Is the Time-To-Build Model Empirically Viable?

Copyright (i) IFAC Computation in Economics. Finance and Engineering: Economic Systems. Cambridge. UK. 1998

IS THE TIME-TO-BUILD MODEL EMPIRICALLY VIABLE? Paul J. Zak, Leonard Tampubolon, David Young Department of Economics, Claremont Gmduate University Claremont, CA 91711-6165

Abstract: The time-to-build model is one possible equilibrium alternative to the standard technology shock real business cycle model. The time-to-build model is appealing because it incorporates the time required to produce, deliver and install capital into standard models in such a way that these models produce endogenous cycles. Should economists take this model seriously when seeking to explain business cycle frequency data? This paper presents a Solow model with a time-to-build lag and proves that this model produces endogenous cycles that, for reasonable parameter values, can generate empirically relevant cyclic variations in output. ©1998 [FAG Keywords: Optimal Growth, Endogenous Cycles, Delay Differential Equations

1

INTRODUCTION

THE SEARCH IS ON for a better theory of business cycles. Following the work of Basu (1996)and Galf (1996) showing that changes in technology only account for 25% of the Solow residual, the relevance of the standard real business cycle models of Kydland & Prescott (1982), Hansen (1985) among many others, in which cycles are driven by the Solow residual is being called into question. 1 Leading candidates in the search for equilibrium alternatives to aggregate technology shock models include the factor intensity choice and factor hoarding model of Burnside & Eichenbaum (1996), the sunspot models of Benhabib & Farmer (1996), Schmit-Grohe (1997), Farmer & Guo (1994), the multi sector technology shock models of Horvath (1997a, 1997b), Basu, 1 Gal! (1996) shows that when one controls for accounting for variable intensity in the use of inputs and markups over cost, technology shocks induce negative comovements with output. Recent critical analyses of technology shock business cycle models can be found in Hairault, Langot & Portier (1997) and Burnside & Eichenbaum (1996) .

Fernald & Horvath (1997), the monetary model of Eichenbaum (1997) , the informational friction model of Hairault, Langot & Portier (1997) and the vintage capital (or time-to-build) models of Benhabib & Rustichini (1991), Boucekkine, Germain & Licandro (1997) and Asea & Zak (1997a, 1997b). With the exception of vintage capital models, the stochastic structure that generates business cycles is added to the models in an ad hoc manner. The ability of sunspot models to match aggregate data using empirically reasonable parameter values has been questioned by Schmit-Grohe (1996), although Benhabib & Farmer (1996) surmount this issue by using a multi-sector model. Whether sunspot models will emerge as a generally accepted theory of business cycles is open to debate and will require a much larger number of studies than have currently been undertaken. Time-to-build models, in which there is a delay during which capital is produced, delivered and installed, are essentially general equilibrium versions of Kalecki's (1935) linear lag model. These models have the ability to gen-

397

assume that r is constant. 3 Since capital purchased at time t does not come on-line until period t+r, it does not begin to depreciate until it is put into proIn this paper, we ask whether the time-to-build duction. Let 0 E [0,1] denote the proportion of the model is an empirically reasonable model of busi- capital stock that depreciates due to production. A ness cycles by identifying parameter values required model with this type of delay structure simplifies to generate endogenous cycles. The criterion we use that of Benhabib & Rustichini (1991) in that proto determine the empirical viability of the time-to- duction only depends on the stock of capital availbuild model is whether the parameters that gener- able; there are no differences in the productivity in ate cycles are both empirically reasonable and pro- different vintages of capital. This model could be duce cycles in aggregates that have a periodicity written in discrete time, but it would result in a similar to that observed in the data. This is done high dimensional dynamical system. Alternatively, using a Solow-type model with a time delay for the the model can be written in continuous time with a delivery of new capital goods. The constant savings discrete time-to-build delay. Such a dynamical sysrate is not necessary for our analysis, it is used only tem is known as a delay differential equation and, as a simplification so that the methodology is as as will be shown, is quite tractable analytically. To transparent as possible. In fact, the constant sav- further simplify the model, assume that population ings rate significantly ties our hands numerically as is constant and normalized to unity so that k repwe show that there is a single value of the deliv- resents per capita capital. ery lag that generates endogenous cycles. Asea & Zak (1997a) show that for an optimal growth model Output, y, at time t is given by with an embedded time-to-build structure there exy{t) = f{k{t - r)), (I) ists a sequence of values for the time-to-build delay that generates endogenous cycles. Thus, if the Solow variant of the time-to-build model is empiri- where f{k) is a standard neoclassical production cally reasonable, the more general Cass-Koopmans function satisfying the Inada conditions. Embedmodel should also generate empirically viable cy- ding this production structure into a Solow model where a large number of identical individuals save a cles. constant proportion s E (O, 1) of their income, leads In this paper, we first prove that a simple Solow- to the following capital market equilibrium condition, type time-to-build model produces Hopf cycles, and then determine the periodicity of these cycles as a ic{t) = sf{k{t - r)) - ok{t - r), (2) function of the parameters of the model. Once this is done, we examine the periodicity of business cycle for some initial function 4 k{t) = cjJ{t) for t E [-r, 0]. frequency data and assess the explanatory power of capital goods delivery lags vis-a.-vis aggregate out- Since the production lag only affects temporal asput. We conclude that time-to-build models can be pects of the model, it has no effect on the steady used to explain business cycle frequency data. state, which is standard. The interior steady state value of capital, k*, is given implicitly by 5 erate endogenous cycles and thus are free from the debate of the source of shocks. 2

