Explaining the Investment Boom of the 1990s

Explaining the Investment Boom of the 1990s Stacey Tevlin Karl Whelan(cid:3) March 7, 2000 Abstract Real equipment investment in the United States has boomed in recent years, led by soaringinvestmentincomputers. We (cid:12)ndthattraditionalaggregateeconometricmodels completely fail to capture the magnitude of this recent growth|mainly because these models neglect to address two features that are crucial (and unique) to the current investment boom. First, the pace at which (cid:12)rms replace depreciated capital has increased. Second,investmenthas been moresensitiveto the costofcapital. We documentthatthesetwofeaturesstemfromthespecialbehaviorofinvestmentincomputers and therefore propose a disaggregatedapproach. This produces an econometric model that successfully explains the 1990s equipment investment boom. (cid:3)DivisionofResearchandStatistics,FederalReserveBoard. Email: kwhelan@frb.govorstevlin@frb.gov. Wewould liketothankRobertChirinko, DavidLebow,Deb Lindner,DanSichel, andPeterTulip fortheir valuable suggestions. We would also like to thank seminar participants at the 2000 AEA meetings, the New York Fed, and the Federal ReserveBoard. The views expressed in this paper are our own and do not necessarily reflect theviews of theBoard of Governors or the sta(cid:11) of theFederal Reserve System.

1 Introduction The behavior of equipment investment in the current U.S. expansion has been remarkable. Growth in real equipment investment over the period 1992-98 averaged 11.2 percent per year, exceeding all other seven-year intervals in the post-War era.1 This development has been of great macroeconomic importance: The investment boom has underpinned the continuing strength of U.S. aggregate demand and has probably also had important supply-side e(cid:11)ects, perhaps playing a role in the unusual late-cycle acceleration in labor productivity. In this paper, we examine whether existing time series models can explain the astounding behavior of equipment investment in the 1990s. We demonstrate that they cannot. Although we examine the traditional, accelerator-style models that previous investment \horserace" studies have found best (cid:12)t the data, we (cid:12)nd that they completely fail to capture the magnitude of the 1990s investment boom.2 We show that the models’ breakdown stems from an important element of investment growth in the 1990s|the surge in real investment in computing equipment. Our analysis of the behavior of computer investment reveals two features that, though crucial to the investment boom of the 1990s, are ignored bystandardaggregate models. Wedemonstratethatadisaggregated approach,whichmodelsinvestmentincomputingandnon-computingequipmentseparately, successfullyexplains the behavior of investment in the 1990s. The (cid:12)rst feature that we document is the sharp increase in the average rate of depreciation in the 1990s. Most econometric models assume a constant depreciation rate and thus a stable relationship between the change in the capital stock and the level of investment. Since the optimal capital stock is a function of the level of output and the cost of capital, this also implies a stable link between investment and changes in output and the cost of capital. However, the increasing rate of depreciation in the 1990s broke this link: Firms neededtoinvestmoretosustainagivenlevelofthecapitalstock. Weshowthattheincrease in the depreciation rate was dueto a shift in the composition of capital towards computers, 1All(cid:12)guresinthispaperreferto1992-basedNationalIncomestatisticsratherthanthe1996-based(cid:12)gures publishedinOctober1999. Theeconometricmodelsinourpaperusecapitalstockdata,andrevisedcapital stocks will not be published untilSpring2000. 2For earlier evaluations of competing investment models, see Clark (1979) and Bernanke, Bohn, and Reiss(1988). Oliner,Sichel,andRudebusch(1995) compareaccelerator-stylemodelswith modelsbasedon Euler equations or Tobin’s Q. 1

which depreciate more rapidly than other types of equipment. Aggregate models do not capture this phenomenon, because, by de(cid:12)nition, they ignore compositional mix-shifts. The second feature that we examine is the role of the cost of capital. Therising average depreciation rate suggests the need to separately model net investment in computing and non-computing equipment. Doing so reveals an important pattern. Computer investment is verysensitive to thecost of capital, far moresothan investment in non-computingequipment. As a result, rapid declines in computer prices played a crucial role in generating the investment boom of the 1990s. This result contrasts sharply with most of the empirical literature on aggregate investment, which typically (cid:12)ndsvery little responseto cost variables. We provide a plausible explanation for the di(cid:11)erent estimates of cost-of-capital elasticities that we observe: Firms respond more to shocks perceived as permanent than to those perceived as transitory, and shocks to computer prices usually result from technological innovations that are unlikely to be reversed. We conclude that the special behavior of equipment investment in the 1990s resulted from the substantial impact of rapid computer price declines on capital accumulation, and the consequent need for higher rates of replacement investment. A simple disaggregated approach, which separately models net and gross investment for computing and non-computing equipment is capable of explaining the recent behavior of investment. The paper is organized as follows. Section 2 reviews the traditional econometric models and documents their poor empirical performance in the 1990s. Section 3 examines the increaseintheaveragerateofdepreciationanditsroleinthebreakdownoftheconventional models. Section 4 discusses why capital accumulation may respond more to the persistent component of the cost of capital than to the less persistent component. Section 5 presents our econometric analysis and documents the performance of our approach in tracking the behavior of aggregate gross investment in the 1990s. Section 6 concludes. 2

2 Traditional Investment Models and Their Recent Failure Traditionalmodelsofinvestmentstartwithatheoryrelatingtheoptimalfrictionlesscapital (cid:3) stock, K , to the production technology and factor prices. If (cid:12)rms could costlessly adjust t (cid:3) the capital stock, they would always set K = K . However, the sluggish behavior of the t t capital stock suggests that there arecosts associated with adjustment. Thetraditional neo- Keynesian investment models used simple ad hoc speci(cid:12)cations of the e(cid:11)ects of adjustment costs, the most common being the partial adjustment approach, which assumed that (cid:12)rms move part of the way towards their optimal frictionless stock each period. Formulating this relationship in terms of the logarithm of the capital stock and using lower case letters to denote the log of variables, the partial adjustment equation is (cid:1)k = (1−(cid:21))(k t (cid:3)−k t−1 ) (1) which can be re-written as: k t = (cid:21)k t−1 +(1−(cid:21))k t (cid:3) (2) Applying repeated substitution to equation (2) gives an equivalent representation for the (cid:3) capital stock, this time as an in(cid:12)nite distributed lag function of past k ’s: t X1 X1 k = (1−(cid:21))(cid:21)rk (cid:3) = γ k (cid:3) (3) t t−r r t−r r=0 r=0 This has been turned into an empirical investment equation by taking the following steps. First, the in(cid:12)nite distributed lag suggested by the partial adjustment theory is replaced with a (cid:12)nite approximation, usually about 8 to 12 quarters. Second, the equation is di(cid:11)erenced to turn it into a net investment equation: XN (cid:3) (cid:1)k = γ (cid:1)k +(cid:15) (4) t r t−r t r=0 Ofcourse, if thecapital stock adjustmentequation iscorrectly speci(cid:12)ed,thenthis di(cid:11)erencingstep isnotnecessary. However, thetraditional literature largely pre-datedcointegration methods and used stationarity-inducing transformations as a precaution against spurious regressions. In our empirical work, we will examine this issue formally. (cid:3) Thisequationisoperationalizedbyassumingaformfork . SpecifyingaCESproduction t function, K (cid:3) is proportional to Yt , where Y is output, C is the cost of capital, and (cid:27) is t Ct (cid:27) t t 3

