On the Nature of Capital Adjustment Costs Cooper, Russell W;Haltiwanger, John C The Review of Economic Studies; Jul 2006; 73, 256; ProQuest Central pg-611 Review of Economic Studies (2006) 73. 611-63.1 0034-6527/06/0023061 l$02.(X) © 2 Spike rate: positive investment 18-67r (0 12> Spike rate: negative investment lX«7f (0-04) Serial correlation of investment rates 0-05K (0-1X13) Correlation of profit shocks and investment 0143 (0-003I LRD. Longitudinal Research Database. The S.E. of these moments are provided in parentheses in Table I.6 Given the size of the data-set (these moments are all based on more than l(X),(KK) plant-year observations), these moments are all very precisely estimated. Such precision should not be interpreted as reflecting little dispersion at the plant level. At the micro-level, there is substantial dispersion. For example, the average investment rate is 12-2. but the S.D. of micro-investment is 33-7. Nevertheless, with a very large sample we estimate the mean and other moments of micro-investment very precisely. In what follows, we exploit the micro-heterogeneity explicitly as we use the estimated dispersion of profit shocks from the micro-data. 6. The S.E. reported in Table I for the variables reported as percentages are in the same units. Later in the paper we use these same variables as fractions with appropriate adjustment of units of S.E. © 2006 The Review of Economic Studies Limited Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 616 REVIEW OF ECONOMIC STUDIES We estimate adjustment cost parameters by using a simulated method of moments approach. This approach requires selecting moments to match. Theoretical and practical considerations suggest that the moments should be relevant in identifying the adjustment cost parameters and are also precisely estimated. We regard the four moments in Table 1 as capturing key features of the behaviour of investment at the micro-level. The first moment is the serial correlation in investment. It is well established in the adjustment cost literature (see, for example, Caballero and Engel, 2003; and CHP) that the serial correlation of investment is sensitive to the structure of adjustment costs. The second moment is the correlation between investment and profit shocks as it reflects the covariance structure between investment and the shocks to profits. This moment and the others we match are quite sensitive to the adjustment cost parameters and in this sense satisfy the relevance criterion. The other two moments capture key features of Figure 1 that have been emphasized in the literature—namely, the investment distribution at the micro-level is very asymmetric and has a fat right tail. To capture these features, we use the positive and negative spike rates in Table 1. Each of these four moments captures key features of investment behaviour at the micro-level, but the exact choice of moments is an open question. In what follows, we use these four moments in our empirical analysis and then present some analysis of robustness of our findings to the choice of alternative moments. Before proceeding, it is worth noting that one moment we choose not to match directly is the fraction of observations with inaction. While Table 1 shows some range of inaction, the more robust finding in Figure 1 and Table 1 is that the distribution of investment is skewed and kurtotic with a fat right tail. Identifying inaction precisely at the micro-level is difficult because in practice there is substantial heterogeneity in capital assets with associated heterogeneity in adjustment costs. For example, buying a specific tool gets lumped into capital equipment expenditures in the same way as retooling the entire production line. Explicitly analysing the role of capital heterogeneity is beyond the scope of this paper, but we discuss this as an area of future research in the concluding remarks. 3. MODELS AND QUANTITATIVE IMPLICATIONS Our most general specification of the dynamic optimization problem at the plant level is assumed to have both components of convex and non-convex adjustment costs as well as irreversibility. Formally, we consider variations of the following stationary dynamic programming problem: V(A,K) = m9xn(A,K)-C(l,A,K)-p(I)I+fiEA>iAV(A',Kf) V(A,K)t (3) where Yl(A,K) represents the (reduced-form) profits attained by a plant with capital K, a profitability shock given by A, I is the level of investment, and K' = K(\ —8) + I. Here unprimed variables are current values and primed variables refer to future values. In this problem, the manager chooses the level of investment, denoted /, which becomes productive with a one period lag. The costs of adjustment are given by C(/, A, K). This function is general enough to have components of both convex and non-convex costs of adjustment. Irreversibility is encompassed in the specification if the price of investment, p(I), depends on whether there are capital purchases or sales. Current profits, for given capital, are given by U(A, K), where the variable inputs (L) have been optimally chosen, a shock to profitability is indicated by A, and K is the current stock of capital. That is, n(/4, K) = maxR(A, K, L) - Lu>{L), © 2006 The Review of Economic Studies Limited Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. COOPER & HALTIWANGER ON THE NATURE OF CAPITAL ADJUSTMENT 617 TABLE 2 Parameterization of illustrative models Model 7 F '■ Ps Ph No AC 0 0 1 1 1 CON 2 0 1 1 1 NC-P 0 001 1 1 1 NCv. 0 0 0-95 1 1 TRAN 0 0 1 0 75 ! TABLE 3 Moments from illustrative models Moment LRD No AC con nc-F nc-;. TRAN Fraction of inaction 0081 ()•() 0-038 0616 0 588 0-69 Fraction with positive investment bursts 018 0298 0-075 0 212 0-213 0 120 Fraction with negative investment bursts 0018 0 203 00 0172 0198 0-024 Corr ]A V(Af, K'). In this second optimization problem, as in CHP, there are two types of fixed costs of adjustment. Both, importantly, are independent of the level of investment. © 2006 The Review of Economic Studies Limited Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. COOPER & HALTIWANGER ON THE NATURE OF CAPITAL ADJUSTMENT 619 The first adjustment cost, / < 1, represents an opportunity cost of investment. If there is any capital adjustment, then plant productivity falls by a factor of (1 — k) during the adjustment period. Studies by Power (1998) and Sakellaris (2001) provide evidence that