sf{k*) = ok* .

A SIMPLE TIME-TO-BUILD MODEL 2

Consider a production technology in which it takes ~ periods to receive and install capital before it becomes productive. Thus, at time t, the productive capital stock is k{ t - r). In this model, it is not possible to reduce the delay r as new capital can not be produced instantaneously and for simplicity

r

Characterizing the dynamics of nonlinear DDEs is quite difficult and closed-form solutions can not generally be found. We explore whether there are

°

2The relationship between Kalecki's work and modern time-t<>-build models is put into perspective in Zak (1997a, 1997b).

(3)

3Asea & Zak (1997b) build a model in which the delivery lag is state-dependent while Asea & Zak (1997c) examine the case when firms can pay to reduce T. 4Due to the lag structure, more than an initial point value is required to start the system in motion, i.e. an initial function must be defined over the range of time delimited by the delay before the system begins in motion. On this, see Asea & Zak (1997a). sThere is also a trivial steady state, k(t) = 0, "It, but this fixed point is not of interest.

398

there cyclic dynamics in this model by characterizing the dynamics in a neighborhood of the steady state given by (3). To do this, we perform a change of variables to shift the relevant steady state to zero, z(t) = k(t) - k*, and take a first order approximation of the new system about the origin. After the change in variables, the characteristic equation is given by

h(>') = >. - Ae->''T,

(4)

where A == s!,(k*) - () < O. For nontrivial values of A and T, there are an infinite number of complex roots to equation (4).

Why? Since there are an infinite number of complex roots to the characteristic equation (4), generically there is a large set of eigenvalues with positive real parts and another large set with negative real parts. For the particular value of the delivery lag, T = T*, there is a corresponding initial function that limits the dynamics of the model to lie on the center manifold, a two-dimensional subspace of the eigenspace which contains a pair of purely imaginary roots to the characteristic equation.

