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Medium-Term Business Cycles

By DIEGO COMIN AND MARK GERTLER*

Over the postwar period, many industrialized countries have experienced significantmedium-frequency oscillations between periods of robust growth versus relativestagnation. Conventional business cycle filters, however, tend to sweep these oscil-lations into the trend. In this paper we explore whether they may, instead, reflect apersistent response of economic activity to the high-frequency fluctuations normallyassociated with the cycle. We define as the medium-term cycle the sum of the high-and medium-frequency variation in the data, and then show that these kinds offluctuations are substantially more volatile and persistent than are the conventionalmeasures. These fluctuations, further, feature significant procyclical movements inboth embodied and disembodied technological change, and research and develop-ment (R&D), as well as the efficiency and intensity of resource utilization. We thendevelop a model of medium-term business cycles. A virtue of the framework is that,in addition to offering a unified approach to explaining the high- and medium-frequency variation in the data, it fully endogenizes the movements in productivitythat appear central to the persistence of these fluctuations. For comparison, we alsoexplore how well an exogenous productivity model can explain the facts. (JEL E3, O3)

Over the postwar period, many industrializedcountries have tended to oscillate between pe-riods of robust growth and relative stagnation.The U.S. economy, for example, experiencedsustained high-output growth during the early tolate 1960s. From the early 1970s to the early1980s, however, output growth was low on av-erage. Since the mid-1990s, there has been, forthe most part, a return to strong growth. Theeconomies of Europe and Japan have had sim-ilar experiences, though the precise details dif-fer.1 A common feature of these oscillations isthat they occur over a longer time frame than istypically considered in conventional business

cycle analysis. Put differently, conventionalbusiness cycle detrending methods tend tosweep these kind of oscillations into the trend,thereby removing them from the analysis.

At the same time, these medium-frequencyoscillations may be intimately related to thehigh-frequency output fluctuations normally as-sociated with the business cycle. The long U.S.expansion of the 1960s featured periods of rapidgrowth and was free of any significant down-turn. Similarly, the high growth period over thepast ten years has been interrupted by only onerecession, considered modest by postwar stan-dards. By contrast, the stagnation of the 1970sand early 1980s featured a number of majorrecessions, including the two considered theworst of the postwar period. All this raises thepossibility that medium-frequency oscillationsmay, to a significant degree, be the product ofbusiness cycle disturbances at the high fre-quency. Put differently, business cycles may bemore persistent phenomena than our conven-tional measures suggest. It follows that we mayneed to reexamine how we both identify andmodel economic fluctuations.

The objective of this paper is twofold: ourfirst goal is to construct an alternative trend/cycledecomposition that includes in the measure of

* Comin: Department of Economics, New York Univer-sity, 269 Mercer Street, New York, NY 10003 (e-mail:[email protected]); Gertler: Department of Economics,New York University, 269 Mercer Street, New York, NY10003 (e-mail: [email protected]). We appreciate thehelpful comments of two anonymous referees, Olivier Blan-chard, Michele Boldrin, Larry Christiano, Jon Faust, JordiGalı́, Pierre-Olivier Gourinchas, Gary Hansen, LouiseKeely, Tim Kehoe, Per Krusell, David Laibson, John Leahy,Rodolfo Manuelli, Ed Prescott, Richard Rogerson, and Gi-anluca Violante. Antonella Trigari and Viktor Tsyrennikovprovided outstanding research assistance. Financial assis-tance from the C.V. Starr Center and the National ScienceFoundation is greatly appreciated.

1 See, e.g., Olivier J. Blanchard (1997).

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the cycle both the high- and medium-frequencyvariation in business activity. We refer to thiscombined high- and medium-frequency varia-tion as the “medium-term business cycle.”2 Wethen present evidence based on a variety ofpostwar U.S. time-series data that the medium-term cycle differs from the conventionally meas-ured cycle in a number of interesting and im-portant ways. Our second goal is to develop, asa first pass, a quantitative model capable ofexplaining the facts. Here we attempt to captureboth high- and medium-frequency business cy-cle dynamics within a unified framework.

In Section I, we present evidence on medium-term fluctuations. Roughly speaking, we con-struct measures of medium-term cycles byrunning a much smoother trend through the datathan is typically used in the conventional high-frequency analysis. We show for a number ofkey variables that the medium-term cycle isconsiderably more variable and persistent thanthe conventionally measured cycle. These find-ings, further, are statistically significant and notan artifact of a small sample size. In addition,there are a number of interesting patterns. Forexample, over the medium term there is a strongcomovement between output and both disem-bodied and embodied technological change.Total factor productivity (TFP) moves procycli-cally over the medium term while the relativeprice of capital (a measure of embodied tech-nological change) moves countercyclically.R&D moves procyclically. The medium-termcycle, however, is not only a productivity phe-nomenon. As we show, measures of both theefficiency and intensity of resource utilizationalso play a role.

In Section II, we develop our model. Weconsider a simple real business cycle modelmodified to allow for endogenous productivity,countercyclical markups, and variable factorutilization.3 We incorporate endogenous pro-

ductivity in order to provide a unified explana-tion for the comovements of TFP, the relativeprice of capital, and R&D over the mediumterm. Another consideration leading us in thisdirection is that many authors have questionedthe importance of high-frequency technologyshocks based on the view that much of thehigh-frequency variation in the Solow residualreflects factors such as unmeasured input utili-zation and imperfect competition as opposed totrue technology shifts (e.g., Craig Burnside etal., 1995; Susanto Basu, 1996). Endogenousproductivity, however, provides an avenuethrough which nontechnological shocks thatmay drive business cycles at the high frequencycan generate the strong medium-frequencymovements in productivity observed in the data.That is, through this approach, one can captureinteresting cyclical variation in both embodiedand disembodied technological change withouthaving to rely exclusively on exogenous shiftsin technology.

To endogenize productivity dynamics, weuse a variation of Paul M. Romer’s (1990)model of R&D expanding the variety of inter-mediate goods. We extend the Romer frame-work to allow for an endogenous rate ofadoption of new technologies. Julio J. Rotem-berg (2003), for example, has argued that inmodeling cyclical productivity dynamics, it isimportant to take into account the significant lagin the diffusion of new technologies.4 In thisregard, we are able to calibrate our model toallow for realistic steady-state time lags be-

2 While similar in spirit, our definition differs from Blan-chard’s (1997), Ricardo J. Caballero and Mohamad L. Ham-mour’s (1998) and Robert M. Solow’s (2000), whose notionof the medium term is confined mainly to the medium-frequency variation in the data. We include both the high-and medium-frequency variation because we want toemphasize the interrelation between high- and medium-frequency fluctuations.

3 Our modeling approach is similar to that of George W.Evans et al. (1998), who also develop an endogenous

growth model to study business fluctuations that take placeover medium-term horizons. They emphasize how theirframework may give rise to sunspot fluctuations in thegrowth rate. In addition to differing considerably in detail,our model has a unique steady-state growth rate: thecomplementarities in our model then work to magnify theeffects of business cycle shocks.

4 Rotemberg uses this fact to argue that technologyshocks affect only low-frequency movements in output. Hethen proceeds to develop a framework with exogenoustechnological change, where nontechnology shocks drivethe high-frequency business cycle but have no effect onlower frequencies. Our approach differs by having feedbackbetween the high- and medium-frequency variation in cy-clical activity, owing to endogenous productivity. Somesupport for our approach is that, as we show, measures ofthe efficiency and intensity of resource utilization showconsiderable medium-frequency variation and are thus notsimply high-frequency phenomena.

524 THE AMERICAN ECONOMIC REVIEW JUNE 2006

tween the creation and diffusion of new tech-nologies. At the same time, because adoption aswell as R&D intensities vary endogenouslyover the cycle, the framework can produce thekind of procyclical movement in productivityover the medium term that we document.5

As with many conventional quantitative mac-roeconomic models (e.g., Christiano et al.,2005; Frank Smets and Raf Wouters, 2005),countercyclical movements in both price andwage markups provide the main source of fluc-tuations at the high frequency. These conven-tional frameworks typically endogenize markupbehavior by introducing nominal rigidities orother types of market frictions. We also endo-genize markup behavior but opt for a very sim-ple mechanism that is meant as a stand-in forthe kind of richer structure that is typicallymodeled. Thus, within our framework, an exog-enous driving force affects the economy ini-tially by generating countercyclical markupmovement. By proceeding with a simplemarkup structure, however, we are able to avoidadditional complexity and keep the focus on theendogenous productivity propagation mecha-nism, which is the unique feature of ourframework.

Section III presents some model simulationsand considers how well the framework can cap-ture the broad patterns in the data. Overall, ourmodel does a reasonably good job in character-izing the key features of the medium-term cy-cle. Even with a nontechnological shock as themain driving force, the model captures most ofthe cyclical variation in productivity, both atthe high and medium frequencies. There areseveral caveats that we discuss. For comparison,we also explore how well an exogenous tech-nology shock model can explain the data. Forthis kind of framework to capture movements inboth embodied and disembodied technologicalchange, it is necessary to allow for exogenousshocks to the relative price of capital, as well asexogenous shocks to total factor productivity.This approach has some strengths, but alsosome shortcomings, as we discuss. Concludingremarks are in Section IV.

I. Measuring the Medium-Term Cycle

In this section we develop some facts aboutmedium-term business fluctuations. To detrendthe data, we use a band pass filter, which isbasically a two-sided moving average filter,where the moving average depends on the fre-quencies of the data that one wishes to isolate.A popular alternative is the Hodrick-Prescott(HP) (1997) filter. This filter estimates a trendby maximizing a criterion function that penal-izes both deviations of the data from the trendand variation in the trend, depending on therelative weights on each objective in the crite-rion function. We opt for the band pass filterbecause it allows us to be precise (in frequencydomain terms) about how our measure of thecycle compares with more conventional mea-sures. Our results, however, do not depend onthis choice of filters.

Conventional decompositions of the datawith the band pass filter associate the businesscycle with the frequencies between 2 and 32quarters (e.g., Marianne Baxter and Robert G.King, 1999, and Christiano and Terry J. Fitzger-ald, 2003). This decomposition produces a rep-resentation of the cycle that is very similar tothe one that the standard HP filter generates. Aswe show, however, in each of these cases, thetrend exhibits considerable variation about asimple linear trend, reflecting the presence ofsignificant cyclical activity at the medium fre-quencies. Our measure of fluctuations, accord-ingly, adds back in this medium-frequencyvariation. We do this, simply put, by applying afilter that identifies a much smoother trend inthe data than does the conventional filter.

In particular, we define the medium-term cy-cle as including frequencies between 2 and 200quarters. The trend accordingly consists of vari-ation in the data at frequencies of 200 quartersand below. We allow for a very smooth nonlin-ear trend, as opposed to simply employing alinear trend for several reasons. First, over thesample period there have been a number offactors, such as demographics, that are likely tohave introduced variation in the data, mainly atthe very low frequencies. Rather than try toincorporate all these factors in our model, weconcentrate mainly on explaining the high- andmedium-frequency variation in the data. Sec-ond, with a linear trend, estimates of some of

5 Comin and Gertler (2004) provide some evidence thatadoption rates are procyclical based on a sample of 22individual technologies described in Stephen Davies (1979)in the United Kingdom during the postwar period.