(cid:3) the elasticity of substitution between capital and labor. Taking logs of K we get t XN XN (cid:1)k t = γ r (cid:1)y t−r −(cid:27) γ r (cid:1)c t−r +(cid:15) t (5) r=0 r=0 P 1 Since γ = 1, the sum of the coe(cid:14)cients on output should approximately equal one r=0 r while the coe(cid:14)cients on the cost of capital should sum to the elasticity of substitution, (cid:27). These sums have an intuitive interpretation since they describe the predicted long-run response of the capital stock to permanent unit shocks to output and the cost of capital. Models of the form of equation (5) have been estimated by Bernanke, Bohn, and Reiss (1988). However, this approach, which describes the determination of the capital stock, only gives us a model of net investment. For macroeconomists interested in business cycle modelling and forecasting, the variable of interest is gross investment, which includes both the change in capital stock and the replacement of depreciated capital. Most empirical models assume a constant average rate of depreciation and estimate an equation for gross investment. In this case, approximating the log-di(cid:11)erence of the capital stock with the growth rate, we get (cid:1)K I (cid:1)k (cid:25) t = t −(cid:14) (6) t K t−1 K t−1 where (cid:14) is the depreciation rate. This gives an equation for gross investment XN XN I t = (cid:14)+ γ r (cid:1)y t−r −(cid:27) γ r (cid:1)c t−r +(cid:15) t (7) K t−1 r=0 r=0 We will label this regression the \traditional model". Note that this approach estimates the depreciation rate as the intercept in the It regression. Kt−1 Previous empirical implementations of this model, estimated on data priorto the1990s, have found that it provides a fairly good description of the cyclical behavior of investment. Indeed, Oliner, Rudebusch, and Sichel (1995) have shown that models of this form provide superior forecasting performance to popular alternative speci(cid:12)cations based on Euler equations or Tobin’s Q. This is not to say that these models are without problems. For instance, despite microeconomic evidence that the elasticity of substitution is close to one, regressions usually reveal a small and often insigni(cid:12)cant role for the cost of capital. Indeed, comparisons of forecasting power have often favored the pure accelerator formulation ((cid:27) = 0) over models including the cost of capital. Summarizing these results, Chirinko’s comprehensive1993surveyconcludedthat\onbalance,theresponseofinvestmenttoprices tends to be quite small and unimportant relative to quantity variables." 4

Our estimation of equation (7) con(cid:12)rms these previous results, revealing a small longrun cost-of-capital elasticity of -0.34. (Our data are described in Appendix A.) However, Figure 1, which shows the in-sample (cid:12)t, reveals a far more serious problem. The model fails completely to capture the 1990s’ increase in investment relative to the capital stock. After 1991, the model underpredicts by larger and larger amounts, with these residuals principally o(cid:11)set by large negative residuals over the early part of the sample.3 By 1997:4, the actual level of investment relative to the capital stock is 7 percentage points higher than can be explained by the model; this translates into a 31 percent error on the level of investment. A Chow test for parameter stability con(cid:12)rms that the model has gone o(cid:11) track in the 1990s, with the null hypothesis of stable coe(cid:14)cients resoundingly rejected.4 In the rest of the paper, we explore the reasons for the traditional model’s complete breakdown in the 1990s. 3 The Unstable Aggregate Depreciation Rate The most obvious simplifying assumption made in the derivation of the traditional model is that the average rate of depreciation is constant. We can easily check the validity of this assumption by solving for the aggregate depreciation rate obtained from re-arranging the perpetual inventory equation (K t = (1−(cid:14) t )K t−1 +I t ) to get: I −(cid:1)K t t (cid:14) = (8) t K t−1 Figure 2 shows this series as the solid line (rather obscurely labeled \Using Chain-Weight Investment and Capital", for reasons that will become apparent in a moment). It shows that the aggregate depreciation rate has not been constant, but has increased substantially in recent years, rising from 0.13 in 1989 to 0.16 in 1997. 3This is also true for popular alternative versions of this equation such as the Jorgenson\neoclassical" model and an augmented version that includes cash flow. Models based on Tobin’s Q, although predicting stronginvestmentoverthelastfewyearsofoursample,alsodonottrackthebehaviorof It inthe1990s Kt−1 particularly well. 4The(cid:12)gurealsorevealsaproblemreportedinprevioushorseracepapers. Evenwhenresidualsaresmall during the middle part of our sample, they tend to be positively autocorrelated. We believe this is due to the (cid:12)nite-lag approximation to the true in(cid:12)nite-lag capital stock adjustment formula, equation (3). When (cid:21) is large (and empirical estimates suggest it is), this approximation will omit autocorrelated terms. Our regressions in Section 5 are not based on the(cid:12)nite-lag approximation. 5

The main cause of this uptrend is straightforward. Di(cid:11)erent types of equipment depreciate at di(cid:11)erent rates and Oliner (1989, 1994) has shown that computers depreciate signi(cid:12)cantly faster than other types of equipment. The National Income and Product Accounts (NIPA) capital stocks used in our analysis are constructed under the assumption of separate, constant, depreciation rates for each of 27 underlying equipment categories, with the depreciation rate for computers taken directly from Oliner’s research.5 Thus, it should come as nosurprisethat the recent explosion in computer investment has led to an increase in the average rate of depreciation for total equipment. Before we move on to discuss the implications of a varying pace of depreciation for econometric modelling, we need to note a surprising pattern in our calculated series for the aggregate depreciation rate. 3.1 A Depreciation Puzzle While we had expected that the high rates of computer investment would have raised the aggregate depreciation rate in the 1990s, we were surprised to (cid:12)nd that this was not just a recent phenomenon but rather an acceleration of a long-running trend. In fact, our calculatedseriesfortheaggregatedepreciationrate doubles over1965-1997. Themagnitude of this apparent mix-shift seems very large, particularly as it suggests that variations in the average rate of depreciation have had an important e(cid:11)ect on aggregate gross investment throughout the past 30 years, something not found by previous researchers. The solution to this puzzle turns out to be a change in the NIPA methodology for constructing real aggregates. Since1996, allNIPArealexpenditureaggregates, includingrealGDP,havebeenderived using a Fisher chain-aggregation methodology.6 Since 1997, real capital stock aggregates have been constructed using the same methodology. Rather than aggregating all quantities according to their base-year prices, as in the traditional Laspeyres index, the growth rate of a chained aggregate reflects a mix of old and new prices. Given a series of quantities and prices for n goods, q (t) and p (t), the gross growth rate for the Fisher chain-aggregate i i quantity is de(cid:12)ned as: s P P n p (t)q (t) n p (t−1)q (t) G(t) = P i=1 i i P i=1 i i (9) n p (t)q (t−1) n p (t−1)q (t−1) i=1 i i i=1 i i 5SeeKatz and Herman (1997) for a description of theNIPAstocks. 6SeeLandefeld and Parker(1997) for a discussion of this methodology. 6

