Firm-specific capital, nominal rigidities and the business cycle

That the assumptions macro modellers use to generate inflation inertia are “either implausible on a priori grounds or directly in conflict with micro data” (Section 1, Fed IFDP 990, p. 2). In many New Keynesian models firms index non-optimised prices to lagged inflation, and such models “account for inflation inertia by assuming that firms re-optimize their prices every six quarters or even less often” – the authors cite Eichenbaum and Fisher’s estimate of roughly six quarters and note that “Smets and Wouters’ (2003) estimated model implies that firms reoptimize prices on average once every nine quarters.” Models without lagged indexation, when estimated, imply price changes “once every two years or less often.” The conflict is with Bils and Klenow, Golosov and Lucas, and Klenow and Kryvtsov, “who argue that firms change prices more frequently than once every two quarters” – Golosov and Lucas calibrate to once every 1.5 quarters.

Inflation is inertial not because firms rarely reset prices but because, when they do reset, they change them by only a small amount – and they do so because each firm’s short-run marginal cost curve rises in its own output (Section 1, pp. 2-3). “In our model, a firm’s capital is pre-determined and can only be changed over time by varying the rate of investment. These properties follow from our assumption that capital is completely firm-specific.” The argument then runs through the firm’s decision: “consider a firm that contemplates raising its price. The firm understands that a higher price implies less demand and less output. A lower level of output reduces marginal cost, which other things equal, induces a firm to post a lower price. Thus, the dependence of marginal cost on firm-level output acts as a countervailing influence on a firm’s incentives to raise price.”

Not much, and the authors say so themselves (Section 1, p. 3). “Anything, including firm-specificity of some other factor of production or adjustment costs in labor, which causes a firm’s marginal cost to be an increasing function of its output works in the same direction as firm-specificity of capital. This fact is important because our assumption that the firm’s entire stock of capital is predetermined probably goes too far from an empirical standpoint.” The load-bearing property is the upward-sloping firm-level marginal cost curve, not firm-specific capital as such.

The two models’ log-linearised equilibrium equations differ only in how structural parameters map into the reduced-form inflation-marginal cost coefficient; parameterised in that coefficient they are identical for aggregate data (Section 1, pp. 3-4). “The form of this equation is identical in both models: the change in inflation at time t is equal to discounted expected change in inflation at time t + 1 plus a reduced form coefficient … multiplying time t economy-wide average real marginal cost. The difference between the two models lies in the mapping between the structural parameters and” that coefficient. In the homogeneous capital model it depends only on the discount rate and the fraction of firms re-optimising within the quarter; in the firm-specific model it also falls with the cost of varying capital utilisation and with the elasticity of the firm’s demand curve – “the more elastic is a firm’s demand, the greater is the reduction in demand and output in response to a given price increase. A bigger fall in output implies a bigger fall in marginal cost which reduces a firm’s incentive to raise its price.” Two consequences are drawn: “we can estimate the model in terms of [the coefficient] without taking a stand on whether capital is firm-specific or homogeneous,” and “we cannot assess the relative plausibility of the homogeneous and firm-specific capital models using macro data.” The authors flag the scope condition in a footnote: “In non-linear framework, however, it is not true that the solutions to the homogeneous and firm-specific capital models are observationally equivalent with respect to macro data,” citing Levin, Lopez-Salido, Nelson and Yun.

Estimating the parameters that determine whether firm-specific capital actually does the work, rather than conditioning on assumed values (Section 1, pp. 5-6). That firm-specific capital can rationalise more frequent price re-optimisation “was first demonstrated by Sbordone (1998, 2002) and further discussed by Gali, Gertler and Lopez-Salido (2001) and Woodford (2003),” in papers where the firm’s capital stock is fixed; Woodford (2005) allowed investment. But “whether firm specific capital actually does rationalize a lower estimate of [the Calvo parameter] depends critically on the other parameters characterizing firms’ environments. To the extent that capital utilization rates can easily be varied, the assumption of firm specific capital loses its ability to rationalize low values … A maintained assumption of the papers just cited is that firms cannot vary capital utilization rates.” Similarly, those papers condition on a particular assumed demand elasticity. “The key contribution of this paper is to estimate the key parameters governing the operational importance of firm specific capital and to assess the importance for firm specific capital in an estimated dynamic stochastic general equilibrium model.”