525VOL. 96 NO. 3 COMIN AND GERTLER: MEDIUM-TERM BUSINESS CYCLES

the key moments in the data can become veryimprecise. We chose a cutoff for the trend in thefrequency domain of 200 quarters (roughly thesample size) because we found that with thisdecomposition we obtain, for the most part,both reasonably smooth trends and estimates ofthe variability of the filtered data that have areasonable degree of precision.

We decompose the medium-term cycle asfollows: we refer to the frequencies between 2and 32 quarters (the standard representation ofcycles) as the high-frequency component of themedium-term cycle, and frequencies between32 and 200 quarters as the medium-frequencycomponent. Two points are worth keeping inmind: first, in general it is not correct to think ofthe medium-frequency variation in the data asorthogonal to the high-frequency variation. Forthis reason, we focus on understanding the over-all fluctuations in the data between frequenciesof 2 and 200 quarters and not simply the medium-frequency component. Second, it is important tobe careful about the mapping between the fre-quency domain and the time domain: eventhough our measure of the cycle includes fre-quencies up to 50 years, as we show, its repre-sentation in the time domain leads to cycles onthe order of a decade, reflecting the distributionof the mass of the filtered data over the fre-quency domain. For the conventional businesscycle decomposition, as we show, the cycles aremuch shorter.

We proceed as follows. Because most of ourseries are nonstationary, we first convert all thedata into growth rates by taking log differences.We then apply the band pass filter to the growthrate data, obtaining a measure of trend growthrates that corresponds to frequencies 200 quar-ters and below. We then cumulate the trendgrowth rates to obtain measures of the trends inlog levels.6 We also perform several robustnesschecks. Because the band pass filter provides anapproximation only for finite series, we “pad”the series by forecasting and backcasting, tominimize biases that are likely to arise, espe-cially at sample endpoints. In addition, as wediscuss, we also perform a simple Monte Carlo

exercise to assess the statistical significance ofthe trends we measure.

The data are quarterly from 1948:1 to 2001:2,except noted otherwise. We consider two sets ofvariables. The first includes “standard” businesscycle series, including output, consumption, in-vestment, hours, and labor productivity.7 Thesecond includes “other” variables useful forcharacterizing the movements in both produc-tivity and resource utilization over the mediumterm. These include TFP, the quality adjustedrelative price of capital, private R&D, themarkup of price over social marginal cost, andcapacity utilization. The series on TFP, the rel-ative price of capital, and private R&D areavailable only at an annual frequency.8

We measure the markup as the ratio of themarginal product of labor to the household’smarginal rate of substitution between consump-tion and leisure (where the latter reflects thesocial marginal cost of labor). To do so, weassume that production is Cobb-Douglas, andthat both the coefficient of relative risk aversionand the Frisch labor supply are unity. As JordiGalı́ et al. (2002) note, this total markup can bedecomposed into the product of a price markup(the ratio of the marginal product of labor to thereal wage) and a wage markup (the ratio of thereal wage to the household’s marginal rate ofsubstitution).9 For capacity utilization we use

6 For series where the growth rate or the log level arestationary, such as per capita output, we obtain virtuallyidentical results by filtering the data in log levels (after firstremoving a linear trend).

7 Output, hours, and labor productivity are for nonfarmbusiness. Consumption includes nondurables and services.Investment is nonresidential. We normalize all variables bythe working-age population (ages 16–65).

8 TFP is from the Bureau of Economic Analysis (BEA).R&D is nonfederally funded R&D expenditures as reportedby the National Science Foundation (NSF). The relativeprice of capital is the quality adjusted price of equipmentand structures from Jason G. Cummins and Giovanni L.Violante (2002) (which is, in turn, based on Robert J.Gordon, 1990) divided by the BEA price of nondurableconsumption and services. While this series conventionallyapplied in the literature, it is the subject of some controversy(e.g., Bart Hobijn, 2001). We thus also considered the BEAmeasure and found that it exhibits nearly identical cyclicalproperties to the Gordon series, though it is only about 60percent as volatile overall. We return to this latter point inSection III. Finally, another issue is that the Cummins-Violante series is based on extrapolating the Gordon seriesafter 1983. We find, however, that the pre-1983 cyclicalproperties are very similar to those of the whole series.

9 See also Robert E. Hall (1997) for an analysis of thecyclical properties of the labor market distortion over thepostwar period.


the Federal Reserve Board’s measure. Givensome limitations of this series, we also consid-ered an alternative measure of utilization basedon the ratio of electricity usage to the capitalstock, and obtained very similar results.10 Fi-nally, we convert all variables to logs.

Figure 1 plots the medium-term cycle forper capita output. The line with circles givesthe percent deviation of per capita outputfrom trend for the medium-term cycle. Thesolid line gives the medium-term component(i.e., the variation in the data at frequenciesbetween 32 and 200 quarters). The differencebetween the two lines is the high-frequencycomponent (the variation at frequencies be-tween 2 and 32 quarters). Put differently, thisdifference is the measure of the cycle in con-ventional analysis.

Overall, the medium-term cycle appears tocapture the sustained swings in economic activ-ity that we described in the introduction. Thereis sustained upward movement in output rela-tive to trend over the 1960s. Output reversescourse in the 1970s through the early 1980s.There is again a sustained upward movementbeginning in the mid-1990s. As the figure

makes clear, these persistent movements reflectmedium-frequency variation in the data that arelarge relative to the high-frequency variation. Inthis regard, the medium-term cycle appearsmuch larger in overall magnitude than the con-ventionally measured cycle. The percentage risein output relative to trend over the 1960s, forexample, is roughly 14 percent and is mirroredby a roughly similar percentage drop during theperiod of the productivity slowdown. Anothergood example involves the period around the1980–1982 recession: the medium-term cyclesuggests a much larger and more protracteddrop relative to trend, than does the conven-tional high-frequency measure.

Figure 2 confirms that our measure of trendper capita output grows at a relatively smoothpace. There is a steady modest decline in thetrend growth rate over the postwar perioduntil the mid-1990s, where it gently reversescourse. From a more formal perspective, theratio of the standard deviation of outputgrowth at the low frequencies (below 200quarters) to the standard deviation of outputgrowth at frequencies between 2 and 200quarters is only 0.08. Thus, our measure oftrend growth is very smooth relative to theoverall variation in the growth rate.

Table 1 presents a set of formal statistics toconfirm the visual impression left by Figure 1that the medium-term cycle in output is consid-

10 In principle, the Board series includes variable laborutilization as well as capital utilization. We expect that theformer, however, may be less important over the mediumfrequencies.

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Medium-term Component Medium-term Cycle

FIGURE 1. NONFARM BUSINESS OUTPUT PER PERSON 16–65


erably more volatile than the conventionallymeasured high-frequency cycle.11 It also pre-sents a similar set of statistics for each of thetraditional quarterly business cycle variables.For each variable, the table reports the percentstandard deviation over the medium-term cycle,and also the standard deviations of both the

high- and medium-frequency components. Be-low each number are 95-percent confidence in-tervals.12 As the first row indicates, the standarddeviation of per capita output over the mediumterm is roughly twice that of the high-frequencycomponent (4.12 versus 2.39). Moreover, thestandard deviation of the medium-frequencycomponent (3.25) is large in comparison to the

11 Interestingly, by examining data only from the firsthalf of the sample, it would not be possible to detect amedium-term cycle that is distinct from the high-frequencycycle. For the period 1948:1–1973:1, for example, the twodifferent filters yield very similar cycles. Only by examiningdata from the full sample does a distinct medium-term cycleemerge.

12 To compute confidence intervals, we make use of thefollowing. Let �X

2 be the variance of a stochastic process{Xt}t�0

T . If Xt is stationary, the sample variance �̂X2 is as-

ymptotically distributed N(�X2, �). We compute � by trun-

cating the lags in the theoretical expression using theNewey-West truncation parameter.

0

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FIGURE 2. LONG-RUN TREND ANNUAL GROWTH RATE IN NONFARM BUSINESS OUTPUT PER

PERSON 16–65

TABLE 1—STANDARD DEVIATIONS: QUARTERLY FREQUENCIES

FrequencyMedium-term cycle

2–200High-frequencycomponent 2–32

Medium-frequencycomponent 32–200

Output per capita 4.12 2.39 3.25(2.71–5.52) (1.93–2.85) (2.41–4.09)

Hours 2.8 1.73 2.2(2.38–3.62) (1.49–2.01) (1.98–2.8)

Labor productivity 2.4 1.13 2.07(1.93–2.87) (1.03–1.23) (1.7–2.44)

Consumption 2.28 0.93 2.2(1.72–2.84) (0.87–0.99) (1.7–2.7)

Investment 10.08 4.88 8.64(2.17–17.99) (3.04–6.72) (2.19–15.09)


latter. A similar pattern exists for hours, laborproductivity, consumption, and investment.There are a few natural qualifications, however:as one might expect, a priori, the ratio of vari-ation over the medium-term cycle to variationover the high frequency is somewhat lower forhours (2.8 to 1.73) and somewhat higher forlabor productivity (2.4 to 1.13).

Importantly, as the last column of the tableshows, the medium-frequency variation in eachvariable is statistically significant. This can beseen by noting that the confidence interval forthe standard deviation of each variable liessafely above zero. In this regard, our measure ofthe medium-term cycle differs in a statisticallysignificant way from the conventional high-frequency measure.

While Table 1 presents standard deviations ofquarterly data, Table 2 presents a similar set ofstatistics for annual data. It does so for both thestandard business cycle series listed in Table1 and for the other variables we consider (TFP,the relative price of capital, R&D, and themarkup). We consider annual data for two rea-sons. First, because we are attempting to modelfluctuations that are considerably more persis-tent than are typically considered, the quantita-tive framework we develop in the next section

aims to explain variation in the annual as op-posed to the quarterly data. Second, some of thenonstandard series we consider are availableonly at the annual frequency.

Two central conclusions emerge from Table2. First, for the standard business cycle series,the standard deviations of the annual data arequite similar to those of the quarterly data, i.e.,the message from annual data about the relativeimportance of the medium-term cycle is virtu-ally the same as from quarterly data. Second,the “other” variables listed in Table 2 similarlyexhibit significantly greater volatility over themedium term than over the high-frequency cy-cle: in each case the standard deviation is morethan double. For the relative price of capital, thestandard deviation is nearly three times largerthan in the high frequency.13

Next, Table 3 reports (again, for the annualdata) the first-order autocorrelation of each vari-able over the medium term and over the highfrequency. Interestingly, for each series, the

13 For the BEA measure of the relative price of capital,the standard deviation is 2.59 for the medium term, 0.82 forthe high frequency, and 2.44 for the medium frequency.Thus, while the overall volatility is lower, the breakdownacross frequencies is similar to the Gordon series.

