Time to Build and Aggregate Fluctuations

The thesis is that the assumption of multiple-period construction of new capital (“time to build”) is crucial for explaining aggregate fluctuations, and the paper builds a general equilibrium model, fitted to post-war U.S. quarterly data, whose co-movements and serial correlations are quantitatively consistent with the corresponding statistics for the U.S. economy (Abstract, Introduction, p. 1345). The approach integrates growth and business-cycle theory by using a representative, infinitely-lived household as in standard growth theory, but having that stand-in consumer value leisure as well as consumption, since fluctuations in employment are central to the business cycle – with the key modification that multiple periods are required to build new capital goods and only finished capital goods are part of the productive stock (Introduction, p. 1345).

The neoclassical technology (with capital and consumption instantaneously and costlessly convertible, as in Jorgenson’s work) implies the relative price of investment and consumption goods must be constant and that the shadow price of capital equals the price of new investment goods – contradicted by a sizable empirical literature finding a strong association between investment and a stock-market-derived shadow price of capital (Tobin’s q) that varies considerably over the cycle. The alternative, single-capital-good adjustment-cost technology is consistent with that q-investment association but implies equal short- and long-run supply elasticities (implausible given specialized, not-instantaneously-transferable resources), is contradicted by cross-sectional state data showing a -0.35 correlation between commercial-construction intensity and price per square foot, and implies (contrary to findings cited) that only current, not lagged, q should matter for investment (Sec. 2, pp. 1346-1348).

“Most destructive of all to the adjustment-cost technology… is the finding that the time required to complete investment projects is not short relative to the business cycle”: Mayer’s survey found an average (project-size-weighted) lag of twenty-one months between the decision to invest and project completion, and Hall found an average lag of about two years between project design and productive use (Sec. 2, p. 1348). Because the adjustment-cost technology effectively treats capital as available almost immediately, it cannot capture a construction lag this long, whereas a technology in which capital literally takes multiple periods to build can.

The technology specifies J stages of construction: sj,t is the number of projects j periods from completion, with the recursive laws of motion kt+1 = (1-delta)kt + s1,t and sj,t+1 = s(j+1),t for j = 1,…,J-1, so a newly initiated project (sJ,t) takes J periods to become part of productive capital kt (Sec. 3, “Technology,” pp. 1349-1350). Inventories yt are treated as a separate, additional type of capital entering the production function alongside k and n; besides the economic rationale that larger inventories economize on restocking labor and let firms run larger, more efficient production batches, the authors note a purely technical reason for including them: without inventories as a factor of production, it would be impossible to locally approximate the economy with a tractable quadratic objective and linear constraints (Sec. 3, p. 1350).

*Utility depends not on current leisure alone but on a(L)l_t, a distributed lag of current and past leisure choices governed by parameters a0 and eta, so that if a household allocated more time to nonmarket activity recently, the marginal value of additional current leisure is lower – a “Beckerian” household-production rationale in which time is allocated first to the highest-return nonmarket projects, leaving only lower-yield projects once recent leisure has been high (Sec. 3, “Preferences,” pp. 1350-1352). As supporting evidence, the authors cite the lumpiness of labor supply (vacations, extended entries and exits from the labor force not explained by wage movements), large seasonal variation in market hours, and Abowd and Ashenfelter’s failure to find a wage premium for jobs with more variable employment – all consistent with high intertemporal substitutability of leisure (p. 1351).

The technology shock Xt is the sum of a highly persistent “permanent” component (an AR(1) with autoregressive coefficient calibrated at 0.95 per quarter) and a “transitory” component; agents do not observe Xt directly when making labor-supply and new-investment decisions but only a noisy indicator equal to Xt plus a third independent shock (Sec. 3, “Information Structure,” eqq. 3.7-3.10, pp. 1352-1353). Labor supply and new-project decisions must therefore be based on the conditional expectation of the unobserved permanent and transitory components given the noisy indicator (computed via Kalman-filter-like updating formulas), while the subsequent consumption/inventory-investment decision can condition on the true current shock once aggregate output is observed and Xt can be deduced from it (Sec. 3, pp. 1352-1354).

Because the model has no externalities, the authors invoke the standard welfare-theorem result that the competitive equilibrium coincides with the Pareto optimum that maximizes the welfare of the (representative) stand-in consumer subject to the technology and information constraints, reducing equilibrium computation to a discounted dynamic programming problem (Sec. 3, “Equilibrium,” pp. 1354-1355). Because no closed-form solution to this dynamic program is available, the authors first compute the model’s deterministic steady state, then construct a quadratic approximation to utility (matched to be exact at points a fixed percentage deviation from steady state in each variable) around that steady state, yielding a linear-quadratic control problem with linear equilibrium decision rules whose implied second moments can be computed analytically (Sec. 4, pp. 1355-1358).

