capacity utilization. Ca-pacity utilization for vintage s at time t is the ratio of actual outputproduced from the capital of that vintage to the level of output thatcould be produced at full capacity,

Xf (X; X )dX∫ 1X W x stsp 1 2 F(z 2 j), (16)t` Xf (X; X )dX∫0 x s

where F(7) denotes the cumulative distribution function of the standardnormal, and

1 1s 2z { (log W 2 log X 1 j ) (17)t t s 2j

measures the productivity difference between the efficiency of the mar-ginal machine in use at time t and the mean efficiency of a vintage smachine. The expression on the right-hand side of equation (16) followsfrom the formula for the expectation of a truncated lognormal randomvariable.10 Conditional on the current wage, the mean output of a vin-tage s machine at time t equals An increase in thes[1 2 F(z 2 j)]X .t s

current wage reduces capacity utilization and hence output from anygiven vintage.

To illustrate these ideas, figure 1 shows the distribution of machineproductivity for the model calibrated to parameters specified below. Thisfigure also shows the distribution of machine productivity for the mostrecent vintage (right scale). Its position on the horizontal axis reflectsboth the current level of embodied technology and the capital intensityof new machines. Owing to trend growth in technology and relativelylong-lived capital, the average productivity of the most recent vintageis substantially higher than the average productivity of existing ma-chines. Note that trend growth in investment—due to population growthand technological change—causes the aggregate distribution to beskewed.

The wage is shown by the vertical line in figure 1. Capacity utiliza-

10 Capacity utilization may be expressed as

E(X FX 1 W )s s t Pr (X 1 W FW ).s t t[ ]E(X )s

We then apply the following formula: If then2log (m) ∼ N(z, j ),

1 2 F(g 2 j)E(mFm 1 x) p E(m),

1 2 F(g)

where (Johnson, Kotz, and Balakrishnan 1994, vol. 1).g p [log (x) 2 z]/j


Fig. 1.—Distribution of machine productivity

tion—the ratio of actual output to output under full utilization—is ob-tained by computing the integral of XH(X) over the shaded region anddividing by the integral of XH(X) computed over the entire curve. Ob-solescence through embodied technological change implies that, onaverage, old vintages have lower utilization rates than new vintages.

When equations (7), (15), and (16) are combined, aggregate outputmay be expressed as a function of current capacity utilization rates andpast investment decisions:

`

t2j jY p [1 2 F(z 2 j)](1 2 d) Q X . (18)Ot t t2j t2jjp1

Given the current wage Wt, the share of vintage s machines in use attime t is

sPr (X 1 WFW) p 1 2 F(z ). (19)s t t t

When equations (8), (15), and (19) are combined, aggregate labor maybe expressed as a function of current labor requirements and past in-vestment decisions:

`

t2j jL p [1 2 F(z )](1 2 d) Q . (20)Ot t t2jjp1

To derive the optimality conditions for investment, we may express


expected net income in period t from a vintage s machine, condi-sp ,t

tional on Wt, in terms of expected utilization rates:

s s s sp p (1 2 d) {[1 2 F(z 2 j)]X 2 [1 2 F(z )]W }.t t s t t

The choice of kt has a direct effect on profitability through its effect onthe expected value of output It also has a potential indirect effectX .tthrough its influence on utilization rates. For any given realization ofmt, a higher choice of kt raises the probability that a machine will beutilized in the future. This increase in utilization raises both expectedfuture output and expected future wage payments. Because the marginalmachine earns zero quasi rents, this indirect effect has no marginaleffect on profitability, however:11

sp 1 1t s s sp (1 2 d) f(z 2 j)X 2 f(z )W p 0.t s t ts [ ]z j jt

Accordingly, the first-order condition for kt may be expressed as

`

j t˜k p aE R (1 2 d) [1 2 F(z 2 j)]X . (21)O{ }t t t,t1j t1j tjp1

The zero-profit condition may now be expressed as

`

j t˜k p E R (1 2 d) [1 2 F(z 2 j)]XOt t t,t1j t1j t{jp1

M

j t˜2 R (1 2 d) [1 2 F(z )]W . (22)O t,t1j t1j t1j}jp1

The first term on the right-hand side of equation (22) equals the ex-pected present discounted value of output adjusted for the probabilitythat the machine’s idiosyncratic productivity draw is too low to profitablyoperate the machine in period The second term likewise equalst 1 j.the expected present value of the wage bill, adjusted for the probabilityof such a shutdown. Equations (21) and (22) jointly imply that, in

11 The condition that the marginal machine earns zero quasi rents applies for a generaldistribution function. For the lognormal case, one may use the formula for the pdf of arandom normal variable to show that follows directly from eq. (17). Rear-s sp /z p 0t t

ranging this expression implies that wheresMPL p W ,t t

sf(z 2 j)tsMPL { Xt ssf(z )t

is the marginal product of labor for vintage s machines, i.e., the increment to averagemachine output divided by the increment to average labor inputs[1 2 F(z 2 j)]X 1 2t s

obtained through a reduction ins sF(z ) z .t t


equilibrium, the expected present value of the wage bill equals 1 2 a

times the expected present value of revenues.In simulations reported below, we compare the effects of shocks to

disembodied technological change to the effects of shocks to embodiedtechnological change. To allow for disembodied technological change,let At denote the current level of economywide total factor productivity,which is assumed to be known at the time current production andinvestment decisions are made. With disembodied technologicalchange, machine output for a vintage s machine at time t satisfies

A machine with efficiency level X is operatedY(X ) p 1{L (X ) p 1}A X .t s t s t s

at capacity if The term in equation (17) is redefined assA X ≥ W . zt t t

1 1s 2z { [log (W ) 2 log (A X ) 1 j ].t t t s 2j

Equation (18) may then be expressed as`

t2j jY p [1 2 F(z 2 j)](1 2 d) Q A X ,Ot t t2j t t2jjp1

and other model expressions are changed accordingly.