TABLE 2—STANDARD DEVIATIONS: ANNUAL FREQUENCIES

Standard variablesMedium-term cycle

0–50High-frequencycomponent 0–8


Output per capita 3.87 2.2 3.15(3.06–4.67) (1.91–2.48) (2.55–3.75)

Hours 2.84 1.59 2.33(2.42–3.26) (1.44–1.74) (2.03–2.63)

Labor productivity 2.37 1.01 2.11(2.04–2.7) (0.96–1.07) (1.83–2.4)

Consumption 2.2 0.8 2.01(1.83–2.57) (0.73–0.87) (1.72–2.3)

Investment 9.59 4.5 8.29(4.5–14.68) (3.34–5.66) (3.64–12.94)

Other variables

TFP 2.39 1.25 2.01(2.07–2.7) (1.15–1.36) (1.76–2.25)

Relative price of capital (Gordon) 4.34 1.54 4.12(3.12–5.39) (1.27–1.49) (3.05–5.14)

R&D 7.65 3.4 5.95(4–8.99) (2.65–4.13) (3.64–8.26)

Total markup 4.35 2.38 3.65(3.27–5.43) (2.01–2.74) (2.74–3.56)

Capacity utilization 4.26 3.11 2.81(3.28–5.24) (2.38–3.84) (2.3–3.32)


high-frequency component shows relatively lit-tle persistence. In this case, for example, annualoutput being one percentage point above trendin year t implies that it is likely to be only 0.16percent above trend in year t � 1.14 Put differ-ently, conventional detrending methods yieldoutput fluctuations at the annual frequency thatexhibit low persistence. The same is true for theother series. In sharp contrast, the medium-termcycle exhibits significantly greater persistencefor each series. The first-order autocorrelationfor output is 0.65, for example, and is similarlylarge for the other series.

Table 4 presents statistics on comovementsamong the variables, again for both the standardbusiness cycle series and the nonstandard ones.In the first column is the cross correlation ofeach annual variable with annual output overthe medium-term cycle, and in the second col-umn is the same statistic for the conventionallymeasured cycle. By and large, the positive co-movements among the standard variables thatare a hallmark of the conventional businesscycle are also a feature of the medium-termcycle. Similarly, the nonstandard variables forthe most part display a similar comovementwith output across the different frequencies.One notable exception is the relative price ofcapital. At the high frequency there is only a

relatively weak negative correlation betweenannual movements in the relative price of cap-ital (�0.24) and output. This negative relation isclearly stronger over the medium-term cycle(�0.56).15 In addition, while for R&D the con-temporaneous relationship with output is simi-larly positive across the different measures ofthe cycle, the lead/lag pattern is distinct. As weshow later (Figure 5) there appears a lead ofR&D over output over the medium-term cycle.This pattern is absent at the high frequency.

To fully appreciate the behavior of the pro-ductivity-related variables in the medium-termcycle, it is useful to examine the respectivedetrended time series. The three panels in Fig-ure 3 plot the medium-term cycle along with theassociated medium-term component for TFP,the relative price of capital, and R&D, respec-tively. The TFP series is clearly procyclicalover the medium term, exhibiting about two-thirds of the volatility of output. Interestingly,the medium-term cycle in the relative price ofcapital is almost the mirror image of that ofTFP. To the extent that declines in the relativeprice of capital reflect embodied technologicalchange, the figure suggests that there is a strongpositive comovement of both disembodied andembodied technological change. The productiv-ity slowdown of the 1970s and early 1980s, forexample, featured a slowdown of both types of

14 This point estimate is slightly smaller than the pointestimate for HP-filtered output (around 0.4), but this differ-ence is not statistically significant.

15 For the BEA measure, the cross correlation over themedium-term cycle is �0.46 and over the high frequency is�0.17, suggesting a very similar cyclical pattern to theGordon series.

TABLE 3—FIRST-ORDER AUTOCORRELATIONS: ANNUAL

FREQUENCIES

Standard variablesMedium-term

cycle 0–50High-frequencycomponent 0–8

Output per capita 0.65 0.16Hours 0.65 0.11Labor productivity 0.79 0.19Consumption 0.84 0.4Investment 0.76 0.22

Other variables

TFP 0.7 0.16Relative price of capital

(Gordon)0.86 0.33

R&D 0.83 0.07Total markup 0.66 0.09Capacity utilization 0.51 0.19

TABLE 4—CONTEMPORANEOUS CORRELATION WITH

OUTPUT: ANNUAL FREQUENCIES

Standard variablesMedium-term

cycle 0–50High-frequencycomponent 0–8

Hours 0.92 0.85Labor productivity 0.67 0.77Consumption 0.75 0.87Investment 0.67 0.84

Other variables

TFP 0.8 0.83Relative price of capital

(Gordon)�0.56 �0.24

R&D 0.31 0.3Total markup �0.82 �0.65Capacity utilization 0.67 0.93


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C. Nonfederally Funded R&D

B. Relative Price of Capital (Gordon)

A. TFP

FIGURE 3. A. TFP B. RELATIVE PRICE OF CAPITAL (GORDON) C. NONFEDERALITY FUNDED R&D


technological change. Conversely, both types oftechnological change were features of the pro-ductivity boom in the latter part of the 1990s.Finally, as the bottom panel shows, the move-ment in R&D over the medium term is highlyprocyclical. Note the strong inverse relationwith the relative price of capital. This is a sig-nificant finding, since the R&D data are mea-sures of the resources spent to develop newmanufacturing goods, which should enhanceembodied technological change.

Thus, overall, procyclical movement of disem-bodied and embodied technological change and ofR&D, particularly at the medium frequencies, ap-pears to be a salient feature of the medium-termcycle. As we noted, however, the medium-termcycle is not only a productivity phenomenon,since medium-frequency movements in themarkup and capacity utilization appear impor-tant as well, as Tables 2, 3, and 4 suggest.

II. Model

We now develop a model of medium-termbusiness fluctuations. The model is annual asopposed to quarterly, as we noted earlier, be-cause we are interested in capturing fluctuationsover a longer horizon than is typically studied.To this end, we abstract from a number ofcomplicating factors that otherwise might beuseful for understanding quarterly dynamics,such as money and nominal rigidities.

We consider a two-sector version of a rea-sonably conventional real business cycle model,modified to allow for, among other things, en-dogenous productivity and endogenous counter-cyclical markups.16 Procyclical entry and exitby final goods firms induces procyclical varia-tion in the degree of competition and, hence,countercyclical variation in (final goods) pricemarkups, in the spirit of Rotemberg and Mi-chael Woodford (1995). The precise formula-tion we use, though, is based on Galı́ andFabrizio Zilibotti (1995).

Final goods producers use intermediategoods as inputs, along with capital and labor. Asin Romer (1990), creation of new specializedintermediate goods stemming from R&D is the

source of technological change. As we noted inthe introduction, we modify the Romer frame-work to allow for an endogenous rate of adop-tion of new technologies, along with endogenousR&D. By doing so, we are able to allow forempirically reasonable diffusion lags but stillgenerate endogenous medium-term swings inproductivity. Roughly speaking, endogenousR&D permits disturbances to have permanenteffects on productivity movements, while en-dogenous adoption rates accelerate the transi-tion of productivity to the new steady state.Because the model has two sectors, consump-tion goods versus capital goods, we are able todistinguish between embodied and disembodiedtechnological change.

Households are conventional, except we al-low them a bit of market power in labor supplyin order to motivate a wage markup. We thenallow for exogenous variation in this marketpower. This mechanism combined with the en-try/exit mechanism in the goods market permitsthe model to generate countercyclical move-ments in both price and wage markups that areconsistent with the data. As we alluded to ear-lier, this simple structure is meant as a stand-infor a richer model of markup variation.

We first describe final goods firms, and alsocharacterize the entry and exit/markup mecha-nism. We next characterize the creation andadoption of new intermediate goods. Then weturn to households, and finally characterize thecomplete equilibrium.

A. Final Goods Output

Final Output Composite.—There are twosectors: a capital goods sector (k) and a con-sumption goods sector (c). Within each sectorthere is a final output composite, Yx,t; x � k, c.We take the price of the consumption goodcomposite as the numeraire.

The composite for each sector is a CES ag-gregate of the output of Nx,t final goods firms,each producing a differentiated product. Let Yx,t

j

be the output of final goods firm j in sector x.Then:

(1) Yx,t � ��0

Nx,t

(Yx,tj )1/�x,t dj��x,t

,16 For a recent characterization of the real business cycle

framework and its variants, see Sergio Rebelo (2005).


with �x,t � 1. The time-varying parameter �x,tis inversely related to the price elasticity ofsubstitution between the differentiated goods. Inthe symmetric equilibrium, it is the grossmarkup of price over marginal cost that finalgoods producers charge. Individual firms takeas given the time paths of Nx,t and �x,t. As wediscuss shortly, the entry/exit process jointlydetermines these two variables in equilibrium.

Each final goods firm j produces a specializedgood using the following inputs: capital ser-vices, Ux,t

j Kx,tj , where Kx,t

j is capital (of firm j insector x) and Ux,t

j is the capital utilization rate;labor, Lx,t

j ; and an intermediate good composite,Mx,t

j . The production function is Cobb-Douglas,as follows:

(2) Yx,tj � ��Ux,t

j Kx,tj ��Lx,t

j �1 � �1 � ��Mx,tj �,

where � is the intermediate goods share and � isthe capital goods share of value added. Follow-ing Jeremy Greenwood et al. (1988), we assumethat the depreciation rate of capital is increasingin the utilization rate, i.e., for each firm thedepreciation rate is given by �(Ux,t

j ), with�� 0.

Intermediate Goods Composite.—The inter-mediate good composite used by firm j, in turn,is the following CES aggregate of differentiatedintermediate goods, where Ax,t is the total num-ber of specialized goods in use within eachsector:17

(3) Mx,tj � ��

0

Ax,t

(Mx,tj,k)1/� dk��

,

with � � 1. Within each sector, each producerk is a monopolistic competitor who has acquiredthe right to market the respective good via theproduct development and adoption process thatwe describe below. In the symmetric equilib-rium, � is the producer’s gross markup. TheCES formulation implies gains from expandingvariety. Thus, as in Romer (1990) and others,creation and adoption of new intermediate prod-

ucts (i.e., increasing Ax,t) provides the ultimatesource of productivity growth.

Observe that with fixed values of Nx,t and Ax,t,the model dynamics would mimic those of a(two-sector) real business cycle model. In ourframework, however, entry/exit behavior will in-duce stationary procyclical fluctuations in Nx,t thatwill induce countercyclical fluctuations in �x,t. Inturn, endogenous technology development andadoption will induce permanent adjustments ofAx,t, inducing long procyclical swings in both em-bodied and disembodied productivity.

We now turn to the entry/exit and endoge-nous productivity mechanisms.