In the base-year (1992 in our data), all price indexes are set equal to one and the level of each aggregate is set equal to its nominal value. For all subsequent and previous years, the real level series are simply \chained" forward and backwards using the Fisher-aggregation growth rates. For NIPA aggregate real equipment investment and stocks, the Fisher chain procedure aggregates 27 component series. This chain aggregation procedure helps to reduce biases due to valuing goods at prices that become irrelevant once we move away from the base year. However, a complexity it introduces is that the level of the constructed real aggregate is no longer the additive sum of its real components, with this lack of additivity being most evident when there are large relative price shifts within a bundle of goods (as is the case with the equipment bundle because of the substantial declines in the price of computing equipment). This non-additivity invalidates the calculation of the aggregate depreciation rate. To illustrate, consider the following simple example. There are two types of capital, A and B. Suppose now that the aggregate capital stock is constructed according to the traditional Laspeyres (cid:12)xed-weight formula. In this case, the real aggregates for investment and the capital stock are the simple sum of their real components: IFW = IA+IB and KFW = KA+KB. Itis easy to usethis fact to show that the aggregate depreciation rate is a weighted average of the two underlying depreciation rates, with the weights given by the real quantities for the two stocks: ! ! IFW −(cid:1)KFW KA KB (cid:14)FW = t t = (cid:14)A t−1 +(cid:14)B t−1 (10) t KFW KA +KB KB +KA t−1 t−1 t−1 t−1 t−1 Note, however, that the strict additivity of the (cid:12)xed-weight formula was necessary to obtainthisweighted-average expressionfortheaggregate depreciation rate. Oncethisadditivity breaks down, only in the base year can we interpret the depreciation rate calculated from equation (8) as a weighted average; this is because in the base year all real series are equal to their nominal counterparts and so for this year additivity does hold. In fact, as we show in Appendix B, moving away from the base year, the aggregate depreciation rate calculated from equation (8) with chain-weighted data di(cid:11)ers systematically from a weighted-average depreciation rate, displaying a long-run upward trendeven in the absence of mix shifts towards faster depreciating equipment. The explanation for this result is fairly subtle; a full derivation is available in Appendix B. However, the intuition is as follows. The growth rate of a chain-weighted aggregate 7

e(cid:11)ectively equals a weighted-average of the growth rates of its components, where the weightsaregivenbythecomponents’nominalshares. Itturnsoutthatwhenrealinvestment in one type of capital grows faster than others because its relative price is declining, then the nominal shareof this typeof capital in investment will behigher than its nominalshare in the capital stock, implying that the aggregate for real investment grows faster than the aggregate for the real capital stock. As a result, the ratio of the level of real aggregate investment to the level of the real aggregate capital stock, will trend upwards. Hence, the series calculated from equation (8) will also trend upwards. Todemonstratethee(cid:11)ectontheaggregatedepreciationrateofthechangeinaggregation methodology, we constructed (cid:12)xed-weight aggregates for equipment investment and the equipment capital stock by adding up the underlying real series for the 27 equipment investment categories. We then calculated a depreciation rate for these (cid:12)xed weight series, exactly as in equation (10). This (cid:12)xed-weight depreciation rate series is shown as the thick dashed line on Figure 2 (labeled \Using Fixed-Weight Investment and Capital"). It rises steeply over the past few years but increases only very slightly prior to the 1990s, remaining for most of the sample in the range of 0.13, the value most commonly used in studies that construct equipment stocks from a constant aggregate depreciation rate. In contrast, the corresponding series for chain-weighted data climbs steadily from the mid- 1960s on. Thus, although the increasing aggregate depreciation rate in the 1990s mainly reflects a composition shift, the uptrend evident prior to the 1990s is mainly an artifact of chain aggregation.7 Since mix-shifts towards equipment-types that are faster depreciating and declining in relative price can explain the substantial rise over time in our perpetual-inventory estimate of the aggregate depreciation rate, an obvious question is whether removing computing equipment (which depreciates rapidly and has the largest price declines) will result in a stable depreciation rate. The (cid:12)nal series on Figure 2, the thin dashed line (labeled \Non- Computing Equipment, Chain-Weight Investment and Capital"), tells us that the answer is: Almost. This series was calculated by applying equation (8) to newly-calculated chainaggregates for investment and capital stock for all equipment except computers, and it shows a very slow and modest upcreep over time.8 7Becausepreviousresearchinthis(cid:12)eldusedtheold(cid:12)xed-weightdata,thisexplainswhyotherresearchers did not note this curious pattern. 8AppendixA describes how we calculated these aggregates for non-computingequipment. 8

3.2 Implications for Aggregate Investment Modelling Returning to the recent failure of the traditional model, we have seen that the actual depreciation rate required to convert aggregate net investment ((cid:1)k = It −(cid:14) ) into It t Kt−1 t Kt−1 is not a constant that can be proxied by the intercept, as is assumed when we directly estimate equation (7), but in fact has been rising rapidly in recent years. If our aim is a stable econometric model, then a solution to this problem is to instead directly model the behavior of net investment by estimating equation (5). Figure 3 illustrates how this step radically improves in-sample (cid:12)t. It compares the residuals from our estimation of the gross investment ( It ) model, equation (7), with the residuals from estimation of the Kt−1 net investment ( It −(cid:14) ) model, equation (5). Once we do not have to account for the Kt−1 t variations intheaggregate depreciation rate, wenolonger haveresidualsthat trendupover time and the recent net investment residuals, though still positive and relatively large, are not historically unprecedented. This is something of a hollow victory, however, if our ultimate goal is a model of gross investment expenditures. Worse still, these aggregate models cannot explain the source of the increasing aggregate depreciation rate{the explosion in net investment in computing equipment. A complete model of gross investment expenditures in the 1990s must account for the di(cid:11)erent behavior of investment in computing and non-computing equipment. In the next section, we present an alternative to the partial adjustment model that provides intuition for such an approach by illustrating why computer prices may have a di(cid:11)erent e(cid:11)ect on investment than other elements of the cost of capital. 9

4 Cost of Capital Shocks and Capital Accumulation Empirical tests of the traditional models that we have focused on thus far (cid:12)nd only a small role for price variables. Therefore, they imply that rapidly declining computer prices have had little impact on investment in computers. The sheer magnitude of the increase in computer investment in recent years suggests that this may be incorrect. Consider now an alternative theoretical approach, previously presented by Nickell (1979) and Kiyotaki and West (1996), that explains why computer price declines may a(cid:11)ect capital accumulation more than other shocks. The models we have looked at thus far rely on very simple modelling of the e(cid:11)ects of adjustment costs. An alternative is to explicitly modelthe implications of adjustmentcosts for an optimizing (cid:12)rm with rational expectations. To capture only the essential features of theinvestmentproblem,weuseaquadraticapproximationtotheunderlyingpro(cid:12)tfunction: Changes in the capital stock and deviations fromthe frictionless optimal stock both lead to costs which increase according to a simple quadratic function. For a given expected path (cid:3) of k , (cid:12)rms choose the current capital stock to solve " # X1 n(cid:0) (cid:1) o Min E t (cid:18)m k t+m −k t (cid:3) +m 2 +(cid:11)(k t+m −k t+m−1 )2 (11) m=0 where (cid:18) is the (cid:12)rm’s discount rate. The model’s (cid:12)rst-order conditions are: (cid:20) (cid:18) (cid:19) (cid:21) 1 1 1 1 E t −k t+1 + 1+ (cid:18) + (cid:11)(cid:18) k t − (cid:18) k t−1 − (cid:11)(cid:18) k t (cid:3) = 0 (12) Letting L be the lag operator, F be the lead operator, and using the fact that the char- (cid:16) (cid:17) acteristic equation x2 − 1+ 1 + 1 x + 1 = 0 has two roots such that one root ((cid:21)) is (cid:18) (cid:11)(cid:18) (cid:18) between zero and one while the other equals 1 , this can be re-expressed as (cid:18)(cid:21) (cid:20) (cid:18) (cid:19) (cid:21) 1 1 E −(F −(cid:21)) F − Lk − k (cid:3) = 0 t (cid:18)(cid:21) t (cid:11)(cid:18) t Implying a solution " # X1 (cid:21) k t = (cid:21)k t−1 + (cid:11) E t ((cid:18)(cid:21))nk t (cid:3) +n (13) n=0 An intuitive re-formulation of this equation that illustrates the model’s fundamental property, comes from using the fact that (cid:21) = (1−(cid:21))(1−(cid:18)(cid:21)) (which comes from re- (cid:11) 10