E. The Solow Vintage Model

When the restriction that ex post capital-labor ratios are fixed is relaxed,the model described above collapses to the vintage capital model initiallyintroduced by Solow (1960). For the Solow vintage model, define thecapital aggregator Kt:

`

1/a jK { v (1 2 d) IOt t2j t2jjp1

1/ap (1 2 d)K 1 v I , (23)t21 t21 t21

where It denotes gross aggregate capital investment. In the Solow vintagemodel, embodied technological change simply enters the modelthrough the capital accumulation equation in the form of a reductionin the relative cost of new capital goods.

Aggregate production in period t is given by21 2 aj

a 12aY p exp A K L , (24)t t t t( )a 2

where Lt denotes aggregate labor input, At represents disembodied tech-nological change, and is a scale correction that2exp {[(1 2 a)/a](j /2)}results from aggregating across machines with differing levels of idio-syncratic productivity. In this economy, a mean-preserving spread to


idiosyncratic productivity causes an increase in disembodied productivityat the aggregate level. Equations (6), (23), and (24) describe the pro-duction technology of the Solow vintage model.

III. Model Dynamics

In this section we describe the model’s dynamic response to persistentshocks to factor prices; in Section IV we turn our attention to permanentshocks to technology. We focus primarily on three fundamental featuresof the business cycle: the comovement between output and hours, thepersistence of output growth, and the asymmetric response to positiveversus negative shocks. As documented below, the putty-clay model pos-sesses a powerful internal propagation mechanism that yields substantialimprovements over the Solow model in all three dimensions.

A. The Short-Run Marginal Cost Curve

Insight into the dynamic responses of the putty-clay model is providedby the short-run marginal cost (SRMC) curve, which shows the rela-tionship between the level of output and the marginal cost of production(or, equivalently, the inverse of the marginal product of labor). In de-riving this relationship, we are holding the wage and the stock of capitalconstant. Figure 2 shows the SRMC curve, in log deviations from thedeterministic steady state, for the Solow vintage model and three ver-sions of the putty-clay model with different degrees of idiosyncraticuncertainty (measured by j). The slope of the marginal cost curve equalsthe percentage increase in labor input necessary to obtain a 1 percentincrease in aggregate output. For the Solow vintage model with theCobb-Douglas production function, the SRMC curve is linear in logs.

The SRMC curve of the putty-clay model, however, is distinctly non-linear. In the short run, an expansion of output occurs through in-creased utilization of marginal machines. At low levels of output, thereare many idle machines with relatively high efficiency, and output canbe expanded at relatively low cost. At high levels of output, the efficiencylevel of the marginal machine drops sharply, and an expansion of outputcauses a rapid increase in marginal cost.

The degree of curvature displayed by the SRMC curve depends onthe degree of idiosyncratic uncertainty j. For very low values of j, theputty-clay SRMC curve becomes vertical for levels of output a few percentabove steady state. As j increases, the SRMC curve of the putty-claymodel becomes less sharply curved and approximates that of the Solowvintage model as j becomes large. In this way, the model developed inthis paper embeds both the reverse L-shaped marginal cost curve tra-


Fig. 2.—Short-run marginal cost curve

ditionally associated with putty-clay technology and the log-linear mar-ginal cost curve of the Cobb-Douglas production function.

B. Calibration

The models are calibrated using parameter values taken from Kydlandand Prescott (1991) and Christiano and Eichenbaum (1992), except forthe trend growth rates, which are U.S. averages over 1954–96. We assumethat a period is one quarter of one year. In annual basis terms, thecalibrated parameters are b p 0.956, g p 1, z p 3, g p 0.018, n p

and The results reported in this paper are0.015, d p 0.084, a p 0.36.not sensitive to reasonable variations in these parameters. In our cali-bration of the model, the only parameter for which we do not have astrong prior is the variance of the idiosyncratic component of a ma-chine’s productivity, j2. As discussed above, for large j, the SRMC curveis very close to Cobb-Douglas. By lowering j, we increase the curvatureof the SRMC curve and increase the degree to which the model displaysdynamics unique to the putty-clay structure. To make clear distinctions


between the Solow vintage model and the putty-clay model, we set12j p 0.15.

C. Capital Cost Shocks

We start by characterizing the effect of a temporary but persistent shockto 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.13 We assume that embodied tech-nology follows the process

(1 2 r L) ln v p (1 2 r L)t(1 2 a) ln (1 1 g) 1 u ,v t v t

where ut is an independently and identically distributed innovation, Lis the lag operator, and represents trend productivityt(1 2 a) ln (1 1 g)growth. We set the autocorrelation coefficient of the shock process at

implying a half-life for the shock of just under 14 quarters.r p 0.95,v

Figure 3 shows the impulse responses of output and labor hours forboth the putty-clay model (upper panel) and the Solow vintage model(lower panel) to a one-percentage-point positive shock to embodiedtechnology.

