TIME TO BUILD AND THE BUSINESS CYCLE∗ Matthias Meier† November 24, 2017

Abstract Investment is central for business cycles and a defining characteristic of investment is time to build. Whereas existing business cycle models assume constant time to build, I document that time to build is volatile and largest in recessions. Identified capital supply shocks, which lengthen time to build, are contractionary and explain up to one third of business cycle volatility. Toward explaining these facts, I show that capital misallocation is a quantitatively powerful mechanism for the transmission of changes in time to build. I embed this mechanism in a general equilibrium model with heterogeneous producers and capital adjustment costs. When calibrating the model to US micro data, capital supply chain disruptions that lengthen time to build by one month lower investment GDP by one percent. JEL classification: C32, C68, D92, E01, E22, E32. Keywords: Time to Build, Business Cycles, Investment, Misallocation.

∗

I am extremely thankful to my advisors Christian Bayer and Keith Kuester for their continuous guidance and support. I wish to convey special thanks to Gianluca Violante for conversations on this project and for facilitating my visits at NYU. I thank Benjamin Born, Katka Borovičkovà, Jarda Borovička, Daniel Greenwald, Boyan Jovanovic, Greg Kaplan, Moritz Kuhn, Ben Moll, Morten Ravn, Markus Riegler, Felipe Valencia, Venky Venkateswaran, and Jürgen von Hagen, and numerous seminar participants for useful comments at various stages of this project. Financial support from the Bonn Graduate School of Economics, the Institute for Macroeconomics and Econometrics, and the Institute for International Economic Policy, both at the University of Bonn, is gratefully acknowledged. † Universität Mannheim, Department of Economics, Block L7, 3-5, 68161 Mannheim, Germany; E-mail: [email protected].

1

2

MATTHIAS MEIER

1. INTRODUCTION Capital goods are complex and manufactured to the specific needs of a producer. An assembly line, for example, consists of many elements that need to fit together; think of conveyor belts, robotic arms working along these belts, and the concrete foundation that supports the machines. Complexity and specificity of capital goods drive a wedge between the time when capital goods are ordered and their delivery. This time wedge is commonly referred to as time to build and is assumed constant in modern business cycle theory.1 My paper first documents substantial variation in time to build, with peak values in recessions. I provide vector autoregression (VAR) evidence that identified capital supply shocks, which lengthen time to build, have sizeable, contractionary effects. Second, I ask whether capital supply chain disruptions are of first-order importance for business cycles. To address this question, I develop a heterogeneous producer DSGE model. Firms in my model face persistent shocks to their own productivity and their investment is partially irreversible. Capital supply chains are characterized by search frictions, which imply time to build. Disruptions in the supply chain immediately result in time to build variation. Calibrating the model to US manufacturing data, I find that time to build fluctuations are quantitatively important. A one month increase in time to build lowers output by one percent. It is due to two channels that a lengthening in time to build is contractionary. First, later delivery of outstanding investment orders, as follows from longer time to build, mechanically reduces contemporaneous investment and thus production. Second, and this channel is both novel and quantitatively central, longer time to build worsens the allocation of capital across firms. While the efficient allocation dictates that more productive firms use more capital, longer time to build weakens this alignment. At the core of the mechanism, later delivery of an investment good increases the ex-ante uncertainty about productivity in the usage period of the investment good. In turn, firms invest less frequently and, if they invest, ex-post their investment tends to be further away from optimal. A lengthening of time to build therefore means capital is more misallocated across firms, meaning aggregate productivity is lower and so are output, investment, and consumption. To measure time to build, I use the US Census M3 survey of manufacturing firms. The Census provides publicly accessible time series for order backlog and shipment 1

While Kydland and Prescott (1982) assume four quarters time to build, the standard assumption in real business cycle models quickly shifted to one quarter, see Prescott (1986) for example.

TIME TO BUILD AND THE BUSINESS CYCLE

3

in the non-defense equipment goods sector since 1968. These time series allow me to estimate time to build as the duration that new capital good orders remain unfilled in capital producers’ order books. I document that time to build exhibits substantial variation. It fluctuates between three and nine months. Importantly, time to build tends to be largest at the end of recession periods.2 While time to build may endogenously respond to any shock affecting capital demand or supply, I focus on the role of capital supply shocks. In particular, I study shocks that directly lengthen time to build and that I want to think of as capital supply chain disruptions. In the context of a medium-scale VAR, I identify capital supply chain disruptions as structural shocks that contemporaneously only affect time to build. Conversely, all other structural shocks, albeit not explicitly identified, may affect time to build contemporaneously. I find that disruption shocks significantly and persistently lower GDP, investment, and consumption. The identified shock explains more than 20% of the forecast error variance of GDP and consumption. Moreover, the forecast error variance of time to build explained by the identified shock is almost 50%. The model I develop is a real business cycle model. Households consume and supply labor. The model distinguishes between firms that supply capital and firms that demand capital. On the capital demand side, firms that produce consumption goods by combining labor and specific capital. To invest in specific capital, they sign an order contract with an engineering firm. In turn, the engineering firm devises a blueprint for the investment project and searches for capital suppliers to produce the required intermediate goods. The ordered capital goods are produced and delivered when the engineering firm matches with a capital supplier. Shocks to the matching technology cause fluctuations in time to build. These shocks may be seen as shortcut for capital supply chain disruptions, which make it more difficult to supply the required components. The model further features lumpy firm-level investment in line with the micro-level evidence on capital adjustment. The lumpiness arises because investment is partially irreversible. To solve the model, I build on the algorithms in Campbell (1998), Reiter (2009) and Winberry (2016b). Conceptuallly, the idea is to combine global projection with local perturbation solution methods. Compared to Winberry (2016b), the model in 2

This paper is not the first to document the countercyclicality of the backlog ratio, see, e.g. Nalewaik and Pinto (2015). To the best of my knowledge, however, my paper is the first to relate fluctuations in the backlog ratio to time to build in the context of modern business cycle models.

4

MATTHIAS MEIER

this paper is computationally more involved because the idiosyncratic state additionally consists of outstanding capital good orders. A feature of the model that is important for computational tractability, is that the delivery probability of outstanding orders is duration-independent. To evaluate the quantitative importance of capital supply chain disruptions, I calibrate the model to US data. The strategy is to jointly target moments of the investment rate distribution and aggregate fluctuations in time to build. In the calibrated model, disruption shocks that raise time to build by one month cause a sharp 8% drop in investment and a more persistent drop in output up to 1%. I show that the direct effects of later delivery are important for the short-term responses while increased capital misallocation explain about half of the mediumterm responses. To understand the role of time to build in historical business cycles, I suppose all time to build variation in the data is explained by capital supply chain disruptions. This allows me to back out a series of disruption shocks from the calibrated model. I then compute the evolution of GDP under the shock series and absent other aggregate shocks. Under this exercise, supply chain disruptions explain about half of the decline in output and investment during the early 1990s recession and the Great Recession. Related Literature This paper contributes to several literatures. First, this paper contributes to the literature studying the macroeconomic implications of lumpy investment. There is ample evidence for investment lumpiness, see Doms and Dunne (1998), and structural explanations are investment irreversibilities or fixed costs of capital adjustment, see Cooper and Haltiwanger (2006). Recent work has investigated the macroeconomic implications of capital adjustment costs for the response to aggregate productivity shocks, see Khan and Thomas (2008) and Bachmann et al. (2013), and, for the response to uncertainty shocks, see Bloom (2009), Bachmann and Bayer (2013), and Bloom et al. (2014). In my model, the interaction between time to build and investment irreversibilities is key for the transmission of shocks to the matching technology. The transmission mechanism shares the real options effect prominent in the uncertainty literature, albeit without inducing the volatility effect that higher uncertainty eventually realizes and leads to reversals and overshooting, see Bloom (2009). To the extent that longer time to build increases the effective forecast uncertainty, this paper also contributes to the endogenous uncertainty literature, see

TIME TO BUILD AND THE BUSINESS CYCLE

5

Bachmann and Moscarini (2011) and Fajgelbaum et al. (2014). Second, my paper relates to recent work studying the interaction between time to build fluctuations and investment irreversibility. Studying time to build for residential housing, Oh and Yoon (2016) document a time series pattern fairly similar to the one for equipment capital goods documented in this paper. In their model, higher uncertainty increases time to build because residential construction occurs in stages and each stage involves irreversible investment. Kalouptsidi (2014) studies the bulk shipping industry and shows that procyclical fluctuations in time to build dampen the volatility of investment into ships. Third, in modeling a frictional market for capital goods, I build on the search literature. Since Mortensen and Pissarides (1994) search frictions are popular in labor market models. For capital markets, Kurmann and Petrosky-Nadeau (2007) and Ottonello (2015) show that search frictions amplify business cycle shocks. Tightness on the capital goods market governs the intensive margin of investments, while in my setup search frictions also affect the extensive margin of investment. Shocks to the matching technology in my model build on the labor market search and matching literature, see Krause et al. (2008), Sedláček (2014), and Sedláček (2016) for example. The remainder of this paper is organized as follows: Section 2 documents time to build fluctuations and provides VAR evidence. Section 3 presents the central model mechanism and Section 4 develops the quantitative business cycle model. I discuss the calibration in Section 5 and results in Section 6. Finally, Section 7 concludes and an Appendix follows.

2. DOES TIME TO BUILD VARY OVER THE CYCLE? 2.1. A measure of time to build My goal is to estimate time to build using data on the order books of capital good producers. I show that time to build exhibits substantial variation between three and nine months with peak values during recessions. Data on order books of capital good producers is obtained from the publicly available US Census Manufacturers’ Shipments, Inventories, and Orders Survey (M3). The M3 covers two third of manufacturers with annual sales above 500 million USD and some smaller companies to improve industry coverage. M3 participants are selected from the Economic Census and the Annual Survey of Manufacturers and the

6

MATTHIAS MEIER

M3 is benchmarked against these datasets. In support of the survey quality, the Bureau of Economic Analysis uses M3 data to compute US quarterly investment.3 The Census provides publicly available data for shipments and order backlog at the sectoral level. I use data for the non-defense equipment goods sector, which is available at monthly frequency from 1968 through 2015.4 M3 data satisfies a stock-flow equation for equipment good orders, where O denotes new orders net of cancellations, S shipments, and B the beginning-of-period stock of order backlog5 (2.1)

Bt+1 = Bt + Ot − St .

My baseline measure of time to build, also called backlog ratio, is (2.2)

T T Bt ≡

Bt . St

It measures the intensity of flows (shipments) out of the stock of backlogged orders. The unit of T T Bt is months. Figure 1 shows the evolution of time to build. It fluctuates between three and ten months. Importantly, time to build tends to start increasing before recessions and peaks at the end of recession periods. The correlation of annualized real GDP growth with log time to build is -0.3. When detrending the slow-moving trend from time to build using an HP filter with smoothing parameter 810,000, the correlation is -0.4. The finding of a countercyclical backlog ratio coincides with previous studies, see Nalewaik and Pinto (2015) for example. Under two conditions this time to build measure equals the expected waiting time of a new equipment good order: First, the shipment protocol is first-in firstout, i.e. new orders are shipped only after backlogged orders are shipped. Second, shipments are expected to be unchanged in the future. While the first condition is plausibly satisfied, the second one is roughly satisfied given that shipments are highly persistent. In Appendix A, I show that an alternative measure of time to build, based on ex-post shipment realizations, closely resembles my baseline measure. Additionally, I show the evolution of all individual series of (2.1) in the Appendix. 3

See Concepts and Methods of the U.S. National Income and Product Accounts (2014, ch. 3). For more disaggregated two-digit equipment goods sectors, the distinction between defense and non-defense goods is not always available. 5 A new order is defined as a legally binding intention to buy for immediate or future delivery, and the survey does not ask separately for order cancellations. Shipments (net sales) measure the net value of goods sold in a given period, while order backlog measures the value of orders that have not yet fully passed through the sales account. 4

7

TIME TO BUILD AND THE BUSINESS CYCLE

Figure 1: Time to Build

2

4

6

8

10

Months

1970

1980

1990

2000

2010

Notes: Time to build is measured as the ratio of order backlog to monthly shipments, for non-defense equipment goods. Shaded, gray areas indicate NBER recession dates.