arranging the characteristic equation). Making this substitution we get (cid:1)k t = (1−(cid:21))(k t (cid:3)(cid:3)−k t−1 ) (14) where " # X1 k (cid:3)(cid:3) = (1−(cid:18)(cid:21))E ((cid:18)(cid:21))nk (cid:3) (15) t t t+n n=0 (cid:3)(cid:3) Thus,eachperiod,thelogofthecapitalstockadjuststowardsthemovingtarget, k ,which t (cid:3) is a weighted average of expected future k ’s. It can be shown that (cid:21) depends positively t on (cid:11), implying that higher adjustment costs lead to a slower speed of adjustment towards k (cid:3)(cid:3) .9 t (cid:3) The model is completed by a speci(cid:12)cation of the process for k . Pro(cid:12)t maximization t (cid:3) (using a generalized CES production function) will give us a (cid:12)rst order condition: k = t (cid:17) +y −(cid:27)c , where y and c are as before and (cid:17) summarizes the e(cid:11)ects of capital-biased t t t t technological change. To give a concrete example of what (cid:17) means, the stock of computing t capital may tend to rise independently of output and the cost of capital if the structure of production changes in ways that facilitate increased usage of computers. Toimplement thismodelempirically, weneedtospecifytimeseries processesforoutput and the cost of capital. Let y t = (cid:30)(L)y t−1 +(cid:15) t and c t = (cid:25)(L)c t−1 +(cid:23) t where (cid:30) and (cid:25) are m-thorderdistributedlagpolynomialsand(cid:15) and(cid:23) arewhitenoise. Giventheseprocesses, t t (cid:3)(cid:3) we can solve for the e(cid:11)ects of output and the cost of capital on k in terms of empirically t observable variables by using the following formula of Hansen and Sargent (1980): " # X1 E ((cid:18)(cid:21))ny = (cid:20)(L)y (16) t t+n t n=0 where 2 0 1 3 1 mX−1 Xm (cid:20)(L)= 4 1+ @ ((cid:18)(cid:21))r−k(cid:30) A Lk5 (17) 1−(cid:30)((cid:18)(cid:21)) r k=1 r=k+1 Similarly, letting 2 0 1 3 mX−1 Xm (cid:22)(L)= 1 4 1+ @ ((cid:18)(cid:21))r−k(cid:25) A Lk5 (18) 1−(cid:25)((cid:18)(cid:21)) r k=1 r=k+1 9Thistypeofcapitalstockprocess canalsobederivedfrommoregeneralassumptionsabouttechnology and adjustment costs: SeeAuerbach (1989). 11

the process for the capital stock is now k t = (cid:21)k t−1 +(1−(cid:21))(1−(cid:18)(cid:21))((cid:20)(L)y t −(cid:27)(cid:22)(L)c t )+(cid:17)^ t (19) where (cid:17)^ depends on a weighted average of current and future values of the capital-biased t technological change term. Suppose now we estimate equation (19). The technology-bias variable, (cid:17)^, cannot be observed, so this ends up in the error term. Thus, our estimating equation is XN XN k t = (cid:11)+(cid:21)k t−1 + (cid:12) i y t−i + γ i c t−i +u t (20) i=0 i=0 where the model predicts that (cid:12)(L) = (1−(cid:21))(1−(cid:18)(cid:21))(cid:20)(L) γ(L) = (cid:27)(1−(cid:21))(1−(cid:18)(cid:21))(cid:22)(L) (cid:11)+u = (cid:17)^ t t Equation (20) bears a close resemblance to the capital stock equation under partial adjustment. However, the coe(cid:14)cients on y and c now depend on the variables’ own timeseries processesandthediscountrate, (cid:18),aswellas ontheunderlyingproductiontechnology and the adjustment speed, (cid:21). Speci(cid:12)cally, consider the long-run elasticities with respect to y and c, de(cid:12)ned as the sum of coe(cid:14)cients on these variables divided by (1−(cid:21)). These values depend positively on the persistence of the explanatory variables. In Appendix C, we show that if c is an I(1) series, then (1−(cid:18)(cid:21))(cid:22)(1) = 1. But, if c is an I(0) series, then this term is less than 1, and will be approximately zero if c is white noise. The reason for this result is intuitive: Firms are less likely to react to shocks to the \frictionless optimal" stock that they perceive as being temporary than to shocks perceived to be permanent.10 10Notethattheseareconditional elasticities,notlong-runimpulseresponsesofamultipleequationsystem: They describe the behavior of the capital stock conditional on the paths of output and the cost of capital. Thiscontrasts withthework ofKiyotakiandWest(1996). Theyhavealso notedthatthismodelcanallow thecapital stocktohavedi(cid:11)erentelasticities with respecttooutputandthecost ofcapital. However,their empirical implementation imposed the assumption that the cost of capital was an I(1) series, thus ruling out this possibility. Their implementation of this model instead focused on long-run impulse responses of the (k;y;c) system. Their (cid:12)nding of smaller long-run impulse responses to shocks to c comes from their estimated process for c being a less persistent I(1) process than the I(1) process for output (for instancealthough bothareI(1)processes, y t =1:5y t−1 −0:5y t−2 +(cid:15) t implieslargerimpulseresponsesthan y t =0:5y t−1 +0:5y t−2 +(cid:15) t ). It does not come from smaller conditional elasticities for k with respect to c than with respect to y. 12

