ISSN (Print) 0473-453X

Discussion Paper No. 1178

ISSN (Online) 2435-0982

BUSINESS CYCLES WITHCYCLICAL RETURNS TO SCALE

Jay Hyun
Ryan Kim
Byoungchan Lee

June 2022

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Business Cycles with Cyclical Returns to Scale∗Jay Hyun†

Ryan Kim‡

Byoungchan Lee§

HEC Montréal

Johns Hopkins University

HKUST

October 21, 2021

Abstract
We study business cycles with cyclical returns to scale. Contrary to tightly parameterized production
functions (Cobb-Douglas and Constant Elasticity of Substitution), we empirically identify strong
input complementarity that leads to procyclical returns to scale. We therefore propose a flexible
translog production function that allows complementarity-induced procyclical returns to scale, andwe integrate this function into a standard medium-scale dynamic stochastic general equilibrium(DSGE) model. The estimated model with the procyclical returns to scale (i) features procyclicalprice markups, (ii) better matches the cyclicality of factor shares, and (iii) decreases by nearly half
the contribution of markup shocks to output fluctuations.
JEL Codes: C11, E23, E31, E32
Keywords: Business cycles, Translog production function, DSGE model, Returns to scale, Markup,
Markup shocks∗

We are grateful to David Andolfatto, David Berger, Mark Bils, Chris Conlon, Andres Drenik, Heitor S. Pellegrina(discussant), Katherine Ho, Yang Jiao, Loukas Karabarbounis, Amit Khandelwal, Jennifer La’O, Jae Won Lee, Andreas
Mueller, Monica Morlacco (discussant), Seunghoon Na, Emi Nakamura, Hyunseung Oh, Zhesheng Qiu (discussant),
Ricardo Reis, Bernard Salanie, Stephanie Schmitt-Grohé, Jón Steinsson, Martin Uríbe, Carlos Vegh, Jonathan Vogel,David Weinstein, Michael Woodford, Mark Wynne (discussant), Pierre Yared, Tack Yun, and the seminar participantsat the Columbia colloquia, Asian Econometric Society Meeting, EconCon, EGSC, Midwest Macro Meeting, Tsinghua
Workshop in Macroeconomics, WEAI, YES, KERIC, and HKUST-Jinan Macro Workshop for their helpful comments
and suggestions. This paper subsumes previous papers circulated under the titles “Price-Cost Markup Cyclicality:
New Evidence and Implications” and “Business Cycles with Input Complementarity”. All errors are our own.
†

jungsik.hyun@hec.ca, Jungsik Hyun. Department of Applied Economics, HEC Montréal, 3000, chemin de la Côte-Sainte-

Catherine Montréal, QC H3T 2A7, Canada‡
rkim59@jhu.edu, Sung Ryong Kim. Johns Hopkins University, School of Advanced International Studies, 1717 Massachusetts Ave NW, Office 749, Washington DC 20036§
bclee@ust.hk. Department of Economics, Hong Kong University of Science and Technology, 6068 Lee Shau Kee Business
Building, Clear Water Bay, Kowloon, Hong Kong