B. Entry/Exit and Markups

To characterize the joint determination Nx,tand �x,t, for each sector, we use a simple ap-proach based on Galı́ and Zilibotti (1995) (GZ)where procyclical competitive pressures associ-ated with endogenous procyclical net entry in-duce countercyclical movements in the markup.18

Within the GZ framework, competitive pres-sures are increasing in the number of firms inthe market. In particular, the elasticity of sub-stitution among final goods is increasing in thenumber of active firms. Because the markupvaries inversely with the elasticity of substitu-tion, the model implies the following reduced-form relation between the markup and the totalnumber of firms:

(4) �x,t � ��Nx,t �; �� 0.

As in GZ, we assume that final output firmsin sector x must pay a per-period operating costbx�t, which the firm takes as given. Let �(�x,t,Px,tYx,t

j ) be firm profits (which depend on themarkup and output value) where Px,t is the priceof the output composite in sector x (with Pc,t �1). Then, for any firm j to be in the market, itmust be the case that profits are not less thanoperating costs:

(5) ��x,t , Px,t Yx,tj � bx�t .

17 Production is “roundabout,” in the respect that spe-cialized intermediate goods are made by costlessly trans-forming final consumption goods. See, e.g., Basu (1995).

18 It is also possible to generate countercyclical markupsby allowing for money and nominal price rigidities (see,e.g., the discussion in Galı́ et al., 2002). To keep the modeltractable, however, we abstract from money and nominalrigidities.


In a symmetric competitive equilibrium, equa-tion (5) will hold with equality. This zero-profitcondition along with equation (4) will thenjointly determine the markup and the quantityof firms. As we show, in the general equilib-rium, entry will vary procyclically within eachsector and the markup, countercyclically.

In order to ensure a balanced growth equilib-rium with stationary markups in each sector, weassume that the time-varying component of op-erating costs, �t, drifts up over time proportion-ately with the social value of the aggregatecapital stock, as follows:

(6) �t � PtIKt,

where PtI is the social price of the capital (i.e.,

the competitive price) and Kt is the aggregatecapital stock. Note that an individual firm takes�t as given, since it depends only on aggregatefactors. One possible interpretation of this for-mulation is that operating costs are proportion-ate to the sophistication of the economy, asmeasured by the social value of the capitalstock. It is important to note that, because �tdrifts quite slowly in relative terms, its variationdoes not significantly affect medium-termdynamics.

C. R&D and Adoption

As we discussed earlier, we modify Romer(1990) to allow for two stages: R&D and adop-tion. In addition, we distinguish between inno-vation in the capital goods sector versus theconsumption goods sector. Doing so allows usto capture the medium-term variation in therelative price of capital. We first characterizeR&D and then adoption.

R&D (Creation of New Intermediate Goods).—Innovators within each sector develop new in-termediate goods for the production of finaloutput. They then sell the rights to this good toan adopter who converts the idea for the newproduct into an employable input, as we de-scribe in the next section.

We assume a technology for creating newspecialized goods that permits a simple decen-tralization of the innovation process, as in Ro-mer (1990). We introduce some modifications,however, that permit us to calibrate the model ina manner that is reasonably consistent with the

data and also that enhance the ability of themodel to generate procyclical movements inR&D activity.

Specifically, each innovator p in sector x con-ducts R&D by using the final consumption goodcomposite as input into developing new prod-ucts. He finances this activity by obtaining loansfrom households. Let Sx,t

p be the total amount ofR&D by innovator p in sector x and let Zx,t

p behis total stock of innovations. Let x,t be aproductivity parameter that the innovator takesas given, and let 1 � � be the probability thatany existing specialized good becomes obsoletein the subsequent period (whether or not it hasbeen adopted). Then the innovation technologyis given as follows:

(7) Zx,t � 1p � x,t Sx,t

p � �Zx,tp ,

where � is the implied product survival rate.Allowing for a (small) rate of obsolescence oftechnologies is not only realistic, it also gives usflexibility in calibrating the model to match thebalanced growth evidence.

We assume that the technology coefficientx,t depends on aggregate conditions which aretaken as given by individual innovators. In par-ticular, we assume

(8) x,t � x Zx,t ��t� � Sx,t

1 � ��1,

with 0 � � 1 and where x is a scale param-eter. As in the Romer model, there is a positivespillover of the aggregate stock of innovations,Zx,t. In addition, we relax the restriction of a unitelastic response of new technology develop-ment to R&D by introducing a congestion ex-ternality via the factor [�t

� � Sx,t1��]�1. This

congestion effect raises the cost of developingnew products as the aggregate level of R&Dintensity increases (since the factor is decreas-ing in Sx,t). It is straightforward to show that inequilibrium the R&D elasticity of new technol-ogy creation becomes � under this formulation.We are thus free to pick this elasticity to matchthe evidence. As a way to ensure that thegrowth rate of new intermediate product isstationary, we also assume that the congestioneffect depends positively on the scaling factor�t. This is tantamount to assuming that, ev-erything else equal, the marginal gain fromR&D declines as the economy becomes more


sophisticated, as measured by the aggregatecapital stock.

Let Jx,t be the value of an “unadopted” spe-cialized intermediate good. This is the price atwhich an innovator can sell a new product to anadopter. Given the linearity of the R&D tech-nology (as perceived by the individual productdeveloper) and given free entry, each new prod-uct developer must exactly break even. As aconsequence, the following arbitrage conditionsmust be satisfied in any equilibrium with aninterior solution for innovation:

(9) 1/x,t � �Et ��t � 1 Jx,t � 1�,

where �t�1 is the stochastic discount factor.The right side of equation (9) is the dis-

counted marginal benefit from developing anew product, while the left side is the marginalcost.

Equation (9) strongly hints at how the frame-work generates procyclical R&D. During aboom, for example, the expected value of a new“unadopted” product, EtJx,t�1, increases. Thatis, since the profit flow from specialized inter-mediate goods rises, the benefit to creating newtypes of these goods goes up. R&D spendingwill increase in response.

Adoption (Conversion of Z to A).—We nextcharacterize how newly developed intermediategoods are adopted over time (i.e., the process ofconverting Zx,t to Ax,t). Our goal is to capture thenotion that adoption takes time on average, butallow for adoption intensities to vary procycli-cally, consistent with the evidence described inComin and Gertler (2004). In addition, wewould like to characterize the diffusion processin a way that minimizes complications fromaggregation. In particular, we would like toavoid having to keep track, for every availabletechnology, of the fraction of firms that have orhave not adopted it.

These considerations lead us to the followingformulation: a competitive set of “adopters”converts available technologies into use. As wediscussed earlier, they buy the rights to thetechnology from the original innovator. Theythen convert the technology into usable formby employing a costly and potentially time-consuming process. To fund adoption expenses,they obtain loans from households. Once the

new product is in usable form, the adopter isable to manufacture it and sell it to final goodsproducers who use it as part of the intermediategoods composite in the production function (seeequations (2) and (3)).

The pace of adoption depends positively onthe level of adoption expenditures in the follow-ing simple way: whether or not an adopter suc-ceeds in making a product usable in any givenperiod is a random draw with (success) proba-bility �x,t that is increasing in the amount theadopter spends, Hx,t (in units of the final con-sumption goods composite). If the adopter is notsuccessful, he may try again in the subsequentperiod. Thus, under our formulation there isslow diffusion of technologies on average (asthey are slow on average to become usable).The diffusion rate, however, varies positivelywith the intensity of adoption expenditures. Atthe same time, aggregation is simple since oncea technology is in usable form, all firms mayemploy it.

In particular, let Vx,t be the value to anadopter of successfully bringing a new productinto use, i.e., the present value of profits theadopter would receive from marketing the spe-cialized intermediate good, given by:

(10) Vx,t � �m,x,t � �Et ��t � 1 Vx,t � 1�

where �m,x,t is profits at t in sector x earned ona specialized intermediate input. Then we mayexpress the value of an “unadopted” product tothe adopter as

(11)

Jx,t � maxHx,t

��Hx,t � �Et��t � 1��x,tVx,t � 1

� �1 � �x,t )Jx,t � 1)}

where �x,t is the following increasing functionof adoption expenditures, Hx,t, and a scalingparameter parameter �x,t that is exogenous tothe adopter:

(12) �x,t � ��Ax,t

�t� Hx,t� ,

with � � 0, �� 0, and where the adoptertakes the scaling factor Ax,t/�t as given. We


introduce the (slow-moving) scaling factor tokeep the adoption rate stable along a balanced-growth path. Otherwise, the pace of adoptioncould increase without bound as the economygrows. Our formulation assumes a spillovereffect from aggregate adoption, Ax,t, to indi-vidual adoption (i.e., one can learn from theadoption activities of others) that diminishesas the economy becomes more complex, asmeasured by the value of aggregate capitalstock �t.

In period t the adopter spends Hx,t. Withprobability �(�x,tHx,t) he succeeds in makingthe product usable in the subsequent period,enabling him to generate a profit stream worthRt�1

�1 �Vx,t�1 in current value. With probability1 � �(�x,tHx,t), he fails and must start over. Atthe margin, each adopter adjusts conversion ex-penditures until the marginal expenditure (oneunit of final goods consumption output) equalsthe discounted marginal benefit:

(13) 1 �Ax,t

�t� � �Et ��t � 1 �Vx,t � 1 � Jx,t � 1�.

It is now easy to see why adoption expenditureswill move procyclically. During booms, thevalue of having a marketable product rises rel-ative to one still in the development phase, i.e.,Vx,t�1 rises relative to Jx,t�1. In this case, Hx,twill increase since it becomes worth more at themargin to invest in adoption. The reverse, ofcourses, will happen in downturns.

Overall, the following partial adjustmentequation characterizes the adoption rate by anyadopter q:

(14) Ax,t � 1q � �x,t��Zx,t

q � Ax,tq � �Ax,t

q .

The term in brackets is the stock of productsat t that the adopter owns but has not yetconverted. He is able to successfully adoptthe fraction �x,t (which depends on his expen-diture intensity Hx,t).19 The business cycleaffects this evolution in two ways. First, thestock of new products created moves procy-clically due to procyclical movement in R&D

intensity, raising Zx,tq on average across adopt-

ers. Second, adoption expenditures vary pro-cyclically, causing the procyclical variation inadoption rates.

D. Households and Government

Households.—Our formulation of the house-hold sector is reasonably standard. One distinc-tive feature, however, is that we allow eachhousehold to have a bit of market power in laborsupply as a way to motivate a wage markup. Aswe noted earlier, time variation in the wagemarkup will provide the source of exogenousdisturbances in our baseline model.

There is a continuum of households of mea-sure unity. Each household supplies a differen-tiated type of labor Lt

h. Labor input Lt is thefollowing CES aggregate of all the differenttypes of household labor:

(15) Lt � ��0

1

(Lth)1/�w,t dh��w,t

,

where �w,t governs the elasticity of substitutionacross different types of labor and, in equilib-rium, will correspond to a markup of wagesover each household’s cost of supply labor (inperfect analogy to the price markup for eachsector �w,t). Let Wt

h be the wage of household hand Wt the wage index. Then cost minimizationby firms implies that each household faces thefollowing demand curve for its labor:

(16) Lth � �Wt

h

Wt��w,t /�1��w,t �

Lt

with

(17) Wt � ��0

1

(Wth)1/�1 � �w,t� dh� 1 � �w,t

.