By a limited-information strategy matching model impulse responses to those from an identified ten-variable VAR on quarterly US data for 1982:1-2008:3 (Sections 3-4, pp. 14-17). The vector contains the growth rates of the relative price of investment, labour productivity and the GDP deflator; capacity utilisation; log hours; the log real wage less log labour productivity; the consumption and investment shares of GDP; the federal funds rate; and log velocity of MZM. Identification uses restrictions satisfied by the model: only neutral and capital-embodied technology innovations affect labour productivity in the long run; only capital-embodied technology innovations affect the relative price of investment in the long run (following Fisher); and monetary policy shocks move the interest rate contemporaneously but not aggregate quantities or the price of investment. Most data come from FRED, with the investment price series from Fisher (2006). The authors justify using MZM on two grounds: it “is constructed to be a measure of transactions balances, so it is a natural empirical counterpart to our model variable,” and “our statistical procedure requires that the velocity of money is stationary. The velocity of MZM is reasonably characterized as being stationary,” which is “more problematic for the velocity of aggregates like the base, M1 and M2.”

Yes, with an explicit acknowledgement that the asymptotic serial-correlation tests reject and a bootstrap argument that they over-reject (Section 4.2, p. 17). With four lags, “the Akaike, Hannan-Quinn and Schwartz criteria support a choice of q = 2, 2, 1.” The multivariate Portmanteau statistics at lags 4, 6, 8 and 10 are 262, 475, 680 and 880, which “all have a p-value very close to zero, indicating a rejection of the null hypothesis. However, we find evidence that the asymptotic sampling theory rejects the null hypothesis too often. When we simulate the Q statistic using repeated artificial data sets generated from our estimated VAR, we find that the p-values of our Q statistics are 97, 87, 90 and 95 percent.” On that basis “we do not strongly reject the null hypothesis that the disturbance terms in a VAR with p = 4 are serially uncorrelated.”

Standard monetary transmission patterns plus two technology shocks with distinct signatures (Section 4.2, pp. 17-18, Figures 1-3). After a one-standard-deviation monetary policy shock (roughly 30 basis points), the effect on money growth and the interest rate is complete within about a year; there is a significant liquidity effect; “inflation responds very weakly to the policy shock”; output, consumption, investment, hours and capacity utilisation all display hump-shaped responses peaking about a year after the shock, except hours, which peak after about two years, with the peak output response about 0.15 percent; velocity co-moves with the interest rate; and the real wage does not respond significantly. A positive neutral technology shock produces a persistent output rise peaking at roughly 0.35 percent, with hours, investment and consumption rising – though “these rises are only marginally statistically significant” – and a sharp initial fall in inflation. A positive capital-embodied shock significantly raises output, hours, capacity utilisation, investment and the funds rate, cuts the price of investment by roughly 0.2 percent on impact with an ongoing decline, and produces “a marginally significant decline in real wages.”

Roughly 60 percent of the cyclical variance of aggregate output, with capital-embodied technology the largest single contributor (Section 4.3, pp. 18-19, Table 1). Business cycle frequencies are defined as components with periods of 8 to 32 quarters, and contributions are computed from the estimated VAR’s spectral density using the Christiano-Fitzgerald technique, with bootstrap standard errors; HP-filter analogues “yielded essentially the same results.” “The three shocks together account for a substantial portion of the cyclical variance in the aggregate quantities. For example, they account for roughly 60 percent of the variation in aggregate output, with the capital-embodied technology shock playing the largest role. Indeed the capital-embodied technology shock is the largest contributor to the cyclical variation in all of the variables included in the VAR. Intriguingly, the capital embodied technology shock accounts for nearly 30% of the cyclical variation in the real wage, a variable whose cyclical variation is typically difficult to account for empirically.”

Preference/technology constants and steady-state growth rates are fixed; the frictions and shock processes are estimated (Section 5.1, pp. 19-21, Tables 2-3). Fixed: a discount factor implying a 3 percent annualised steady-state real rate; a capital share of 0.36; quarterly depreciation of 0.025 (10 percent annual, roughly the Christiano-Eichenbaum estimate); the fixed cost parameter set to make steady-state profits zero; a steady-state wage markup parameter of 1.05 as in CEE. For the technology growth rates the authors depart from their own estimation sample and say why: “We use data over the sample period 1959II - 2001IV … to estimate the parameters” for capital-embodied technology growth and money growth, because “if we use the sample period 1982:1-2008:3, then the implied point estimate of [neutral technology growth] is less than one, a value that seems implausible to us. It seems reasonable to extend the sample back in time because the value of [neutral technology growth] should not be affected by any change in the monetary policy regime that occurred in the early 1980’s.” The estimated set covers the steady-state price markup, the wage Calvo parameter, the inflation-marginal cost coefficient, the capital utilisation cost parameter, habit, the investment adjustment cost curvature and the money demand elasticity, plus the shock processes.