plant productivity is lower during periods of large investment.8 All else the same, this form of adjustment cost implies that investment bursts are less costly during periods of low profitability as adjustment costs are low. But this need not imply a negative correlation between investment and profitability, since there is a gain to investment in high productivity states if there is sufficient serial correlation in the profitability shocks. The second fixed cost of adjustment, denoted F, is independent of the level of activity at the plant. It is proportional to the level of capital at the plant to eliminate any size effects. Similar results are obtained if the cost is proportional to the plant-specific average capital stock. Thus this cost will naturally produce a positive correlation between investment rates and profitability. The intuition for optimal investment policy in this setting comes from CHP. In the absence of profitability shocks, the plant would follow an optimal stopping policy: replace capital if and only if it has depreciated to a critical level. Adding the shocks creates a state-dependent optimal replacement policy but the essential characteristics of the replacement cycle remain: there is frequent investment inactivity punctuated by large bursts of capital purchases/sales. Thus, the model is able to produce both the inaction and bursts highlighted in Table 1. Relative to the partial adjustment of the convex model, the model with non-convex adjustment costs provides an incentive for the firm to "overshoot its target" and then to allow physical depreciation to reduce the capital stock over time. 3.4. Transactions costs Finally, as emphasized by Abel and Eberly (1994, 1996). it is reasonable to consider the possibility that there is a gap between the buying and selling price of capital, reflecting, inter alia, capital specificity and a lemons problem. This is incorporated in the model by assuming />(/) = pi, if / > 0, and /?(/) = ps if / < 0 where pi, > ps. In this case, the gap between the price of new and old capital will create a region of inaction. The value function for this specification is given by V(j4, K) — max{Vh(A, K), VV(A, K), Vl(A, K)) V(/4, K), where the superscripts refer to the act of buying capital "h", selling capital V. and inaction "i". These options, in turn, are defined by Vh(A.K) = maxYl(A,K)-phl+{lEA>lAV(A',K(\-S) + I), V(A,K) = ma\U(A.K) + pyR + /3EAlAV{A',K(\-d)-R\ ft and V (A, K) = n(A, K) + (1EA>\A V{A', K(1 - <*)). Here, we distinguish between the purchase of new capital (/) and retirements of existing capital (/?). As there are no vintage effects in the model, a plant would never simultaneously purchase and retire capital.9 8. Incorporating this into our model implies some potential misspecification of profitability shocks since it is necessary to distinguish /. and low realizations of profitability shocks. We return to this point in detail later. 9. Though, as suggested by a referee, time aggregation may in fact generate observations of both sales and purchases over a period of time. © 2006 The Review of Economic Studies Limited Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 620 REVIEW OF ECONOMIC STUDIES The presence of irreversibility will have a couple of implications for investment behaviour. First, there is a sense of caution: in periods of high profitability, the firm will not build its capital stock as quickly since there is a cost of selling capital. Second, the firm will respond to an adverse shock by holding on to capital instead of selling it in order to avoid the loss associated with pa < pt,. 3.5. Evaluation of illustrative models As indicated by Table 2, we explore the quantitative implications of five models. While these parameterizations are not directly estimated from the data, they provide some interesting benchmark cases that highlight the key issues arising between models with convex and non-convex costs of adjustment. The first, denoted "No AC" is the extreme model in which there are no adjustment costs. The second row, denoted "CON", corresponds to a specification in which there are only convex costs of adjustment. The case labelled "NC-F' assumes that there are only non-convex costs of adjustment with F > 0, X = 0. The case labelled "NC-A" is a second non-convex cost of adjustment case with F = 0, X = 0-95. Finally, the case labelled "TRAN" imposes a gap of 25% between the buying and selling price of capital. Our quantitative findings for the specifications in Table 2 along with data from the LRD are summarized in Table 3. As noted earlier, there is evidence of lumpiness and inaction in the LRD. In addition, there is low autocorrelation in plant-level investment, which is noteworthy, given that both aggregate and idiosyncratic shocks to profitability exhibit substantial serial correlation. Comparing the columns of Table 3 pertaining to the illustrative models with the column labelled LRD, none of the models alone fits these key moments from the LRD. The extreme case of no adjustment costs (labelled No AC) is given in the second column. This model produces no inaction but is capable of producing bursts in response to variations in the idiosyncratic profitability shocks. It also generates a large fraction of observations with negative investment bursts. Overall, the model without adjustment costs is very responsive to shocks. Evidently, the empirical role of adjustment costs is to temper the response of investment to fundamentals. The quadratic adjustment cost model (denoted CON) adds convex adjustment costs to the No AC model. This specification mutes the response of investment to profitability shocks as the fraction of positive and negative spikes is significantly reduced.10 Further, the convex cost of adjustment model, through the smoothing of investment, creates serial correlation in investment relative to the shock process. Consequently, the correlation of investment and profitability is higher than in the No AC case. Both treatment of non-convex costs of adjustment (NC-F and NC-,1) and/or the model with irreversibility (TRAN) are able to create investment inactivity at the plant level. However, the non-convex models create negative serial correlation in investment data and a lower correlation between investment rates and profitability. The negative serial correlation of the non-convex adjustment cost models is analogous to the upward sloping hazards characterized by CHP. All of the models are able to produce both positive and negative spikes but, naturally, the asymmetry in spike rates is most prominent in the irreversibility specification. 