In light of these results, it is informative to examine the persistence properties of the cost of capital for computing and non-computing equipment. We de(cid:12)ne the cost of capital according to the standard Hall-Jorgenson rental rate formula: !(cid:18) (cid:19) P_ 1−ITC −(cid:28) (cid:3)DEP C = P R +(cid:14)− t (21) t t t P 1−(cid:28) t where P is the price of capital relative to the price of output, R is the real interest rate, t t ITC is the investment tax credit, DEP is the present value of depreciation allowances per dollar invested, and (cid:28) is the marginal corporate income tax rate. Expressedinlogs, thecostofcapitalisthesumoftwoseries{therelative priceofcapital, and the non-relative-price component, which measures the tax-adjusted gross required rate of return on investment. As Figure 4 shows, these two components a(cid:11)ect the computer and non-computer cost of capital series in very di(cid:11)erent ways. The upper panels show that the computer cost of capital is highly non-stationary, exhibiting continuous rapid declines as a result of the remarkable pattern of falling purchase prices. The lower panels show that the relative stability of the non-computer cost of capital comes from a combination of an uneven decline in the relative price of this equipment and a choppy pattern for the non-price component. Even looking within speci(cid:12)c categories, the cost of capital combines components that appear to have very di(cid:11)erent persistence properties. For instance, the relative priceof computers appearsto beavery persistentseries; the relative priceof non-computingequipment seems to have a downward trend, although one that is less dominant than for computers; the non-price components for both variables seem to be relatively stable, mean-reverting series. More formal econometric characterizations of the persistence of these series, using simple autoregressions and unit root tests, con(cid:12)rm the intuition implied by these graphs. These tests suggest that the relative price series for both computing and non-computing equipmentalmostcertainlyhaveunitroots,whilethenon-relative pricecomponentsappear more likely to be stationary series. Therearealsogoodeconomic reasonstobelievethatthepriceandnon-pricecomponents of thecost ofcapital have di(cid:11)erentpersistenceproperties. Thepattern of decliningrelative prices for equipment comes from technological innovations in the equipment-producing industries, and it seems likely that once prices have fallen as a result of innovations, these price reductions will be permanent. In contrast, real interest rates will, in the long-run, 13

be related to the marginal productivity of capital, which will be a stationary variable in any general equilibrium model. Similarly the Hall-Jorgenson tax term is bounded and has tended to be mean-reverting. Tosummarize,explicitlymodellingthee(cid:11)ects ofadjustmentcosts tells usthatthee(cid:11)ect on investment of shocks to the cost of capital depends on the perceived persistence of the shocks. We have also shown that the persistence of the cost of capital varies substantially across equipment type, with the cost of capital for computers being dominated by the persistent decline in purchase prices. Theseresults suggest usinga disaggregated approach that allows di(cid:11)erent types of equipment to have di(cid:11)erent elasticities with respect to the cost of capital. 5 Econometric Modelling 5.1 Regressions We estimated thecapital stock adjustmentformula, equation (20), for aggregate equipment as well as for computing and non-computing equipment. Because the proposed regressions contain nonstationaryvariables,we(cid:12)rstaddressedwhetherthereisacointegrating relationship. We ran the potential cointegrating regressions and applied Phillips-Ouliaris-Hansen tests for a unit root in the residuals. We could not reject the hypothesis that the error term has a unit root for any of the three categories. (This may be because our error term contains the biasedtechnological change term (cid:17)^, andit is possiblethat this termhasaunit t root.) These results indicate that the conventional approach in the \horserace" literature of di(cid:11)erencing to avoid a spurious regression was probably well-founded.11 We will follow this approach in estimating a di(cid:11)erenced version of equation (20).12 11Forcompleteness, wealso estimated ourregressions in levels; theimportant resultsof thissection were unchanged. 12Note, though, that our approach of directly estimating the capital stock adjustment equation di(cid:11)ers from the approach of the traditional models. These models applied repeated substitution of the lagged k term to transform the theoretical ARMA equation into an MA(1) equation, and then approximated t this equation using an an MA(n) regression. However, if the adjustment cost parameter, (cid:21), is high (and empiricalestimatessuggestthatitis),thentermsomittedinthisMA(n)approximationwillstillhavelarge coe(cid:14)cients. Since these terms are probably positively autocorrelated, we believe that this accounts for the poor autocorrelation properties of the traditional models. 14

The results are shown in Table 1. The aggregate results (column 1) are familiar from previous empirical investment papers. The estimated (cid:21) of 0.93 implies relatively slow adjustment. The sum of the coe(cid:14)cients on output is signi(cid:12)cantly positive and the sum of the coe(cid:14)cients on the cost of capital, though negative as expected, is quite small. The long-run elasticities are shown in the bottom part of the table. For the cost of capital, this elasticity is only -0.18. The second column of Table 1 shows this regression for computing equipment. Limited data availability requires us to estimate over a smaller sample for computing equipment (1980-97), which leads to less tightly estimated coe(cid:14)cients.13 Nonetheless, this column containsanimportantresult: Theestimatedlong-runelasticityofthecomputercapitalstock with respect to the cost of capital is -1.6, nearly 9 times the estimate from the aggregate model. Column 3 reports the results for non-computing equipment; these are similar to the aggregate regression. According to the model in the previous section, regressors with more persistent time series processes should have higher elasticities. Thus, part of the explanation for the larger cost-of-capital elasticity for computing equipment could be that the variance for the computer cost of capital is dominated by persistent shocks (falling computer prices). Columns 4-6 examine this hypothesis and provide con(cid:12)rmation. For both computing and noncomputing equipment, the elasticities with respect to the more persistent components of the cost of capital (the relative price terms) are larger|in the case of computers, significantly so. Moreover, the long-run investment elasticity with respect to computer prices is also statistically signi(cid:12)cantly larger than the non-computer elasticity with respect to non-computer prices.14 In fact, by estimating the persistence properties of the various regressors we can calculate exactly how much higher the elasticities on persistent regressors should be. We estimated processes for price and non-price variables for both computing and non-computing equipment, using a stationary representation for the non-price variables, and imposing the 13Wechosethisstartingdatabecausethestockofcomputingequipmentwasverysmallbefore1980. None of theresults reported here are sensitive to thechoice of sample. 14The results we have shown in this section are robust. Durbin’s h statistics are low indicating that the regressions arefreeofresidualautocorrelation. Speci(cid:12)cationchanges(suchasincludingatrendandadding extra lags) did not signi(cid:12)cantly alter any of our results. Furthermore, the regressions show no evidence of parameter instability in the 1990s. 15

assumption that the processes for the price variables are I(1). Using these processes along with equations (17), (18), and (20), we (cid:12)nd that the cross-equation restrictions implied by the model tell us that, for both computing and non-computing equipment, the conditional elasticity of thecapitalstockwithrespecttothenon-pricevariables shouldequalabouthalf the elasticity with respect to the price variables. A Wald test of these cross-equation restrictions reveals thattheycannotberejected. However, becauseof therelative imprecision of the estimates we are reluctant to place too much emphasis on these tests. Our assumption that the relative price series are I(1) also implies that the estimated long-run elasticities with respect to these variables should equal the elasticities of substitution for each type of capital. The implied elasticity of substitution for non-computing equipment is -0.33, in line with standard estimates from previous investment studies, although still perhaps surprisingly low. For computing equipment, the implied elasticity of substitution of -1.83 is extremely large. A possible interpretation of this result is that computer technologies are more easily substitutable for other factors. 5.2 Implications of Computer Price Measurement Error One question about our large estimate of the elasticity of computer net investment with respect to its relative price is whether it could be a(cid:11)ected by errors in the measurement of computer prices. The reasons to suspect that measurement error may be a(cid:11)ecting this coe(cid:14)cient are twofold. First, the NIPA computer price index is a constant-quality series. This price is constructed from so-called \hedonic" price regressions, and there is certainly room for mis-speci(cid:12)cation and mis-measurement in these regressions. Second, like almost all NIPA expenditure categories, real investment in computing equipment is constructed by deflating the nominal expenditure series by the price index. Thus, any measurement error in the price index will a(cid:11)ect both the right- and left-hand sides of our net investment regression. While such measurement error may a(cid:11)ect our regressions, we believe that consideration ofthisfactorpointstoapriceelasticityforcomputingequipmentthatislarger inmagnitude than our estimate. This is because this type of measurement error biases the estimated long-run elasticity with respect to prices towards minus one and our estimate is -1.83. To illustrate this result, consider a simpli(cid:12)ed version of our theoretical investment equation, 16