An inflation-marginal cost coefficient of 0.014, wage re-optimisation once every 4.5 quarters, habit of 0.76, and an investment elasticity of 0.66 with respect to the price of installed capital (Section 5.1, pp. 21-22). “Our point estimate of [the wage Calvo parameter] implies that wage contracts are re-optimized, on average, once every 4.5 quarters.” The estimated coefficient implies a Calvo price parameter of 0.896 under homogeneous capital, “This implies that firms re-optimize prices roughly every 9.36 quarters … This value is much larger than the value used by Golosov and Lucas (2007).” Under firm-specific capital the same estimate implies “firms re-optimize prices on average once every 1.8 quarters. So the assumption that capital is firm-specific has a very large impact on inference about the frequency at which firms re-optimize price.” The elasticity of capital utilisation with respect to the rental rate is 0.08 percent, “larger than the value estimated by CEE (2005) and indicates that it is relatively costly for firms to vary the utilization of capital.” Habit at 0.76 is “reasonably close to the point estimate of 0.66, reported in CEE (2005).” The investment elasticity with respect to a one percent temporary rise in the price of installed capital is 0.66, with the authors noting that “the more persistent is the change in the price of capital, the larger is the percentage change in investment,” because adjustment costs make agents forward-looking. A caveat on the markup: the lower bound of unity binds on the steady-state price markup, so it is set to 1.01, but “the estimation criterion displays very little curvature” with respect to it, and the paper reports re-estimates at 1.05 and 1.20.

Substantially less volatile but more persistent than Solow-residual estimates (Section 5.1, pp. 22-23). The estimated processes are an AR(1) in the growth rate of capital-embodied technology with coefficient 0.55 (s.e. 0.12) and innovation standard deviation 0.21 percent (s.e. 0.04), and an AR(1) in neutral technology growth with coefficient 0.42 (s.e. 0.27) and innovation standard deviation 0.17 percent (s.e. 0.08). “A one-standard deviation neutral technology shock drives [neutral technology] up by 0.17 percent in the period of the shock and by 0.29 percent in the long run. A one-standard-deviation shock to embodied technology drives [it] up by 0.21 percent immediately and by 0.47 percent in the long run.” Against Prescott (1986), who found a near-random-walk shock with a growth rate standard deviation of roughly 1 percent after renormalisation, “our estimates imply that the unconditional standard deviation of the growth rate of neutral technology is roughly 0.19 percent. So we find that technology shocks are substantially less volatile but more persistent than those estimated by Prescott.” The authors give two candidate reasons: “Prescott’s estimate of technology confounds technology with variable capital utilization,” and the analyses use different data and identifying assumptions.

Well overall, with specific failures the authors enumerate (Section 5.2, pp. 23-24). For the monetary policy shock, “most (but not all) of the model responses lie within the two-standard deviation confidence interval computed from the data. This is true even though firms in the firm-specific capital version of the model change prices on average once every 1.8 quarters.” Output responds persistently, positive for over three years, peaking at about a year; the interest rate declines sharply then returns within a year, though “the model does not account for the overshooting pattern of the interest rate in the data,” nor the overshooting in transaction balances; the real wage stays essentially unaffected, as in the data; consumption, investment and hours show hump-shaped rises consistent with the VAR; velocity falls, but “this fall is nearly as strong as the VAR based response”; and “capacity utilization in the model rises by only a very small amount, and understates the estimated rise in the data.” For the neutral technology shock the model accounts for the rises in output, hours, investment, consumption and the real wage “however, the model does not capture the extent of the fall in inflation that occurs immediately after the shock.” For the capital-embodied shock the model “does very well … except that it does not account for the rises in capacity utilization and the federal funds rate,” and money growth is high relative to the VAR. On that last point the authors run a diagnostic: holding money growth at its steady-state level, “output and hours worked rise by much less, while inflation falls compared to what happens when monetary policy is accommodative. We conclude that the model requires accommodative monetary policy to match the expansionary effects of a positive capital embodied technology shock.”

Because the data on inflation changes and marginal cost show at best a weak relation, and a steeper line fits drastically worse (Section 6, pp. 24-25, Figures 4a-4b). Plotting the change in inflation less its discounted expected future change against the log of marginal cost (measured by labour’s share in GDP, which they note “is the correct measure if fixed costs are zero” and “is approximately correct here, since our estimate of [the fixed cost] is close to zero”), “the distribution … is at best weakly related to the magnitude of” marginal cost. The line with slope equal to the point estimate of 0.014 “passes through the central tendency of the data,” while a line with slope 0.68 – the value that in the homogeneous capital model would imply re-optimisation once every 1.8 quarters – “leads to a drastic deterioration in fit.” The authors are careful about the inferential status of the plot: the residuals “cannot be used as a formal measure of model fit”; one should focus on the projection onto date-t information, “because then [the equation] implies that least squares consistently recovers the true value.” The projected version, Figure 4b, looks very similar.