4. ESTIMATION None of these extreme models is rich enough to match key properties of the data. The model we estimate includes convex and non-convex adjustment processes as well as irreversible investment. 10. There is a small amount of inaction, which reflects the discrete state space approximation of our quantitative approach. © 2006 The Review of Economic Studies Limited Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. COOPER & HALTIWANGER ON THE NATURE OF CAPITAL ADJUSTMENT 621 This combining of adjustment cost specifications may be appropriate for a particular type of capital (with say installation costs and some degree of irreversibility) and/or may also reflect differences in adjustment cost processes for different types of capital. As the data is not rich enough to study a model with heterogeneous capital, our approach is to consider a hybrid model with all forms of adjustment costs and. in turn, to estimate the key parameters of this specification by matching the implications of the structural model with relevant features of the data. The adjustment cost parameters are estimated using the following routine. For arbitrary values of the vector of parameters, 0 = (F, y, X, ps), the dynamic programming problem is solved and policy functions are generated. Using these policy functions, the decision rule is simulated, given arbitrary initial conditions, to create a simulated version of the LRD." We then calculate a set of moments from the simulated data, which we denote as 4^(0). This vector of moments depends on the vector of structural parameters, 0, in a non-linear way. The estimate 0 minimizes the weighted distance between the actual and simulated moments. Formally, we solve £(0) = minll"7 - Vs(Q)\'W[Vii - ^v(0)], (6) e where W is a weighting matrix. This simulated method of moments procedure will generate a consistent estimate of 0. We use the optimal weighting matrix given by an estimate of the inverse of the variance-covariance matrix of the moments.12 Of course, the 4^(0) function is not analytically tractable. Thus, the minimization is performed using numerical techniques. Given the potential for discontinuities in the model and the discretization of the state space, we used a simulated annealing algorithm to perform the optimization. For the estimation, we consider two specifications of non-convex adjustment costs. In one, which we term the fixed cost case, the costs are represented by a lump-sum cost of adjustment F > 0 without any opportunity costs of adjustment X = 1. In the second, which we term the opportunity cost case. F = 0 and /. < 1. These are taken as leading specifications in the literature, and thus our estimation provides insights into which is more capable of capturing relevant features of the data.13 In addition to the adjustment cost parameters, the dynamic optimization problem is also parameterized by the curvature of the profitability function and the process governing the shocks to profitability. The two specifications of adjustment costs, particularly the presence of disruption effects, require different approaches to uncovering the underlying shocks to profitability and characterizing the profitability function. 4.1. Estimation with fixed costs: F > 0 and X = I Assume that the dynamic programming problem for a plant is given by V(A, K) = max[V''(,4, AO, V*{A,K), v'(A,K)) V(A,K), (7) where, as above, the superscripts refer to the act of buying capital "b'\ selling capital *\v", and inaction "/". These options, in turn, are defined by [ ]. The simulation is lor 5(k) periods. The initial 15 periods are not used in calculating moments so that the results are independent of the assumed initial conditions. The moments are not sensitive to adding more periods to the simulation or to dropping more of the initial periods. 12. See Smith (1993) for details of methodology and measurement of the weighting matrix. In our case, given the large micro data-set we use we estimate the moments that we are attempting to match very precisely. As such, most of the moments we arc attempting to match receive a very large weight in (6). 13. Caballero and Engel (1999) consider / < 1 and F — 0. while Thomas (2002) assumes /. = I. and F is random. © 2006 The Review of Economic Studies Limited Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 622 REVIEW OF ECONOMIC STUDIES Vb(A, K) = maxII(A, K)-FK-I- y-{I/K)2 K + flEA>lA V(A', K{\-5) +1), VS(A, K) = max Tl(A, K)-FK + p,R--(R/K)2K + pEAnA V(A\ K(\-S)- R), R 2 and V'(A, K) = U(A, K) + pEA,\AV(A', K(\-#)). We have specified some parameters of the model (fi = 0-95, S = 0-069, pt = 1) for the functional forms discussed above. Further, we retain our specification of the profit function, I\(A,K) = AK9. For the structural estimation, we focus on three parameters, 0 = (F, y,ps), which characterize the magnitude of the non-convex and the convex components of the adjustment process and the size of the irreversibility of investment. As explained next, we estimate the curvature of the profit function and the A process independently of the dynamic programming problem. 4.1.1. Estimates of $ and measuring profitability shocks. Using the assumption of I = 1, profits at plant / in period t are given by n(Ail,Kil) = Ai,Kft, (8) regardless of the level of investment activity. Suppose that au = ln(A,,) has the following structure ai,=b,+£it, (9) where b, is a common shock and e,-, is a plant-specific shock. Assume e,-r = /?,.