1Introduction

Standard business cycle models make strong a priori structural assumptions on the shape of the
production function. The most widely used production function in macroeconomics is the CobbDouglas production function. Despite its convenient tractable features, this production function
imposes an excessively restrictive structure on how firms substitute their inputs (elasticity of
substitution), the productivity of each input (marginal product of input), and the productivity of all
inputs together (returns to scale). This production function was often justified by the Kaldor (1957)
growth facts, but the recent decline in the labor share (e.g., Karabarbounis and Neiman 2014) calls
this justification into question. Many researchers acknowledge this limitation and have started toadopt a more general constant elasticity of substitution (CES) production function, but even thisproduction function has important restrictions: one single constant parameter governs the elasticity
of substitution among inputs, and returns to scale is typically assumed to be constant and fixed overtime.
We empirically assess the plausibility of these restrictions by imposing and estimating a flexible
translog production function (Christensen et al. 1973, 1975). Compared to a CES, the translog is
another generalization of a Cobb-Douglas production function that allows more flexibility in input
substitution, the marginal product of input, and returns to scale. Similar to the non-parametricproduction function estimation technique developed in the industrial organization literature (Gandhi
et al. 2020), we utilize the first-order condition of firms to estimate the marginal product of input
and to assess the variability in returns to scale. We employ standard panel data techniques withdetailed industry-level panel data for the estimation.
Through our estimation, we find strong complementarity between labor and energy that leads
to time-varying procyclical returns to scale. The idea of time-varying returns to scale is striking
yet simple. It reflects the idea that when firms employ more factors during boom periods, there
are synergies among these factors that lead to the larger aggregate marginal product of inputs andreturns to scale compared to periods of recession. The procyclical movement in returns to scale also
induces a procyclical wedge between the marginal product of input and the real input price, which isthe price markup in standard macroeconomic models.
Motivated by our empirical evidence, we estimate a medium-scale dynamic stochastic generalequilibrium (DSGE) model as in Smets and Wouters (2007), incorporating a flexible translog
production function. Given the empirical importance of complementarity between labor and energy
in generating procyclical returns to scale, we include energy input and allow a translog substitution
parameter between labor and energy. Alongside the model with the translog production function, we
estimate two other models with nested production functions: (i) a standard two-factor Cobb-Douglas
production function with labor and capital which was originally used by Smets and Wouters (2007)
and (ii) a three-factor Cobb-Douglas production function that additionally includes energy input.
1

The estimated model with the translog production function generates procyclical returns toscale and procyclical price markups, consistent with our empirical findings. We confirm significant
complementarity between labor and energy, which induces procyclical returns to scale. In turn, theprocyclical returns to scale leads to procyclical price markups in the otherwise standard DSGE model;
the larger returns to scale in expansion decreases marginal costs and allows price markups to rise.
To the contrary, the models with the Cobb-Douglas production functions—regardless of including or
excluding energy input—generate countercyclical returns to scale and acyclical price markups, which
are inconsistent with our empirical findings.1
Furthermore, we document that the model with a translog structure better matches the empirical
cyclicality of input shares compared to other models with Cobb-Douglas production functions. In
the data, labor shares are countercyclical, and energy shares are procyclical. However, under the
Cobb-Douglas production function with fixed costs, all input shares are perfectly positively correlated
and cannot match different cyclicality of input shares. We break this tight link between input sharesand generate data-consistent labor and energy share cyclicality with the flexible translog production
function. At the same time, our model generates procyclical profit shares, as in Smets and Wouters(2007) but for a different reason; our model relies on the procyclical price markups, whereas the theirmodel relies on the countercyclical returns to scale.
Finally, in our model with the procyclical returns to scale, the contributions of price and wage
markup shocks to output fluctuations are substantially smaller than those in the models with the
Cobb-Douglas production function. As in Smets and Wouters (2007), we conduct the forecast
error variance decomposition exercise for each of the three different models with the corresponding
production functions. Although integrating the energy input into the conventional Cobb-Douglas
production function has a negligible effect on the decomposition results, adding the translog structurewith the procyclical returns to scale reduces nearly half of the contribution of the markup shocks on
output. The comparisons of the Bayesian estimation results and the impulse responses to price andwage markup shocks across the different models reveal the importance of procyclical returns to scalein suppressing the markup shocks. The procyclical returns to scale changes Calvo parameters andamplifies responses of real variables, which in turn reduce the residual variations that are previously
attributed to the price and wage markup shocks.
To the best of our knowledge, this paper is the first to investigate the role of procyclical returns
to scale in a business cycle framework. Our paper is closely related to previous studies that go beyond
constant returns to scale in business cycle analysis (Benhabib and Farmer 1994, 1996; Schmitt-Grohé1
We focus on the unconditional returns to scale and markup cyclicality when comparing the model predictions withthe data. Although the Smets and Wouters (2007) model generates countercyclical markups conditional on demand
shocks, it features acyclical markups unconditionally. In addition, the conventional model features countercyclicalreturns to scale because of the fixed cost of production. In our model with translog production, however, thecountercyclical effects of the fixed costs on returns to scale are dominated by the procyclical effects of the input
complementarity. See Section 3.3.1 for details.