As a statement about the size of the inflation-marginal cost coefficient rather than only about marginal cost dynamics (Section 6, p. 25). Solving the inflation equation forward makes the change in inflation a discounted sum of expected future marginal costs scaled by that coefficient. “This relation makes clear why many authors incorporate features like variable capital utilization and sticky wages into their models. These features can reduce the response of expected marginal cost to shocks. Relation (6.1) reveals another way to account for inflation inertia: assign a small value to [the coefficient]. The evidence in Figure 4a and 4b indicates that a small value of [the coefficient] must be part of any successful resolution of the inflation inertia puzzle.” The dilemma is then stated squarely: “to get the macro data right (i.e., a low [coefficient]) we must make assumptions about the frequency at which firms re-optimize prices that seem implausible in light of the micro data. In contrast, suppose we adopt the more plausible assumption that firms re-optimize prices on average once every 1.8 quarters. Then the homogeneous capital model implies [a coefficient of] 0.68. But this means that the model gets the macro data wrong.”

The cross-firm distribution of production after a monetary policy shock, which the homogeneous capital model gets implausibly wrong (Section 7, pp. 28-29, Figures 6-7). Starting from a symmetric steady state with an expansionary shock in period 1, firms are sorted by when they last re-optimised. Under homogeneous capital, in period 4, “a small fraction of the firms are producing a disproportionate share of the output. Indeed, roughly 70% of the firms who did not re-optimize their prices in periods 2, 3 and 4 produce 100 percent of output. The remaining firms effectively shut down. A key factor driving this result is the high elasticity of demand for a firm’s output … in the estimated benchmark model.” Under firm-specific capital, “the dramatic degree of inequality of production associated with the homogeneous capital model no longer obtains. Still, there is some inequality”: average production by group in period 4 is 1.8, 1.3, 1.0 and 0.8 for firms that last optimised in periods 1 through 4, with Gini coefficients of 0.12, 0.15 and 0.26 in periods 4, 8 and 16. Raising the steady-state markup to 1.05 softens the homogeneous model’s implication – the 70 percent of firms then produce “only a bit larger than 70 percent” of output – but at that markup the homogeneous model still implies re-optimisation once every 9 quarters against roughly 3 quarters in the firm-specific model, “on this basis we would still prefer the firm specific capital model.” The authors also record a shortcoming of their own benchmark: it “implies that firm-level output is too volatile, relative to firm-level prices.”

As one candidate among several, with the adjudication explicitly left to micro data (Section 1, pp. 5-6). The related mechanisms they list are firm-specific labour (Woodford), sector-specific labour (Gertler and Leahy), strategic complementarities from a demand elasticity increasing in the firm’s price (Kimball; Eichenbaum and Fisher), cross-sector heterogeneity in price-change frequency (Bils and Klenow; Carvalho; Nakamura and Steinsson), intermediate inputs (Basu; Huang), rational inattention (Sims; Mackowiak and Wiederholt; Woodford) and sticky information (Mankiw and Reis; Reis). Their own verdict is deliberately unfinished: “The merits of these alternative propagation mechanisms is a subject of an ongoing, vigorous debate. A detailed assessment of their empirical strengths is beyond the scope of this paper. It is clear however that the debate will be settled on the field of microeconomic data … While we emphasize the importance of firm specific capital in this paper, we leave open the possibility that the other propagation mechanisms discussed above may be at least as important.”

A strong preference for firm-specific capital, conditional on taking two-quarter price re-optimisation as the target (Section 8, pp. 29-30). “If we assume that capital is homogenous we can account for inflation inertia. However, this version of the model has micro implications that are implausible: firms re-optimize their prices on average once every 9.4 quarters and a monetary policy shock induces extreme dispersion in prices and output across firms. These considerations lead us to strongly prefer the firm-specific capital model.” The conditional is then made explicit: “in this paper, we have take[n] as given that firms re-optimize prices roughly once every two quarters. If we take the position that firms re-optimize prices on average roughly once a year, then we can reconcile the micro - macro pricing puzzle with a value of [the steady-state markup] around roughly 1.10,” and at such values “firm specific capital still plays a critical role.” The authors note the tradeoff this opens – higher markups mean less elastic demand, with “important implications for the micro implications of the model, like the volatility of firm level output” – and leave that to future research.