without dynamics or non-price cost-of-capital terms: (cid:1)k = (cid:11)+(cid:12)(cid:1)y −γ(cid:1)p +(cid:15) t t t t (cid:3) Suppose now that the NIPA price, p , is measured with error so that (cid:3) (cid:1)p = (cid:1)p +u t t t The measured real net investment series is the nominal series divided by the measured price: (cid:1)k (cid:3) = (cid:1)k +(cid:1)p −(cid:1)p (cid:3) t t t t = (cid:11)+(cid:12)(cid:1)y −γ(cid:1)p +(cid:15) +(cid:1)p −(cid:1)p (cid:3) t t t t t = (cid:11)+(cid:12)(cid:1)y −γ(cid:1)p (cid:3) +(1−γ)((cid:1)p −(cid:1)p (cid:3) )+(cid:15) t t t t t = (cid:11)+(cid:12)(cid:1)y −γ(cid:1)p (cid:3)−(1−γ)u +(cid:15) t t t t Note now that Cov(−(cid:1)p (cid:3) ;−(1−γ)u ) = (1−γ)(cid:27)2 t t u Thus, the sign of the bias in the estimate of γ depends on the value of γ itself. If γ < 1, then the bias is positive, while if γ > 1 the bias is negative. Since our estimate of the coe(cid:14)cient on the relative price of computing equipment is greater than one in magnitude, this suggests that, if measurement error is a factor, then the true coe(cid:14)cient is greater in magnitude than our estimate.15 5.3 Out-of-Sample Forecasting Our interpretation of the results in Table 1 is that they are broadly consistent with the theoretical approach outlined in the previous section. However, what of the fact that prompted this exploration, the investment boom of the 1990s? To test whether our twoequation procedure for predicting net investment helps to explain the recent behavior of the capital stock, we estimated our preferred equations for computing and non-computing equipment (Columns 5 and 6 of Table 1) through 1989:4. We then simulated them out 15In any case, we believe the evidence on NIPA price deflators suggests a sanguine interpretation of the measurement errorproblem. Recentresearch byDoms(1999) hasshownthatpricedeclinesmeasuredfrom matchedmodels(followingthepriceofthesamemachineovertime)aresimilartotheNIPAmeasuresbased on hedonic regressions. 17

of sample, taking the realized paths of output and the cost of capital as given, to obtain simulated capital stock series for computing and non-computing equipment. Applying chain aggregation to our two simulated capital stock series, we obtained a simulated series for the aggregate capital stock. As shown in Figure 5, the two-equation system produces a series (the dotted line) that tracks the actual behavior of the equipment capital stock (the solid line) in the 1990s much better than the out-of-sample simulated series fortheaggregate version of thesameregression (thedashedline). Theseries generated by the aggregate regression, like the in-sample residuals from the aggregate net investment model in Figure 2, fall further and further behind observed capital stock growth as the 1990s proceed. In contrast, while the disaggregated system underpredicts actual capital stock growth somewhat for a number of periods from 1993 on, it moves back in line by the end of our sample (1997:4). The reason for the superior tracking performance of the disaggregated system is intuitive: This approach allows the massive decline in computing prices to feed through to capital accumulation far more than aggregate econometric regressions. More important than the system’s ability to track the aggregate capital stock, however, is its ability to explain the behavior of gross equipment investment. As the perpetual inventorydepreciationratesforcomputingandnon-computingarerelativelystableover our sample, we can use a simple out-of-sample forecasting procedure for gross investment: We convert the disaggregated out-of-sample forecasts for capital stocks into forecasts for gross investment using the most recently observed depreciation rates. Applying this procedure to our system estimated through 1989:4 produces gross investment series for computing and non-computing investment. Aggregating these series, we obtain a good description of the recent behavior of aggregate equipmentinvestment: Oursimulated out-of-sample series for aggregate gross investment grows 6.9 percent per year over 1990-97, pretty close to the observed value of 7.5 percent. Moreover, as shown in Figure 6, our simulated series (the dotted line) captures the move to rapid investment growth in 1992 and the sustained high rate of growth thereafter. In contrast, an aggregate model|using the same speci(cid:12)cation andthe1989aggregatedepreciationrate|wouldhaveaveraged about3.1percentagepoints too low over the period 1990-97 (the dashed line). 18

6 Conclusions Boosted by exploding investment in computing equipment, the behavior of equipment investment in the U.S. in the 1990s has been unprecedented. Thus, it should not be too surprising that the traditional econometric models of investment, based as they are on historical correlations, have completely failed to explain the boom. We conclude that these developments provide three important lessons for macroeconomists: (cid:15) Prices Matter: Many previous studies have found limited roles for price variables, stressing the ability of an accelerator model to explain the cyclical behavior of investment. In contrast, we (cid:12)nd an important role for equipment prices. Speci(cid:12)cally, falling computer prices played a crucial role in the investment boom of the 1990s. (cid:15) Depreciation Matters: Most empirical studies have tended to ignore the role played by the replacement of depreciated capital. We have shown that an increasing depreciation rate was of (cid:12)rst-order importance in the extraordinary behavior of equipment investment inthe1990s. Moreover, wehavepointedtoanimportantissueinthemeasurement of depreciation rates: Methodological changes to the NIPAs have made the standard measure of the average depreciation rate based on aggregate data invalid. (cid:15) Aggregation Matters: Depreciation rates vary widely across di(cid:11)erent types of equipment. Also, a model with rational expectations and adjustment costs tells us that the e(cid:11)ects of cost of capital shocks will not be uniform across all types of equipment. We show that a two-equation system for net and gross investment in computing and non-computing equipment, estimated through 1989, is capable of explaining the magnitude and pattern of the U.S. equipment investment boom of the 1990s, while aggregate models completely fail. Put simply, our explanation of equipment investment in the 1990s is that declining computer prices had a very large e(cid:11)ect in boosting the accumulation of computer capital. Consequently, this led to even greater rates of replacement investment. Ultimately, of course, the true test of any model is its ability to forecast future developments. We hope that the future does not turn out to be as unkind to our empirical approach as the 1990s proved to be to the traditional econometric models. 19

