Investment spikes: New facts and a general equilibrium exploration

The debate is whether plant-level investment lumpiness has aggregate consequences; the paper contributes new facts about spikes, a recalibration of Thomas’s model to match them, and a re-examination of the irrelevance result under that calibration. The authors set out the two sides directly, quoting Caballero’s Handbook survey that “it turns out the changes in the degree of coordination of lumpy actions play an important role in shaping the dynamic behavior of aggregate investment,” and Thomas’s contrary finding that “in contrast to previous partial equilibrium analyses, [the] model results reveal that the aggregate effects of lumpy investment are negligible. In general equilibrium, households’ preference for relatively smooth consumption profiles offsets changes in aggregate investment demand implied by the introduction of lumpy plant-level investment.” They note that this “irrelevance result” prompted Prescott to argue that “partial equilibrium reasoning to an inherently general equilibrium question cannot be trusted.”

U.S. Census establishment tabulations for 1972-1998 and a Chilean plant census for 1981-1999, capital-weighted, with spikes defined as investment above 20 percent of beginning-of-period capital. The U.S. capital expenditure data come from the Annual Survey of Manufactures with capital stocks built by perpetual inventory following Becker et al.; importantly, the authors “do not have access to the underlying micro data for the U.S. Census, but instead have tabulations that group plants according to their current investment rates.” The Chilean data are a census of manufacturing plants with ten or more employees from the National Statistics Institute, “an unbalanced panel which has on average 1780 plants per year, from 1981 to 1999.” Three measurement choices are argued for. Very small nonzero investment rates are grouped with exact zeros, on the reasoning that they “represent some sort of maintenance or replacement investment (for which the fixed cost presumably does not apply).” The 20 percent spike threshold is adopted for comparability, with the 35 percent threshold as a check – the two capital-weighted U.S. series correlate at 0.95. And observations are capital-weighted: “at a sufficiently fine level of aggregation every decision is lumpy… Conversely for the entire economy there are never any zeros and spikes are rare,” so equal weighting makes measured lumpiness an artefact of the size distribution; capital weighting partly offsets this, has the bonus that “a capital-weighted average of investment rates delivers a measure that equals the aggregate investment in the sample divided by the total capital in the sample,” and matches the interest in general equilibrium effects that “operate through prices [and] depend on aggregate indicators of lumpiness.” Trends are removed by regression on a linear trend, with the authors noting an HP filter “delivers very similar results for all of our findings.”

Spikes are strongly procyclical and near-zeros strongly countercyclical, in both countries. The correlation between the capital-weighted spike share and the aggregate investment rate, both detrended, “is 0.87 for the US sample and 0.96 for the Chile sample,” and the correlation between capital-weighted near zeros and the aggregate rate “is -0.94 for the U.S. sample and -0.56 for the Chilean sample.” The authors note a difference between the countries flagged by Fuentes, Gilchrist and Rysman – the unweighted “percentage of establishments with exactly zero investment is two to three times higher in the Chilean sample” – and that capital weighting shrinks the measured zeros sharply, “as would be expected,” since “fewer large firms report literally zero investment,” while making much less difference to the spikes.

Almost all of the variation in the aggregate investment rate is variation in investment by spiking plants. Spikes account for about half the level of total investment in each country, but their share of the variation is much larger: the investment rate constructed for spiking firms “tracks the movements in the aggregate investment rate closely; the correlations between the de-trended series is 0.99 for each sample,” and “the share of variance of Itot/K accounted to by I20/K (as opposed to the residual) is 97 percent for the U.S. sample and 86 percent for the Chile sample.” The authors report an alternative exact three-way decomposition in a footnote which is somewhat less dramatic (0.964 for the U.S., 0.759 for Chile, with the remainder in a covariance term). The converse, they note, is that “there is little variation in total investment explained by the firms investing between zero and 20 percent,” so “for the purposes of modeling investment fluctuations it is critical to understand the timing of the investment spikes.”