In addition to supplying labor, each householdconsumes and saves. We allow for perfect con-sumption insurance (implying in our case thatall households will consume the same amount).A households may save by either accumulatingcapital or lending to innovators and adopters. Italso has equity claims on all monopolistically

19 Note that the total (per period) investment in adoptionundertaken by the qth adopter is Hx,t � [Zx,t

q � Ax,tq ].


competitive firms. It makes one-period loans toinnovators and adopters and also rents capitalthat it has accumulated directly to firms.

Let Ct be consumption. Then the householdmaximizes the present discounted utility asgiven by the following expression:

(18) Et �i � 0

�

�i� ln Ct � i ��Lt � i

h �1 � �

1 � � �,

with 0 � 1 and � � 0. The budget con-straint is as follows:

(19) Ct � WthLt

h � �t � �Dt � PtkKt

� PtkKt � 1 � Rt Bt � Bt � 1 � Tt ,

where �t reflects the profits of monopolisticcompetitors paid out fully as dividends tohouseholds, Bt is total loans the householdmakes at t � 1 that are payable at t, and Ttreflects lump-sum taxes. Rt is the (possiblystate-contingent) payoff on the loans. Thehousehold’s decision problem is simply tochoose the wage rate, consumption, labor sup-ply, capital, and bonds to maximize equation(18) subject to (19), as well as the labor demandcurve (16).

Observe that because a household has somemarket power, the wage it chooses is a markup�w,t over its marginal rate of substitution be-tween labor and consumption, (Lt

h)�Ct:

(20) Wth � �w,t � �Lt

h��Ct .

Further, in the symmetric equilibrium, allhouseholds will charge the same wage and sup-ply the same amount of labor. In equilibrium,thus, �w,t is the markup of the wage index overthe household marginal rate of substitution.

As we noted in the introduction, a number ofauthors have argued that countercyclical move-ments in (something like) �w,t provides themain source of high-frequency business fluctu-ations, although there is considerable disagree-ment as to what truly underlies the fluctuationsin this variable. As we noted earlier, variouslabor market frictions that give rise to wagerigidity can generate countercyclical move-ments in �w,t. Rather than adding further to thecomplexity, however, we simply explore theimplications of exogenous shifts in �w,t, keep-

ing in mind that these shifts may reflect labormarket frictions that we have not explicitlymodeled.

Government.—Finally, government spendingis financed with lump-sum taxes:

(21) Gt � Tt .

E. Symmetric Equilibrium Conditions

The economy has a symmetric sequence ofmarkets equilibrium. The endogenous statevariables are the aggregate capital stock, Kt, thetotal stocks of intermediate goods invented ineach sector, Zc,t and Zk,t, and the total stock ofintermediate goods adopted in each sector, Ac,tand Ak,t. The following system of equationscharacterizes the equilibrium.

Resource Constraints and Technology.—LetYt denote aggregate net value added output.This quantity equals the sum over sectors of thevalue of gross output, Px,tYx,t, net expenditureson specialized intermediate goods, (Ax,t)

1��Mx,t,and operating costs, �x,t:

(22) Yt � �x � c,k

�Px,tYx,t � �Ax,t�1 � �Mx,t � �x,t.

(Recall, Pc,t � 1.) Note that net expenditures onintermediate goods in a sector is the total pro-duced Mx,t normalized by the per-unit cost ofproducing (Ax,t)

��1. The latter reflects the pro-ductivity gains from diversity implied by theCES aggregator (equation (3)).

The uses of output, in turn, are divided be-tween consumption, Ct, investment, Pk,tYk,t,government spending, Gt, and the total costs ofR&D and adoption, ¥x�c,k [Sx,t � (Zx,t� j �Ax,t)Hx,t]:

(23) Yt � Ct � Pk,t Yk,t � Gt

� �x � c,k

�Sx,t � �Zx,t � j � Ax,t �Hx,t .

In addition, capital evolves over time accord-ing to:

(24) Kt � 1 � �1 � ��Ut ��Kt � Yk,t .


The gross final output composite in each sec-tor, Yx,t, is given by

(25) Yx,t � �Nx,t��x,t�1��UtKt

Lt��

�Lx,t��1��

� �Mx,t��,

where the term (Nx,t)�x,t�1 reflects the efficiency

gains to diversity implied by the CES aggrega-tor for final goods in each sector.

Factor Markets.—The labor market in eachsector satisfies the requirement that the marginalproduct of labor equals the product of the(sector-specific) markup and the household’smarginal rate of substitution between leisureand consumption:

(26) �1 � ��1 � ��Px,t Yx,t

Lx,t� �x,t�w,t Lt

�Ct ,

where aggregate labor input Lt equals the sumof labor input across sectors:

(27) Lt � Lc,t � Lk,t .

Observe that the total markup of the marginalproduct of labor over the household’s marginalrate of substitution in each sector corresponds tothe product of the price and wage markups:�x,t�w,t. This latter expression, in turn, corre-sponds to the labor market wedge that we usedearlier to compute the time series for themarkup.

It is straightforward to show that the capital/labor ratios and the capital utilization rates areuniform across sectors. The equilibrium condi-tions for the sectoral allocations of capital, thesectoral utilization rates, and the sectoral allo-cations of the intermediate goods are, then, re-spectively:

(28) ��1 � ��Px,t Yx,t

Kx,t� �x,t �Dt � ��Ut �Pt

K,

(29) ��1 � ��Px,t Yx,t

Ut� �x,t��Ut �Pt

KKx,t ,

(30) �Px,t Yx,t

Mx,t� �x,t Px,t

M ,

where Px,tM is the relative price of the intermediate

goods composite in sector x (see below), and with

(31) Kx,t � �Kt /Lt �Lx,t .

Consumption/Saving.—We can express theintertemporal Euler equation as

(32) Et��t � 1�Dt � Pt � 1K

PtK �� 1,

with �t�1 � �Ct/Ct�1.Arbitrage between acquisition of capital and

loans to adopters and innovators implies

(33) �Et�t � 1Rt � 1�

� Et��t � 1�Dt � Pt � 1K

PtK ��.

We now characterize the variation in finalgoods firms, Nx,t, and in specialized intermedi-ate goods, Ax,t, beginning with the former.

Entry and Exit.—Free entry by final goodsproducers in each sector yields the followinginverse link between the sector’s markup,�x,t(Nx,t), and per-firm output, (Nx,t)

1��x,tYx,t/Nx,t � (Nx,t)

��x,tYx,t.

(34)�x,t �Nx,t � � 1

�x,t �Nx,t ��Nx,t �

��x,tYx,t � bx�t

where the scaling factor, �t, drifts slowly overtime according to equation (6).

Evolution of Productivity.—The R&D tech-nology (equations (7) and (8)) implies that inthe symmetric equilibrium the gross growth rateof new intermediate products in each sectordepends positively on R&D (scaled relative tothe social value of the capital stock), as follows:

(35)Zx,t � 1

Zx,t� x�Sx,t

�t� �

� �.

Similarly, the optimality condition for adoption,along with the adoption technology (equations(12) and (14)), imply that the gross rate ofadoption of new intermediate goods in equilib-rium obeys the following partial adjustmentmechanism, where the rate of adjustment de-pends positively on adoption expenditures, Hx,t:


(36)Ax,t � 1

Ax,t� ��Ax,t

�tHx,t��Zx,t

Ax,t� 1� � �.

Free entry into research and development(see equations (7), (8), and (9)) implies that thediscounted value of the gain from R&D mustequal the total cost

(37) �Et ��t � 1 � Jx,t � 1� � �Zx,t � 1 � �Zx,t � � Sx,t .

Similarly, free entry into adoption (see equa-tions (12) and (13)) implies that adoption ex-penditures will be increasing in the differencebetween the value of an adopted project Vx,t�1,versus the value of an unadopted one Jx,t�1:

(38)Ax,t

�t� � �Et ��t � 1 �Vx,t � 1 � Jx,t � 1 � � 1,

with Vx,t given by the discounted stream ofearnings from a specialized intermediate good:

(39) Vx,t � ��x,t� 1�1 � 1/��

Px,t Yx,t

Ax,t�

� �Et��t � 1Vx,t � 1�,

where the first term in brackets is profits at timet and where the value of an unadopted technol-ogy, Jx,t, obeys

(40)

Jx,t � �Hx,t � �Et��t � 1�Jx,t � 1

� ��x,tHx,t��Vx,t � 1 � Jx,t � 1)}.

Relative Prices and Productivity Dynamics.—Finally, we present expressions for the equilib-rium-relative prices of intermediate goods andof capital. We also draw the connection of theseprices to movements in both disembodied andembodied technological change.

From the CES aggregator for intermediategoods (equation (3)), the relative price of theintermediate good composite in the symmetricequilibrium is given by

(41) Px,tM � � �Ax,t �

1 � �.

Observe that Px,tM declines as the number of

specialized intermediate products, Ax,t, in-creases, since � exceeds unity. Technologicalprogress is thus manifested in falling prices ofintermediate goods.

To see the link between intermediate goodsprices and disembodied productivity, considerthe expression for net value added output withineach sector, Yx,t

v (obtained by netting out mate-rials and overhead costs):

(42) Yx,tv � �x,t Kx,t

� Lx,t1 � �

where �x,t is the Solow residual in sector x andis given by

(43) �x,t � �1 ��

�x,t��Nx,t

�x,t�1

� � �

�x,tPx,tM��/(1 � �)

Ut��

�x,t

Kx,t� Lx,t

1 � � .

Observe that �x,t increases as Px,tM declines. The

strength of the effect, further, depends posi-tively on the intermediate goods share parame-ter �.

Movements in Px,tM generate variation in �x,t

that reflects true technological change. Weshould expect the influence of Px,t

M on �x,t to playout mainly over the medium and low frequen-cies, since the force underlying this relationshipis the variation of Ax,t, which, in turn, is aproduct of the endogenous research and tech-nology adoption processes. Observe also thatunderlying the Solow residual are several othersfactors that do not reflect true technologicalchange and that are likely to be important at thehigh frequency. The capital utilization rate, Ut,enters �x,t because it reflects unmeasured inten-sity of input usage. The term Nx,t

�x,t�1 reflects theagglomeration effect present in the CES com-posite for final output. The markup �x,t also enters�x,t due to the roundabout input-output nature ofproduction. On balance, Ut, Nx,t

�x,t�1, and �x,tcontribute to procyclical variation in �x,t, par-ticularly at the high frequencies, while Px,t

M con-tributes to procyclical variation mainly at themedium frequencies and below.