A caveat of estimating time to build using the M3 is that it excludes structure capital and imported equipment capital, which on average account for about a third of total non-residential private fixed investments in the US.6 For residential housing, Oh and Yoon (2016) document countercyclical time to build. Another caveat of my time to build measure is that compositional changes may affect measured time to build without changing actual time to build. The problem arises if capital goods with long time to build constitute a larger share of total order backlog during recessions than during booms. I address this caveat in two ways. First, I use M3 data to compute time to build for the four two-digit industries that produce equipment goods. Figure 7 in the Appendix shows that the industrylevel series are countercyclical. Hence, countercyclical fluctuations in aggregate time to build are not due to cyclical shifts in the purchase of broadly defined types of capital goods. Second, I use annual Compustat data to compute the average time to build across firms. Assessing various data treatments (weighting, mean vs. median) Figure 8 shows that time to build is robustly countercyclical. 6

Out of total private non-residential fixed investment, structure capital constitutes on average 25% over the last 40 years, declining over time with 10% in 2015. Imported equipment capital is on average 10% of total investments, increasing over time with 20% in 2015.

8

MATTHIAS MEIER

2.2. Macroeconomic evidence The evolution of time to build may reflect a number of macroeconomic shocks, in particular any shock that affects capital demand or supply. The focus of my paper is on the role of capital supply shocks. More specifically, I study shocks that directly lengthen time to build and I want to think of these shocks as capital supply chain disruptions. An example for such a disruption shock is the 2011 Japanese earthquake; the consequences of which have been studied for the Japanese economy in Carvalho et al. (2016) and for the US economy in Boehm et al. (2015). Toward identifying the dynamic effects of supply chain disruptions, I fit a mediumscale, eight-variable VAR model. The model allows for rich dynamic interactions between time to build and several macroeconomic series, prominent in both structural and empirical business cycle models, see, for example, Christiano et al. (1999). The vector of endogenous variables is:          Y =        

Time to Build



Real GDP

                

Real Consumption Real Investment Consumer Prices Real Wage Federal Funds Rate

Total Factor Productivity

I use data at quarterly frequency that covers 1968Q1 through 2014Q4. Total factor productivity is from Fernald (2014) and the remaining macroeconomic series are sourced from FRED.7 All variables but the federal funds rate are transformed by the natural logarithm. Notice that I use non-durable consumption goods, because durable consumption goods include equipment goods that time to build shocks may directly affect. Similarly, my preferred investment time series is nonresidential investments because only firms invest in my model. The results are robust against using total consumption and total investment instead. The baseline structural VAR model is estimated in levels with four lags and linear 7

The FRED series names are GDPC96 (Real GDP), DNDGRA3Q086SBEA (Real Personal Consumption Expenditures: Nondurable goods), B008RA3Q086SBEA (Real Private Fixed Investment: Nonresidential), CPI, AHETPI/CPI (Average Hourly Earnings of Production and Nonsupervisory Employees: Total Private; deflated by CPI ), FEDFUNDS (Effective Federal Funds Rate).

9

TIME TO BUILD AND THE BUSINESS CYCLE

time trend. The baseline identification assumption is that time to build increases in response to a capital supply chain disruptions while all other macroeconomic time series do not respond contemporaneously. This identification scheme is conservative in the following sense. Except for supply chain disruptions, all structural shocks may affect time to build contemporaneously, while the time to build shock may affect all variables except time to build only through a one-quarter lag.8 Figure 2 shows the impulse responses to a supply chain disruption that raises time to build by one month at peak. The shock has a persistent, significant effect on time to build. More interestingly, GDP, investment, and consumption significantly fall in response to a disruption. Not only are the responses statistically significant, but their magnitudes are also economically relevant: GDP and consumption fall by up to 2%, and investment by up to 6% within the first three years. Figure 2: Impulse responses to a one month capital supply chain disruption Time to build (in months)

GDP (in %)

7

1

6.5

0 -1

6 -2 5.5

-3

5

-4 0

4

8

12

16

20

24

0

Investment (in %)

4

8

12

16

20

24

Consumption (in %)

2

1

0

0

-2

-1

-4 -2

-6

-3

-8 -10

-4 0

4

8

12

16

Quarters

20

24

0

4

8

12

16

20

24

Quarters

Notes: Solid, blue lines show (selected) responses to a match efficiency shock, under the baseline identification scheme. Shaded, gray areas illustrate the associated 90% confidence intervals.

Complementary to IRF analysis, Table I shows the shares of forecast error variance explained by capital supply chain disruptions. Such disruptions explain an important 8

The identification strategy resembles Christiano et al. (2005) for monetary policy shocks.

10

MATTHIAS MEIER

fraction of macroeconomic fluctuations: more than 20% of GDP and consumption, and 7% of investment. This provides further evidence in support of this paper’s suggestion that capital supply chain disruption are crucial for a better understanding of business cycle fluctuations. Importantly, at business cycle frequency disruptions explain almost 50% of the forecast error variance of time to build itself. That is, other structural shocks explain only 50%. This further motivates my modeling choice in Section4, in which I focus on the effects of capital supply chain disruption, instead of conventional business cycle shocks, such as a shock in aggregate productivity. TABLE I Forecast error variance decomposition

GDP Investment Consumption Time to build

1 year

2 years

3 years

4 years

5 years

∞

0.2 0.3 0.8 73.4

7.6 0.9 9.8 57.0

18.1 2.8 22.2 48.8

22.6 4.9 26.9 44.8

23.4 6.6 28.2 42.2

18.2 6.7 24.6 31.1

Note: The shares of forecast error variance explained by capital supply chain disruptions are expressed as percentages for different forecast horizons ranging from 1 year to infinity.

Appendix B provides robustness on the VAR results. First, I examine the baseline model specification in levels with linear time trend. When estimating the VAR in first differences, the results are broadly robust. GDP, investment, and consumption respond significantly and with similar magnitude to a disruption shock. Second, I study an alternative identification scheme, in which capital supply chain disruptions can affect all variables contemporaneously, but no other structural shock can affect time to build contemporaneously. While the responses to a disruption tend to be larger under this alternative scheme, the differences are small. Third, using the methodology developed in Gafarov, Meier, and Montiel Olea (2016), I replace zero restrictions with elasticity bounds. Instead of contemporaneous zero restrictions as in the baseline scheme, I constrain the elasticity of the contemporaneous response to a disruption of variables other than time to build to be bounded by ±1%. I show that my baseline results are robust against such relaxation. Fourth, I show that the identified capital supply chain disruption do not mistakenly capture other structural shocks. Beyond the robustness in Appendix B, the results are also robust against estimating a monthly VAR, in which I replace GDP by IP and investment by new orders for non-defense capital goods. Further, the results are robust against dropping the Great Recession period.

11

TIME TO BUILD AND THE BUSINESS CYCLE

3. CAPITAL MISALLOCATION This section discusses a novel, and quantitatively important, mechanism. In short, firms facing longer time to build have (ex-ante) less information about the profitability of an investment project. In turn, capital becomes more misallocated when time to build lengthens, which depresses real economic activity.

3.1. Analytical introspection of mechanism To introspect capital misallocation as mechanism for time to build fluctuations, I introduce a stylized model economy, which permits analytical solution. The economy consists of a fixed unit mass of perfectly competitive firms, indexed by j, that produce a homogeneous consumption good (3.1)

α ν yjt = xjt kjt `jt ,

using firm-specific capital, kjt , labor, `jt , and with decreasing returns to scale, 0 < α + ν < 1. Idiosyncratic productivity, xjt , follows an AR(1) process in logs (3.2)

log(xjt+1 ) = ρx log(xjt ) + σx jt+1 ,

0 ≤ ρx ≤ 1,

iid

jt+1 ∼ N (0, 1).

I denote by wt the wage rate and by rt the user cost of capital. Labor is adjusted every period to maximize profits (3.3)

α ν πjt = xjt kjt `jt − wt `jt − rt kjt .

For analytical tractability, I assume that capital is adjusted subject to time to build and focus on the long-run effects of exogenous changes in time to build.9 The profitmaximizing capital policy of firm j in period t for period t + τ is (3.4)

1−ν

ν

1

1−ν

1−ν kj,t,t+τ = (α/rt ) 1−α−ν (ν/wt ) 1−α−ν E[xjt+τ |xjt ] 1−α−ν .

In words, kj,t,t+τ is the optimal capital policy subject to τ periods time to build. The aggregate capital stock used in period t + τ with τ periods time to build is Kt,t+τ = 9

R1 0

kj,t,t+τ dj, and aggregates Lt,t+τ , Yt,t+τ are computed analogously. Total

The main model of this paper, introduced in Section 4, extends this setup to allow for a richer set of adjustment frictions and studies the dynamic responses to supply chain disruptions.

12

MATTHIAS MEIER

factor productivity is defined as TFPt,t+τ ≡ log Yt,t+τ − α log Kt,t+τ − ν log Lt,t+τ . After some tedious derivations, provided in Appendix C, this simplifies to (3.5)

TFPτ =

1 1 1 α 1 − ρ2τ 1 x 2 σ − σ2 . 2 1 − α − ν 1 − ρ2x x 2 (1 − ν)(1 − ν − α) 1 − ρ2x x

Equation (3.5) shows that longer time to build (τ ) unambiguously lowers aggregate TFP, which is due to increased capital misallocation.10 The quantitative strength of the time to build-TFP nexus depends on three factors: a) If α increases, capital misallocation becomes more important for TFP. b) The closer technology gets to CRS (α + ν % 1), the more optimal firm size responds to productivity, amplifying the capital misallocation of time to build. c) The last term captures the conditional variance of productivity shocks while waiting for capital good delivery V[log xjt+τ | log xjt ] =

1 − ρ2τ x σ2 1 − ρ2x x

ρx →1

= τ σx2 ,

which drives capital misallocation and increases in ρx and σx . Further note that a capital-labor substitution elasticity below unity would strengthen the quantitative impact of time to build on TFP.11 3.2. Quantitative bite of mechanism To understand whether capital misallocation is quantitatively important for the transmission of time to build fluctuations, I need to attach values to α, ν, ρx , and σx . In the literature, a wide range of estimates exist and instead of focusing on a specific estimate, I review the literature. Estimates differ in the micro data employed, which implies differences in time period, frequency (quarterly or annual), unit of observation (plants or firms), and scope (public or private firms, manufacturing or 10

In general equilibrium, wt and rt may be a function of τ . However, equilibrium prices shift input factors in equal proportion to output, thus TFP is not a function of equilibrium prices. 11 The empirical evidence surveyed in Chirinko (2008) centers around a capital-labor substition elasticity of about 0.5, substantially below unit elasticity as implied by Cobb-Douglas technology.

13

TIME TO BUILD AND THE BUSINESS CYCLE

all sectors). Further, the calibration or estimation strategy differs. One approach is to estimate all parameters directly from the micro data, another approach is indirect via targeting moments of the investment rate distribution. Table II summarizes prominent estimates in the literature, see the table notes for details. To make parameters of the productivity process comparable, I have imputed the parameters at quarterly frequency. Using equation (3.5), I compute the loss in aggregate TFP if time to build increases from five to six months. The TFP losses range from moderate 0.01% up to above 1%, and centered around quantitatively important TFP losses in between 0.2% and 0.3%. In other words, the capital misallocation mechanism clearly has quantitative bite.