Next, we derive the period of the cycle. Generically, the period of a cycle is ~. For the given value of T* , we can use equations (5) and (6) to find the corresponding w*, the frequency of the cycle. Performing Does this model exhibit endogenous cycles? Let us this calculation, one finds that the period of the cydecompose the characteristic equation into real and cle, call it P = A.~, where A = s!,(k*) - () < 0 imaginary parts by setting>. = J.L + iw. Applying as above and N = 1,2,3, .... The N appears in Euler's formula, complex roots of (4) can be written the determination of the period of the Hopf cycle as sines and cosines, producing the pair of transcenbecause of the periodicity of the sine function in dental equations, equation (5) which was used to solve for w*. Note J.L - Ae-~'T sin(Tw) = 0 (5) that though w* depends on N, the delay that produces cycles, T* does not. There is only a single w + Ae-~'T COS(TW) = o. (6) value of the delay that causes periodic dynamics, A Hopf cycle arises if there is a valid solution for T but the period of the ensuing cycle varies with N. when there are purely imaginary roots to (4), sub- The two-dimensional dynamics in the phase space ject to a technical condition being met which is dis- appear as concentric ellipses about the steady state. cussed below. Setting J.L = 0 and solving for a value of the delay that satisfies (5) and (6), we find that In order to compare the period of the cycle with the data, a functional form for f(k) is chosen. As is -211" standard, we use a Cobb-Douglas production func(7) T* = A >0. tion, f(k) = k Ot , for a E (0,1). In this case, the period P = 6(l::jN. Observe that even though This is a valid solution since A < o. Thus, there is a Solow constant savings rate s is used to derive a single candidate value of the delay at which cycles the law of motion for the capital stock, s does not appear. affect the period of the cycle when production is Cobb-Douglas. Given the functional form for proThe value of T given in (7) will produce Hopf cycles duction, we are able to perform numerical simulaif the transverse crossing condition holds at T = T*. tions of the model. These simulations are not part This condition requires that roots cross from the of a calibration exercise, for the model in this paper complex to the imaginary plane with positive speed. is much too simple to be realistically used for busiFormally, the transverse crossing condition is, ness cycle analysis. Rather, given accepted ranges of the parameter in the model, we want to deterRe[h(>'(T»] 0 d (8) mine if a more complicated (i.e. realistic) time-todT >. build model could be built to explain business cyUsing (4) directly, one can show that dRe[h£;('T)}) = cle frequency data. As a first step, we verify that -Awe-'T~ which is strictly positive. Therefore, the model does produce cycles. By using a version we have proven that the Solow model with time of the Solow model, the ensuing dynamics are suflags produces endogenous cycles about the interior ficiently simple that a commercial dynamical systems solver can be used to perform the numerics. 6 steady state. We use a software package called Phaseplane (ErFor values of T which are different than T* the steady state dynamics constitute a high6Boucekkine, Licandro & Paul (1997) discuss the diffidimensional saddle for a most initial functions 4>(t). culty in nwnerically simulating vintage capital models.

399

mentrout (1990» to simulate the delay differential equation (2) for T = T* when production is CobbDouglas. The figure is not reproduced here due to space constraints, but it shows that the model rapidly converges to a stable cycle.

3 COMPARING THE MODEL TO THE DATA Our next task is to empirically characterize business cycles and compare these data to the model's output. If the time-to-build model is to be used to explain business cycle frequency data, there will have to be some source of uncertainty in the model since we do not live in a perfect foresight world. This will add noise to the cycles produced by the model, but the variation due to this noise will be small if lags in the delivery of capital goods are the primary source of business cycles. One natural way to add uncertainty to the model is to use stochastic capacity constraints on the production of capital goods, as in Asea & Zak (1997b), which is equivalent to making the time delay T stochastic (and state-dependent). Asea & Zak (1997b) prove that such a construct in an optimal growth model generates Hopf cycles of varying periods, capturing the nonconstant periodicity observed in business cycle frequency data. Nevertheless, that paper does not perform empirical tests of the model. If the cycles produced by the time-to-build model in this paper are not "too far" from the cycles seen in the data, then the time-to-build model may capture a reasonable amount of the variation in output over the business cycle. The average duration of U.S. business cycles beginning in 1854 with the peaks and troughs determined by the NBER dating committee. The mean cycle length peak to peak of all 31 cycles is 53 months, with the post-war mean being 61 months. The range of cycle durations runs from 17 months to 116 months.

[.004, .01251 for monthly data. The other parameter which determines the cycle's periodicity is et, the proportion of output paid to capital. This is typically estimated to vary from .25 up to a value of 1 if human capital is included in the definition of the capital stock. This parameter does not vary with the frequency of observation.8 Using the parameter values for 6 and et and various values of N, the period P of cycles that the model produces is well within the range of cycles seen in the data. Table presents representative values for the period P of the Hopf cycle. 9 It is not surprising that the model is able to reproduce observed business cycle frequency data since N acts as a free parameter. Unless we have a theory for which particular Hopf cycle corresponds to observed cycles in the data, the model is not inconsistent with the data, but without further modifications, is of questionable predictive power. It is important to note that business cycles may have many causes-we do not claim that delivery lags for capital goods are the only explanation for business cycles, only that lags should be considered among the set of possible explanations. In particular, some business cycles may have etiologies completely unrelated to delivery lags. For example, the two shortest post-war U.S. recessions were from January to July, 1980, and from July, 1981 to November, 1982. These recessions are widely credited to the tight money regime put in place by then Federal Reserve Board Chairman Paul Volker in order to reduce the high rates of price inflation in the 1970s. This suggest that a general theory of business cycles may have to look at multiple causative factors if it is to span the set of forces that drive the cyclicality in aggregates.