Lastly, we consider the relative price of cap-ital and embodied technological change. Inequilibrium, Pt

K obeys

(44) PtK � ��k,t

�c,t� � �Nk,t

1 � �k,t

Nc,t1 � �c,t� � Pt

I,

where PtI is the social price of capital (i.e., the

price that would arise in competitive markets),given by

(45) PtI � �Ak,t

Ac,t� �1 � ��

� �Pk,tM

Pc,tM � �

.

Movements in the social price of capital dependon the state of technology in the capital goodssector relative to the consumption sector, asmeasured by the ratio of intermediate goodssector prices. Note that Pt

I is likely to varycountercyclically over the medium term be-cause the pace of endogenous technologicalchange in the capital goods sector relative to theconsumption goods sector is likely to vary pro-cyclically: in particular, R&D and adoptionexhibit stronger procyclical variation in the invest-ment sector as a consequence of the stronger pro-cyclical variation of profitability in this sector.

As with the effect of Px,tM on �x,t, the impact of

Pk,tM /Pc,t

M on PtK reflects true technological

progress and is also likely to play out over themedium and low frequencies. There are simi-larly nontechnological influences on Pt

K that arelikely to be particularly relevant at the highfrequency. Specifically, Pt

K depends on the rel-ative markups in the two sectors and also on therelative number of firms operating in each mar-ket (the latter reflects the relative strength of theagglomeration effect from the CES aggrega-tors). Note that the high-frequency variation inPt

K is also likely to be countercyclical becauseentry exhibits greater procyclical variation inthe capital goods sector and, conversely, themarkup exhibits greater countercyclical varia-tion. Thus, forces at both the high and mediumfrequency contribute to countercyclical varia-tion in Pt

K over the medium term.

III. Model Simulations

In this section we explore the ability of themodel to generates medium-term cycles. Wesolve the model by loglinearizing around thedeterministic balanced growth path and then

employing the Anderson-Moore code, whichprovides numerical solutions for general first-order systems of difference equations. We firstdescribe the calibration and then turn to somenumerical exercises.

A. Model Calibration

The calibration we present here is meant as abenchmark. We have found that our results arerobust to reasonable variations around this bench-mark. To the extent possible, we use the restric-tions of balanced growth to pin down parametervalues. Otherwise, we look for evidence else-where in the literature. There is a total of 18parameters. Ten appear routinely in other studies;the eight others relate to the R&D and adoptionprocesses and also to the entry/exit mechanism.

We begin with the standard parameters. Weset the discount factor � equal to 0.95 to matchthe steady state share of nonresidential invest-ment to output. Based on steady-state evidence,we also choose the following number: (the cap-ital share) � � 0.33; (the materials share) � �0.50; (government consumption to output)G/Y � 0.2; (the depreciation rate) � � 0.08; and(the steady-state utilization rate) U � 0.8. Weset the inverse of the Frisch elasticity of laborsupply � at unity, which represents an interme-diate value for the range of estimates across themicro and macro literature. Similarly, we set theelasticity of the change in the depreciation ratewith respect the utilization rate, (��/�)U at0.33, which lies within the range that otherstudies suggest.20 Finally, based on evidence inBasu and John G. Fernald (1997), we fix thesteady-state gross valued added markup in theconsumption goods sector, �c, equal to 1.1 andthe corresponding markup for the capital goodssector, �k, at 1.2.

We next turn to the “nonstandard” parame-ters. The steady-state growth rates of GDP andthe relative price of capital in the model arefunctions of the growth rate of new technologiesdeveloped in the consumption and capital goodssectors. By using the balanced growth restric-

20 We set U equal to 0.8 based on the average capacityutilization level in the postwar period as measured by theBoard of Governors. We set (��/�)U equal to 1⁄3 based onthe range of values used in the literature that vary from 0.1used by Robert G. King and Rebelo (1999) to unity used byBaxter and Dorsey Farr (2001).


tions and matching the average growth rate ofnonfarm business output per working-age per-son (0.024), the average growth rate of theGordon quality adjusted price of capital relativeto the BEA price of consumption goods andservices (�0.026), and the average share ofprivate research expenditures in GDP during thepostwar period (0.009),21 we can identify theannual survival probability of new technologies,�, and the productivity parameters in the tech-nologies for creating new intermediate goods ineach sector, c and k. Accordingly, we set � �0.97 (implying an annual obsolescence rate ofthree percent), c � 2.27, and k � 32.36.

There is no direct evidence on the grossmarkup � for specialized intermediate goods.Given the specialized nature of these products,it seems that an appropriate number would be atthe high range of the estimates of markups inthe literature for other types of goods. Accord-ingly, we choose a value of 1.6, but emphasizethat our results are robust to reasonable varia-tions around this number.

There is also no simple way to identify theparameter �, the elasticity of new intermediategoods with respect to R&D. Zvi Griliches(1990) presents some estimates using the num-ber of new patents as a proxy for technologicalchange. The estimates are noisy and range fromabout 0.6 to 1.0, depending on the use of panelversus cross-sectional data. We accordingly set� equal to 0.8, the midpoint of these estimates.

We now consider the parameters that governthe adoption process. To calibrate the steady-state adoption frequency �, we use informationon diffusion lags. Given that � is the probabilitya technology is adopted in any given period, theaverage time to adoption for any intermediategood is 1⁄� .Mansfield (1989) examines a sampleof embodied technologies and finds a mediantime to adoption of 8.2 years. Comin andGertler (2004) examine a sample of British datathat includes both disembodied and embodied

innovations. They find median diffusion lags of9.8 and 12.5 years, respectively. Assuming thatthe median is not too different from the mean, areasonable value for � is 0.1, implying an av-erage time to adoption of ten years.

Another parameter we need to calibrate is theelasticity of the frequency of adoption with re-spect to adoption expenditures. Doing so is dif-ficult because we do not have good measures ofadoption expenditures, let alone adoption rates.One partial measure of adoption expenditureswe do have is development costs incurred bymanufacturing firms trying to make new capitalgoods usable (which is a subset of the overallmeasure of R&D that we used earlier). A simpleregression of the rate of decline in the relativeprice of capital (the relevant measure of theadoption rate of new embodied technologies inthe context of our model) on this measure ofadoption costs and a constant yields an elasticityof 0.95. Admittedly, this estimate is crude,given that we do not control for other determi-nants of the changes in the relative price ofcapital. On the other hand, we think it providesa plausible benchmark value.

We next turn to the entry/exit mechanism.We set the elasticity of the price markup withrespect to the number of firms in each sector(i.e., �log �x,t/�log Nx,t). By doing so, the over-all medium-term variation in the number offirms is roughly consistent with the data. In thedata, the percent standard deviation of the num-ber of firms relative to the standard deviation ofthe total markup over the medium-term cycle is0.65. With our parameterization, the model pro-duces a ratio of 0.15. Given, however, that thedata do not weight firms by size, and given thatmost of the cyclical variation is due to smallerfirms, the true size-weighted variation in thenumber of firms (relative to the markup) isprobably closer to 0.15 than 0.65. As a check,we show that our results are largely unaffectedby varying the price markup elasticity over therange from 0.5 to 1.5.

Finally, we fix the autocorrelation of the pref-erence/wage markup shock so that the modelgenerates an autocorrelation that approximatelymatches that of the overall gross markup,�t � �t

w, as measured by Galı́ et al. (2002). Thisresults in a value of 0.6. We then choose astandard deviation of the shock, so that the uncon-ditional variance of the overall gross markupmatches the data (for the medium-term cycle).

21 The NSF divides R&D into three components: basic,applied, and development. The first two are clearly includedin our notion of R&D. Development is defined by the NSFas expenses devoted to improving existing products or pro-cesses. Therefore, some of what the NSF classifies as devel-opment expenses may belong to our investments in technologyadoption. We take this view in the calibration and assume thatonly half of the private expenses in development as reported bythe NSF are R&D. That brings the average private R&Dexpenses in the postwar period to 9⁄10 of 1 percentage point.


B. Some Numerical Experiments

To be clear, the exercises that follow aremeant simply as a first pass at exploringwhether the mechanisms we emphasize havepotential for explaining the data: they are notformal statistical tests. As we discussed earlier,we treat innovations to the wage markup, �t

w,stemming from exogenous fluctuations inhousehold labor market power as the mainsource of disturbances to the economy. We keepin mind, though, that this simple mechanism ismeant as a shortcut for a richer description ofcountercyclical wage markup behavior.22

Ideally, we would like to evaluate the modelagainst moments of the data that are conditionalon a shock to �t

w. There are several difficultieswith this approach, however. First, while iden-tified VAR methods may be useful for measur-ing the near-term effects of shocks to theeconomy, they usually are very imprecise atmeasuring the kind of medium-term effects thatare the focus of our analysis.23 Second, identi-fication of orthogonalized movements in �t

w ishighly problematic, particularly given that inpractice there is likely a strong endogenouscomponent to this variable.

Rather than attempt to match impreciselyestimated conditional moments, we follow thelead of the traditional real business cycle lit-erature by exploring how well a single shock,presumed to be the principle driving force,can explain the unconditional patterns in thedata. We view this kind of exercise as a checkon the plausibility of the scenario we offer, asopposed to a clear test against alternatives.While the RBC literature focused on the abil-ity of technology shocks to account for thehigh-frequency variation in the data, we insteadconsider the ability of our model driven byshocks to �t

w to explain the combined high- andmedium-frequency variation, as well as each ofthe components. We also explore how well anexogenous technology shock model can explainmedium-term fluctuations, as compared to ourbenchmark model.

The Baseline Model with Endogenous Pro-ductivity.—Before confronting the data, we firstgain some intuition for the workings of themodel by examining the model impulse re-sponses to a shock to �t

w. In Figure 4, the solidline in each panel is the response of the respec-tive variable within our full-blown model. Thedotted line corresponds to the simple bench-mark with neither endogenous productivity norendogenous markup variation (though allowingfor endogenous factor utilization). The periodlength is a year.

The rise in �tw effectively raises the price of

labor, reducing labor demand and output. Boththe initial impact and the impact over time onoutput is larger in our full-blown model than inthe benchmark model. Within the full-blownmodel, the decline in output leads to a rise inboth the consumption and capital goods pricemarkups (due to a decline in entry within eachsector), enhancing the initial drop in output.Over time, output climbs back to trend but doesnot make it back all the way due to the endog-enous permanent decline in productivity, whichwe describe shortly. After a decade or so, itlevels off at a new steady state. The permanentdecline in output is about one-half of the initialdecline. In the benchmark model without en-dogenous productivity, output simply reverts toits initial steady state.

Even though the shock is not technological,the rise in �t

w generates a persistent decline inTFP and labor productivity and a persistentincrease in the relative price of capital. Theinitial decline in measured TFP and labor pro-ductivity results mainly from variable factorutilization (in conjunction with overhead costs)and from the rise in the price markups.24 Overtime, there is a decline in true productivityrelative to trend. In particular, the initial con-traction in economic activity induces a drop inboth R&D and the rate of technology adop-

22 As discussed in Galı́ et al. (2002), a model with eithernominal or real wage rigidities can generate a countercycli-cal wage markup.