£/,,_ i + //„. Taking logs of (8) and quasi-differencing yields itu =p(:Ku-\ +9kit -pi:8kit-\ +b, -pi:b,-i + r}i,. (10) We estimate this equation via generalized method of moments (GMM) using a complete set of time dummies to capture the aggregate shocks and using lagged and twice-lagged capital and twice-lagged profits as instruments. To implement this estimation, real profits and capital stocks are calculated at the plant level. A more detailed discussion of the measurement of real profits and capital as well as the estimation of the profit function and associated robustness issues are provided in the Appendix. Our specification of the relatively simple AR(1) process for the idiosyncratic shocks is motivated by the need to keep the state space relatively parsimonious for the downstream numerical analysis and estimation. The results give us an estimate of 9 and an estimate of the process for the idiosyncratic components of the profitability shocks. From the plant-level data, 0 is estimated at 0-592 (0006) and pe is estimated at 0-885 (0004).14 The estimate of 6 is significantly below 1, and this is interesting in its own right. Using the LRD plant-level data on cost shares we estimate = 0-72, which implies a demand elasticity of —6-2 and a mark-up of about 16%.15 Having estimated 8 we recover an from (8) and decompose it into aggregate and idiosyncratic components using (9). This latter step amounts to measuring the aggregate shock as the mean of a,-, in each year and the idiosyncratic shock as the deviation of a-xt from the year-specific 14. The S.E. are in the parentheses. The R2 of the regression was 0-58. 15. While we do not estimate a production function in this paper, the existing plant-level literature suggests that constant returns to scale (CRS) is a reasonable assumption (see, for example, Baily, Hulten and Campbell, 1992; Olley and Pakes, 1996). If denotes labour's coefficient in the Cobb-Douglas technology and £ is the elasticity of the demand curve, thenfl = ((1 -aL)(\ +£))/( 1 -aL(\ +{)). © 2006 The Review of Economic Studies Limited Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. COOPER & HALTIWANGER ON THE NATURE OF CAPITAL ADJUSTMENT 623 TABLE 4 Parameter estimates: ). = I Spec. Structural parameter estimates (S.E.) Moments }' F Ps Corr (/', j_|) Corr (i.a) Spike+ Spike £(0) LRD 0-058 0143 0-186 0018 all 0 049 0039 0-975 0-086 0-31 0127 0-030 6399 9 (0.002) (0-001) (0004) y only 0455 0 1 0 605 0-540 0-23 0028 53,182 6 (0-002) Ps only 0 0 0795 0 113 0-338 0-132 0033 7673-68 (0-002) F only 0 0-0695 1 -0-004 0-213 0 105 0-0325 7390-84 <0-00046) LRD. Longitudinal Research Database. mean. Using this decomposition, we find that b, has an S.D. of 008, and with an AR(I) specification, the relevant AR( 1) coefficient (denoted as pf, in what follows) is given by 0-76 with an S.E. of 019. We also find that the S.D. of e,-, is 0-64. To sum up, in what follows, we use the following key estimates from the data in our estimation of adjustment costs: 0 = 0-592, oi: = 0-64, pi: = 0 885, ab = 008, ph = 0-76. These were the parameter values used in Section 3.5. These statistics imply that the innovations to the aggregate shock process have an S.D. of 0-05 and the innovations to the idiosyncratic shock process have an S.D. of 0-30. Neither process is estimated to have a unit root. These moments of the shock processes are critical for understanding the nature of adjustment costs since key moments, such as investment bursts, reflect the variability of profitability shocks, the persistence of these shocks, and the adjustment costs associated with varying the capital stock. Moreover, the characterization of these processes provide the necessary information for the solution of the plant-level optimization problem, which requires the calculation of a conditional expectation of future profitability. 4.1.2. Estimates of (F,y,ps). Table 4 reports our results for different specifications along with S.E.16 The last column presents £(0) from (6), a measure of fit for the model. The first row estimates the complete model with three structural parameters used to match four moments. We are able to come fairly close to matching the moments with a vector of structural parameters given by y = 0 049, F = 0039, p„ = 0-9752, where X = 1 throughout. These parameter estimates imply relatively modest, but statistically and economically significant, convex, and non-convex adjustment costs. The estimates indicate that a model, which mixes the various forms of adjustment costs, is able to best match the moments. This model can produce the low serial correlation in investment as well as the muted response of investment to shocks. Further, with the non-convex adjustment and the irreversibility, the model produces both positive and negative investment bursts of the frequency found in the data. Restricted versions of the estimated model are also reported for purposes of comparison. Note how poorly the estimated quadratic adjustment cost does as it creates excessive serial correlation as well as a large contemporaneous correlation between investment and the shocks. Interestingly, the quadratic adjustment cost model can produce both positive and negative investment 16. In this table, corr(i, /'_|) is the serial correlation of the plant-level investment rate and corr(i , 0, k = 1. This is seen by comparing the values of £(0). The row in the table labelled 'U only" focuses solely on disruption costs. There is evidence here of disruption costs but the model does not fit the moments as well as the "all" specification. This result indicates that it is important to have all forms of adjustment costs in the specification. Relative to previous results, Caballero and Engel (1999) estimate a model in which the disruption costs are random. Further, they do not allow any other forms of adjustment costs. Caballero and Engel (1999) report a mean adjustment cost of 16-5%, which, in our notation, is a value of k — 0-835. This latter value is quite close to ours, which is striking given that they estimate their model with industry-level data, while we estimate ours using micro-data. There are a number of subtle additional differences in methodology that may be at work as well. Caballero and Engel assume that capital becomes immediately productive and also has stochastic adjustment costs. Both of the latter imply lower average adjustment costs, which is consistent with the pattern of the estimated k values across the studies. To obtain a better sense of the magnitude of adjustment costs in this model, we simulated the estimated policy functions and calculated the resulting costs of adjusting the capital stock. The average adjustment cost paid relative to the capital stock was 0-0091 and was 0-031 as a fraction of profits.24 Though not reported in the table, we also estimated all four adjustment cost parameters, © = (F,y, k, ps). We were unable to improve upon the fit summarized in Table 5: allowing F > 0 did not enable us to better match the moments. 