TABLE II Lengthening in time to build from 5 to 6 months: aggregate TFP loss

Calibration

α

ν

ρx

σx

TFP loss

data source: annual manufacturing plant-level LRD data, 1972-1998

Cooper and Haltiwanger (2006) Khan and Thomas (2008) Khan and Thomas (2013)

0.235 0.256 0.270

0.604 0.640 0.600

0.970 0.963 0.901

0.062 0.012 0.068

0.21% 0.01% 0.30%

data source: annual manufacturing plant-level ASM/CMF data, 1972-2009

Kehrig (2015)

0.290

0.650

0.622

0.138

1.14%

0.900

0.150

0.23%

0.940

0.026

0.04%

data source: quarterly firm-level Compustat data, 1973-2012

Gilchrist et al. (2014)

0.255

0.595

data source: quarterly firm-level IRS data, 1997-2010

Winberry (2016a)

0.210

0.640

Notes: A period is a quarter and I increase τ from 5/3 to 6/3. Whenever the original calibration 1/4 is at annual frequency, I impute quarterly persistence ρx using ρx,annual , and quarterly dispersion σx accordingly. Cooper and Haltiwanger (2006) estimate the revenue production function yjt = θ x ˜jt kjt , and θ = α/(1 − ν) = 0.592 together with the labor cost share aL = ν/(α + ν) = 0.72 pins down α and ν. When considering the production function as maximizing out labor, xjt = x ˜1−ν jt , which allows me to impute σx . Khan and Thomas (2008) assume ν = 0.64 following Prescott (1986), and calibrate α = 0.256 to match an aggregate capital-to-output ratio of 2.3. They calibrate ρx and σx by targeting the share of LRD plants with (large) positive/negative investment or inaction. Khan and Thomas (2013) include as target the cross-sectional investment rate dispersion. Kehrig (2015) estimate all parameters from the data while including six-digit industry fixed effects. Gilchrist et al. (2014) assume a 30% capital cost share and estimate the remaining parameters from the data. Winberry (2016a) assumes ν = 0.64, α + ν = 0.85, and targets moments of the investment rate distribution to calibrate ρx and σx .

14

MATTHIAS MEIER

4. MODELING CYCLICAL FLUCTUATIONS IN TIME TO BUILD This section develops a model which extends the basic real business cycle model in two ways. First, producers of consumption goods vary in their productivity and use producer-specific capital. Second, investment is partially irreversible and specific capital are supplied by engineering firms, which require intermediate goods purchased on a frictional market. Disruptions to the supply chain of intermediate goods, modeled as shocks to the matching technology on the intermediate goods market, cause fluctuations in time to build. 4.1. Households Households value consumption and leisure. I assume the existence of a representative household with separable preferences (4.1)

U (Ct , Lt ) =

Ct1−σ − ψLt , 1−σ

where Ct is consumption and Lt labor supply in period t. I denote by σ the intertemporal substitution elasticity, ψ parametrizes the disutility of working. This period utility function is common in the related literature, see Khan and Thomas (2008) and Bloom et al. (2014) for example, and may be justified by indivisible labor, see Hansen (1985) and Rogerson (1988). The household owns all firms and receives aggregate profits denoted Πt . The problem of the household is (4.2)

max U (Ct , Lt ) s.t. Ct ≤ wt Lt + Πt ,

Ct ,Lt

where wt is the wage. Due to household ownership, firms discount future profits by (4.3)

Qt,t+1 = β

pt+1 , pt

where pt = Ct−σ denotes the marginal utility of consumption. 4.2. Engineering firms and intermediate good suppliers Producers of consumption goods invest in specific capital by signing an order contract with an engineering firm. Engineering firms devise a blueprint which describes the intermediate goods required for the specific capital good. The specific intermediate goods are purchased through a frictional market on which engineering firms

15

TIME TO BUILD AND THE BUSINESS CYCLE

need to search for a matching supplier (think of a single supplier as shortcut for a network of suppliers). Once matched, I assume the engineering firm produces the ordered capital and delivers it at the end of the period. Formally, I assume a continuum of capital sub-markets exogenously differentiated by cost parameter ξ, with cdf G defined for support R+ . Consumption good producers randomly access a sub-market ξ. As will become clear later, this setup serves useful to introduce stochastic fixed capital adjustment costs. The remainder of this subsection focuses on an arbitrary submarket ξ. I assume a large mass of engineering firms (short: engineers) and intermediate good suppliers (short: suppliers). The mass of active engineers be Et , the mass of active suppliers be St . Formally, the matching technology between engineers and suppliers is (4.4)

Mt = mt Etη St1−η .

Stochastic matching efficiency mt follows an AR(1) process in logs (4.5)

log(mt ) = (1 − ρm ) log(µm ) + ρm log(mt−1 ) + σ m m t ,

iid

m t ∼ N (0, 1),

and captures capital supply chain disruptions, which drive time to build fluctuations. I define market tightness as θt = Et /St . The order filling probability for an engineer is qt = mt θtη−1 , and the matching probability for a supplier is θt qt . Once matched, the probability of match separation is exogenously given by χ. Suppliers and engineers need to hire ξ overhead workers to operate. Workers are mobile across sectors so equilibrium wages are equal across sectors. When matched for any given investment order, denoted it , a quantity it of intermediate goods is delivered to the engineer for unit price pSt . Suppliers transform consumption goods into intermediate goods with unit marginal costs. Given the stochastic discount factor in (4.3), the value of an unmatched and matched supplier is (4.6)

S VtS = −ξwt + θt qt JtS + (1 − θt qt )Et [Qt,t+1 Vt+1 ],

(4.7)

S JtS = pSt it − it + (1 − χ)Et [Qt,t+1 Jt+1 ],

respectively. The market for capital good orders is a competitive spot market, and engineers can perfectly commit to a capital good order contract. Further, a consumption good producer can only hire one engineering firm, thus the number of engineers equals the number of orders. Engineers receive price pE t per unit of capital

16

MATTHIAS MEIER

goods delivered. To produce capital, the engineer needs to find a matching supplier. The value of an unmatched and matched engineer is, (4.8)

E VtE = −ξwt + qt JtE + (1 − qt )Et [Qt,t+1 Vt+1 ],

(4.9)

S E JtE = pE t it − pt it + (1 − χ)Et [Qt,t+1 Jt+1 ],

respectively. In equilibrium, engineers make zero profits on the spot market for investment orders, and I assume that capital suppliers satisfy a free entry condition (4.10)

VtE = VtS = 0.

When matched, engineer and capital supplier split the match surplus by Nash bargaining over the unit price pSt , where φ is the engineer’s bargaining weight (4.11)

max(JtE − VtE )φ (JtS − VtS )1−φ . pS t

The two equations in (4.10) together with the solution to (4.11) jointly define the equilibrium values of θt , pSt , pE t . Assumption: Matches are formed for a single period, χ = 1. This assumption is technically convenient because any engineer without order contract is identical, i.e. unmatched. The assumption also follows directly from a strict notion of specific capital goods and intermediate goods. Conversely, under χ < 1 consumption good producers could hire a matched engineer to deliver next period using the same intermediate goods. Further, χ = 1 considerably simplifies equilibrium conditions. The solution to (4.11) is pSt = φ + (1 − φ)pE t and the price engineers receive is pE t = 1+

ξwt 1 φqt it .

Investment expenditures pE t it = it + ft consist

of a size-dependent component with unit price, and a fixed cost component (4.12)

ft =

ξwt . φqt

It further follows that equilibrium tightness is constant θ =

φ 1−φ .

Hence, supply

chain disruptions, i.e. lower matching efficiency mt , unambiguously lower delivery probability qt and thus lengthen average time to build.

TIME TO BUILD AND THE BUSINESS CYCLE

17

4.3. Consumption good producers The consumption good producers in my model resemble the firms in Section 3. The economy is populated by a fixed unit mass of consumption good producers, indexed by j, that are perfectly competitive. Technology is DRS and combines labor and specific capital to produce a homogeneous consumption good (4.13)

α ν yjt = zt xjt kjt `jt ,

where zt denotes aggregate productivity, and xjt idiosyncratic productivity. Idiosyncratic productivity follows a log AR(1) process described by parameters ρx and σx . Aggregate productivity has a deterministic trend, and the model is formulated along the balanced growth path. Along the trend, zt follows a log AR(1) process described by ρz and σz . Productivity shocks are assumed to be independent of each other. Labor adjustment is frictionless and I define the gross cash flow as (4.14)

cfjt ≡ max

`jt ∈R+

n

o

α ν zt xjt kjt `jt − wt ljt .

Capital adjustment is not frictionless. Firm-specific capital evolves over time according to γkjt+1 = (1 − δ)kjt + ijt , where δ denotes the depreciation rate, ijt is investment, and γ denotes constant, aggregate growth of labor productivity. I assume three types of capital adjustment frictions. (1) To invest, consumption good producers need to sign a contract for quantity iojt of (ordered) investment goods with an engineer. The engineer produces specific capital using intermediate goods purchased from suppliers on a frictional market. As a result, investment orders are not delivered instantaneously, but with probability qt . Thus average time to build is 1/qt . I assume that re-adjusting an outstanding order before delivery is prohibitively costly. (2) Investment entails a fixed sunk cost, ft (ξjt ), which naturally arises from the setup of the capital supply side, see equation (4.12). The fixed cost is a function of capital sub-market ξ. Consumption good producers randomly draw ξjt every period, unless they have signed an order contract. Order contracts specify ξ. (3) I assume resale losses of capital, which reflects the notion of specific capital.12 The resale loss of divestment is captured by investment price function pi (io ), which equals 0 ≤ p¯i ≤ 1 if investment io < 0, and which equals one else. Total investment expenditure is 12

I assume reselling is also subject to time to build: Disinvesting producers need to hire an engineer that searches for a capital supplier that transforms the capital into consumption goods.

18

MATTHIAS MEIER

summarized by function ac (4.15)

act (iojt , ξjt ) = (1 − pi (iojt ))iojt + ft (ξjt ).

In the dynamic firm problem, I distinguish between consumer good producers with and without outstanding orders. The idiosyncratic state of firms without outstanding order is described by (kjt , xjt , ξjt ) with probability distribution µV defined for space S V = R+ × R+ × R+ . Firms with outstanding order have idiosyncratic state (kjt , iojt , xjt , ξjt ) with distribution µW defined for S W = R+ × R × R+ × R+ . The V W joint cross-sectional distribution is µt = (µVt , µW t ) defined for S = S × S . The

economy’s aggregate state is denoted by st = (µt , zt , mt ). In the following, I drop time and firm indices and use 0 notation to indicate subsequent periods. The value of a firm without outstanding order is given by V n

o

(4.16)

V (k, x, ξ, s) = max V O (k, x, ξ, s), V N O (k, x, s) ,

(4.17)

V N O (k, x, s) = cf (k, x, s) + E Q(s, s0 )V ((1 − δ)k/γ, x0 , ξ 0 , s0 ) x, k, s ,

(4.18)

V O (k, x, ξ, s) = max W (k, io , x, ξ, s) , o





n



o

i ∈R

where V N O is the value conditional on not ordering investment goods and V O is the firm value conditional on ordering io . The order is chosen to maximize the value of a firm with outstanding order, given by (4.19)

W (k, io , x, ξ, s) = cf (k, x, s) h

i

+ q(s) − ac(io , ξ, s) + E Q(s, s0 )V ((1 − δ)k + io )/γ, x0 , ξ 0 , s0 x, s




h 

i

+ (1 − q(s)) E Q(s, s0 )W (1 − δ)k/γ, io /γ, x0 , ξ, s0 k, io , x, s .