Rather than formally estimating the model’s parameters econometrically, the authors calibrate nearly all of them from steady-state national-accounts ratios and from findings in other applied literatures, leaving only about seven free parameters (with two severely constrained a priori) to be chosen by matching simulated second moments to the data (Sec. 5, “Model Calibration,” pp. 1360-1363). Given survey evidence of construction periods averaging near two years for plants, they set the construction period to J=4 quarters, with one-fourth of the investment value put in place each quarter; labor’s income share is set to theta=0.64, based on compensation plus proprietary income (adjusted for consumer-durable depreciation and indirect business taxes) as a share of GNP; the depreciation rate is set to 10 percent per year as a compromise across asset types; and the subjective discount rate is set to 4 percent per year, implying a steady-state capital-to-annual-output ratio of 2.4 (Sec. 5, pp. 1361-1362).

The model’s estimated output autocorrelations over six lags closely track the corresponding U.S. sample values (Table II); the model reproduces the U.S. economy’s pattern of investment being roughly three times as volatile as output and consumption about half as volatile, along with strong positive correlations of consumption and hours with output and a negative (though somewhat smaller in magnitude) correlation between the capital stock and output (Table III vs. Table IV) (Sec. 5, “Results,” pp. 1363-1365). The model’s smoothed (trend) output series also matches the number, spacing, and amplitude of peaks and troughs observed in the actual U.S. smoothed series over the 118-quarter sample (Sec. 5, “The Smoothed Series,” p. 1366).

A striking empirical fact is that, in percentage terms, cyclical output variation comes mainly from variation in hours of employment, not from variation in labor productivity or the capital stock – yet there is no large apparent movement in the real wage over the cycle that would obviously explain why market-goods consumption and leisure consumption move in opposite directions. The paper’s non-time-separable leisure specification is “of particular importance” here: with the calibrated parameters, the standard deviation of hours worked is 18 percent greater than that of productivity, matching the data’s qualitative pattern; but under the special case a0=1 (ordinary time-separable utility), with all other parameters unchanged, the standard deviation of hours becomes 24 percent less than that of productivity – the wrong pattern (Sec. 5, “Sensitivity of Results to Parameter Selection,” pp. 1366-1367).

Replacing time-to-build with a single-period capital good subject to a quadratic adjustment cost – even at the small magnitude parameter zeta=0.5, which implies only about a one-percent-of-GNP increase in gross investment for a one-percent increase in the relative price of investment goods – produces “grossly inconsistent” covariance properties: most output fluctuation now comes from productivity changes rather than hours (the standard deviation of hours falls to 0.60 versus 1.29 for productivity, the reverse of the U.S. pattern), consumption’s volatility nearly doubles, investment’s volatility is roughly halved, and capital-stock volatility falls by more than half (Sec. 5, “Importance of Time to Build,” pp. 1367-1368). The polar case of zeta=0 (equivalent to J=1, neither time to build nor adjustment costs) is also rejected: it produces a large positive correlation between capital stock and output and a negative correlation between inventories and output, both contrary to the data (p. 1368). By contrast, the time-to-build model’s fit is not sensitive to the exact construction period assumed – three- or five-quarter versions fit about as well as the calibrated four-quarter version (p. 1368).

Most of the variance in technology must come from the persistent (permanent) component, rather than the transitory component, for the model’s serial-correlation properties to match the U.S. data, and the variance of the noisy indicator shock cannot be too large relative to the permanent shock’s variance, since that would make cyclical employment vary less than cyclical productivity – again the wrong pattern (Sec. 5, “Sensitivity of Results to Parameter Selection,” pp. 1366-1367). More broadly, the authors report that the model’s implied covariation statistics are “surprisingly insensitive” to many parameter values across broad ranges, which they suggest implies that even economies with somewhat different structural parameters could still display qualitatively similar business cycles (p. 1366).

The authors flag several refinements that could improve the model: introducing hours-per-week of capital utilization as a decision variable; allowing multiple types of capital with different construction periods and resource-use patterns (which would also allow modeling tax-system effects on different asset classes); and improving the estimation procedure, since – despite recent advances by Hansen and Sargent – “further advances are needed before formal econometric methods can be fruitfully applied to testing this theory.” (Sec. 6, pp. 1368-1369). They also note a computational limitation for policy analysis: because their solution method relies on the equivalence between competitive equilibrium and a planning problem, evaluating feedback policy rules (which depend on the aggregate state and would generally not coincide with the welfare-maximizing allocation) requires different methods than those developed in the paper (Sec. 6, p. 1369).