23. For these results we set the serial correlation of the idiosyncratic (aggregate) shock at 0-92 (0-82) and the S.D. of the innovation to the shock was at 0-22 (0 05). At these parameter values, we are able to reproduce the serial correlation and S.D. for the shocks reported using the methodology described above. In effect, this constitutes another indirect inference procedure. Further, when we re-estimated the profitability function using the simulated data using the same techniques as in Section 4.1.1 (but ignoring the effect of k), we obtained a value of 0 quite close to the value assumed in the analysis. 24. We appreciate conversations with Anil Kashyap on these calculations. In the calculations reported here, when we refer to the average adjustment costs as a fraction of profits, we are referring to the ratio of the expected value of adjustment costs to the expected value of profits. An alternative would be to take the average of the ratio of adjustment costs to profits. The latter is equal approximately to (1 — k) times the fraction of periods with adjustment. Back-of-the-envelope calculations suggest that these two alternatives yield roughly similar results. © 2006 The Review of Economic Studies Limited Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. COOPER & HALTIWANGER ON THE NATURE OF CAPITAL ADJUSTMENT 627 TABLE 6 Sectoral parameter estimates Parameter estimates (S.E.) Moments Sector F Ps corr(/. i_|) corr(i'.tf) spike+ spike urn 331-LRD 0-086 0-133 0-064 0-02 n/a 331 -est. 00 0-07 1 0-946 -0-03 0-249 0-076 0-022 62-19 «MKX>3> (0-0017) (0.0046) 0015 0 0-70 0-76 0-041 0-146 0-073 0-018 8-13 < 0-0037) (0 0344) (00167) 371-LRD 0-082 0132 0-204 0-013 n/a 371-est. 0-012 0 069 1 0 962 ()(>69 0-338 0112 0037 452-96 (0-0026) (0.0057 > (00106) 371-est. 0-051 0 0-679 0-8082 0 251 0 132 0-096 0027 318-04 (0-0068) (00398 > (00218) LRD. Longitudinal Research Database; est., estimates. 4.3. Sectoral results Our analysis of plant-level investment imposes the equality of various parameters (e.g. depreciation rates, mark-ups, transition matrices for shocks, etc.) across sectors. Further, the overall fit of the model, as measured by the £(0) statistic leaves some room for improvement. Thus, it is interesting to study some specific sectors in detail. In this subsection, we re-estimate the model for some leading sectors: steel (331) and transportation (371). For this exercise, we re-estimated the curvature of the profit function and recalculated depreciation rates and transition matrices for profitability shocks by sector.-5 The estimation results for the structural parameters of the adjustment processes for the two sectors are also reported in Table 6. Once again we find support for the presence of both convex and non-convex adjustment costs along with irreversibility. Further, the specification in which the non-convexity takes the form of disrupting the production process again fits the data best. We again find small quadratic adjustment costs and evidence of irreversibility. 4.4. Robustness to alternative moments In this subsection, we consider the robustness of our results to the choice of alternative moments. As stated in Section 2.2, the moments we have chosen to match capture key features of investment behaviour at the micro-level. We consider two alternative moments that capture the shape and dispersion of investment illustrated in Figure I.26 Specifically, rather than using the positive and negative spike rates to capture the asymmetry and fat right tail of the investment distribution, we use the 90th percentile and the 10th percentile of the investment rate distribution for each year averaged over time. The average 90th percentile from the LRD is 0-299 and the 10th percentile is given by —0014. These moments capture the asymmetry and fat right tail of the investment distribution. When we use these two moments along with the two correlation moments in our analysis, the estimated adjustment cost parameters are qualitatively and quantitatively similar to the parameter 25. The rates of physical depreciation were 0-076 for sector 331 and 0-063 for sector 371. The curvature of the profit functions differs across these sectors and was 0-66 for 331 and 0-78 for 371. respectively. The AR(1) coefficients for idiosyncratic and aggregate (sectoral) shocks, respectively, were 0-69 and 0-85 for 331 and 0-85 and 0-62 for 371. The S.D. of the innovations for idiosyncratic and aggregate (sectoral) shocks, respectively, were 0-28 and 0-23 for 331 and 0 32 and 0-16 for 371. 26. These were suggested to us by the reviewers and the editor. @ 2006 The Review of Economic Studies Limited Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 628 REVIEW OF ECONOMIC STUDIES estimates reported in Table 5. That is, matching these alternative moments requires a relatively modest convex cost component and substantial disruption adjustment costs and transaction adjustment costs. To be specific, the best fit for matching this alternative set of moments implies an estimate of y = 0 042, k = 0-86, and ps — 0-80. These parameter estimates yield simulated moments of corr(j,/_|) = 0-033, corr(/,a) = 0133, the average 90th percentile investment rate of 0-308, and the average 10th percentile investment rate equal to 0-00. As these alternative moments are estimated precisely in the actual data, the adjustment cost parameters are as tightly estimated as those in Table 5. 5. AGGREGATE IMPLICATIONS The estimation results reported in Table 5 indicate that a model, which mixes both convex and non-convex adjustment processes can match moments calculated from plant-level data quite well. An issue for macroeconomists, however, is whether the presence of non-convexity at the micro-economic level "matters" for aggregate investment. In particular, there are economic forces, such as smoothing by aggregation and relative price movements, which imply that non-convexities at the micro-level may be less important for aggregate investment. This issue of aggregate implications has already drawn considerable attention in the literature. CEH find that introducing the non-linearities created by non-convex adjustment processes can improve the fit of aggregate investment models for sample periods with large shocks. CHP similarly find that there are years where the interaction of an upward sloping hazard (investment probability as a function of age) and the cross-sectional distribution of capital