Spike investment varies because the number of spiking plants varies, not because the size of a typical spike varies. Total investment by spiking plants is decomposed into investment per adjuster and the capital-weighted number of adjusters, an approach “analogous to the one proposed by Klenow and Kryvstov (2005) for studying price dynamics.” Shares are computed as covariance ratios that by construction sum to one: “for the U.S. sample ShareADJ20 is 0.87, while for the Chilean sample it is 0.925.” (The exact three-way decomposition gives 0.850 and 0.903 for the variance ratios themselves.) The authors confirm that “the dominant role of the extensive margin also appears when the threshold for identifying spikes is 35 percent instead of 20 percent” and under different detrending. This is the moment the paper subsequently asks a model to match.

The prevalence of spikes forecasts aggregate investment beyond what past investment and sales convey, and with the sign a fixed-cost model implies. The authors add the capital-weighted share of adjusters to “an otherwise standard accelerator type investment equation” of the kind repeatedly shown to win forecasting horse-races. “For the U.S. sample, the coefficients on both the first and second lags of ADJ20 are significant, whereas in the Chilean data, only the second lag is consistently significant,” and with a 35 percent threshold “both lags one and two are significant in both samples.” The key reading is the sign: “the estimated signs of the [coefficients] are all negative, suggesting that investment is depressed in the period after an investment surge. This correlation is to be expected based on fixed costs models (and would be of the opposite sign if the past ADJ20 variable was standing in for productivity shocks or other factors that raise investment demand).” On magnitude, the standard deviation of the spike variable is 0.046 in the U.S. and 0.093 in Chile, against investment-rate standard deviations of 0.017 and 0.054; in the one-lag specification the paper states that a one standard deviation move in the spike variable shifts the predicted investment rate by 0.7 of a standard deviation in the U.S. and 0.57 in Chile. (As printed, that sentence describes the move as an increase, which sits oddly with the negative coefficients reported two paragraphs earlier and with the paper’s repeated statement – including in the conclusion – that investment is depressed after a surge; the direction the paper argues for throughout is the depressing one.)

A DSGE model in which a unit measure of plants with decreasing-returns technology must pay a randomly drawn fixed cost, in units of labour, to adjust capital at all. Output is Cobb-Douglas in capital and labour with the two exponents summing to less than one, there is no entry or exit, labour is freely variable, and the fixed cost is i.i.d. across time and plants from a distribution with finite support and maximum B. Adjusters “bear no marginal adjustment costs: they can buy or sell capital at price 1.” The model’s tractability comes from symmetry: all firms that choose to invest at a given date pick the same new capital level, so “firms are distinguished by the time since their last investment.” Investment follows a cutoff rule – in each vintage there is a threshold fixed cost below which a firm invests – and the combination of a fixed depreciation rate with a finite upper bound on the fixed cost “guarantees that all firms will eventually find it optimal to invest,” delivering a maximum vintage J. TFP follows an AR(1) around a deterministic trend. Preferences are log consumption with linear disutility of labour, and “when the upper bound of fixed costs, B, is set to 0, all firms adjust their capital each period… there is a representative firm, and the model collapses to a standard RBC model with decreasing return to scale.” The model is calibrated annually “because the plant-level evidence is based on annual surveys,” and solved by log-linearisation around the steady state, which the authors note is advantageous because the state space includes the whole cross-sectional distribution of capital.

Thomas chose the fixed cost to match average spike prevalence, not the cyclical decomposition, and the resulting model gets the variance shares badly wrong. B was chosen to match two Doms-Dunne facts – “in the average year, 8 percent of plants raise their real capital stocks by 30 percent or more” and “these plants account for 25 percent of aggregate investment” – so it is unsurprising that “the model also matches the prevalence of spikes in our sample.” But “spikes only account for about 62 percent of the total variance of investment and the extensive margin accounts for only 51 percent of the variance of spikes; in the data both these percentages are roughly 90 percent.”