An outstanding order io is delivered with probability q. I assume that investment expenditures, ac, are payed upon delivery. This assumption is conservative. Discounting implies that under earlier payment, the cost of investment would increase in time to build, and thereby amplify the effects of supply chain disruptions. The extensive margin of the capital adjustment decision is described by the thresˆ x, s) that satisfies hold value ξ(k, (4.20)

ˆ x, s), s) = V N A (k, x, s). V A (k, x, ξ(k,

19

TIME TO BUILD AND THE BUSINESS CYCLE

ˆ x, s). Adjustment is optimal whenever fixed adjustment costs ξ < ξ(k, Note that my formulation of the firm problem nests the conventional firm problem with one period time to build whenever q(s) = 1 ∀s. 4.4. Recursive Competitive Equilibrium A recursive competitive equilibrium is a list of value functions (V, W, V S , J S , V E , J E ), ˆ prices (w, ps , pE , f ), market tightness (θ), and the policy functions (C, L, `, io , ξ), cross-sectional distribution (µ) that satisfy ˆ are (i) Consumption good producers: V and W solve (4.16)–(4.19) and (`, io , ξ) the associated policy functions. (ii) Engineers and suppliers: V S , J S , V E , J E solve (4.6)–(4.9). (iii) Households: C and L solve (4.2). (iv) Labor market clearing: Z

Z

`(k, x, s)dµ +

L(s) =

SV

S

1{ξ<ξ(k,x,s)} ξdµV + ˆ

Z

ξdµW

SW

where 1{·} equals one if · is true and zero otherwise. (v) Intermediate good market and capital good market clearing: Free entry and zero profit conditions (4.10) hold and pS is bargained according to (4.11). (vi) Consumption good market clearing: C(s) =Y (s) − I(s) Z

Y (s) =

zxk α `(k, x, s)ν dµ

ZS

I(s) = S VZ

1{ξ<ξ(k,x,s)} q(s)ac(io (k, x, s), ξ, s)dµV ˆ

+

q(s)ac(io , ξ, s)dµW .

SW

(vii) Consistency: The evolution of µ is consistent with policy functions. 4.5. Solution The recursive competitive equilibrium is not computable, because the solution depends on the infinite-dimensional distribution µ and its law of motion. Instead, I solve for an approximate equilibrium adopting the algorithm in Campbell (1998) and Reiter (2009) to my model. The general idea is to use global approximation

20

MATTHIAS MEIER

methods with respect to the individual states, but local approximation methods with respect to the aggregate states. I solve the steady state of my model using projection methods and perturb the model locally around the steady state to solve for the model dynamics in response to aggregate shocks. Compared to the Krusell-Smith algorithm, see Krusell et al. (1998), the perturbation approach does not require simulating the model dynamics in order to update the parameters of the forecasting rules. Further it can handle a large number of aggregate shocks. Terry (2015) compares the Krusell-Smith algorithm with the Campbell-Reiter algorithm for a Khan and Thomas (2008) economy. He finds that the Campbell-Reiter algorithm is more than 100 times faster. Ahn et al. (2016) combine the Campbell-Reiter algorithm to compute aggregate dynamics for a general class of heterogeneous agent economies in continuous time. More closely related to this paper, Winberry (2016a) uses (and extends) the Campbell-Reiter algorithm to solve a variation of the Khan and Thomas (2008) economy. In Appendix D, I show that equilibrium conditions can be simplified to reduce the idiosyncratic state space. I further explain in more detail the Campbell-Reiter algorithm applied to my model and contrast it with the Krusell-Smith algorithm, likewise applied to my model. Finally, I explain how the projection method is applied to the firm problem. 5. CALIBRATION The central mechanism for the transmission of supply chain disruptions is the allocation of capital across firms. At the core of my calibration strategy is to match salient features of establishment-level investment dynamics. While the establishmentlevel data is reported annually, the length of a period in the model should reflect the frequency at which economic agents make decisions, see Christiano and Eichenbaum (1987). I set the length of a period to a month. I set the discount factor β to match an annual risk-free rate of 4%. I assume logutility in consumption, σ = 1. The parameter governing the household’s disutility from work, ψ, is calibrated to match one third of time spent working. On the intermediate good markets, I assume symmetric Nash bargaining between engineers and suppliers, φ = 0.5.13 The delivery probability under φ = 0.5 is qt = mt , thus I do not need to calibrate matching function elasticity η. To calibrate mean, persistence, and 13

In fact, I can choose any φ ∈ (0, 1) without any impact on my results, as long as I re-calibrate µm to match average time to build.

TIME TO BUILD AND THE BUSINESS CYCLE

21

variance of mt , I target the corresponding first and second moments of the empirical baseline measure of time to build. In the model, I compute time to build consistent with the data by using the backlog ratio. Investment corresponds to shipments and I compute aggregate order backlog according to Z

B(s) = SV

1{ξ<ξ(k,x,s)} ac(io (k, x, s), ξ, s)dµV + ˆ

Z

ac(io , ξ, s)dµW .

SW

I set the mean matching efficiency to match an average time to build of 5.5 months in line with the data. Given that the backlog ratio has a weak, non-linear time trend, I detrend the monthly time series using a low-frequency HP filter with λ = 8, 100, 000 and fit persistence and standard deviations to the cyclical component.14 To calibrate the parameters of the production technology, notably α ν, ρx , σx , I target moments from the manufacturing plant-level Longitudinal Research Database (LRD) based on Cooper and Haltiwanger (2006). For the details I refer to the notes of Table II. The calibrated parameter values are well within the range of estimates surveyed in Table II and imply moderate long-run losses in aggregate TFP. I calibrate δ to match an annual depreciation rate of 10% and γ to satisfy an annualized aggregate productivity growth of 1.6%, see Khan and Thomas (2008). I assume that G, the distribution fixed capital adjustment costs (ξ), is uniform ¯ To calibrate ξ¯ as well as resale losses p¯i , with zero lower bound and upper bound ξ. I target the share of plants with investment rates above 20% and below -20% in the data, respectively. For consistency with the calibration of pararmeters describing the productivity process, I use the share of spike investment rates as documented in Cooper and Haltiwanger (2006) based on the LRD. To account for differences in time frequency, I aggregate simulated data to annual frequency when computing investment spike shares. The two adjustment cost parameters allow me to match the observed 19% share of plants with positive spikes and the 2% share of negative spikes. The fixed cost is important to generate fat tails, while the resale loss is particularly important in generating the large difference between positive and negative spikes. Appendix E provides more details on the calibration and shows that the calibrated model closely matches several non-targeted moments. 14

The low-frequency HP filter, translated to quarterly frequency with λ = 100, 000, has also been used, for example, in Shimer (2005) and Shimer (2012). This filter is a much lower-frequency filter than the common specification (quarterly λ = 1, 600) in business cycle analysis. The common filter, however, seems to remove much of the cyclical volatility in the variable of interest.

22

MATTHIAS MEIER TABLE III Quarterly model calibration

Description

Parameter

Value

Households Discount factor Intertemporal elasticity Preference for leisure

β σ ψ

0.99 1.000 2.250

Engineers and suppliers Bargaining power Mean matching efficiency Persistence of matching efficiency Dispersion of matching efficiency

φ µm ρm σm

0.500 0.546 0.959 0.144

Consumption good producers Output elasticity of capital Output elasticity of labor Depreciation rate Aggregate productivity growth Idiosyncratic persistence Idiosyncratic dispersion Aggregate persistence Aggregate dispersion Capital resale loss Fixed adjustment cost upper bound

α ν δ γ ρx σx ρz σz p¯i ξ¯

0.235 0.604 0.025 1.004 0.970 0.062 0.950 0.007 0.850 0.001

6. MACROECONOMIC EFFECTS OF SUPPLY CHAIN DISRUPTIONS This section discusses the quantitative impact of capital supply chain disruptions. In short, disruptions that lengthen time to build by one month lower GDP by up to one percent. Furthermore these shocks explain about half of the decline in output and investment during the early 1990s recession and the 2007-09 Great Recession. Figure 3 shows the macroeconomic responses to a disruption that lengthens time to build from 5.5 months to 6.5 months on impact. The one-month increase corresponds to one standard deviation of time to build. The shock causes substantial fluctuations in output, investment, and consumption. Investment is most directly affected by the disruption. It falls by 8 percent on impact and remains depressed by 2 percent two years after the shock. Output falls by 1 percent on impact and converges back reaches its trough of 0.55 percent five quarters later. Measured aggregate total factor

23

TIME TO BUILD AND THE BUSINESS CYCLE

productivity declines gradually and reaches its trough at below 0.2 percent 6 quarters after the shock. Figure 3: Responses to a supply chain disruption 7

2 0

6.5

-2 6

-4

5.5

-6

5

0

4

8

12

16

20

24

0.8

0.8

0.4

0.4

0

0

-0.4

-0.4

-0.8

-0.8

-1.2

Total response Direct channel

-8

0

4

8

12

16

20

24

0.5

-1.2

0

4

8

12

16

20

24

0

4

8

12

16

20

24

0.2

0 0

-0.5 -1

-0.2

-1.5 -2

0

4

8

12

16

20

24

-0.4

0

4

8

12

16

20

24

Notes: The sub-figures show impulse responses to a supply chain disruption that lengthens time to build by one month starting from steady state. ‘Direct channel’ denotes the impulse responses when aggregate TFP changes are eliminated through opposing aggregate productivity (z) shocks. Aggregate TFP is computed as T F P = log(Yt ) − α log(Kt ) − ν log(Lt ).

The aggregate effects of adverse shock to the matching technology are explained

24

MATTHIAS MEIER

by a direct and an indirect channel. The direct channel captures that longer time to build delays delivery of outstanding investment orders and thus reduces investment and output. The indirect channel captures that longer time to build affects firm-level investment policies: firms invest less frequently and, if they invest, their investment reflect less their contemporaneous productivity. In turn, the alignment between firm-level capital and productivity weakens. Thus, longer time to build lowers measured aggregate total factor productivity. A more detailed discussion of the indirect channel is provided in Section 3. To understand the relative quantitative importance of the two transmission channels, I suggest a simple exercise. While the indirect channel affects measured aggregate total factor productivity, the direct channel has no impact on measured productivity. To isolate the direct channel, I compute a series of exogenous shocks to aggregate productivity that exactly offset the effects on measured aggregate productivity. Measured aggregate TFP then remains at its steady state level. The effects of the direct channel are the macroeconomic responses to the joint occurrence of the initial match efficiency shock and the series of productivity shocks. The resulting ‘direct channel’ responses are shown as dotted lines in Figure 3. Note that this exercise is an approximation, which gives an upward bias to the effects of the direct channel. The reason is that the series of productivity shock only offsets the effects on realized aggregate productivity, but not the effects on expected future aggregate productivity. The direct channel is central to understand the immediate responses, while the medium-term effects are to a larger extent explained by the indirect channel operating through capital misallocation. In particular, the two-year response of output and consumption is importantly shaped by misallocation.15 The direct channel is most important on impact of the shock because in subsequent periods firms adjust their investment policies. Firms prepone investment orders as delivery takes longer, see Figure 12 in Appendix F. Capital adjustment frequency falls consistent with wait-and-see behavior. Note that I evaluate the quantitative impact of shocks to the matching technology in general equilibrium. Accounting for general equilibrium effects is important, because household consumption smoothing motives may substantially dampen the investment and output responses that would arise in partial equilibrium, see Khan and Thomas (2008). The initial increase in consumption reflects a general equilibrium 15

The prominent role of aggregate productivity in my model may be linked to the finding in Chari et al. (2007) on the efficiency wedge.