vintages matters in accounting for aggregate investment. We study the contribution of non-convex adjustment costs to aggregate investment (defined by aggregating across the plants in our sample) in two ways. First, we compare aggregated and plant-level moments. We use the term aggregated to indicate that the results pertain to aggregation over our sample and thus may not accord with aggregate investment from, say, the National Income and Product Accounts (NIPA). Second, we compare the aggregate time series implications of our estimated model, termed the best fit, against a model with quadratic adjustment costs. For moments, we calculate the serial correlation of investment rates and the correlation of investment rates and profitability from aggregated investment using the panel of manufacturing plants. To be precise, we compute aggregate statistics from the LRD by creating a measure of the aggregate investment rate and, using the series of profitability shocks described in Section 4.1.1, a measure of aggregate profitability. The results stand in stark contrast with the moments reported in Table 1. The serial correlation of aggregate investment is 0-46, and the correlation between investment and the profitability shock is 0-51 for the aggregated data. In contrast, the plant-level data exhibits much less serial correlation, 0-058, and much less contemporaneous correlation between investment and shocks, 0143. How well does the best-fit model match these aggregate facts? If we compute aggregate investment and the aggregate shock from a simulation using the best-fit model, the serial correlation of aggregate investment is 0-63 and the correlation between investment and the profitability shock is 0-54. Thus, aggregation of the heterogeneous plants alone substantially increases both the serial correlation of investment and its correlation with profitability. In fact, the aggregate moments reported above seem to be much closer to the prediction of a quadratic cost of adjustment model: from Table 3, a model with quadratic adjustment costs implies high serial correlation and high contemporaneous correlation of investment and shocks. This suggests a second exercise in which we ask how well a quadratic adjustment cost model can match the aggregate data created by the estimated model. To study this, we created a time series © 2006 The Review of Economic Studies Limited Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. COOPER & HALTIWANGER ON THE NATURE OF CAPITAL ADJUSTMENT 629 simulation (periods) for the estimated model. We then searched over quadratic adjustment costs models to find the value of )• to maximize the R1 between the series created by the best-fit model and that created by the quadratic model. A value of y — 0-195 solved this maximization problem, and the R1 measure was 0859. Thus, the quadratic model explains most but not all of the time series variation from the best-fit model. This analysis of the aggregate implications is potentially incomplete in that we ignore two important factors. First, we do not explicitly consider general equilibrium effects.27 As emphasized by Caballero (1999), Thomas (2002). Veracierto (2002), and Kahn and Thomas (2003), it is likely that there is further smoothing by aggregation due to the congestion effects that are potentially present in the capital goods supply industry and/or due to endogenous interest rate fluctuations. Second, we work with a subset of manufacturing plants. In particular, we have selected a balanced panel and have thus ignored entry and exit, including the investment associated with that decision. Thus, our "aggregate results" refer to the aggregation over a fixed set of plants. Nonetheless, the analysis uncovers strong effects of smoothing over heterogeneous plants without variations in relative prices. While identifying the mechanisms that smooth out plant-specific non-convexities is of interest, it should be clear that both smoothing by aggregation and variations in factor prices are important to the smoothing process. That said, it is also clear that the non-convexities at the plant level are not totally smoothed by aggregation: ourgoodness-of-fit measure is 0-859 not I! 6. CONCLUSIONS The goal of this paper is to analyse capital dynamics through competing models of the investment process: what is the nature of the capital adjustment process? The methodology is to take a model of the capital adjustment process with a rich specification of adjustment costs and solve the dynamic optimization problem at the plant level. Using the resulting policy functions to create a simulated data-set, the procedure of indirect inference is used to estimate the structural parameters. Our empirical results point to the mixing of models of the adjustment process. The LRD indicates that plants exhibit periods of inactivity as well as large positive investment bursts but little evidence of negative investment. The resulting distribution of investment rates at the micro-level is highly skewed even though the distribution of shocks is not. A model, which incorporates both convex and non-convex aspects of adjustment, including irreversibility, fits these observations best. In particular, a model of adjustment in which the non-convex cost entails the disruption of production fits the data best. In terms of further consideration of these issues, we plan to continue this line of research by introducing costs of employment adjustment. This is partially motivated by the ongoing literature on adjustment costs for labour as well as the fact that the model without labour adjustment costs implies labour movements that are not consistent with observation (see. e.g. Caballero. Engel and Haiti wanger. 