The difference in output dynamics between the two models is striking.For the putty-clay model, output rises very little initially, steadily in-creases for a period of four years, and gradually returns to steady state.For the Solow vintage model, the peak response occurs at the onset ofthe shock, after which output declines monotonically: output dynamicssimply mirror shock dynamics with no evidence of any cyclical pattern.In the putty-clay model, output exhibits a clear hump-shaped responsethat creates a slowly unfolding and long-lasting business cycle.

The dynamic response of hours also exhibits a strong cyclical patternin the putty-clay model. Labor hours expand at the onset of the shockas the economy seeks to increase its investment rate. The initial expan-sion in hours is muted, however, owing to the sharply increasing slopeof the SRMC curve embedded in the putty-clay model. As more capitalis brought on-line, the marginal cost curve shifts out and labor expandsfurther. As a result, the peak labor response occurs two years after theonset of the shock.

12 Lowering j to 0.1 does not alter model results in any substantial manner; lowering jmuch further causes numerical problems for the dynamic solution methods. On the otherhand, raising j to 0.5 for almost all essential purposes replicates the Solow vintage modeldynamics, whereas intermediate values provide results that are a combi-(j p 0.2–0.25)nation of the Solow and putty-clay models with low j.

13 It is, not, however, the same as a distortionary shock to the cost of capital goods froma change in taxes or a shock that might result from a monetary disturbance in a modelwith nominal rigidities. Such shocks affect the equilibrium allocation but not the feasibleallocation for the economy.

Fig. 3.—Persistent cost of capital shock


The slow but sustained rise in hours above steady-state levels occursbecause the benefits to building new capital goods are much greater ifexisting, efficient capital is not scrapped in the process. With ex postLeontief technology, labor cannot be reallocated across machines toequate marginal products. To benefit from new machines without losingthe productive services of existing capital, the economy must hire newworkers to operate these machines. Eventually, as the productive valueof these machines falls, labor returns to steady state. We refer to thisdynamic linkage between labor and machines as the putty-clay effect.

In this exercise, the capital cost shock has important implications forthe dynamic link between investment and productivity—both in theaggregate and at the machine level. Aggregate movements in total factorproductivity, conventionally measured as the Solow residual, are dis-played in figure 3 for both the putty-clay and the Solow vintage models.In both models, movements in the Solow residual reflect the rapidexpansion in investment that occurs as the economy installs new capitalgoods that embody the latest technology. For the putty-clay model, thisrapid expansion in investment is shown in the upper panel of figure 4.

Figure 4 also decomposes aggregate investment into the quantity ofnew machines, Q, and the capital intensity of each new machine, k. Atthe onset of the shock, Q rises and k falls as a large number of low–capitalintensity machines are produced. This rapid expansion of inexpensive(in terms of forgone consumption) machines facilitates the increase inlabor input. This reliance on low efficiency capital does not persist,however. As the real interest rate falls and the real wage rises, firmssubstitute into high–capital intensity capital goods.

The capital intensity of new machines remains at elevated levels fora number of years as households store the benefits of the temporaryshock through increased saving, implying capital deepening. As shownin the lower panel of figure 4, rising capital intensities dramatically offsetthe exponential rate of decay in technology and consequently providea sustained increase in the efficiency levels of new machines for a num-ber of years after the shock has occurred. Because machines are long-lived, this sustained increase in efficiency levels of new machines trans-lates into a sustained increase in aggregate labor productivity over along horizon.

D. Nonlinear Dynamics

The nonlinearities embedded in the SRMC curve also imply an asym-metric response to positive versus negative shocks. For small shocks,these asymmetries are negligible, and the response to positive and neg-ative shocks is roughly the same. For larger shocks, however, these asym-

Fig. 4.—Persistent cost of capital shock: putty-clay model


metries can be substantial: the response to negative shocks is more rapidand larger than that to positive shocks.

One plausible source of persistent movements in factor costs sizableenough to generate measurably asymmetric responses is a change inmarginal tax rates associated with a tax reform. To display these asym-metries, we consider the differential effect of a 10 percent increaseversus decrease in the marginal cost of labor. We embed the labor costshock by inserting a wedge between the wage paid by firms and thewage received by workers. Formally, firms pay for each unit(1 1 h )Wt t

of labor employed, where ht is a payroll tax that is rebated lump-sumto the household. We assume

ln (1 1 h ) p r ln (1 1 h ) 1 e ,t h t21 t

where et is an independently and identically distributed innovation. Forthe simulations reported here, we set Figure 5 reports ther p 0.98.h

outcome of these simulations. For purposes of comparison, the responseof an increase in labor cost is displayed with the opposite sign.

An increase in labor cost causes an immediate shutdown of machinesas the economy moves down the relatively flat portion of the SRMCcurve. This shutdown produces a sharp contraction in output and hoursin both the initial and subsequent periods. In contrast, a decrease inlabor cost has only a limited effect on either output or hours until newcapital is put in place. Evidence of nonlinearity diminishes after fourto six years as capacity eventually adjusts. As a result, in the putty-claymodel, large contractionary shocks cause steep immediate declines inoutput and hours, whereas large expansionary shocks generate a hump-shaped dynamic response even more pronounced than in the case ofsmall expansionary shocks. Of particular interest here is the fact thatthe asymmetries are the most pronounced for labor, a result supportedby Neftci’s (1984) nonlinear time-series analysis.