TIME TO BUILD AND THE BUSINESS CYCLE

25

mechanism. Since prices are flexible in my model, the intratemporal household optimality condition dictates that consumption has to increase initially in response to the initial decrease in investment, because the capital input in production is predetermined and labor demand falls. Figure 13 shows the macroeconomic responses to a supply chain disruption in a setting identical to that of Figure 3 except that goods market do not clear. While the baseline model is a closed-economy model, this setup may be justified as small open economy. For the US economy, the truth should be somewhere in between. Figure 13 shows substantially larger responses, with GDP falling by up to 1.5%. In addition, this model features more propagation and the trough output response is obtained only 2 years after the shock hits the economy. The responses in Figure 3 show quantitatively important and persistent effects of capital supply chain disruptions. Next, I assess the importance of supply chain disruptions to understand past business cycles. To this end, I compute a matching technology shock series that fits the empirical time to build series. This confines my analysis to the period from 1968 through 2015. Using the model, I compute the time series for output, investment, consumption, and employment. To be clear, fluctuations in these series are only driven by disruption shocks. To make the quarterly series comparable to the data, I HP filter both the simulated series and their empirical counterparts using the same low-frequency filter I use in the calibration. More details on the empirical time series are provided in Section 2.2. Figure 4 plots the model-implied time series against their empirical counterparts. Three observations stand out. First, NBER recession periods (grey-shaded areas) are matched by periods where shocks to the matching technology induce below-trend output growth. Second, supply chain disruptions explain an important share of the observed business cycle variations. These shocks alone explains a drop in investments of more than 8% during the Great Recession and the early 1990s recession, compared to a drop of 16% in the data. For output, the model also explains about half of the empirically observed drop during these two recessions and for consumption it is almost a quarter. Third, in the model GDP tends to start falling before the recession, which reflects the corresponding pattern in the time to build series. Outside the present model, a possible explanation for this discrepancy may be that time to build also increases in response to increased investment demand, so attributing all the precrisis increase in time to build to supply chain disruptions might be misleading.

26

MATTHIAS MEIER

Figure 4: Role of supply chain disruptions in understanding past business cycles

16

4

8 0 0 -4

Model Data

-8 1970 1980 1990 2000 2010

-8 -16 1970 1980 1990 2000 2010

4

4

0

0

-4

-4

-8 1970 1980 1990 2000 2010

-8 1970 1980 1990 2000 2010

Notes: Above time series are computed matching the empirically observed (filtered) movements in time to build through capital supply chain disruption shocks and otherwise using the baseline model calibration. Grey-shaded areas indicate NBER recession dates.

7. CONCLUSION This paper contributes to our understanding of business cycles by establishing countercyclical time to build as a new stylized fact. It then asks: What are the business cycle implications of fluctuations in time to build? To address this question, I first provide time series evidence, which suggests quantitatively large responses to supply chain disruptions. Second, I develop a dynamic stochastic general equilibrium model, in which capital good markets are characterized by search frictions. Fluctuations in time to build are driven by shocks to the matching technology, which reflect capital supply chain disruptions. Calibrating the model to US data, I show that the empirically observed fluctuations in time to build are quantitatively of first-order importance for business cycles. Of particular quantitative importance

TIME TO BUILD AND THE BUSINESS CYCLE

27

is the interaction of time to build and firm investment policies leading to capital misallocation. It should be a direct consequence of this paper to investigate the sources of fluctuations in time to build. In particular, it may be useful to study capital good supply networks. Small changes at critical points in such networks, for example the exit of an important supplier, could have non-trivial aggregate implications for time to build. A complementary explanation may revolve around trade credit. While the empirical evidence rejects an important role for aggregate financial conditions, trade credit in capital good production networks might be important to understand the observed time to build fluctuations. For example, suppose capital suppliers produce subject to cash-in-advance constraints. During recessions short-run liquidity in the form of trade credit may become scarce. As a result, suppliers may need to slow down production despite long order books.

REFERENCES Ahn, S., G. Kaplan, B. Moll, and T. Winberry (2016): “No More Excuses! A Toolbox for Solving Heterogeneous Agent Models with Aggregate Shocks,” Sed presentation slides. Bachmann, R. and C. Bayer (2013): “‘Wait-and-See’ business cycles?” Journal of Monetary Economics, 60, 704 – 719. Bachmann, R., R. J. Caballero, and E. M. R. A. Engel (2013): “Aggregate Implications of Lumpy Investment: New Evidence and a DSGE Model,” American Economic Journal: Macroeconomics, 5, 29–67. Bachmann, R. and G. Moscarini (2011): “Business Cycles and Endogenous Uncertainty,” 2011 Meeting Papers 36, Society for Economic Dynamics. Baumeister, C. and J. D. Hamilton (2015): “Sign Restrictions, Structural Vector Autoregressions, and Useful Prior Information,” Econometrica, 83, 1963–1999. Bloom, N. (2009): “The Impact of Uncertainty Shocks,” Econometrica, 77, 623–685. Bloom, N., M. Floetotto, N. Jaimovich, I. Saporta-Eksten, and S. J. Terry (2014): “Really Uncertain Business Cycles,” Working Paper 14-18, US Census Bureau Center for Economic Studies. Boehm, C. E., A. Flaaen, and N. Pandalai-Nayar (2015): “Input Linkages and the Transmission of Shocks: Firm-Level Evidence from the 2011 Tohoku Earthquake,” Working Papers 15-28, Center for Economic Studies, U.S. Census Bureau.

28

MATTHIAS MEIER

Campbell, J. (1998): “Entry, Exit, Embodied Technology, and Business Cycles,” Review of Economic Dynamics, 1, 371–408. Carvalho, V. M., M. Nirei, Y. U. Saito, and A. Tahbaz-Salehi (2016): “Supply Chain Disruptions: Evidence from the Great East Japan Earthquake,” CEPR Discussion Papers 11711, C.E.P.R. Discussion Papers. Chari, V. V., P. J. Kehoe, and E. R. McGrattan (2007): “Business Cycle Accounting,” Econometrica, 75, 781–836. Chirinko, R. S. (2008): “σ: The long and short of it,” Journal of Macroeconomics, 30, 671 – 686, the CES Production Function in the Theory and Empirics of Economic Growth. Christiano, L. J. and M. Eichenbaum (1987): “Temporal aggregation and structural inference in macroeconomics,” Carnegie-Rochester Conference Series on Public Policy, 26, 63–130. Christiano, L. J., M. Eichenbaum, and C. L. Evans (1999): “Chapter 2 Monetary policy shocks: What have we learned and to what end?” Elsevier, vol. 1 of Handbook of Macroeconomics, 65 – 148. ——— (2005): “Nominal Rigidities and the Dynamic Effects of a Shock to Monetary Policy,” Journal of Political Economy, 113, 1–45. Coibion, O. (2012): “Are the Effects of Monetary Policy Shocks Big or Small?” American Economic Journal: Macroeconomics, 4, 1–32. Conley, T. G., C. B. Hansen, and P. E. Rossi (2012): “Plausibly Exogenous,” The Review of Economics and Statistics, 94, 260–272. Cooper, R. W. and J. C. Haltiwanger (2006): “On the Nature of Capital Adjustment Costs,” Review of Economic Studies, 73, 611–633. Doms, M. E. and T. Dunne (1998): “Capital Adjustment Patterns in Manufacturing Plants,” Review of Economic Dynamics, 1, 409–429. Fajgelbaum, P., E. Schaal, and M. Taschereau-Dumouchel (2014): “Uncertainty Traps,” Working Paper 19973, National Bureau of Economic Research. Fernald, J. G. (2014): “A quarterly, utilization-adjusted series on total factor productivity,” Working Paper Series 2012-19, Federal Reserve Bank of San Francisco. Gafarov, B., M. Meier, and J. L. Montiel Olea (2016): “Delta-Method Inference for a Class of Set-Identified SVARs,” mimeo, Columbia University. Gali, J. (1999): “Technology, Employment, and the Business Cycle: Do Technology Shocks Explain Aggregate Fluctuations?” American Economic Review, 89, 249–271.

TIME TO BUILD AND THE BUSINESS CYCLE

29

Gilchrist, S., J. W. Sim, and E. Zakrajšek (2014): “Uncertainty, Financial Frictions, and Investment Dynamics,” Working Paper 20038, National Bureau of Economic Research. Hansen, G. D. (1985): “Indivisible labor and the business cycle,” Journal of Monetary Economics, 16, 309 – 327. Justiniano, A., G. E. Primiceri, and A. Tambalotti (2010): “Investment shocks and business cycles,” Journal of Monetary Economics, 57, 132 – 145. Kalouptsidi, M. (2014): “Time to Build and Fluctuations in Bulk Shipping,” American Economic Review, 104, 564–608. Kehrig, M. (2015): “The Cyclical Nature of the Productivity Distribution,” Working papers, Center for Economic Studies, U.S. Census Bureau. Kehrig, M. and N. Vincent (2016): “Do Firms Mitigate or Magnify Capital Misallocation? Evidence from Plant-Level Data,” mimeo, University of Texas at Austin. Khan, A. and J. K. Thomas (2008): “Idiosyncratic Shocks and the Role of Nonconvexities in Plant and Aggregate Investment Dynamics,” Econometrica, 76, 395–436. ——— (2013): “Credit Shocks and Aggregate Fluctuations in an Economy with Production Heterogeneity,” Journal of Political Economy, 121, 1055 – 1107. Kilian, L. and D. P. Murphy (2012): “Why Agnostic Sign Restrictions Are Not Enough: Understanding the Dynamics of Oil Market VAR Models,” Journal of the European Economic Association, 10, 1166–1188. Krause, M. U., D. Lopez-Salido, and T. A. Lubik (2008): “Inflation dynamics with search frictions: A structural econometric analysis,” Journal of Monetary Economics, 55, 892–916. Krusell, P., A. A. Smith, and Jr. (1998): “Income and Wealth Heterogeneity in the Macroeconomy,” Journal of Political Economy, 106, 867–896. Kurmann, A. and N. Petrosky-Nadeau (2007): “Search Frictions in Physical Capital Markets as a Propagation Mechanism,” Cahiers de recherche 0712, CIRPEE. Kydland, F. E. and E. C. Prescott (1982): “Time to Build and Aggregate Fluctuations,” Econometrica, 50, 1345–70. Mertens, K. and M. O. Ravn (2011): “Understanding the aggregate effects of anticipated and unanticipated tax policy shocks,” Review of Economic Dynamics, 14, 27 – 54, special issue: Sources of Business Cycles. Mortensen, D. and C. A. Pissarides (1994): “Job Creation and Job Destruction in the Theory of Unemployment,” Review of Economic Studies, 61, 397–415.

30

MATTHIAS MEIER

Nalewaik, J. and E. Pinto (2015): “The response of capital goods shipments to demand over the business cycle,” Journal of Macroeconomics, 43, 62 – 80. Oh, H. and C. Yoon (2016): “Time to build and the real-options channel of residential investment,” mimeo, Vanderbilt University. Ottonello, P. (2015): “Capital Unemployment, Financial Shocks, and Investment Slumps,” Tech. rep., University of Michigan. Prescott, E. C. (1986): “Theory ahead of business-cycle measurement,” Carnegie-Rochester Conference Series on Public Policy, 25, 11 – 44. Ramey, V. A. and M. D. Shapiro (1998): “Costly capital reallocation and the effects of government spending,” Carnegie-Rochester Conference Series on Public Policy, 48, 145–194. Ramey, V. A. and D. J. Vine (2010): “Oil, Automobiles, and the U.S. Economy: How Much have Things Really Changed?” Working Paper 16067, National Bureau of Economic Research. Reiter, M. (2009): “Solving heterogeneous-agent models by projection and perturbation,” Journal of Economic Dynamics and Control, 33, 649 – 665. Rogerson, R. (1988): “Indivisible labor, lotteries and equilibrium,” Journal of Monetary Economics, 21, 3 – 16. Romer, C. D. and D. H. Romer (2004): “A New Measure of Monetary Shocks: Derivation and Implications,” American Economic Review, 94, 1055–1084. Schmitt-Grohe, S. and M. Uribe (2004): “Solving dynamic general equilibrium models using a second-order approximation to the policy function,” Journal of Economic Dynamics and Control, 28, 755–775. Sedláček, P. (2014): “Match efficiency and firms’ hiring standards,” Journal of Monetary Economics, 62, 123–133. ——— (2016): “The aggregate matching function and job search from employment and out of the labor force,” Review of Economic Dynamics, 21, 16–28. Shimer, R. (2005): “The Cyclical Behavior of Equilibrium Unemployment and Vacancies,” American Economic Review, 95, 25–49. ——— (2012): “Reassessing the Ins and Outs of Unemployment,” Review of Economic Dynamics, 15, 127–148. Terry, S. J. (2015): “Alternative Methods for Solving Heterogeneous Firm Models,” mimeo, Boston University. Winberry, T. (2016a): “Lumpy Investment, Business Cycles, and Stimulus Policy,” mimeo, Chi-

TIME TO BUILD AND THE BUSINESS CYCLE

31

cago Booth. ——— (2016b): “A Toolbox for Solving and Estimating Heterogeneous Agent Macro Models,” mimeo, Chicago Booth.