1997). Further, it would be insightful to utilize this model to study the effects of investment tax subsidies. Here, those subsidies enter quite easily through policy induced variations in the cost of capital. Clearly, one of the gains to structural estimation is to use the estimated parameters for policy analysis. An interesting aspect of that exercise will be a comparison of the estimated 27. Recall that we have been ahle to identity the adjustment costs using cross-sectional differences in investment dynamics across plants having controlled for aggregate shocks. However, even though we have been able to identify the adjustment costs, aggregate variation in investment will reflect the complex interaction of shocks, endogenous factor prices, and adjustment dynamics. © 2(K)6 The Review of Economic Studies Limited Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 630 REVIEW OF ECONOMIC STUDIES model and a quadratic adjustment cost model in terms of their predictions of the aggregate effects of an investment tax credit. Finally, there are some methodological issues worth exploring further. For this exercise, we have chosen to estimate the model using a simulated method of moments approach. There are, of course, competing approaches, including maximum likelihood estimation (MLE) as well as estimation from the Euler equations, including periods of inaction. In future research, we plan on exploring these competing approaches. These alternative approaches have the potential advantage in that they do not rely on matching specific moments but rather can confront the micro-data directly. Still, it is useful to emphasize that the simulated method of moments approach we use here has a number of distinct advantages especially in this context. First, structural models of investment with non-convex adjustment costs obviously imply a range of inaction. In the actual micro-data, while we do observe some range of inaction as we report in Table 1, the more robust finding is that the distribution of investment is skewed and kurtotic with a mass around 0 and a fat right tail. Identifying inaction precisely at the micro-level is difficult because in practice there is substantial heterogeneity in capital assets with associated heterogeneity in adjustment costs. For example, buying a specific tool gets lumped into capital equipment expenditures in the same way as retooling the entire production line. The former presumably has little or no adjustment costs while the other is subject to presumably high adjustment costs. The implication is that in pursuing MLE or Euler equation estimation using the actual micro-data a researcher must define "inaction" without observing the underlying capital heterogeneity. Specifically, should a researcher identify inaction as literally zero investment or small investment expenditures that reflect "buying a wrench?"28 In contrast, we exploit robust moments of the investment and profit distributions. While we believe the moments we match are robust to capital heterogeneity, this discussion reminds us of the limitations of even high-quality micro-data such as the LRD. The class of adjustment cost models we focus on in this paper are best interpreted as applying to variations in capital expenditures that are relatively homogeneous in type. It is unlikely that we will have a rich longitudinal micro data-set on establishments with annual data on capital expenditures by detailed asset class; unobserved heterogeneity in capital is a feature of reality that estimation of investment dynamics with micro-data on plants needs to confront, regardless of the methodology used. Another strength of our approach is that analysis and estimation does not require direct and continuous access to the micro-data. Given that virtually all of the high-quality longitudinal micro data-sets are proprietary, it is very useful to have methods available that can take advantage of moments from the micro-data that can be analysed off-site. The research community at large has a much better chance of exploring alternative models of investment using the simulated method of moments approach given limited direct access to the micro-data. Our findings suggest there would be considerable value to statistical agencies producing summary measures of the distributions of micro-investment behaviour and its auto covariance and cross-covariance with other micro-measures. In the end, there are many dimensions to improve the match between the models we specify and estimate and the full richness of the actual micro-data. Despite these limitations, we have identified features of the micro-data that can only be reconciled with models that contain both convex and non-convex adjustment costs. In particular, a modest convex component and substantial transaction and disruption costs are required to capture key features of micro-investment. 28. In related earlier work, CHP took a stand on this by defining investment spikes of investment greater than 20% as the investment that is subject to fixed adjustment costs. They investigated the robustness of this admittedly ad hoc threshold but noted that this was a limitation of their analysis. Note as well that Goolsbee and Gross (1997) consider a model with heterogeneous capital goods. © 2006 The Review of Economic Studies Limited Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. COOPER & HALTIWANGER ON THE NATURE OF CAPITAL ADJUSTMENT 631 APPENDIX In this section, we discuss the measurement of key variables and the details and robustness of estimation for the profit function. Real variable profits are measured as revenue less variable costs (labour and material) deflated by the gross domestic product implicit price deflator for consumption. We make the assumption that measured real variable profits provide an estimate of n{A, K) = ).AK° in the model. The question then is how adjustment costs in the model are captured in the measured real variable profits or other cost measures in the data. We think it is reasonable to assume that some part of capital adjustment costs are reflected in real variable measured profits either due to the disruption of productivity and/or perhaps equivalently that some of the labour that might have normally been used for production is used for installing capital. It is precisely these arguments that motivate us to consider including an adjustment cost factor via /. We also think it is reasonable to assume that some adjustment costs take the form of purchased services or contract work that are not captured in measured real variable profits. It is precisely these arguments that motivate us to include adjustment costs that are additivcly separable from real variable profits and may be either convex or non-convex in nature—hence we consider adjustment costs in the form of both y or F. In terms of measurement in the LRD, there are no annual data on such purchased services or contract work and so we cannot directly measure these costs. However, we can infer the presence of these costs through the dynamics of