References [1] Auerbach, Alan (1989). \Tax Reform and Adjustment Costs: The Impact on Investment and Market Value", International Economic Review, 30, 939-962. [2] Bernanke, Ben, HenningBohn, andPeter Reiss (1988). \Alternative Non-Nested Speci(cid:12)cation Tests of Time Series Investment Models", Journal of Econometrics, 37, 293- 326. [3] Chirinko, Robert (1993). \Business Fixed Investment Spending: A Critical Survey of Modelling Strategies, Empirical Results, and Policy Implications", Journal of Economic Literature, 1875-1911. [4] Clark, Peter K. (1979). \Investment in the 1970s: Theory, Performance, and Prediction", Brookings Papers on Economic Activity. [5] Doms,Mark(1999).HedonicandMatchedModelIndexesofPCPrices,mimeo,Federal Reserve Board. [6] Feldstein, Martin and Michael Rothschild (1974). \Towards an Economic Theory of Replacement Investment", Econometrica, 42, 393-423. [7] Gravelle, Jane (1994). The Economic E(cid:11)ects of Taxing Capital Income, Cambridge: MIT Press. [8] Hansen, Lars Peter and Thomas Sargent (1980). \Formulating and Estimating Dynamic Linear Rational Expectations Models", Journal of Economic Dynamics and Control, 2, 7-46. [9] Katz, Arnold and Shelby Herman (1997). \Improved Estimates of Fixed Reproducible Tangible Wealth, 1929-95", Survey of Current Business, May, 69-92. [10] Kiyotaki, Nobuhiro and Kenneth West (1996). \Business Fixed Investment and the Recent Business Cycle in Japan", in NBER Macroeconomics Annual, 1996, eds., Ben Bernake and Julio Rotemberg. [11] Landefeld, Steven and Robert Parker (1998). \BEA’s Chain Indexes, Time Series, and Measures of Long-Term Economic Growth ", Survey of Current Business, May, 58-68. 20

[12] Nickell, Steve (1979). Expectational Investment Models, London School of Economics, manuscript. [13] Oliner, Stephen (1989). \Constant-Quality Price Change, Depreciation, and Retirement of Mainframe Computers", in Price Measurements and Their Uses, ed. Allan Young, Murray Foss, and Marilyn Manser, Chicago: University of Chicago Press. [14] Oliner, Stephen(1994). MeasuringStocks of ComputerPeripheralEquipment: Theory and Application, mimeo, Federal Reserve Board. [15] Oliner, Stephen, Glenn Rudebusch,and Daniel Sichel (1995). \New and Old Models of Business Investment: A Comparison of Forecasting Performance", Journal of Money, Credit, and Banking, 27, 806-826. [16] Shapiro, Matthew (1986). \The Dynamic Demand for Capital and Labor", Quarterly Journal of Economics, 101, 513-542. [17] U.S. Dept. of Commerce, Bureau of Economic Analysis (1998). \Fixed Reproducible Tangible Wealth in the United States: Revised Estimates for 1995-97 and Summary Estimates for 1925-97", Survey of Current Business, September, 36-46. 21

Appendices A The Data Our dataset consists of quarterly series over 1950:1-1997:4 for real output, as well as real investment, realcapital stock, andthecostof capital fortotal equipment, computingequipment, and non-computing equipment. Our output series is real 1992 dollar output of the private business sector, which is de(cid:12)ned as GDP minus output from government and nonpro(cid:12)t institutions and the imputed income from owner-occupied housing. Our series on real investment for total equipment is private nonresidential producers’ durableequipmentexpendituresfromNationalIncomeandProductAccounts(NIPA)Table 5.5. The data for real computer expenditures is the Computers and Peripherals series from thesamesource. Forrealcapitalstockseriesfortotalequipmentandcomputingequipment, we started with the annual NIPA capital stock data, which are available through 1997 and published in Department of Commerce (1998). These annual data, which represent yearend stocks, were then converted to quarterly series using an interpolation routine that sets the growth rate for each quarter according to its share in the annual total for investment expenditures. Our series on real equipment investment and real capital stock excluding computing equipment were not created by subtracting the real series for computing equipment from the realaggregates. Thelack of additivity of thechain-aggregation formulameans that this isanincorrectcalculation. Rather,intheory,weneedtoconstructanewaggregatefromthe 26 disaggregated non-computing equipment categories. In practice, a \chain-subtraction" procedurewhichappliesequation(9)toaggregate equipmentandthenegativeforcomputer investment works just as well and does not require data on 26 investment series. The cost of capital is measured using the Hall-Jorgenson rental rate formula: !(cid:18) (cid:19) P_ 1−ITC −(cid:28) (cid:3)DEP C = P R +(cid:14)− t t t t P 1−(cid:28) t where P is the price of capital relative to the price of output, R is the real interest rate, t t ITC is the investment tax credit, DEP is the present value of depreciation allowances, and (cid:28) isthemarginalcorporateincometaxrate. Therelativepriceseries,P ,arede(cid:12)nedrelative t 22

to the deflator for private business output. The \capital gains" term is implemented as a three-year moving average of the percentage change in P . To construct the real interest t rate, R , we subtracted expected inflation { proxied by the average inflation rate of the t private businessoutput deflator over the previous (cid:12)ve years - from the nominal rate on Baa corporate bonds. We then added a constant \risk premium" that normalized this required rate of return so that its average equalled the average rate of return on physical capital in our sample (6.8 percent), where this is measured as the ratio of nominal capital income to the nominal capital stock. The tax term was constructed using data on investment tax credits and service lives (used in the calculation of depreciation allowances) from Gravelle (1994). Weused(cid:14) = 0:31forcomputingequipment,(cid:14) = 0:13fornon-computingequipment, and a nominal-capital-stock weighted average of these two rates for aggregate PDE. We use a nominal capital stock weighted average because of the problems with aggregate perpetual inventory depreciation rates discussed in Section 3. B Depreciation Rates with Chained Aggregates Whatwillthecalculated aggregate depreciation ratefromequation (8) looklikewithchainaggregateddata? Tokeeptheanalysistransparent,wewilllookatasimplecase. TheFisher chain formula is somewhat cumbersome, so instead we will use the Tornqvist aggregation formula. This procedure weights the growth rate of each category according to its share in the nominal aggregate, and produces aggregates with almost identical properties to the Fisher procedure. We will also make the following assumptions. Both types of capital depreciate at the same rate (cid:14); the price of type-A capital falls at rate γ relative to the price of type-B capital and output, which are both normalized to equal one. Finally, (cid:12)rms (cid:16) (cid:17) (cid:16) (cid:17) (cid:11) 1−(cid:11) produce with a Cobb-Douglas production function (Q = A KA KB ) and there t t t t are no adjustment costs. Now, assuming no taxes, the cost of capital for type A simpli(cid:12)es to PA(r+(cid:14) +γ). The cost of capital for type B is (r+(cid:14)). Given our assumptions, (cid:12)rms accumulate capital according to the (cid:12)rst-order conditions: (cid:11)Q KA = t t PA(r+(cid:14)+γ) t (1−(cid:11))Q KB = t t (r+(cid:14)) 23