IV. Permanent Technology Shocks

An important unresolved issue in macroeconomics is the extent to whichpermanent innovations in technology can explain output fluctuationsat the business cycle frequency. While past research has claimed varyingdegrees of success, more recent work has tempered enthusiasm for busi-ness cycle theories based on permanent technology shocks. Cogley andNason (1995) and Rotemberg and Woodford (1996) demonstrate thatthe standard Solow model with permanent disembodied productivityshocks is unable to match the persistence and comovement propertiesof key aggregate variables. Christiano and Eichenbaum (1992) showthat such models predict excessive contemporaneous correlation be-tween output growth and labor productivity growth. In this section, we


Fig. 5.—Asymmetries in response to labor cost shocks. Impulse responses to a 10 percentincrease in the cost of labor are displayed with the reverse sign.

extend this literature in two directions by analyzing the effects of per-manent embodied, as well as disembodied, technology shocks and al-lowing for putty-clay technology. Greenwood, Hercowitz, and Krusell(1997) argue that the evidence supports embodied technology as theprimary source of technological change, making the analysis of suchshocks of particular interest. In what follows, we focus on three featuresof the business cycle that the Solow model has difficulty replicating:persistence in output growth, positive comovement between forecastablecomponents of output and hours, and the correlation between thegrowth of output and productivity. As shown below, the putty-clay model


TABLE 1Output Persistence with Permanent Shocks

Moments

Solow Vintage Model Putty-Clay Model Data

Disembodied(1)

Embodied(2)

Disembodied(3)

Embodied(4)

Estimate(5)

StandardError(6)

r(Dy , Dy )t t21 .01 .07 .04 .80 .31 .08ˆj(Dy )/j(Dy )t,4 t,4 .06 .25 .14 .83 .64 .06ˆj(Dy )/j(Dy )t,8 t,8 .08 .30 .17 .77 .71 .07ˆj(Dy )/j(Dy )t,16 t,16 .09 .31 .18 .67 .65 .06

Note.—In the columns labeled “disembodied” (“embodied”), the only source of disturbances is permanent disem-bodied (embodied) technology shocks. The variable y denotes the log of output, and where expec-ˆDy { E (y 2 y ),t,j t t1j t

tations are based on date t information; The terms j(x) and r(x, y) denote the asymptotic values of theDy { y 2 y .t,j t1j t

standard deviation of x and the correlation of x and y, respectively. Data sample pertains to 1960–97.

generates dynamic responses to permanent technology shocks that ac-cord well with these key properties of the data.

Table 1 reports two types of statistics summarizing the persistenceproperties of the two model economies in response to permanent tech-nology shocks. In this table and table 2 below, j(x) and r(x, y) denotethe asymptotic values of the standard deviation of x and the correlationof x and y, respectively. For purposes of comparison, the point estimatesand standard errors for U.S. data from 1960 to 1997 are shown incolumns 5 and 6. (See the Appendix for a description of the data andempirical methodology.)

The putty-clay model delivers significantly more persistence in outputgrowth than the Solow model. This finding holds for the two alternativemeasures of persistence that we consider: the autocorrelation of outputgrowth and the ratio of standard deviations of forecastable outputgrowth to total output movements, Here, forecastableˆj(Dy )/j(Dy ).t,k t,k

refers to the conditional expectation Not surprisingly,ˆDy p E {y 2 y }.t,k t t1j t

embodied technology shocks generate much more persistence in outputgrowth than disembodied shocks do. This is especially true for the putty-clay model, which, when driven solely by permanent embodied tech-nology shocks, actually overpredicts the persistence in output growthobserved in U.S. postwar data.

The intuition behind these results is provided by the impulse re-sponses to permanent technology shocks, plotted in figure 6. In boththe putty-clay and Solow vintage models, the long-run effect of a 1 2

percent increase in technology is to raise output, consumption, anda

investment by 1 percent while leaving the long-run level of labor un-changed. In the case of disembodied shocks (shown in the panels onthe right), total factor productivity increases immediately, causing animmediate expansion of output. For both models, this initial increasein output in response to a disembodied shock to technology representsa large fraction of the permanent increase; nearly all of the output


Fig. 6.—Permanent productivity shocks

movement is unforecastable, and output growth displays very littlepersistence.

If technology shocks are embodied in capital (shown in the panelson the left), productivity does not increase until the economy investsin new capital goods. In the Solow vintage model, a positive shock toembodied technology still causes a large immediate expansion in output,owing to a surge in hours in response to the high rate of return to


TABLE 2Permanent Technology Shocks

Moments

Solow Vintage Model Putty-Clay Model Data

Disembodied(1)

Embodied(2)

Disembodied(3)

Embodied(4)

Estimate(5)

StandardError(6)

Unconditional:j(Dc)/j(Dy) .55 1.13 .70 2.87 .36 .02j(Dh)/j(Dy) .36 1.57 .09 .91 .66 .04j(Di)/j(Dy) 2.15 6.03 1.76 8.14 1.82 .11j(Dp)/j(Dy) .65 .69 .91 .77 .68 .04r(Dc , Dy )t t .99 2.83 .99 2.12 .52 .06r(Dh , Dy )t t .98 .95 .96 .68 .74 .04r(Di , Dy )t t .99 .97 .99 .53 .60 .05r(Dp , Dy )t t .99 2.71 1.00 .50 .76 .03r(Dc , Dc )t t21 .10 .05 .10 .06 .34 .08r(Dh , Dh )t t21 2.03 2.03 .20 .20 .60 .06r(Di , Di )t t21 2.02 2.02 2.01 2.02 .50 .07r(Dp , Dp )t t21 .06 .18 .03 .73 .60 .06