Compression means many firms drawing nearly the same fixed cost, which makes it cheap for a shock to move many of them from inaction to action. The authors’ intuition: “increasing the number of plants doing positive investment requires marginal plants to switch from inaction to action; this decision depends on the fixed costs for the indifferent plants. If marginally inactive plants face the same fixed cost as marginally active plants, increasing the number of plants investing is inexpensive.” Under a uniform distribution – Thomas’s choice, following Caballero and Engel – “increasing the number of plants investing requires activating plants that have substantial differences in the fixed costs they are facing… In this case it will be efficient to rely more on intensive adjustment.” Substituting a compressed distribution (keeping average adjustment costs comparable) raises the extensive margin’s share “to 92.6 percent and the variance of Itot/K due to I20/K rises to 99.9 percent.” The authors isolate the mechanism cleanly: re-running with a uniform distribution whose B is chosen so that “the average adjustment costs faced by firms is the same,” the extensive-margin share “drops back towards the level in the baseline Thomas specification.”

No, and the paper tests this specifically. Two alternative distributions are examined. One “has much more variance than the one we use in our preferred calibration, since about 50% of firms draw a fixed cost that is roughly uniformly distributed between 0.03 and 0.06”; the other “has only one point of compression rather than two,” with virtually all firms drawing B, which puts the model “closer to the first generation of Ss models.” Under both, “the extensive margin remains dominant.” The authors’ conclusion is stated with an explicit confidence level: “based on other cases we considered we are convinced that compression is a necessary ingredient for delivering substantial extensive adjustment, but the exact nature of the compression is not critical,” and they single out the high-variance case as “especially important because [it shows] that allowing for not trivial heterogeneity does not necessarily overturn the basic intuition.”

Because the original calibration implies adjustment costs an order of magnitude smaller than the empirical estimates, and a profit function flatter than later estimates support. Under Thomas’s calibration “total expenditure due to adjustment costs is roughly 1/5 of one percent of total investment spending,” which the authors call small “if we think of the costs of the planning, budgeting, and committee work that accompany most investments,” and against cases like “the re-tooling of a factory, or the temporary closure of a retail store to redesign it.” Cooper and Haltiwanger’s simulated model puts adjustment spending at 0.91 percent of capital against investment of 12.2 percent of capital, so “adjustment costs average roughly 7.5 percent of investment; in other words, they find adjustment costs roughly 40 times the size assumed by Thomas”; Abel and Eberly find 1.1 to 9.7 percent. On curvature, “subsequent to Thomas’ paper a large empirical literature has estimated this curvature to be between 0.5 and 0.7, markedly lower than one.” The authors note the theoretical stake plainly: “it is hardly surprising that lumpiness is quantitatively irrelevant when fixed costs are small.”

Raising B alone makes firms wait far too long; adding curvature gives them a stronger reason to adjust, so the two are combined. With B at 0.03, adjustment spending reaches “nearly two percent of investment” and the maximum vintage rises to 24, “because as the costs become higher, firms tolerate larger deviations from their target capital before adjusting.” Doubling B to 0.06 pushes the maximum vintage to 45 and adjustment spending “to just over three percent of investment,” but then “roughly 96 percent of the plants do not invest.” Setting returns to scale to 0.6 “doubles the resources spent on adjustment costs, and reduces the maximum vintage J, so that firms adjust faster. The extensive margin remains dominant.” The preferred calibration combines B = 0.06 with returns to scale of 0.6. The authors are explicit that “this calibration is not fully optimized” and that better matching is likely possible, while maintaining that the two lessons – compression drives the extensive margin, and high fixed costs plus curvature give non-trivial adjustment spending – would survive.

Under the preferred calibration nearly all investment is spikes, because the model has no maintenance motive; adding one fixes the cross-section without changing the dynamics. Maintenance is modelled as a “breakdown shock” hitting a fraction of plants at the start of the period, which “must invest immediately 8% of their capital to compensate for the capital destruction,” free of the fixed cost; the fraction is set to 30 percent. This brings spikes down to 77 percent of total investment and “does not affect our other results noticeably (and it does not affect the impulse responses).” The authors draw a methodological point from this in a footnote: since maintenance investment “will not change over the business cycle it will have almost no effect on aggregate dynamics,” so “calibrating the model to match the cross-sectional distribution of investment rates is not informative about the business cycle behavior” – which is why they target the capital-weighted business-cycle statistics of the cross-section rather than its average properties.