These conditions imply capital stock growth rates gA = gQ +γ and gB = gQ. Using the Tornqvist formula, the growth rate for the chain-aggregated capital stock is gCW = (cid:18)(gQ+γ)+(1−(cid:18))gQ = gQ+(cid:11)γ where (cid:18) is the share of capital of type A in the aggregate nominal capital stock. Note that these nominal stocks are de(cid:12)ned as the \replacement value" of the capital stock and are obtained by reflating the real capital stock for each category by the current-period price of new capital. Now consider the behavior of a chain-aggregate for real investment. Re-arranging the expressions for the growth rate of the capital stock we get: IA t = gQ+γ+(cid:14) KA t−1 IB t = gQ +(cid:14) KB t−1 Thus, for each type of capital, the ratio of real investment to the real capital stock is a constant. So, real investment for capital of types A and B also grow at rate gA and gB. To calculate the growth rate of the chain aggregate, we need nominal shares of investment: ! ! ! ! ! PAIA IA KB KB KA PAKA t t = t t−1 t t−1 t t PBIB KA IB KB KA PBKB t t t−1 !t t−1 ! t !t t gQ +(cid:14)+γ gQ+1 PAKA = t t gQ+(cid:14) gQ +1+γ PBKB ! t t PAKA > t t PBKB t t The share of capital of type A in nominal investment is larger than its share in the nominal capital stock. The reason for this is intuitive. The real capital stock of type A is growing faster than the real stock of type B. This means that, measured in today’s dollars at replacement cost, there is more investment relative to the capital stock for type A than there is for type B; as a result the nominal share of investment for type A is higher. Since real investment of type A grows at rate g +γ while real investment of type B grows at Q rate g , the growth rate of the Tornqvist chain aggregate for real investment places more Q weight on the faster growing category than does the corresponding growth rate for the aggregate capital stock. Hence, the chain aggregate for investment will always grow faster 24

than the chain aggregate for the capital stock. This example, in which relative price shifts cause the fast growing category to have a larger share in nominal investment than in the nominal capital stock, lines up precisely with reality: Computers currently have a much larger share in nominal equipment investment (14 percent in 1997) than in the nominal equipment capital stock (5 percent in 1997). Now, supposewe solve for theaggregate depreciation rate fromthe chain-aggregates for investment and the capital stock: ICW (cid:14)CW = t −gCW t KCW t t−1 Then this value will equal (cid:14) onlyin thebase year. SinceICW grows faster than KCW in each period, this \depreciation rate" gets larger each period. More generally, if we allowed the two types of capital to have varying depreciation rates, the depreciation rate estimated fromthisequation wouldonlyequalaweighted average of theunderlyingdepreciation rates in the base year, as we move forward from the base year this measure would eventually be higher than each of the underlying depreciation rates. 25

C Omitted Proof Proof that (cid:22)(1)(1−(cid:18)(cid:21))= 1 when (cid:25)(1) = 1: Inserting the expression for (cid:22)(1) we need 2 0 13 mX−1 Xm (1−(cid:18)(cid:21)) 4 1+ @ ((cid:18)(cid:21))r−k(cid:25) A5 = 1−(cid:25)((cid:18)(cid:21)) r k=1 r=k+1 Re-arranging the left-hand-side of this equation we get 2 0 13 2 0 1 3 mX−1 Xm mX−1 Xm (1−(cid:18)(cid:21)) 4 1+ @ ((cid:18)(cid:21))r−k(cid:25) A5 = (1−(cid:18)(cid:21)) 4 1+ @ (cid:25) A ((cid:18)(cid:21))k5 r r k=1 r=k+1 k=1 r=k+1 Now use (cid:25)(1) = 1: 2 0 13 " ! # mX−1 Xm mX−1 Xk (1−(cid:18)(cid:21)) 4 1+ @ ((cid:18)(cid:21))r−k(cid:25) A5 = (1−(cid:18)(cid:21)) 1+ 1− (cid:25) ((cid:18)(cid:21))k r r k=1 r=k+1 k=1 r=1 Expanding this expression we get 1+(1−(cid:25) 1 )((cid:18)(cid:21))+(1−(cid:25) 1 −(cid:25) 2 )((cid:18)(cid:21))2+::::::+(1−(cid:25) 1 −(cid:25) 2 −::−(cid:25) n−1 )((cid:18)(cid:21))m−1 −(cid:18)(cid:21)−(1−(cid:25) 1 )((cid:18)(cid:21))2−(1−(cid:25) 1 −(cid:25) 2 )((cid:18)(cid:21))3−:::::::::::−:(1−(cid:25) 1 −(cid:25) 2 −::−(cid:25) n−1 )((cid:18)(cid:21))m = 1−(cid:25) ((cid:18)(cid:21))−(cid:25) ((cid:18)(cid:21))2−:::−(cid:25) ((cid:18)(cid:21))n 1 2 n = 1−(cid:25)((cid:18)(cid:21)) as required. 26

Table 1 Capital Stock Growth Regressions (Standard errors in parentheses) Total Computers Excluding Total Computers Excluding Computers Computers (50-97) (80-97) (50-97) (50-97) (80-97) (50-97) (cid:21) 0.93 0.90 0.92 0.94 0.86 0.93 (.02) (.04) (.02) (.02) (.05) (.02) Sum of the Coe(cid:14)cients on: Output 0.10 0.18 0.11 0.10 0.21 0.10 (.01) (.13) (.02) (.01) (.10) (.02) Cost of capital -0.02 -0.17 -0.01 (.01) (.09) (.005) Relative prices -0.03 -0.26 -0.03 (.01) (.10) (.02) Cost of capital -0.01 -0.08 -0.01 without prices (.005) (.08) (.006) Long-run Elasticities: Output 1.49 1.75 1.33 1.46 1.44 1.33 (.36) (1.41) (.31) (0.37) (0.84) (.32) Cost of capital -0.18 -1.59 -0.13 (.09) (.75) (.083) Relative price of capital -0.45 -1.83 -0.33 (.26) (.45) (.31) Cost of capital -0.18 -0.53 -0.14 without relative price (.11) (.50) (.10) 27

Figure 1 The Demise of the Traditional Investment Regression Investment relative to the Capital Stock Data Fitted Values 0.24 0.22 0.20 0.18 0.16 0.14 0.12 0.10 0.08 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 28

Figure 2 Perpetual Inventory Depreciation Rates for Aggregate Equipment Using Chain-Weight Investment and Capital Using Fixed-Weight Investment and Capital Non-Computing Equipment, Chain-Weight Investment and Capital 0.17 0.16 0.15 0.14 0.13 0.12 0.11 0.10 0.09 0.08 0.07 1966 1968 1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 29

Figure 3 Residuals from Gross and Net Aggregate Equipment Investment Regressions Errors Expressed As Percentages of the Capital Stock Gross Investment Residuals Net Investment Residuals 0.08 0.06 0.04 0.02 0.00 -0.02 -0.04 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 30

Figure 4 The Cost of Capital In Logs Computer Cost of Capital Relative Price of Computers Non-Price Component - Computers 5 5 -0.3 4 4 -0.4 3 3 -0.5 2 2 1 1 -0.6 0 0 -0.7 -1 -1 -2 -2 -0.8 1970 1980 1990 1970 1980 1990 1970 1980 1990 Cost of Capital Excluding Computers Relative Price of Non-Computing Equipment Non-Price Component - Excluding Computers -1.2 0.15 -1.2 -1.3 -1.3 0.10 -1.4 -1.4 0.05 -1.5 -1.5 -1.6 -1.6 0.00 -1.7 -1.7 -0.05 -1.8 -1.8 -1.9 -0.10 -1.9 1950 1960 1970 1980 1990 1950 1960 1970 1980 1990 1950 1960 1970 1980 1990 31

Figure 5 Out-of-Sample (Post-1989) Forecasts of Growth in the Stock of Equipment Annualized Growth Rates Data Aggregate Model (Accelerator with Cost of Capital) Disaggregated Model (Same Specification) 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.00 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 32

Figure 6 Out-of-Sample (Post-1989) Forecasts of Growth in Real Equipment Investment Year-over-Year Percentage Changes Data Aggregate Model (Accelerator with Cost of Capital) Disaggregated Model (Same Specification) 20 15 10 5 0 -5 -10 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 33