Forecastable :ˆ ˆr(Dc , Dy )t,4 t,4 1.00 1.00 1.00 1.00 .95 .07ˆ ˆr(Dh , Dy )t,4 t,4 21.00 21.00 .44 .44 .86 .06ˆ ˆr(Di , Dy )t,4 t,4 21.00 21.00 2.98 2.98 .76 .07ˆ ˆr(Dp , Dy )t,4 t,4 1.00 1.00 .95 .95 .76 .10ˆ ˆb(Dc , Dy )t,4 t,4 3.18 3.18 1.61 1.61 .22 .07ˆ ˆb(Dh , Dy )t,4 t,4 21.68 21.68 .14 .14 .59 .09ˆ ˆb(Di , Dy )t,4 t,4 24.44 24.44 2.49 2.49 1.70 .24ˆ ˆb(Dp , Dy )t,4 t,4 2.68 2.68 .86 .86 .41 .09

Note.—In the columns labeled “disembodied” (“embodied”), the only source of disturbances is permanent disem-bodied (embodied) technology shocks. The variables c, y, h, i, and p denote the log of consumption, output, hours,investment, and output per hour, respectively. The term where expectations are based on date tˆDy { E (y 2 y ),t,j t t1j t

information; The terms j(x), r(x, y), and b(x, y) denote the asymptotic values of the standard deviationDy { y 2 y .t,j t1j t

of x, the correlation of x and y, and the regression coefficient of x on y, respectively. The data sample pertains to1960–97.

investment. Because of the rapid expansion in production, the initialoutput response represents a large fraction of the permanent response,and output movements are mostly unforecastable. In the putty-claymodel, however, a positive shock to embodied technology causes onlya small initial expansion. Output and hours rise over time as new capitalis brought on-line, with labor reaching its peak response a number ofyears after the shock occurs. As a result, most of the output dynamicsare forecastable, and output growth displays a high degree of persistencein response to embodied shocks to technology.

In addition to the limited degree of internal propagation, Rotembergand Woodford (1996) criticize the high negative correlations betweenforecastable movements in output and hours implied by the Solow vin-tage model in response to permanent technology shocks. This point isformalized in table 2, which reports a variety of statistics summarizingthe dynamic properties of the two models. For the Solow vintage model,the negative correlation between forecastable movements in output and


hours follows from the fact that the peak response in hours occurs atthe onset of the shock, whereas output continues to grow as capitalaccumulation proceeds. Thus hours are falling and output is rising. Theputty-clay model’s slow expansion of output and hours reverses thiscorrelation and provides a closer match with the data on this dimension.

Finally, we consider the model’s predictions regarding the contem-poraneous correlation between output and productivity growth. If hourswere held constant, both models would predict a perfectly positive cor-relation between growth in output and productivity. As can be seen fromtable 2, with disembodied shocks, this correlation is nearly unity as theeffect of the movements in hours on productivity is dwarfed by the effectsof the shock itself.

In the case of embodied technology shocks, the model’s predictionsdiffer greatly. The Solow vintage model predicts a highly negative cor-relation between growth in output and productivity. This negative cor-relation results from the immediate expansion of hours, which producesa large decline in productivity at the same time as the largest increasein output. In contrast, the putty-clay model predicts a positive correlationbetween output and productivity growth. This difference lies in themuted expansion of hours, which does less to offset the positive co-movement in productivity and output directly resulting from the shock.The putty-clay model thus provides an explanation for why the corre-lation between growth in output and productivity may be positive butless than unity, even in the absence of shocks to demand.

Although embodied shocks help explain a number of empirical reg-ularities, some results reported in table 2 are not consistent with thebusiness cycle. In particular, embodied shocks imply negative comove-ment between consumption growth and output growth and excess vol-atility of consumption and investment relative to the data.14 This excessvolatility occurs because factor cost movements create investment pat-terns that overwhelm the usual desire to smooth consumption.

V. Estimation

The results in Section IV highlight the putty-clay model’s ability to ex-plain key business cycle facts such as persistence in output growth andpositive forecastable comovement between output and hours. In thissection we provide a more formal evaluation of the empirical relevance

14 The response of consumption to an innovation to embodied technology dependscrucially on the specification of preferences. Using a putty-putty model, Greenwood, Her-cowitz, and Huffman (1988) show that if there is no intertemporal substitution effect onlabor effort, consumption can immediately rise in response to a positive shock to embodiedtechnology.


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 nestsboth the putty-clay model and the Solow vintage model within the sameeconometric framework. We assume that sector 1 output is producedusing the putty-clay production process described above, whereas sector2 output is produced using the Solow vintage technology. Letting At

denote the level of disembodied technology, we assume that final-goodsoutput is a Cobb-Douglas function of sectoral output: l 12lY p A Y Y .t t 1t 2t

We then estimate l, the share of output obtained from the putty-claysector. If our estimate of l is close to unity, the data effectively put alarge weight on putty-clay production in order to match the vector ofmoments that we consider. If our estimate of l is close to zero, the datasuggest little if any role for putty-clay in explaining the moments wechoose to match.