23 In particular, long-run effects of impulse responsefunctions from identified VARs are measured much lessprecisely than are short-run effects.

24 The formal definition of TFP that we adopt is the fol-lowing: TFPt � YtAk,t

��(��1)/Kt�Lt

1��, where � is the fraction ofthe variety externality in the creation of new capital not cap-tured by the linear aggregation methodology used by the BEAto compute the capital stock. We calibrate � to 0.56 to matchthe long-run growth rate of TFP implied by the model to thepostwar data. In case the variety externality was completelymismeasured, annual TFP growth would be 1.72 percent in-stead of the actual 1.4 percent. In addition, this definitionassumes that the Bureau of Labor Statistics measures of TFPdo not correct properly for capacity utilization.


tion.25 The initial decline in R&D slows the rateof creation of new intermediate products in boththe goods and capital sectors, ultimately leading

to a permanent drop relative to trend in TFPand labor productivity. The slowdown in theadoption rate leads to a near-term decline indisembodied productivity growth. Absent thisendogenous slowdown in technology adoption,the decline in productivity following the drop inR&D would take much longer, assuming real-istic diffusion lags.

Similarly, variable price markup behavior ac-counts for the initial rise in the relative price ofcapital, while the increase over time is due to a

25 We are assuming that that R&D involves the same factorintensities as goods production. It is thus not as sensitive towage behavior as in the Schumpeterian models of PhilippeAghion and Peter Howitt (1992) and Aghion and Gilles Saint-Paul (1991). Barlevy (2004) shows that if productivity shocksin the final goods sector are sufficiently persistent, R&D can beprocyclical even in a Schumpeterian model.

0 10 20−1

−0.5

0

Y

0 10 20−1

−0.5

0

K

0 10 20−1

−0.5

0

L

0 10 20−0.5

0

0.5TFP

0 10 200

0.5

1

PK

0 10 20−1

−0.5

0

Y/L

0 10 20−3

−2

−1

0S

0 10 20−3

−2

−1

0λ

0 10 20−1.5

−1

−0.5

0

0.5U

0 10 200

0.5

1µw

FIGURE 4. IMPULSE RESPONSE FUNCTIONS FOR A UNIT SHOCK TO WAGE MARKUP


decline in productivity in the capital goods sec-tor relative to the consumption goods sector.Because investment initially falls proportion-ately more than output, the capital goodsmarkup rises relative to the consumption goodsmarkup, leading to a jump in the relative priceof capital. The disproportionate drop in invest-ment also implies that expected profits in thecapital goods sector fall relative to the con-sumption goods sector. As a consequence, thereis a relatively greater drop in both R&D andtechnology adoption in the capital goods sector,leading to a slowdown in the creation and adop-tion of specialized intermediate goods, relativethe consumption goods sector. This leads to apermanent increase in the relative price ofcapital.

We next explore how well the model pro-duces medium-term fluctuations. We do so bycomparing moments of artificial data generatedby the model economy (driven by shocks to �t

w)with the unconditional moments of the actualdata. Table 5 reports the standard deviationsof the variables for the medium-term cycleand also for the respective high- and medium-frequency components. As we noted earlier, we

normalize the exogenous shock to �tw so that the

unconditional variance of the total markup overthe medium term generated by the model ex-actly matches the data (4.35).

Overall, the model does reasonably well. Forseven of the nine variables (other than themarkup), the model standard deviations for themedium term are within the 95-percent confi-dence intervals for the data. For output andhours, the model closely matches both the over-all variation in the data and the breakdown ofthe variation between the high and mediumfrequencies. Even though the exogenous drivingforce is not technological, the model capturesreasonably well the variation in TFP over themedium term (2.42 versus 2.39), and does par-ticularly well at the medium frequencies (2.15versus 2.01). The model estimate is below the95-percent confidence interval for TFP at thehigh frequency, but still is about 82 percent ofthe mean in the data. The model is within theconfidence interval for the variation in laborproductivity over the medium-term cycle andover the medium frequencies, in particular. It istoo low at the high frequency. However, allow-ing for unobserved labor effort (which is likely

TABLE 5—MODEL EVALUATION: STANDARD DEVIATIONS

Frequency Medium-term cycle 0–50High-frequencycomponent 0–8


Variable Data Model Data Model Data Model

Output 3.87 4.18 2.20 2.22 3.15 3.46(3.06–4.67) (1.91–2.48) (2.55–3.75)

TFP 2.39 2.42 1.25 1.03 2.01 2.15(2.07–2.7) (1.15–1.36) (1.76–2.25)

Labor productivity 2.37 2.22 1.01 0.46 2.11 2.17(2.04–2.7) (0.96–1.07) (1.83–2.4)

Hours 2.84 3.13 1.59 1.87 2.33 2.49(2.29–3.09) (1.42–1.74) (1.88–2.48)

Relative price of capital 4.34 2.55 1.54 0.84 4.12 2.39(3.12–5.39) (1.27–1.49) (3.05–5.14)

R&D 7.65 11.69 3.40 6.4 5.95 9.62(4–8.99) (2.65–4.13) (3.64–8.26)

Investment 9.59 13.13 4.50 6.38 8.29 11.35(4.5–14.68) (3.34–5.66) (3.64–12.94)

Consumption 2.20 2.45 0.80 1.07 2.01 2.17(1.83–2.57) (0.73–0.87) (1.72–2.3)

Total markup 4.35 4.35 2.38 2.7 3.65 3.11(3.27–5.43) (2.01–2.74) (2.74–3.56)

Capacity utilization 4.26 4.86 3.11 2.45 2.82 4.11(3.28–5.24) (2.38–3.84) (2.3–3.32)

Note: For Tables 5–7, model moments are averages over 1,000 simulations of a sample size corresponding to the data.


more important at the high than the mediumfrequencies) would improve performance onthis dimension.

The two variables where the model does lesswell are the relative price of capital and R&D.The model generates roughly 60 percent of thevariability in the relative price of capital ob-served in the data (2.55 versus 4.34) and about50-percent greater volatility of R&D relative tothe data (11.69 versus 7.65). In each case, how-ever, measurement error in the data could be afactor. First, error in the quality adjustment forthe Gordon relative price of capital series couldbe magnifying the measure of volatility. As wenoted in Section I, the BEA measure of therelative price of capital, which employs a sim-pler method for quality adjustment, has a co-movement with output over the medium termthat is very similar to the Gordon series. How-ever, the standard deviation of the BEA seriesover the medium term is only 2.59, which isclose to the model’s prediction. Since the qual-ity adjustment in the Gordon series is the sub-ject of some controversy, a case can be madethat the true volatility lies somewhere in be-tween the Gordon and BEA estimates, implyingthat the model may not be as far off on therelative price of capital as the Gordon numbersmight suggest. Second, the discrepancy be-tween the model and the data regarding R&Dvolatility may be in part due to the very impre-cise measure of private R&D that is available.This measure, based on the NSF definition ofR&D that basically includes expenditures tocreate new goods, is just a subset of total privateR&D.26 It is also likely to contain a significantoverhead component, implying that variableR&D input is likely to be more volatile thanmeasured total input. Another possibility is thatthe returns to R&D investments diminish morerapidly than our baseline calibration implies.Interestingly, when we reduce �, the elasticityof new intermediate products to R&D, from 0.9to 0.7 (a number within the lower bound ofGriliches’s estimates), the volatility of R&Dfalls to the upper bound for the confidence in-terval of the data, with no significant effect onthe other moments of the other variables.

Figure 5 portrays how well the model cap-tures the cyclical comovements of variables inthe data. The top-left panel plots the autocorre-lation of output over the medium-term cycle, atlags and leads of ten years. The other panelsplot the cross correlation of a particular variablewith output at time t. In each case, the dashedline represents the data and the solid line themodel. The two dotted lines are 95-percent con-fidence intervals around the data. For most, themodel matches the comovements in the data: foreight of the ten variables the model cross cor-relations lie mainly within the respective confi-dence intervals. The notable exceptions againare the relative price of capital and R&D. Themodel does capture both the contemporaneousand lagged comovement of each of these twovariables with output. In each case, however, itimplies a smaller lead over output than the datasuggest. In the conclusion we discuss someways to improve the model’s ability to accountfor both the timing and volatility of each ofthese variables.

It is also worth emphasizing that the modelcaptures the variation of output at low frequen-cies relative to the high and medium frequen-cies. The ratio of the standard deviation ofoutput growth at frequencies below 50 years tothe standard deviation of output growth at fre-quencies between 0 and 50 years that the modelgenerates is 0.08, exactly what the data suggest,as we reported in Section I.

Finally, though we do not report the resultshere, in the working paper version of this paperwe show that all the key features of the model,R&D, endogenous technology adoption, and entryand exit are important for explaining the data.Eliminating any one of these features leads to asignificant deterioration in model performance.

An Exogenous Technology Shock Model.—Wenow consider how well a conventional exoge-nous productivity framework can account forthe facts. We thus eliminate endogenous pro-ductivity and endogenous markup movements.What is left is an RBC model with variablefactor utilization. We keep this latter featurebecause it improves the overall model perfor-mance, as noted by King and Rebelo (1999).Because we want the model to be consistentwith the evidence on embodied as well as dis-embodied technological change, we also intro-

26 See, e.g., Comin (2004) and Comin and Sunil Mulani(2005).


duce a shock to the production of new capitalgoods. In particular, we make the creation ofnew capital goods a multiple of the level ofinvestment, where the multiple in an exogenousrandom variable. The inverse of this multiple isthen the relative price of capital in units ofconsumption goods. Where relevant, we use the

same parameter values as in our baseline model.One exception is that to enhance the ability ofthe model to account for hours volatility, weraise the Frisch labor supply elasticity fromunity to two, a value recommended by ThomasF. Cooley and Edward C. Prescott (1995).

We consider three types of experiments: (a) a

−10 −5 0 5 10−1

0

1Output

−10 −5 0 5 10−1

0

1Labor productivity

−10 −5 0 5 10−1

0

1TFP

−10 −5 0 5 10−1

0

1Hours

−10 −5 0 5 10−1

0

1Consumption

−10 −5 0 5 10−1

0

1Investment

−10 −5 0 5 10−1

0

1R&D

−10 −5 0 5 10−1

0

1

Pk

−10 −5 0 5 10−1

0

1Mark−up

−10 −5 0 5 10−1

0

1Capacity utilization

FIGURE 5. CROSS CORRELATION WITH OUTPUT: MODEL (SOLID), DATA (DASHED), AND

95-PERCENT CONFIDENCE BANDS (DOTTED)


shock to TFP; (b) a shock to the relative price ofcapital; and (c) simultaneous shocks to the rel-ative price of capital. We begin with an exoge-nous shock to TFP. We set the auto-correlationof the shock at 0.88, which is consistent withvalue that King and Rebelo (1999) use for quar-terly data. This results in a serial correlation ofTFP over the medium-term cycle generated bythe model of 0.63, which is slightly below theactual number of 0.7, though well within the95-percent confidence interval. Because themodel performance would deteriorate if wewere to use a more persistent process, we stickwith the value proposed by King and Rebelo(1999). We then adjust the standard deviation ofthe shock to match the unconditional standarddeviation of TFP over the medium-term cycle.Since the measure of TFP in the data is notcorrected for utilization, we adjust the exoge-nous process for TFP to match the serial corre-lation and variance of the model Solow residual(which is similarly not corrected for utilization).