22000), and a growing literature that generalizes an aggregate Cobb-Douglas production function (e.g.,
Antras (2004); Chrinko (2008); Karabarbounis and Neiman (2014); Cantore et al. (2015)).2 Most
previous studies reject a Cobb-Douglas production function and find complementarity among inputs
beyond what Cobb-Douglas technology implies, similar to our analysis.3 To integrate the generalproduction function into the DSGE framework, we normalize the translog production function so thatit preserves dimensionless parameters, as pioneered by De Jong (1967) and first incorporated in the
DSGE framework by Cantore and Levine (2012) and Cantore et al. (2014). Regarding time-varying
parameters, Koh and Santaeulàlia-Llopis (2017) propose a CES production function that features
the time-varying elasticity of substitution. We complement previous studies by proposing a translog
production function with the time-varying returns to scale.
The procyclical returns to scale speaks to the considerable literature that studies the countercyclicality of the price markup, which is a first-order building block in many sub-fields of macroeconomics.4
Despite its importance, existing empirical evidence on price markup cyclicality is mixed.5 This
paper finds that integrating procyclical returns to scale leads to unconditional procyclical pricemarkups in the standard medium-scale DSGE model. Our emphasis on the procyclical returns toscale differs from previous papers that emphasize other sources of procyclical price markup, such as
wage rigidity (Nekarda and Ramey 2020), time-varying demand elasticity (Stroebel and Vavra 2019),
and endogenous assortment (Anderson et al. 2020). The resulting procyclical price markup, in turn,decreases the contributions of markup shocks to output fluctuations compared to what is reported
in Smets and Wouters (2007). These results imply that using a translog production function may
alleviate the concerns raised in previous studies about the excessive importance of the markup shocks
in new Keynesian models (see, e.g., Chari et al., 2009; Justiniano et al., 2010).
The remainder of this paper is structured as follows. Section 2 presents the industry-level data
2
There are important studies that micro-found the aggregate production function with heterogeneous industry or
firm models (e.g., Atalay 2017; Raval 2019; Oberfield and Raval 2021). In particular, Baqaee and Farhi (2021) showthat the aggregate returns to scale can vary across time due to the change in allocative efficiency. Our paper instead
extends the production function itself with the translog structure and infers the aggregate implications of using thismore flexible production function.
3
One notable exception is Karabarbounis and Neiman (2014), who find that labor and capital are substitutes;
however, in estimating their parameters, they study long-term trends rather than business cycle movements. See, e.g.,
Hassler et al. (2019) for a discussion of different substitution patterns across short-run and long-run horizons among
energy and other inputs.
4
In the context of models of nominal rigidity, countercyclical markups conditional on demand changes are necessary
to explain both procyclical wages and countercyclical unemployment (Rotemberg and Woodford 1991, Rotemberg 2013).
In the study of monetary policy, many New Keynesian models suggest that central banks should target constant averagemarkup for price stability (Goodfriend and King 1997). In the scholarship on price dynamics, countercyclical markups
conditional on financial distortion explain missing disinflation during the Great Recession (Gilchrist et al. 2017). Ravn
et al. (2006) find that introduction of deep habit formation substantially affects cyclicality of price-markups. Finally,
Bils et al. (2018) find that unconditional countercyclical markups explain at least half of the cyclicality in the laborwedge.
5
Some studies find that price markups are countercyclical, and other studies find that price markups are procyclicalor acyclical. See, e.g., Bils (1987), Rotemberg and Woodford (1991), Rotemberg and Woodford (1999), Gali et al.
(2007), Bils et al. (2013), Bils et al. (2018) for countercyclicality and Hall (2013), Nekarda and Ramey (2020), Stroebel
and Vavra (2019), and Anderson et al. (2020) for procyclicality.

3and the estimation results with a translog production function. Section 3 presents and estimates a
medium-scale DSGE model with the translog production function and discusses the business cycle
implications. Section 4 concludes.

2

Empirical Analysis

This section estimates production function coefficients under the translog production functional form
assumption. We present the data used in this analysis and our empirical framework, which allows us
to recover the marginal product of input, along with the estimation results. Then, we discuss thevariability in returns to scale.