4

THE EXPLANATORY POWER OF THE THEORY

Given the range of periods of business cycles in the data, we use intervals of values for the parame- If delivery lags on capital goods are a source of outters in the model to determine the range of peri8For discussions of the determination of parameter values ods that the model is able to reproduce. From the real business cycle literature, the depreciation rate from macro data, see Cooley (1995), Rebelo & Stokey (1995), Lucas (1990), Christiano (1988). The case that a = 1 is on capital is estimated between .048 and .15 annu- made by Mankiw (1995), who argues that it is this value for ally, depending on whether or not residential struc- which the model to produces a balanced growth path when tures are included? This translates into a range of k is broadly defined to include both physical and human capital.

7The upper value for depreciation is depreciation plus the rate of population growth which is about 1.5% in the U.S.

9 As a -+ 1, P in the text.

400

-+ 00,

for any values of li in the range given

Table 2: Estimation output

Table 1: Predicted Period of the Hopf Cycle

PARAMETER

PERIOD N= 20

PERIOD N= 30

PERIOD N= 40

a =. 25 , 0=.0040

105 75 168

70 50 112

52 38

a= .50, 0=.0083

a= .85, 0= .0125

84

put variations, then lags should have explanatory power empirically. There is a question, though, as to the proper measure of delivery lags, as a direct measure is, to our knowledge, unavailable. The first indirect measure that comes to mind is the change in unfilled orders for durable goods which, until last year, was one of the components of the Conference Board's Index of Leading Economic Indicators (see The Conference Board (1997». Retrieving this data as well as the Federal Reserve Board's index of industrial production showed that unfilled orders for durables on its own and relative to total orders for durable goods do not have statistically significant explanatory power. More precisely, both unfilled orders and industrial production were Hodrik-Prescott (HP) filtered to remove the growth trend and then were tested for unit roots. Neither series had unit roots, and a number of regressions of unfilled orders and ARMA terms on industrial production were estimated. The inclusion of unfilled orders added an insignificant amount of explanatory power, so a search for a better measure of capital goods delivery lags was undertaken. Another measure of delivery lags is the backlog of capital appropriations (BCA) , a data series collected by The Conference Board. HP filtered industrial production and the backlog of capital appropriations from the first quarter 1953 to the first quarter of 1990 both show increasing variation beginning in the mid-1970s, with the industrial production series appearing to settle down near 1990 (the last year of data available). After verifying that HP filtered BCA does not have a unit root, we again estimated a set of time-series models to determine if BCA has explanatory power vis-a-vis industrial production. Table presents the estimation results of a regression of BCA with zero, one and two lags against industrial production, including an ARMA(1 ,1) error structure. Both the current value of BCA and the two-quarter lag are statistically sig-

VARIABLE Constant BCA BCA(- 1) BCA(-2) AR(1) MA(1)

COEFFICIENT -0.06670 0.00012 0.00003 -0.00014 0.73953 0.53176

OBSERVATIONS

R2

146

.84

T-STATISTIC -0.144 3.084 0.682 -3.709 11.174 6.697

nificant, with the regression explaining 84% of the variation in industrial production. The significance of BCA is robust to variations in the number of lags and included ARMA terms. These results give us confidence that delivery lags on capital goods do have explanatory power for business cycle frequencY data.

5

CONCLUSION

A research program is underway that seeks to open up the black box of technology shocks used in RBC models. One possible avenue is to examine whether bottlenecks in the production, delivery and installation of capital goods might explain business cycle frequency data. This paper took a simple time-tobuild model in which there is a fixed-period delivery lag for capital goods and proved that such a model has endogenous cycles. Moreover, for reasonable parameter values, this model cannot be rejected as a model of business cycles. It was further shown that a measure of backlogs of capital goods has robust and statistically significant explanatory power when analyzing HP filtered industrial production data. The tentative positive answer to the question posed in the title means that there is much work to be done. Bottleneck models with better microfoundations need to be developed, along the lines of Gale (1996), and incorporated into a general equilibrium framework before they can be considered a viable alternative to technology shock models. Standing at a crossroads is both exciting and daunting- we hope that the results presented here will motivate

401

business cycle theorists to veer down this less traveled road.

REFERENCES ASEA, P. K. AND P. ZAK, {1997a}. Time-to-Build and Cycles. NBER Technical Working Paper #211. ASEA, P. K. AND P. ZAK, {1997b}. Bottlenecks.