32

MATTHIAS MEIER

Appendix

33

TIME TO BUILD AND THE BUSINESS CYCLE APPENDIX A: TIME TO BUILD FLUCTUATIONS

My ex-post measure of time to build captures the time which new orders remain in the capital good producers’ order books using ex-post realizations of shipments (instead of current shipments). To be precise, I compute the lowest number of future periods required to deplete the given order backlog

T] T B t ≡ min

bτ c X

τ

!2 St+j + (τ − bτ c)(St+bτ c+1 − St+bτ c ) − Bt

,

j=1

where b·c denotes the floor function. The second term in above formula captures a linear interpolation of shipments between two periods, by which the ex-post time to build measure becomes continuous. Figure 5 compares my baseline measure with the ex-post measure of time to build. Differences between the two series are barely visible, which mainly reflects the high auto-correlation of monthly shipments.

Figure 5: Time to Build

2

4

6

8

10

Months

1970

1980

1990 Baseline

2000

2010

Ex-post

Notes: Time to build is measured as the ratio of order backlog to monthly shipments, for non-defense equipment goods. Shaded, gray areas indicate NBER recession dates.

The two panels of Figure 6 show the individual series defining the order stock-flow equation. The series are plotted in nominal values because the stock-flow equation is defined over nominal values.

34

MATTHIAS MEIER

800

Figure 6: Responses of investment orders to an adverse match efficiency shock Bln. USD

0

200

400

600

Order Backlog

1970

1980

1990

2000

2010

1990

2000

2010

Bln. USD

0

20

40

60

80

Shipments New Orders

1970

1980

Notes: The time series for order backlog, shipments, and new orders refer to the non-defense equipment goods sector and are expressed in nominal values. Shaded, gray areas indicate NBER recession dates.

35

TIME TO BUILD AND THE BUSINESS CYCLE

Figure 7: Sector-level time to build Industrial Machinery and Equipment

Electronic and Other Electrical Equipment Months

4 2

1

2

3

3

4

5

5

6

Months

1970

1980

1990

2000

2010

1970

Transportation Equipment

1980

1990

2000

2010

Instruments and Related Products Months

1

0

1.5

5

2

10

2.5

15

Months

1970

1980

1990

2000

2010

1970

1980

1990

2000

2010

Notes: Time to build is measured as the ratio of order backlog to monthly shipments, for non-defense equipment goods. Shaded, gray areas indicate NBER recession dates.

36

MATTHIAS MEIER

Figure 8: Firm-level time to build

0

0

2

5

4

6

10

8

10

Median months time to build

15

Mean months time to build

1970

1980

1990

Weighted

2000

Unweighted

2010

1970

Continuing

1980

1990

Weighted

2010

Continuing

15 10 5 0

0

5

10

15

20

IQR months time to build

20

Std. months time to build

2000

Unweighted

1970

1980

Weighted

1990

2000

Unweighted

2010

Continuing

1970

1980

Weighted

1990

2000

Unweighted

2010

Continuing

Notes: Firm-level time to build as the ratio of order backlog to sales for equipment goods using annual Compustat data. The figures show the cross-sectional mean, median, standard deviation, and interquartile range, when weighted by firm-level sales, unweighted, and again weighted but only considering continuing firms for which order backlog is observed at least for 20 years. Shaded, gray areas indicate NBER recession dates.

37

TIME TO BUILD AND THE BUSINESS CYCLE APPENDIX B: ROBUSTNESS OF THE STRUCTURAL VAR RESULTS B.1. Alternative identification scheme and first differences

First, I investigate the results under an alternative identification assumption. While the baseline identification scheme tends to be conservative, its restrictions are stronger than the restrictions of the general equilibrium model. As alternative identification, I suggest to have the time to build shock ‘ordered first’. This term refers to the ordering of variables in the VAR. It means that time to build shocks can contemporanously affect all other variables in the VAR, but no shock other than time to build shocks can affect time to build contemporaneously. Figure 9 shows that the baseline identification implies smaller macroeconomic respones to time to build shocks compared to the alternative identification, albeit the differences are not large. Impulse responses under the alternative identification remain significant.

Figure 9: Impulse responses to a one standard deviation time to build shock (model in levels with linear time trend, alternative identification schemes) GDP (in %)

Time to build (in months) 5.8

0.2 0

5.7

-0.2 5.6

-0.4

Baseline Ordered First

5.5

-0.6 0

4

8

12

16

20

24

0

Investment (in %)

4

8

12

16

20

24

Consumption (in %)

0.5

0.2

0

0

-0.5 -0.2 -1 -0.4

-1.5 -2

-0.6 0

4

8

12

16

Quarters

20

24

0

4

8

12

16

20

24

Quarters

Notes: Solid lines show (selected) impulse responses to a time to build shock under the baseline identification scheme. Dashed lines show the impulse responses under the alternative identification scheme, in which time to build is ‘ordered first’. Shaded, gray areas illustrate the 90% confidence intervals associated with the alternative identification scheme.

Figure 10 shows the cumulative impulse responses when estimating a VAR, in which all variables enter in first differences and the linear time trend is dropped. At the same time, the figure compares the two identification schemes. The differences of the impulse responses across identification schemes appears negligible. The important finding is that the impulse responses are similar to the ones in Figure 9. While I assumed a linear time trend for the latter, the findings on time to build shocks

38

MATTHIAS MEIER

appear robust to non-linear time trends.

Figure 10: Cumulative impulse responses to a one standard deviation time to build shock (model in first differences, two alternative identification schemes) Time to build (in months)

GDP (in %) 0.2

6.2

0

6

-0.2

5.8

-0.4 -0.6

5.6

Baseline Ordered First

-0.8

5.4

-1 0

4

8

12

16

20

24

0

Investment (in %)

4

8

12

16

20

24

Consumption (in %) 0.2

0

0 -0.2

-1

-0.4 -0.6

-2

-0.8 -3

-1 0

4

8

12

16

20

24

0

Quarters

4

8

12

16

20

24

Quarters

Notes: Solid lines show (selected) cumulative impulse responses to a time to build shock under the baseline identification scheme. Dashed lines show the impulse responses under the alternative identification scheme, in which time to build is ‘ordered first’. Shaded, gray areas illustrate the 90% confidence intervals associated with the alternative identification scheme.

B.2. Elasticity bounds In this subsection, I propose a new approach to provide robustness for point-identified structural VAR models in a frequentist setup. Structural VAR models, such as Gali (1999), Christiano et al. (2005), and Bloom (2009), impose various zero restrictions on contemporaneous and long-run responses to obtain point identification. As robustness, I propose to replace some or all of the zero restrictions by bounds on the elasticity with respect to the shock of interest.16 For example, instead of assuming an uncertainty shock does not contemporaneously affect GDP, as robustness I would restrict the elasticity of GDP with respect to a change in uncertainty due to an uncertainty shock to be bounded between ±c%. This nests the point-identified model in the limit case when all bounds are zero (c = 0). The structural VAR model is no longer point-identified when replacing a zero restriction with strictly positive bounds on the elasticities (c > 0). 16

Elasiticity bounds have recently gained popularity in the Bayesian structural VAR literature, see, e.g., Kilian and Murphy (2012) and Baumeister and Hamilton (2015).

39

TIME TO BUILD AND THE BUSINESS CYCLE

I implement this robustness exercise using the results in Gafarov, Meier, and Montiel Olea (2016), which provide inference for set-identified structural VAR models. Formally, to apply their results, I need to assume that for a given IRF either the lower and upper elasticity bound may not hold jointly. Notice that confidence sets are estimated based on Delta method inference. In fact, bootstrap inference is not necessarily valid here because the endpoints of the identified sets are not fully differentiable. The suggested robustness is similar to Conley et al. (2012) which proposes as robustness to relax the exclusion restriction when using IV methods. I suggest the following robustness for the conservative baseline identification. Instead of zero restrictions on contemporaneous responses, I constrain the elasticity of all variables (except for the backlog ratio) with respect to the match efficiency shock to be between -1% and +1%, see Table IV. For an increase in the backlog ratio of 2.5%, the contemporaneous responses are bound to be between -0.025% and +0.025%. TABLE IV identification schemes: constraints on contemporaneous elasticities

Baseline Robustness

TTB

GDP

Inv

Con

CPI

Wag

FFR

LaP

+ +

0 ±1%

0 ±1%

0 ±1%

0 ±1%

0 ±1%

0 ±1%

0 ±1%

Notes: +/0/±1% indicate that the elasticity is constrained to be positive/exactly zero/between -1% and +1%, respectively. The contemporaneous elasticity of variable i and time to build in response to time to build shocks is given by (e0i B1 )/(e01 B1 ), where ei is the i-th column of the identity matrix I8 . TTB: Time to build, GDP: Real GDP, Con: Real Consumption, Inv: Real Investment, CPI : Consumer Prices, Wag: Real Wage, FFR: Federal Funds Rate, LaP: Labor Productivity. Figure 11 shows the resulting impulse responses under the robustness identification scheme. Instead of a single impulse response, there is an interval with admissible impulse responses (dotted lines). The confidence set is adjusted accordingly based on Gafarov, Meier, and Montiel Olea (2016). Notice that the main findings of the baseline model in Figure 11 are ‘robust’ in the sense that the declines in GDP, investment, and consumption remain significant.

40

MATTHIAS MEIER

Figure 11: Impulse responses to a one standard deviation time to build shock (model in levels with linear time trend, two alternative identification schemes) GDP (in %)

Time to build (in months) 5.8

0.2 0

5.7

-0.2 5.6

-0.4

Baseline Bounds of Identi-ed Set

5.5

-0.6 0

4

8

12

16

20

24

0

Investment (in %)

4

8

12

16

20

24

Consumption (in %)

0.5

0.2

0

0

-0.5 -0.2 -1 -0.4

-1.5 -2

-0.6 0

4

8

12

16

Quarters

20

24

0

4

8

12

16

20

24

Quarters

Notes: Solid lines show (selected) responses to a time to build shock under the baseline identification scheme. Dashed lines show the bounds of the identified set under elasticity constraints, see Table IV. Shaded, gray areas illustrate the 90% confidence intervals for the identified sets.