To address recent criticisms regarding standard RBC-style moment-matching exercises, our methodology relies on both unconditional mo-ments traditionally emphasized in the RBC literature and forecastablemoments emphasized by Rotemberg and Woodford (1996).15 The un-conditional moments include the standard deviations of the growth ratesof investment and hours relative to the standard deviation of output,the correlations of investment and hours with output, and the first-orderautocorrelations of these variables. The forecastable moments includecorrelations and regression coefficients among forecastable changes inoutput, investment, and hours over a four-quarter horizon. These mo-ments also include the ratio of the standard deviation of the forecastablechange in output relative to the standard deviation of the total changein output at this horizon.

To compute moment estimates for the historical data, we specify avector autoregression (VAR) process for the logs of output,(y , h , i ),t t t

hours, and investment in the data. As shown in the Appendix, the sta-tistical properties and hence all relevant moments of this VAR are sum-marized by an unknown parameter vector y. We estimate y using stan-dard time-series techniques and then use the resulting parameterestimate to compute a set of moments along with the variance ofˆg(y),these moments Vg.

Our estimation strategy is to choose l, the share of output accountedfor by the putty-clay sector, along with other relevant parameters, tominimize the distance between model moments and data moments. For

15 Christiano and Eichenbaum (1992) provide a generalized method of moments pro-cedure for estimating and evaluating business cycle models based on unconditional mo-ments. A contribution of this paper is to provide a generalized method of momentsprocedure that allows for both types of moments within a unified econometric framework.


a given vector w of model parameters, we use our model solution tocompute gM(w), the model’s analogue of g(y). By minimizing

′ 21ˆ ˆL(w) p [g (w) 2 g(y)] V [g (w) 2 g(y)]M g M

with respect to w, we obtain the minimum distance estimator Givenw.T observations of historical data, provides a x2 test for equalityˆT # L(w)between andˆ ˆg (w) g(y).M

For our first moment-matching exercise, we consider two sources offluctuations: disembodied technological change and embodied tech-nological change. We assume that shocks follow an AR1 process andthen freely estimate [rA, rv] , the autocorrelations of the shock processes,and the relative importance of embodied shocks, where jxj /(j 1 j ),v A v

is the standard deviation of the innovation to the AR1 process for ran-dom variable x. To generalize our results beyond technology shocks, wealso consider a specification that includes the labor cost shock, ht, in-troduced above. Under this specification, we also estimate the autocor-relation and the relative importance of labor cost shocks.

Table 3 reports the results of this exercise. Column 1 reports theresults from a benchmark putty-putty model whose parameters are im-posed rather than estimated. For this benchmark model, we assume thatall shocks are disembodied with Column 2 shows the resultsr p 0.95.A

based on estimation using the seven unconditional moments reportedin the middle of the table. Columns 3 and 4 show results based on thefull set of unconditional and forecastable moments, with column 4 alsoallowing for labor cost shocks. This table also reports the test statistic

for the various specifications. For comparison purposes, weT # L(w)report both unconditional and forecastable moments for all specifica-tions. Although they are not used in the estimation, we also report acomparable set of consumption moments.

Estimation results based on unconditional moments imply a largeshare for the putty-clay production technology—on the order of 50percent.16 The estimate of is about 0.2, suggesting a sub-j /(j 1 j )v A v

stantial role for embodied technology shocks in explaining aggregate

16 The putty-clay share is estimated with a standard error of 0.02, although this numbershould be treated with some caution given that we reject a test of equality between modeland data moments. The basic finding of a substantial role for putty-clay technology isrobust, however, to a variety of changes in specification, including changes in preferences,the introduction of convex adjustment costs to investment, and the inclusion of otherdisturbances such as changes in government spending. This finding is also robust toreasonable changes in the horizon used to compute forecastable moments and to a pro-cedure that matches moments computed from consumption data rather than investmentdata. As a final check of robustness, we also considered an alternative to estimating

by fixing this ratio at 0.6, Greenwood, Hercowitz, and Krusell’s (1997) estimatej /(j 1 j )v A v

of the share of postwar technological change embodied in capital. Under this specification,setting and we estimate a putty-clay share of 0.69.r p r p 1 j p 0,A v L


TABLE 3Estimation of the Two-Sector Model

Model Parameters andMoments

Putty-Putty

Benchmark(1)

Two-SectorModel Data

(2) (3) (4)Estimate

(5)

StandardError(6)

Putty-clay share 0 .47 .68 .64j /(j 1 j )v v A 0 .23 .01 .00j /(j 1 j 1 j )h v A h .64rv 1.00 .98rA .95 1.00 .95 .94rh .99Unconditional moments:

j(Dh)/j(Dy) .57 .39 .30 .67 .65 .04j(Di)/j(Dy) 2.85 2.49 2.60 2.67 1.80 .11r(Dh , Dy )t t .99 .69 .96 .75 .74 .04r(Di , Dy )t t 1.00 .87 .99 .99 .61 .05r(Dy , Dy )t t21 2.02 .08 .06 .09 .32 .08r(Dh , Dh )t t21 2.04 .21 .24 .25 .61 .06r(Di , Di )t t21 2.03 .07 .04 .08 .50 .07