Extremely strong, and robust within Thomas’s own parameter family. Comparing the fixed-cost model with the zero-fixed-cost RBC model, “the two models are virtually indistinguishable, with the two lines sitting on top of each other. The response on impact of the fixed cost model is about 99.8 percent of the response of the RBC model.” Moreover “this result holds for many variations of parameter values. For instance, changing the elasticity of labor supply or the source of shocks does not affect the result. Increasing the level of fixed costs (B), while maintaining a uniform distribution, also makes little difference: for instance, when B is multiplied by a factor of 10… the impact response of the fixed cost model is 98 percent of the response of the RBC model.” This is the contrast with partial equilibrium, where fixed-cost models typically generate both smoother aggregate investment and oscillatory “echo” dynamics.

The response to a productivity shock differs qualitatively but modestly; the response to a shift in the cross-sectional distribution differs a great deal. For the TFP shock, “the response is initially smaller in the fixed cost model: on impact the response of the fixed cost model is only 89 percent of the response of the RBC model,” reflecting smoothing, and “the fixed cost model exhibits a noticeable hump 12 periods after the shock,” an echo effect caused by the initial surge shifting the distribution toward recent vintages that are unlikely to invest. The timing follows from the hazard rate, which “is initially steeply convex: the alphas (probability of adjustment) are very small for the first vintages before rising noticeably after 12 periods.” The authors state plainly that “the quantitative differences between the responses of the two models to a TFP shock are modest.” The larger divergence comes from directly perturbing the cross-sectional distribution – a stand-in for the effects of an uncertainty shock in Bloom’s sense or an investment tax credit – by putting more firms in the first two vintages. There the RBC model shows “the usual, monotonic, smooth convergence,” while the recalibrated model shows a smaller investment response (except in the first two periods) and “a magnified ’echo effect’ when firms which had invested recently finally re-invest after 8 to 11 periods.” Crucially, the same experiment run in the baseline Thomas model gives “essentially identical predictions” to the RBC model – “this equivalence for us is proof that general equilibrium effects are not the only reason why Thomas found no aggregate effect of fixed costs.”

They concede general equilibrium matters and that Thomas’s broader point stands; what they deny is that irrelevance is generic. “We conclude, therefore, that although general equilibrium attenuates the differences between the fixed cost model and the RBC model, it does not eliminate these differences. In other words, the irrelevance result is not a generic finding that comes from the general equilibrium, but rather a result that depends on the details of how the model is calibrated, especially regarding the production side.” In the conclusion: “we agree with Thomas that there can be substantial differences between the importance of lumpiness in a GE models and partial equilibrium models. However, many have gone farther and concluded that GE makes fixed costs to investment completely irrelevant for the business cycle. Both our empirical and theoretical work shows this conclusion is premature.” They immediately add: “Given the currently available information our calibration is reasonable, but we recognize much more work needs to be done in this respect to determine how these models should be estimated and calibrated.”

Idiosyncratic productivity shocks, preferences other than log utility, and the possibility that partial-equilibrium panel estimates mislead. On Khan and Thomas’s extension with idiosyncratic productivity shocks, which finds no significant aggregate effect, the authors note it uses relatively low adjustment costs, modest curvature, a uniform fixed-cost distribution and log-normal shocks, so “the marginally inactive firms will not be similar to the marginally active ones.” Their own position is explicitly a conjecture: “we conjecture that our results would hold if the idiosyncratic productivity shocks do not eliminate the compression associated with our parameterization of the fixed costs, but would go away if they did.” They also relay, without dismissing it, Khan and Thomas’s point that general equilibrium feedbacks affect plant-level dynamics, “which would imply that the panel data estimates from partial equilibrium models that we use may be misleading.” On preferences, “all of these results are obtained with log utility,” kept deliberately the same as Thomas’s “since the dispute is about whether general equilibrium offsets are central to this debate”; allowing a higher intertemporal elasticity of substitution, following Bachmann, Caballero and Engel, “we find also more smoothing than in our baseline.” They also note that because their solution is log-linear, it cannot produce the time-varying elasticity of aggregate investment to TFP that Bachmann et al. emphasise.