Working paper, UCLA and Claremont Graduate University. ASEA, P. K. AND P. ZAK, {1997c}. Microbottlenecks, Working paper, UCLA and Claremont Graduate University, in preparation. BASU, S., (1996). Procyclical Productivity: Increasing Returns or Cyclical Utilization. Quarterly Journal of Economics, 61(3), 719-75l. BASU, S., FERNALD, J. AND HORVATH, M., (1996). Aggregate Production Function Failures and Business Cycle Analysis, Working paper, Stan-

ford University. BENHABIB, J. AND FARMER, R. E.A., (1996). Indeterminacy and Sector-Specific Externalities, Journal of Monetary Economics, 37, 421-443. BENHABIB, J. AND RUSTICHINI, A., (1991). Vintage Capital, Investment, and Growth, Journal of Economic Theory, 55, 323-339. BOUCEKKINE, R., GERMAIN , M., AND LICANDRO, 0., {1997}. Replacement Echos in the Vintage Capital Growth Model, Journal of Economic Theory, 74, 333-348. BOUCEKKINE, R., LICANDRO, 0., AND PAUL, C., (1997). Differential-Difference Equations in Economics: On the Numerical Solution of Vintage Capital Growth Models, Journal of Economic Dynamics and Control, 21, 347-362. BURNSIDE, C. AND EICHENBAUM, M., {1996}. Factor- Hoarding and the Propagation of BusinessCycle Shocks, American Economic Review, 86(5), 1154-1176 . CHRISTIANO, L. J., (1988). Why Does Inventory Investment Fluctuate So Much?, Journal of Political Economy, 101, 1042-1067. CONFERENCE BOARD, THE, {1997}. Business Cycle Indicators, The Conference Board. COOLEY, T., ED. (1995). Frontiers of Business Cycle Research, Princeton. EICHENBAUM, M., (1997). Money, Market Frictions and the Business Cycle, Working Paper,

Northwestern University. ERMENTROUT, 8., (1990). Phaseplane, Version 3.0. Brooks/Cole. FARMER, R. E.A. AND Guo, J.-T., (1994). Real Business Cycles and the Animal Spirits Hypothesis, Journal of Economic Theory 63, 42-72.

402

GALE, D., (1996). Delay and Cycles, Review of Economic Studies 63, 169-198. GALf, J., (1996). Technology, Employment, and the Business Cycle: Do Technology Shocks Explain Aggregate Fluctuations?, C. V. Starr Working Paper # 96-28. HAIRAULT, J.-O., LANGOT, F. AND PORTIER, F., (1997). Time to Implement and Aggregate Fluctuations, Journal of Economic Dynamics and Contro~ 22, 109-121. HAMILTON, J. D., (1989). A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle, Econometrica 57(2), 357-384. HANSEN, G., (1985). Indivisible Labor and the Business Cycle, Journal of Monetary Economics, 16, 309-327. HORVATH, M. {1997a}. Cyclicality and Sectoral Linkages: Aggregate Fluctuations from Independent Sectoral Shocks, Stanford University. HORVATH, M. (1997b). Sectoral Shocks and Aggregate Fluctuations, Stanford University. KALECKI, M., {1935}. A Macroeconomic Theory of Business Cycles, Econometrica, 3, 327-344. KYDLAND, F.E. AND PRESCOIT, E.C., {1982}. Time-toBuild and Aggregate Fluctuations Econometrica 50, 1345-1370. ' , MANKIW, N. G., (1995). The Growth of Nations,

Harvard Institute of Economic Research Discussion Paper # 1731. LUCAS, R. E., {1990}. Why Doesn't Capital Flow From Rich Countries to Poor Ones?, American Economic Review Paper and Proceedings, 80(2}, 92-96. STOKEY, N. AND REBELO, S., {1995}. Growth Effects of Flat-Rate Taxes, Journal of Political Economy, 103(3), 519-550. SCHMIIT-GROH/'E, S., {1997}. Comparing Four Models of Aggregate Fluctuations Due to SelfFulfilling Expectations, Journal of Economic Theory 72(1), 96-147. ZAK, P. J., {1997a}. Kaleckian Lags in General Equilibrium, working paper, Claremont Graduate University. ZAK, P. J ., (1997b). Dynamic Feedback Models in Economics, New Directions In Macroeconomic Modeling, Andrew Hughes-Hallett and Peter McAdam, Eds., forthcoming.