investment. As described in the text, we estimate the parameters of the real variable profit function using the quasi-differenced (10). In practice, we use a transformed version of (10) taking advantage of assumptions that the production function is Cobb-Douglas and that the demand function has constant elasticity. Specifically, we assume that the plant faces an inverse demand function given by p = y~'< and so has a revenue function of R(v) = v1-''. Assuming that the production function is given by y = AK" and that K is predetermined, L is the variable factorts). and w is the price of the variable factor(s). The equations that follow are based on one variable factor for exposilional purposes but extend easily to multiple variable factors and in our case there are two: labour and materials. Optimization over the variable factor yields a revenue function R(K, A,w), a profit function n(K,A,m), and variable factor demand of L = h(K. A,m). The implied revenue and profit functions are given by where « = «(I — >;) and 0=0(1- »/). So the coefficient on K in both the revenue and profit functions is the same, given by I) = -^-r. Moreover, the properties of the shocks to revenue and profits are the same up to a factor of proportionality. l-y> So the estimation strategy is to estimate 0 from either a quasi-differenced profit or revenue regression on the capital stock. The latter seems preferred since there is potentially less measurement error involved. There are a small number of observations with negative measured real variable profits but by construction there are no businesses with negative real revenue. While it may be that negative real variable profits are possible we suspect that this largely reflects measurement error. To explore this issue we estimated the log-linear quasi-diffcrenccd real revenue function on all observations and the log-linear quasi-differenced real profit function on only those observations with non-negative real variable profits. We obtained very similar estimates of 0 using both approaches. In the analysis in the paper we use the estimate of 0 from the real revenue quasi-differenced estimation, but this is not critical for the reported results. To obtain the profit shocks, we use the estimate of H and use (12) to infer the shock. We decompose the shock into aggregate and idiosyncratic components (the aggregate shock is the first moment of the profit shock in any given year, and the idiosyncratic shock is the residual after controlling for year effects). We then estimate the properties of the aggregate and idiosyncratic shocks (both the degree of first-order autocorrelation and the variance of the innovations). We note again that the properties of these shocks are quite similar across alternative ways of estimating 0. To provide a sense of the robustness of both the estimates of 0 as welt as the properties of the shocks, we performed a number of cross-checks. One cross-check that we perform is that the above procedures yield two alternative estimates of p,-.—one estimate is as described from inferring the shock from the profit function and the second is from the estimation of the quasi-differenced revenue function. These two estimates of p,: are very similar to each other providing support for this estimation strategy. As a second cross-check, we explored the properties of the innovations to this first-order process. We found that the innovations to the AR( 1) representation of i:if had a serial correlation very close to 0 (-0 OS). In addition, as a check on the robustness of our estimate of li. we also estimated AR( 2) specifications with appropriate second quasi-differencing in our estimation of 0. We found (i was 0-60 when we assumed an AR(2) compared to the estimate of" 0-592 reported in the text with (he AR( I) specification. There was essentially no improvement in the overall lit of the model with an AR(2) specification. Interestingly, even for the ordinary least squares estimation we obtained an estimate of 0 of 0-591. Thus, the estimate of 0 is very robust. As noted above, once we have an estimate of 0 (which is (11) (12) © 2006 The Review of Economic Studies Limited Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 632 REVIEW OF ECONOMIC STUDIES apparently very robust), the properties of the aggregate and idiosyncratic shocks are well captured by an AR(1) process (e.g. the implied innovations are serially uncorrected). We note in closing that is it not uncommon in the firm-level literature to assume a first-order process for the underlying shocks. For example, the Olley and Pakes (1996) and Levinsohn and Petrin (2000), hereafter LP, methods for estimating production functions are based on the assumption of first-order Markov processes. A key difference between the approach in their papers and that pursued here is that they use a more general first-order Markov process. For example, LP specify the first-order process as the productivity shock in period t being a polynomial function of the productivity shock in / — 1. As a further point of comparison, unlike these papers we are not seeking to identify the factor elasticities of both variable and quasi-fixed factors of production. Rather, we seek to identify the coefficient on the quasi-fixed factor in the profit function after having already maximized out the variable factors of production. As such, we are similar to LP's "second stage" where they identify the coefficient on capital. In LP's second stage, they employ a GMM approach with lagged values of inputs as instruments in a manner similar to that employed here (although again with the polynomial specification for the first-order process their estimation is more complex). Acknowledgements. The authors thank the National Science Foundation for financial support. Andrew Figura, Chad Syversons, and Jon Willis provided excellent research assistance in this project. We are grateful to Shutao Cao for his comments on the final version of this paper. Comments and suggestions from referees and the editor of this journal are gratefully acknowledged. We are grateful to Andrew Abel, Victor Aguirregabiria. Ricardo Caballero, Fabio Canova, V. V. Chari, Jan Eberly, Simon Gilchrist, George Hall, Adam Jaffe, Patrick Kehoe, John Leahy, David Runkle, and Jon Willis for helpful discussions in the preparation of this paper. 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