Table 6 reports the results. Overall, the sim-ple TFP shock model does about as well atcapturing volatility over the medium-term cycleas it does for the conventional high-frequencycycle. The model accounts for roughly 80 per-cent of the volatility of output. The volatilitiesof the other variables relative to output arereasonably in line with the data. The main ex-ception involves hours. As in the typical high-

frequency analysis, the model generates onlyabout half the relative volatility of hours thatappears in the data. From our vantage, however,a significant shortcoming of the TFP shockmodel is that, by construction, it cannot accountfor the cyclical pattern of embodied technolog-ical change, which we have argued is a keyfeature of the medium-term cycle. We thus nextconsider a shock to the relative price of capital.

As with the TFP shock, we attempt to feed inan exogenous price that matches the autocorre-lation and standard deviation of the filtered datafor the medium-term cycle. Because the relativeprice of capital is very persistent, to come closeto the autocorrelation in the data it is necessaryto feed in a near unit root process.27 (Our nu-merical simulation procedure requires station-ary forcing processes, thus ruling out a pure unitroot shock.) In particular, we set the autocorre-lation of the shock, �(Pk), at 0.99. Having ahighly persistent process for the relative price of

27 Greenwood et al. (2000) also consider a model of howexogenous shocks to the relative price of capital mightgenerate business fluctuations. They differ by consideringonly equipment prices and also by adding investment ad-justment costs, where the latter is chosen so that the modelvariance matches the data. Also, they just focus on the highfrequency. Finally, they use a forcing process for the rela-tive price of capital that is much less persistent than our datafor the medium term would suggest is reasonable.

TABLE 6—TWO-SECTOR RBC MODEL: STANDARD DEVIATIONS (0–50)

Variable DataTFP

shock

�(Pk) � 0.99 �(Pk) � 0.90

Pk

shockBoth

shocksPk

shockBoth

shocks

Output 3.87 3.20 0.53 2.64 2.77 3.86(3.06–4.67)

TFP 2.39 2.39 1.42 2.39 1.76 2.67(2.07–2.7)

Labor productivity 2.37 2.22 1.52 2.35 1.45 2.35(2.04–2.7)

Hours 2.84 1.22 1.12 1.5 1.58 1.88(2.29–3.09)

Relative price of capital 4.34 — 4.34 4.34 4.34 4.34(3.12–5.39)

Investment 9.59 10.00 2.63 8.48 14.76 16.94(4.5–14.68)

Consumption 2.20 1.84 2.06 2.55 1.07 1.87(1.83–2.57)

Capacity utilization 4.26 2.83 4.26 4.85 5.27 5.74(3.28–5.24)


capital is also plausible on a priori theoreticalgrounds: that is, it seems reasonable to presumethat embodied technological changes are per-manent, implying that a (near) unit root processwith drift is appropriate.28

As the third row of Table 6 shows, the modelwith a highly persistent shock to the relativeprice of capital does not fare well. It has only aminimal impact on output volatility for tworeasons. First, in contrast to a disembodied tech-nology shock which directly affects the produc-tion of all output, this shock has a direct effectonly on the production of new capital goods.Second, because the shock is highly persistent,it creates little incentive for intertemporal sub-stitution of hours.

We next allow for simultaneous shocks to thetwo technology variables. We keep the autocor-relations of each shock unchanged, but adjustthe variance of each shock innovation to matchthe corresponding unconditional standard devi-ations of the filtered data. As the fourth columnof Table 6 shows, there is a clear improvementin model performance, relative to the case withjust shocks to the relative price of capital. None-theless, the two-shock model explains less ofthe overall output volatility than the simple TFPshock case, slightly less than 70 percent, ascompared to roughly 80 percent.29 Otherwise,the relative volatilities of the other variables arelargely similar to the pure TFP shock case. Oneexception is that utilization is significantly morevolatile, due to a direct impact of movements onthe relative price of capital on the incentive toadjust capital utilization.

The reason the two-shock model does notexplain more the 70 percent of the volatility ofoutput involves the high persistence of theshock to the relative price of capital, whichgreatly minimizes intertemporal substitution.

To confirm, we redid the experiment, reducing�(Pk) from 0.99 to 0.90. As the last column ofTable 6 shows, in this case the exogenous tech-nology model exactly matches the volatility ofoutput. There are some shortcomings, however.The enhanced intertemporal substitution leadsto investment volatility that is far greater than inthe data (16.94 versus 9.59).30 On the otherhand, hours volatility remains too low relativeto the data (1.88 versus 2.84).

As we noted earlier, however, deviating sig-nificantly from a (near) unit root process forexogenous embodied technological change isnot appealing from a theoretical perspective. Itis also not appealing from an empirical one. InTable 7 we show the autocorrelations for TFPand the relative price of capital over the medium-term cycle for the different cases of the exoge-nous technology shock that we considered. Forcomparison, we also report these autocorrela-tions for our baseline endogenous productivitymodel. For all the exogenous technology shockcases, the model autocorrelation for the relativeprice of capital is low relative to the data. For

28 A similar argument can be made regarding disembod-ied technology shocks; however, convention in this litera-ture quickly evolved into considering stationary processesthat are not close to being unit root. This complicates theinterpretation of what these shocks actually are.

29 Shocks to the relative price of capital affect measuredTFP by affecting utilization, which in the two-shock casereduces that size of the embodied technology shock requiredto match TFP. It is for this reason that output volatility islower in the two-shock case than in the case with just shocksto disembodied productivity.

30 One way to dampen investment would be to addadjustment costs to investment. It is unclear, though, howimportant these costs may be at the annual frequency. In-vestment adjustment costs also significantly dampen theresponse of hours to exogenous productivity shocks (see,e.g., Neville and Ramey, 2002).

TABLE 7—FIRST-ORDER AUTOCORRELATIONS (0–50)

Variable DataBaselinemodel

RBC, TFPshock

RBC,�(Pk) � 0.99

RBC,�(Pk) � 0.90

Pk

shockBoth

shocksPk

shockBoth

shocks

TFP 0.70 0.75 0.65 0.82 0.71 0.65 0.65(0.51–0.89)

Relative price of capital 0.91 0.83 — 0.74 0.74 0.70 0.70(0.78–1.04)


the highly persistent forcing process, �(Pk) �0.99, it is just below the lower bound for the95-percent confidence interval (0.74 versus0.78). For the less persistent forcing process,�(Pk) � 0.90, it moves farther away from thetarget, dropping to 0.70.

By contrast, as column 2 of Table 1 shows,our baseline endogenous productivity modelsucceeds at capturing the high degree of persis-tence in the relative price of capital at the me-dium frequencies (0.87 versus 0.91 in the data).It similarly captures the persistence of TFP(0.75 versus 0.70).

IV. Concluding Remarks

We have explored the idea that the postwarbusiness cycle should not be associated withonly high-frequency variation in output (as gen-erated by the conventional business cycle filter),but rather should also include the medium-frequency oscillations between periods of ro-bust growth and periods of relative stagnation.We first construct measures of the cycle basedon this notion and demonstrate that the resultingfluctuations are considerably more volatile andpersistent than are the usual high-frequencymeasures. These results suggest that postwarbusiness fluctuations may be more importantphenomena than conventional analysis suggests.

The medium-term cycle we have identifiedfeatures significant procyclical movements inboth embodied and disembodied technologicalchange. These facts, among others, motivatedus to approach modeling the medium-term cy-cle by modifying a reasonably conventionalbusiness cycle framework to allow for R&D,technology adoption, and variation in markups.A virtue of our approach is that we are able tofully endogenize cyclical productivity dy-namics, including both embodied and disem-bodied dynamics. In addition, our endogenousproductivity mechanism offers an avenuethrough which nontechnological disturbancesat the high frequency can have sustained ef-fects on productivity over the medium term.As with many recent quantitative macroeco-nomic frameworks, the volatility of output atthe high frequency is associated with counter-cyclical markup variation. Within our frame-work, however, these high-frequency fluctuationsinfluence the pace of both R&D and adoption,

in turn generating subsequent movements inproductivity. The net effect is a propagationmechanism that generates the kind of persistentbusiness cycles that our filter uncovers.

There are, however, several places where themodel performance could be improved. As inthe data, the model generates a countercyclicalmovement in the relative price of capital overthe medium term and procyclical movement inR&D. The model, however, generates onlyabout 60 percent of the variability in the relativeprice of capital that is observed in the data. Aswe noted, it may be that the conventionally usedGordon series overstates the volatility. The of-ficial BEA series has similar cyclical propertiesover the medium term, but an overall volatilitythat is in line with our model estimates. Anotherpossible solution might be to introduce a shockto the innovation process for the production ofnew capital goods, in order to add an indepen-dent source of volatility to the relative price ofcapital. Conversely, the model R&D variabilityis about 50 percent greater than what we ob-serve in the data. As we discussed earlier, hereit may be important to try to gather improvedmeasures of R&D that are not only more com-prehensive, but also can separate variable fromoverhead costs. Another possibility is that thereturns to R&D investment may diminish morerapidly than our baseline calibration implies.Although we do not report the results here, byusing an elasticity of new products with respectto R&D that is in the lower range of Griliches’sestimates, it is possible to bring R&D volatilitycloser in line with the evidence, without causingthe performance of the model on other dimen-sions to deteriorate.

We also show that, under certain circum-stances, a pure exogenous productivity frame-work can explain a good fraction of thevariation in the data. To capture the movementsin both embodied and disembodied technologi-cal change, it is of course necessary to haveshocks to both total factor productivity and therelative price of capital. To generate sufficientlylarge output and employment fluctuations, how-ever, it appears necessary to allow for exoge-nous shocks to the relative price of capital thatare less persistent than the data suggest. Beyondthis empirical consideration, as we noted, thereare also good theoretical reasons to believe thatmovements in embodied productivity should


not be transitory. On the other hand, it issurely worth exploring exogenous productivitymodels more thoroughly than we have done inthis paper.

Finally, to minimize complexity, we modeledmarkup variation at the high frequency in a verysimple way. By contrast, as we noted in theintroduction, recent quantitative macroeco-nomic frameworks endogenize markup varia-tion by introducing money together withnominal price and wage rigidities. It wouldbe straightforward to enrich our model alongthis dimension. Doing so would yield a ratherdifferent perspective on what’s at stake overthe performance of monetary policy. Withinthese more conventional frameworks, medium-frequency variation is driven mainly by exoge-nous shocks. With our endogenous productivitymechanism, however, (large) high-frequencyfluctuations can have very persistent effects. Tothe extent monetary policy influences this highfrequency variation, it will have ramificationsfor medium-frequency dynamics.

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