Forecastable moments:ˆ ˆr(Dh , Dy )t,4 t,4 .89 2.78 .90 .82 .89 .04ˆ ˆr(Di , Dy )t,4 t,4 .94 2.89 .84 .80 .86 .05ˆ ˆb(Dh , Dy )t,4 t,4 .80 2.49 .43 .53 .64 .10ˆ ˆb(Di , Dy )t,4 t,4 3.60 21.84 2.99 2.80 1.83 .27ˆj(Dy )/j(Dy )t,4 t,4 .27 .17 .25 .26 .62 .07

T # L(w) :Unconditional moments 708 361All moments 1,148 786 444

Consumption moments:j(Dc)/j(Dy) .28 .73 .40 .35 .36 .02r(Dc , Dy )t t .94 .72 .91 .93 .52 .06r(Dc , Dc )t t21 .19 .09 .18 .28 .34 .08

ˆ ˆr(Dc , Dy )t,4 t,4 2.08 .99 .23 .30 .95 .07ˆ ˆb(Dc , Dy )t,4 t,4 2.04 2.15 .19 .27 .22 .07

Note.—See the notes from table 1. The term jx is the standard deviation of the innovation to the AR1 process forrandom variable x. The parameters of the putty-putty benchmark are imposed. The estimate of rh is constrained at theimposed upper bound of .99.

dynamics. Relative to the putty-putty benchmark, allowing for embodiedshocks and a nonzero weight on putty-clay technology provides a sub-stantial gain in terms of fit, reducing by nearly 50 percent.T # L(w)The estimated values of rA and rv reach their upper bound of unity,emphasizing the importance of highly persistent shocks when matchingunconditional moments.

Estimation results based on the full set of moments also imply a largeputty-clay share—on the order of two-thirds—regardless of the shockprocesses. Again, introducing putty-clay technology provides a substan-


tial gain in fit, which is further improved through the introduction oflabor cost shocks. For either specification, the estimated model alsodoes reasonably well at matching consumption moments.

Estimation results based on the full set of moments also imply thatfactor-cost shocks provide an important source of fluctuations. Althoughthe estimate of drops to zero and the persistence of dis-j /(j 1 j )v A v

embodied shocks falls to 0.94, our estimates imply that highly persistentmovements in ht substantially improve the model’s ability to match em-pirical moments. Computing variance decompositions, we find that la-bor cost shocks account for 15 percent of output fluctuations at theone-year horizon and 47 percent at the five-year horizon. Similar toBraun’s (1994) findings in the context of a putty-putty model, theseresults imply that disembodied shocks to technology do well at explain-ing output movements at high frequencies, whereas persistent shocksto factor costs do better at lower frequencies.

VI. Conclusion

By combining investment irreversibilities, capacity constraints, and var-iable capacity utilization, the putty-clay model developed in this paperprovides a rich framework for analyzing a number of issues regardinginvestment, labor, capacity utilization, and productivity. In this paperwe highlight some implications for employment, output, and investmentat business cycle and medium-run frequencies. Compared to standardmodels based on putty-putty technology, the putty-clay model displaysa substantial degree of persistence and propagation for both output andhours in response to shocks to factor costs and technology. And, unlikestandard models, the putty-clay model generates forecastable comove-ments between labor, output, and consumption consistent with the data.Finally, owing to the existence of a nonlinear marginal cost curve, theputty-clay model generates interesting asymmetries with recessionssteeper and deeper than expansions.

Beyond its descriptive appeal, the putty-clay production process is alsofound to be empirically relevant for explaining business cycle and me-dium-run dynamics. Estimates obtained from a two-sector model thatminimize the distance between moments generated by the model andthose obtained from the data place a sizable weight on putty-clay pro-duction—on the order of one-half to two-thirds. This finding is robustto the specification of the shock processes and the choice of momentsused in the estimation. In addition to supporting a major role for putty-clay technology, our results suggest that factor price shocks may be keyto explaining fluctuations, especially at the medium-run frequencies oftwo to eight years.


Appendix

A. Solution Methods

The state space of the model consists of the shocks plus the mean machineefficiency and total investment for each past vintage, X , Q , j p 1, … , `.t2j t2j

We approximate this state space in one of two ways. For the simulation illustratingthe asymmetric response to the labor cost shock shown in figure 5, an extendedpath algorithm based on Fair and Taylor (1983) is used in which the life spanof machines is truncated at 120 quarters (30 years). This method is computa-tionally very expensive. The second method, used for all other simulations andthe estimation exercises, uses a log linearization of the equilibrium conditions,evaluated at the deterministic steady state. We approximate the weighted infinitesum of leads and lags of variables by simple time-series processes. For example,for some variable u and coefficient sequence we approximate the sum`{a } ,j 1

by the process where`O a u V { u /B(L),jp1 j t2j t t21

2 p…B(L) p b 2 b L 2 b L 2 2 b L0 1 2 p

and p is finite. For the analysis in this paper we used and chose valuesp p 1corresponding to b0 and b1 that minimize the weighted squared deviations be-tween the approximate time-series representation and the true lag or lead struc-ture. The resulting approximated linear model is solved at trivial computationalcost using the Anderson and Moore (1985) implementation of the Blanchard-Kahn (1980) method. For simulations of moderately sized shocks, the two meth-ods yield nearly identical answers.