41

TIME TO BUILD AND THE BUSINESS CYCLE B.3. Correlation of identified capital supply chain disruptions with other shocks

The importance of capital supply chain disruptions could potentially reflect other structural shocks that are not well identified in my model. To address this concern, I correlate my identified capital supply chain disruption series with various business cycles shocks, constructed in a number of papers outside my empirical framework. These business cycles shocks include direct estimates of productivity shocks and numerous policy shocks. Table V provides the correlation of the capital supply chain disruption series with leads and lags of the external business cycle shock series. By and large, I find capital supply chain disruptions to be uncorrelated with external shocks. This finding further supports to the importance of exogenous shocks to capital supply. TABLE V Correlogram of time to build shocks with external business cycle shocks

TFP UA-TFP UA-TFP-I UA-TFP-C MP Oil Defense Tax

-4

-3

-2

quarterly lags/leads -1 0 +1

+2

+3

+4

-0.07 -0.09 -0.03 -0.10 0.02 -0.01 -0.12 0.02

-0.05 -0.13∗ -0.15∗∗ -0.08 0.08 0.00 -0.15∗∗ -0.06

0.00 0.04 0.02 0.04 0.04 -0.02 -0.02 0.02

0.00 0.07 0.11 0.03 0.02 -0.02 -0.03 0.04

-0.07 -0.04 0.03 -0.07 0.04 0.01 -0.08 0.09

-0.08 0.00 0.00 0.00 0.09 0.09 -0.01 0.04

0.00 0.07 0.05 0.05 0.11 -0.04 -0.10 -0.02

-0.03 -0.05 0.03 -0.09 -0.01 0.01 -0.16∗∗ 0.01

-0.04 -0.06 -0.03 -0.07 0.06 0.00 -0.04 -0.13

Note: The table shows the correlation of capital supply chain disruptions with various shock series at lags/forwards between -4 and +4 quarters. */**/*** denote 10%/5%/1% significance levels, respectively. Productivity shock series are from Fernald (2014): TFP, Utilization-Adjusted (UA) TFP, UA-TFP in equipment and durables, and UA-TFP in non-durables. Monetary policy shocks (MP) are based on Romer and Romer (2004) and Coibion (2012). Oil price shocks are based on Ramey and Vine (2010). Surprise defense expenditures as fiscal shocks are from Ramey and Shapiro (1998), and tax shocks from Mertens and Ravn (2011). The identified capital supply chain disruption series appears not to reflect investment-specific productivity shocks along the lines of Justiniano et al. (2010) and Justiniano et al. (2010). Beyond the evidence in Table V, this conclusion is supported by the finding that extending my VAR model by the relative price of investment goods only marginally affects the results presented here. By the same argument, identified capital supply chain disruptions appear not to reflect uncertainty shocks. The identified shocks further do not appear to reflect changes in aggregate financial conditions. This conclusion is based on the following result. The VAR exercise in Gilchrist et al. (2014) finds that uncertainty shocks are crucially transmitted through credit spreads. When replacing uncertainty by time to build, I do not find evidence for the transmission of time to build shocks through credit spreads.

42

MATTHIAS MEIER APPENDIX C: CLOSED-FORM RESULTS FOR AGGREGATE TFP

This section provides some details on the derivation of aggregate TFP. Profit-maximizing labor demand and the profit function after maximizing out labor are

 `jt =

α νxjt kjt wt

1  1−ν

π ˜jt (k) = (1 − ν)

,



ν wt

ν  1−ν

1

α

1−ν xjt k 1−ν − rt k.

Capital is adjusted subject to τ periods time to build, i.e. firms choose in period t how much capital to use in period t + τ . Denoting by Et the conditional expectation operator, the firm problem is kj,t,t+τ = arg max Et [˜ πjt+τ (k)]. k

1 1−ν ], the optimal capital policy follows as Defining x ˜jt = Et [xjt+τ

kj,t,t+τ =

1−ν   1−α−ν 

α rt

ν wt

ν  1−α−ν

1−ν 1−α−ν x ˜jt .

The firm’s employment policy can be simplifed to `j,t,t+τ =

α    1−α−ν

α rt

1−α−ν+αν  (1−ν)(1−α−ν)

ν wt

1

α

1−ν 1−α−ν xjt+τ x ˜jt ,

and the firm’s output follows as yj,t,t+τ =

α    1−α−ν

α rt

ν wt

ν  1−α−ν

1

α

1−ν 1−α−ν xjt+τ x ˜jt .

Denoting by E the unconditional expectation operator, I define two moments of productivity



1−ν

1−α−ν ˜1 = E x X ˜jt 1

h



 = exp α

1−α−ν ˜ 2 = E x 1−ν x X jt+τ ˜jt

i

where στ2 = V[log xjt+τ | log xjt ] = Kt,t+τ =

α rt

α   1−α−ν 

 = exp 1−ρ2τ x 1−ρ2 x

ν wt

 ,

1 1 1 α σx2 − στ2 2 (1 − α − ν)2 2 (1 − ν)(1 − α − ν)2



ν  1−α−ν



˜1 X

1−α−ν+αν

From here, it is straightforward to obtain aggregate TFP as a function of time to build TFPt,t+τ =

˜1, =X

σx2 . Aggregate capital, labor, and output follow as

α ν (1−ν)(1−α−ν) ˜ X2 rt wt α ν   1−α−ν   1−α−ν α ν ˜2 = X rt wt

Lt,t+τ = Yt,t+τ

1−ν    1−α−ν

1 1 α 1 σx2 − στ2 2 (1 − α − ν)2 2 (1 − ν)(1 − α − ν)2

1 1 1 α σx2 − στ2 . 21−α−ν 2 (1 − ν)(1 − α − ν)

43

TIME TO BUILD AND THE BUSINESS CYCLE APPENDIX D: SOLUTION ALGORITHM D.1. Simplified consumption good firm problem

To solve the model most efficiently, I rewrite the firm problem. First, I transform the firm problem. Instead of io , the investment order, I let firms choose ko , the new capital stock upon delivery. Computationally, this transformation has the advantage that I can use the same grid for ko as for k, and this grid can be tighter than the one for io . To leave the firm problem unchanged, ko o needs to evolve over time to guarantee the implicitly defined investment order satisfies io0 = iγ . Using the identity, io = γko + (1 − δ)k, the evolution of ko over time (conditional on no delivery) o according to ko0 = kγ − δ(1−δ)k . Second, in slight abuse of notation, I drop the aggregate state γ2 s and instead use time subscripts for functions that depend on the aggregate state. I express the firm value functions in utils, see Khan and Thomas (2008), and redefine the value function such that the expectation with respect to idiosyncratic productivity does not have to be computed within the maximization problem. This raises computational efficiency and it tends to smooth the value functions. More precisely, I define V˜t (k, x, ξ) = pt Ex Eξ V (k, x0 , ξ 0 ), V˜tA (k, x, ξ) = pt VtA (k, x, ξ), ˜ t (k, x, ξ) = Ex W ¯ t (k, x0 , ξ), W ¯ t (k, x, ξ) = pt Wt (k, x, ξ), where Ex (Eξ ) V˜tN A (k, x) = pt V N A (k, x), W 0 0 denotes the expectation with respect to x (ξ ) conditional on x (ξ) and pt = Ct−σ as before. Then equations (4.16), (4.17), (4.18), and (4.19) can be rewritten as:

n

o

V˜t (k, x, ξ) = Ex Eξ max V˜tA (k, x0 , ξ 0 ), V˜tN A (k, x0 )

V˜tN A (k, x) = pt cft (k, x) + βEt V˜t+1 ((1 − δ)k/γ, x, ξ)



V˜tA (k, x, ξ) = max o kt

n

¯ t (k, kto , x, ξ) W

∈R+



o

o

¯ t (k, k , x, ξ) = pt cft (k, x) W

h

+ qt − pt [(1 − pi (k, ko ))(γko − (1 − δ)k) + ftE (ξ)] + βEt V˜t+1 (ko , x, ξ)



h

˜ t+1 (1 − δ)k/γ, ko /γ − δ(1 − δ)k/γ 2 , x, ξ + (1 − qt ) βEt W



i


i

˜ t (k, ko , x, ξ) = Ex W ¯ t (k, x0 , ξ) W where Et denotes the expectation with respect to aggregate state st+1 conditional on st . The net present value of the fixed adjustment cost can be expressed by f act ξ, where f act is defined recursively f act = qt pt

wt + (1 − qt )βEt f act+1 . φqt

44

MATTHIAS MEIER

In turn, this allows me to simplify the firm problem as

n

o

V˜t (k, x) = Ex Eξ max V˜tA (k, x0 ) − f act ξ 0 , V˜tN A (k, x0 ) wt + (1 − qt )βEt f act+1 φqt   V˜tN A (k, x) = pt cft (k, x) + βEt V˜t+1 ((1 − δ)k/γ, x) f act = qt pt

V˜tA (k, x) = max o

kt ∈R+

n

¯ t (k, kto , x) W

o

¯ t (k, ko , x) = pt cft (k, x) W

h

+ qt − pt (1 − pi (k, ko ))(γko − (1 − δ)k) + βEt V˜t+1 (ko , x)



h

˜ t+1 (1 − δ)k/γ, ko /γ − δ(1 − δ)k/γ 2 , x + (1 − qt ) βEt W



i


i

˜ t (k, ko , x) = Ex W ¯ t (k, x0 ) W Importantly, this allows me to compute the extensive margin adjustment policy in closed form, V˜ A (k, x0 ) − V˜tN A (k, x0 ) ξˆt = t . f act Next, I approximate firm values using collocation where Φ denotes basis functions in matrix representation and c denotes vectors of coefficients V˜t (k, x) 'ΦV (k, x)cVt ˜ t (k, ko , x) 'ΦW (k, ko , x)cW W t The approximations are exact at the nk collocation nodes k1 , ..., knk and k1o , ..., kno k . In practice, I choose the same collocation nodes for k and ko . As baseline we use cubic B-splines to approximate the firm value functions. This does not only have the advantage of being computationally fast, but also conditional on the coefficients we know the Jacobian in closed form. In particular, I can write down the optimality condition for intensive margin capital adjustment (kto ) as o W qt pt ps (k, kto )γ =qt βEt ΦVk (kto , x)cVt+1 + (1 − qt )βEt ΦW ko ((1 − δ)kt /γ, kt , x)ct+1 , W o where ΦVk = (∂ΦV )/(∂k) and ΦW ko = (∂Φ )/(∂k ). I approximate the AR(1) process of idiosyncratic productivity using Tauchen’s algorithm. I denote the discrete grid points of x by x1 , ..., xnx consisting of nx grid points and the transition probability from state xj to state xj 0 one period later by πx (xj 0 |xj ). To render the infinite-dimensional distribution µt tractable, I approximate it with a discrete histogram. That is, µt measures the share of firms for each discrete combination of capital stock ki1 , outstanding order kio2 (both correspond to the collocation nodes), and productivity xj . A further distinction is useful: Let µVt denote the cross-sectional distribution of firms without outstanding orders over idiosyncratic states (ki , xj ) and µW the distribution of firms with outstanding orders t o V W over (ki1 , ki2 , xj ). It holds that µt = (µt , µt ).