2.1

Data

This paper uses annual six-digit North American Industry Classification System (NAICS) industrylevel data from the NBER-CES Manufacturing Industries Database. This database records detailed
information on 473 manufacturing industries from 1958 to 2009. The information is compiled fromthe Annual Survey of Manufacturers and the Census of Manufacturers. The variables in this databaseinclude gross output (value of shipment), value added, and 5-factor inputs (production worker,
non-production worker, capital, material, and energy) for each industry over time. These dataalso include industry-specific deflators for output, material, energy, investment, and wage bills for
production workers and total employees. We report our summary statistics in Appendix A, and amore detailed explanation of this database can be found in Bartelsman et al. (2000).
The greatest advantage of the NBER-CES data over aggregate data is that they allow us to
exploit both time-series and cross-sectional variations and corresponding panel data techniques toestimate production function parameters. A large variation in the data is especially important for
our analysis, which seeks to relax strong functional form assumptions. The advantage comes with thecost, as our estimates come from the manufacturing sectors. In Appendix B.8, we use the Integrated
Industry-Level Production Account (KLEMS) data that cover all sectors in the US economy for ashorter time period as a robustness check and confirm the strong complementarity between laborand energy input. The complementarity results are similar when we re-estimate the complementarity
parameter with our DSGE model by using aggregate data that cover the entire US economy in
Section 3.2.

4

2.2

Empirical Framework and Estimation Results

Estimaing a translog production function is challenging because it has an excessive number ofparameters.6 For example, for the five inputs available in the NBER-CES data, we must estimate
twenty parameters with a translog production function, many more than the five parameters in a
Cobb-Douglas production function and the six parameters in a CES production function. Even withthe detailed industry-level data for many years, it is difficult to estimate all twenty parameters in thetranslog production function because of multicollinearity.7
To overcome the challenge of estimating many parameters, we exploit a firm’s first-ordercondition in the spirit of the non-parametric identification method developed in the IO literature(Gandhi et al. 2020).8 Firms’ optimization conditions generate a relationship between the marginal
product of each input and its price. Using this relationship and a panel data estimation technique,
we recover the production part associated with the marginal product of input, which has significantly
fewer parameters.
In estimating the first-order condition, we address potential estimation concerns by choosing
the following three specifications. First, we use energy input to tightly link the marginal productof input (energy) from the real input (energy) price in the first-order condition. Second, we apply
log-linearization and demean the variables to make the equations linear and to address potential
endogeneity concerns. Third, we use lagged input prices as instrumental variables to address nonclassical measurement errors and to rule out other remaining endogeneity issues. We conduct various
robustness checks to address other potential concerns and report the results in Appendix B.
Estimation Framework For simplicity, consider the following translog production function with
only two inputs, labor and capital:
βkk
βllln(Y ) = εa + βl ln(L) + βk ln(K) + βlk ln(L) ln(K) +ln(L) ln(L) +ln(K) ln(K),
|
{z
} |
2
2
{z
}
Cobb-Douglas(2.1)

second-order terms

where Y is output, L is labor, K is capital, and εa is the log of total factor productivity. The
first part of the production function is a conventional Cobb-Douglas function, which is a firstorder approximation of a general production function. A translog production function extends this6

For example, Syverson (2011) wrote that “many researchers also use the translog form... is more flexible, thoughmore demanding of the data”. It is also difficult to calibrate parameters given that there is no previous work that
integrates a translog production function into the business cycle model.
7
For example, running an OLS regression of log output on 20 inputs with industry and time fixed effects makes
more than half of the parameters statistically insignificant at the conventional level, including the labor input. Asidefrom other issues related to the estimation, such as the conditional mean independence assumption, the multicollinearity
of inputs is likely to make the estimates statistically insignificant.
8
The other way to proceed is to impose more structure in the estimation. In Section 3.2, we utilize the full structureof the business cycle model, estimate the key parameters, and confirm the industry-level empirical results.

5

approximation of a general production function to the second order. Assuming βlk = 0, βll = 0, andβkk = 0 recovers a Cobb-Douglas production function.
By generalizing Equation (2.1) to five different inputs available in the NBER-CES data, wehave:