B. Econometric Methodology

We begin by specifying a stochastic process for the log of output, hours, andinvestment in the data that depends on a parameter vector n, which is to beestimated. Our approach follows Rotemberg and Woodford’s (1996) method ofspecifying a tightly parameterized low-order VAR system to characterize the data.Our data consist of chain-weighted real private nonfarm output, total privatehours, and chain-weighted real nonresidential business fixed investment (equip-ment and structures) for the period 1960–97. Relative to gross domestic product,the output and hours series exclude government and farm output as well as theimputed output obtained from owner-occupied housing. The output and hoursseries are thus defined in a mutually consistent manner. With the exception ofinvestment in structures and machinery by the agricultural sector and the house-holds and institutions sector, the investment series is also consistent with theoutput and hours series.

After first removing a linear time trend from the hours series, we assume thatthe first difference of the log of output, the log of the investment-output ratio,and the log of hours, can be represented using a two-lag sta-(Dy , i 2 y , h ),t t t t

tionary VAR representation. Defining

′ ′u p [Dy , i 2 y ,h ,Dy , i 2 y ,h ], E(u u ) p S ,t t t t t t21 t21 t21 t21 t t u

′ y i h ′ ′e p [e , e , e ], E(e e ) p S , E(e e ) p 0 for s ( 0,t t t t t t e t t1s

we express the stochastic process for ut in companion form as


P et ′u p Au 1 v , A p , v p , E(v v ) p S ,t t21 t t t t v[ ] [ ]I 0 0

where P is a matrix of VAR coefficients. The parameter vector y and its associatedvariance is then

21vec(P) S ^ S 0e uy p , V p ,w[ ] [ ]vech(S ) 0 Se 22

where S22 is a function of Se and the duplication matrix Dn satisfying(Hamilton 1994, pp. 301–2).17D vech(S ) p vec(S )n e e

Given this specification for the stochastic process for output, hours, and in-vestment, we compute second moments of the first differences of these variablesas well as second moments of the forecastable components of the kth differences,where the forecast is made conditional on time information. To obtaint 2 kexpressions for these second moments as functions of the underlying VAR pa-rameters, we first express the kth differences and as functionsDy , Di , Dht,k t,k t,k

of data known at time t and shocks that occur between t and Fort 1 k. x pwe havey, i, h

k

x xDx { x 2 x p b u 1 d v ,Ot,k t1k t k t j t1jjp1

wherek

y ′ i i y ′ k h ′ kb p e A , b p b 1 e (A 2 I ), b p e (A 2 I ),Ok 1 k k 2 k 3ip1

k2j

y ′ s i y ′ k2j h ′ k2jd p e A , d p d 1 e A , d p e A .Oj 1 j j 2 j 3sp0

When we take expectations as of time t, the forecastable components of the kthdifference of x are

xˆDx { E {x 2 x } p b u .t,k t t1k t k t

The covariance between the kth differences of x and y isk

x y ′ x y ′j { E(Dx Dy ) p (b )S (b ) 1 (d )S (d )Oxy,k t,k t,k k u k j v jjp1

whereas the covariance between the forecastable components of the kth differ-ences is

x y ′ˆ ˆj { E(Dx Dy ) p (b )S (b ) .ˆˆxy,k t,k t,k k u k

Using these expressions, we define two sets of moments. The first set ofmoments,

′j j j jDi Dh Di,Dy Dh,Dyg p , , , , r , r , r ,1 Dy Di Dh[ ]j j j j j jDy Dy Di Dy Dh Dy

17 The “vec” operator transforms an matrix into an vector by stacking the2n # n n # 1columns. The “vech” operator transforms an matrix into an vectorn # n [n(n 1 1)/2] # 1by vertically stacking those elements on or below the principal diagonal.


includes the standard deviations of Dit and Dht relative to the standard deviationof Dyt; the unconditional correlations of Dit, Dht with Dyt; and the autocorrelationsof Dyt, Dit, and Dht, respectively. The second set of moments,

′j j j j jˆ ˆˆ ˆˆ ˆ ˆ ˆ ˆiy,k hy,k iy,k hy,k y,kg p , , , , ,2 [ ]2 2j j j j j j jˆˆˆ ˆ ˆ ˆy,k y,k i,k y,k h,k y,k y,k

includes regression coefficients and correlations between the predictable com-ponents of Dit,k and Dht,k and the predictable component of Dyt,k. The secondset of moments also includes the ratio of the variance of the predictable com-ponent of Dyt,k relative to the total variance of Dyt,k. Recognizing that g1 and g2

are functions of A, Su, Sv, and hence the VAR parameter vector y, we obtainour moment vector of interest: An estimate of the variance of′ ′ ′g(y) p [g g ] .1 2

this moment vector is

g gV p V .g y′y y

We estimate the model parameter vector

j jv hw p a, r , r , r , ,A v h[ ]j 1 j j 1 j 1 jv A v A h

by minimizing the distance between g(y) and gM(w), where gM is the model’sanalogue of g(y). To obtain gM as a function of w, we rely on the fact that ourmodel solution is linear and may be expressed in the first-order companionform

X p AX 1 BU ,t t21 t

where and Xt is a vector of model variables, with′ ′E(UU ) p I y p e X , c pt t t y t t

and The moment vector gM may then be computed as a function′ ′e X , h p e X .c t t h t

of A and B. Because the matrices A and B are (nonlinear) functions of theunderlying model parameters w, the moment vector gM is also a function of w.Standard errors for w may then be obtained from

g gM M21V p V .w g′( )w w

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Documents

Putty-Clay and Investment: A Business Cycle Analysispeople.bu.edu/sgilchri/research/puttyclay_jpe00.pdf · Putty-Clay and Investment: A ... assistance. We wish to thank Flint Brayton,