45

TIME TO BUILD AND THE BUSINESS CYCLE D.2. Campbell-Reiter algorithm

Using the preceding approximation and simplification steps, the model equilibrium is described by the following non-linear equations: (D.1)

n

ΦV (k, x)cVt = Ex Eξ max V˜tA (k, x0 ) − f act ξ 0 , V˜tN A (k, x0 )

o

ξˆt (k, x) = (V˜tA (k, x) − V˜tN A (k, x))/f act V˜tN A (k, x) = pt cft (k, x) + βEt ΦV ((1 − δ)k/γ, x)cVt+1 ¯ t (k, kto , x) V˜tA (k, x) = W ¯ t (k, ko , x) = pt cft (k, x) W

h

+ qt − pt (1 − pi (k, ko ))(γko − (1 − δ)k) + βEt ΦV (ko , x)cVt+1

h

+ (1 − qt ) βEt ΦW ((1 − δ)k/γ, ko /γ − δ(1 − δ)k/γ 2 , x)cW t+1

i

i

α/(1−ν)

cft (kt , xt ) = (1 − ν) (ν/wt )ν/(1−ν) (zt xt )1/(1−ν) kt wt = ψ/pt qt = mt (φ/(1 − φ))η−1 (D.2) (D.3)

0 ¯ ΦW (k, ko , x)cW t = Ex Wt (k, x ) wt f act = qt pt + (1 − qt )βEt f act+1 φqt

(D.4)

o W qt pt ps (k, kto )γ = qt βEt ΦVk (kto , x)cVt+1 + (1 − qt )βEt ΦW ko ((1 − δ)kt /γ, kt , x)ct+1

(D.5)

1 = Y t − It pt Yt =

X

α/(1−ν)

µt (ki1 , ki2 , xj ) (ν/wt )ν/(1−ν) (zt xj )1/(1−ν) ki1

i1 ,i2 ,j

It =

X

µVt (ki , xj )G(ξˆt (ki , xj ))qt ps (ki , kto (xj )) [γkto (xj ) − (1 − δ)ki ]

i,j

+

X

o s o o µW t (ki1 , ki2 , xj )qt p (ki1 , ki2 ) [γki2 − (1 − δ)ki1 ]

i1 ,i2 ,j

(D.6)

µVt+1 (ki0 , xj 0 )

=

X

πx (xj 0 |xj )µVt (ki , xj )[ωtV,V,A (i, i0 , j) + ωtV,V,N A (i, i0 , j)]

i,j

X

+

o W,V πx (xj 0 |xj )qt µW (i1 , i2 , i0 , j) t (ki1 , ki2 , xj )ωt

i1 ,i2 ,j

(D.7)

µW t+1 (ki01 , ki02 , xj 0 )

=

X

πx (xj 0 |xj )µVt (ki , xj )ωtV,W (i, i01 , i02 , j)

i,j

+

X

W,W πx (xj 0 |xj )µW (i1 , i2 , i01 , i02 , j) t (ki1 , ki2 , xj )ωt

i1 ,i2 ,j

(D.8)

log(mt+1 ) = (1 − ρm ) log(µm ) + ρm log(mt )

(D.9)

log(zt+1 ) = ρz log(zt )

46

MATTHIAS MEIER

With the following auxiliary equations for the law of motion of the distribution:

ωtV,V,A (i, i0 , j) =

 k 0 −kto (xj )  G(ξˆt (ki , xj ))qt kii0 −ki0 −1  G(ξˆt (ki , xj ))qt

  

ωtV,V,N A (i, i0 , j)

=

0

0

(i1 , i2 , i , j) =

=

kto (xj ) ∈ [ki0 , ki0 +1 ]

if

(1 − δ)ki /γ ∈ [ki0 −1 , ki0 ]

(1−δ)k /γ−k [1 − G(ξˆt (ki , xj ))] k 0 i −k 0 i0

if

(1 − δ)ki /γ ∈ [ki0 , ki0 +1 ]

0

else

i +1

i

 k 0 −(1−δ)ki /γ ki0 −kto (xj ) 2 ˆt (ki , xj ))(1 − qt ) i1  G( ξ  ki0 −ki0 −1 ki0 −ki0 −1   1 1 2 2     if kto (xj ) ∈ [ki02 −1 , ki02 ] and (1 − δ)ki /γ ∈ [ki0 −1 , ki0 ]    (1−δ)ki /γ−ki0 ki0 −kto (xj )   2 1 ˆt (ki , xj ))(1 − qt )  G( ξ  ki0 +1 −ki0 ki0 −ki0 −1  1 1 2 2    o  0 −1 , ki0 ] and (1 − δ)ki /γ ∈ [ki0 , ki0 +1 ] if k (x ) ∈ [k  j t i  2 2 o ki0 −(1−δ)ki /γ kt (xj )−ki0

2 G(ξˆt (ki , xj ))(1 − qt ) 1k 0 −k 0 ki0 +1 −ki0  i −1 i  1 1 2 2     if kto (xj ) ∈ [ki02 , ki02 +1 ] and (1 − δ)ki /γ ∈ [ki0 −1 , ki0 ]    (1−δ)ki /γ−ki0 kto (xj )−ki0    G(ξˆt (ki , xj ))(1 − qt ) k 0 −k 0 1 k 0 −k 02  i +1 i i +1 i  1 1 2 2    o  0 , ki0 +1 ] and (1 − δ)k if k (x ) ∈ [k  i /γ ∈ [ki0 , ki0 +1 ] j t i  2 2  

else

 k −k 0  q i i2   t ki0 −ki0 −1   

ωtW,W (i1 , i2 , i01 , i02 , j)

if else

0

ωtW,V

kto (xj ) ∈ [ki0 −1 , ki0 ]

 k −(1−δ)k /γ  [1 − G(ξˆt (ki , xj ))] i0k 0 −k 0 i   i i −1   

ωtV,W (i, i01 , i02 , j) =

kto (xj )−ki0 ki0 +1 −ki0

if

if

ki2 ∈ [ki0 −1 , ki0 ]

k −k 0 qt k 0i2 −ki 0 i +1 i

if

ki2 ∈ [ki0 , ki0 +1 ]

0

else

 ki0 −(1−δ)ki1 /γ  (1 − qt ) 1k 0 −k 0   i i −1  1 (1 − qt )    

(1−δ)ki1 /γ−ki0

1

ki0 +1 −ki0 1

if

(1 − δ)ki1 /γ ∈ [ki01 −1 , ki01 ]

and

i20 = i2

if

(1 − δ)ki1 /γ ∈ [ki01 , ki01 +1 ]

and

i20 = i2

1

0

else

Labeled equations (D.1)–(D.9) are the main equations, and all other unlabeled equations are auxiliary in defining the model equilibrium. Given nk collocation nodes and nx discrete grid points of x, equations (D.1)–(D.9) are nf = 2n2k nx + 3nk nx + 4. I organize these equations in (D.10)

Et [f (xt , xt+1 , yt , yt+1 )] = 0,

z 2 where t = (m t , t ) ∈ R denotes the vector of aggregate shocks. xt denotes predetermined state variables and yt denotes non-predetermined state variables 2

(D.11)

xt = [µt ; log(mt ); log(zt )] ∈ Rnx ≡nk nx +nk nx +2 ,

(D.12)

o ny ≡nk nx +2nk nx +2 yt = [cVt ; cW . t ; log(act ); log(kt ); log(pt )] ∈ R

2

TIME TO BUILD AND THE BUSINESS CYCLE

47

¯, y ¯, y ¯ ) = 0. In the general case, the model The non-stochastic steady state is defined as f (¯ x, x solution is given by (D.13)

yt = g(xt , ζ),

(D.14)

xt+1 = h(xt , ζ) + ζ σ ˜ t+1 ,

where ζ is the perturbation parameter and g : Rnx × R+ → Rny and f : Rnx × R+ → Rnx . The exogenous shocks are collected in t+1 ∈ Rn , and σ ˜ ∈ Rnx ×n attributes shocks to the right equations while also scaling them (by σ m , σ z ). To solve the two policy functions, I use a firstorder approximation. I follow the perturbation algorithm in Schmitt-Grohe and Uribe (2004). This requires to compute the Jacobians of function f (locally) at steady state. Importantly, the algorithm in Schmitt-Grohe and Uribe (2004) checks for existence and uniqueness of a model solution.

D.3. Krusell-Smith algorithm This subsection suggests how the model can be solved using the Krusell-Smith algorithm. Following Krusell et al. (1998), and the adaption for heterogeneous firms by Khan and Thomas (2008), I assume agents in my model only observe a finite set of moments, informative about the entire distribution, instead of observing µ directly. The agents approximate equilibrium prices and the evolution of the observed moments by a log-linear rule. I approximate the distribution µ by the aggregate capital stock,

Z (D.15)

Kt =

kdµ, S

and the stock of investments outstanding from the preceding period (D.16)

Ito =

Z

(γko − (1 − δ)k)dµW .

SW

If time-to-build dropped to zero q = 1, Ito would constitute the investments activated in addition to new orders. I suggest the following log-linear forecast rules (D.17)

log Kt+1 = βk0 (zt , mt ) + βk1 (zt , mt ) log Kt + βk2 (zt , mt ) log Ito ,

(D.18)

o log It+1 = βi0 (zt , mt ) + βi1 (zt , mt ) log Kt + βi2 (zt , mt ) log Ito ,

and the log-linear pricing rule (D.19)

log pt = βp0 (zt , mt ) + βp1 (zt , mt ) log Kt + βp2 (zt , mt ) log Ito .

The forecasting and pricing rules are described by coefficients that depend on the exogenous aggregate shock. For discretized processes of z and m, the equilibrium under bounded rationality with the above rules becomes computable. I use these rules to solve for the optimal policy functions and then simulate the economy and compute equilibrium prices pt in every period t. The simulated economy allows price series are then used to update the coefficients of the log-linear rules. I stop the procedure when the coefficients have converged.

48

MATTHIAS MEIER APPENDIX E: ADDITIONAL INFORMATION ON THE MODEL CALIBRATION

Cooper and Haltiwanger (2006) targets the spike investment shares, but also persistence of investment rates and the correlation of investment rates with idiosyncratic productivity, when estimating a richer specification of capital adjustment costs including convex adjustment costs. I exclude the latter two moments because they may depend sensitively on the specific time to build setup. Nonetheless, the model matches these moments reasonably well with a persistence of 1.6% (empirically 5.8%), and a productivity correlation of 24% (empirically 14%). An alternative strategy to calibrate adjustment costs is to target cross-sectional skewness and kurtosis of investment rates, see Bachmann and Bayer (2013). In fact, our calibrated model closely matches these moments in the data: skewness/kurtosis in the model are 5.1/48.3, while in a balanced panel of Census data these are 6.5/67.4 for total investment and 5.5/47.9 for equipment investment, see Kehrig and Vincent (2016). Since skewness and kurtosis monotonically increase in the adjustment cost parameters, this indicates the calibrated adjustment costs may be too low. TABLE VI calibration targets

Model

Data

18.6% 1.8%

19.1% 1.8%

0.016 0.14

-0.007 0.22

5.1 48.3

4.9 43.4

Targeted (LRD) Positive spikes Negative spikes Non-targeted (LRD) Persistence Productivity correlation Non-targeted (Census) Skewness Kurtosis

Notes: All moments relate to annual investment rates computed as I/K. Positive and negative spikes denote the share of investment rates larger than 20% and smaller than -20%, resp. LRD moments are from Cooper, Haltiwanger (2006), Census moments are from Kehrig, Vincent (2016). Alternative data sources used to calibrate and estimate similar models are the IRS tax data, see, e.g., Winberry (2016b), and Compustat data, see, e.g., Bloom (2009). Both datasets are at the firm-level. The IRS does includes only positive investments, and Compustat is biased to large private firms. The main disadvantage of the LRD dataset is that it covers manufacturing only.

49

TIME TO BUILD AND THE BUSINESS CYCLE APPENDIX F: ADDITIONAL RESULTS FROM THE MODEL SIMULATION

Figure 12: Further responses to a supply chain disruption Order backlog 88 87 86 85

0

4

8

12

16

20

24

0.34 0.32 0.3 0.28 0.26 0.24 0.22 0.2 0

4

8

12

16

20

24

Quarters Notes: The sub-figures show impulse responses to a supply chain disruption that lengthens time to build by one month starting from steady state. ‘Direct channel’ denotes the impulse responses when aggregate TFP changes are eliminated through opposing aggregate productivity (z) shocks. Aggregate TFP is computed as T F P = log(Yt ) − α log(Kt ) − ν log(Lt ). Inaction measures the share of firms without outstanding orders that do not make a new order in a given period. The order backlog is the total of investment orders outstanding for delivery.

50

MATTHIAS MEIER

Figure 13: Partial equilibrium responses to a supply chain disruption 7

5

6.5

0 -5

6 -10 5.5 5

Total response Direct channel

-15 0

4

8

12

16

20

24

0.5

-20

0

4

8

12

16

20

24

0.2

0 0

-0.5 -1

-0.2

-1.5 -2

0

4

8

12

16

20

24

-0.4

0

4

8

12

16

20

24

Notes: The sub-figures show impulse responses to a supply chain disruption that lengthens time to build by one month starting from steady state. ‘Direct channel’ denotes the impulse responses when aggregate TFP changes are eliminated through opposing aggregate productivity (z) shocks. Aggregate TFP is computed as T F P = log(Yt ) − α log(Kt ) − ν log(Lt ).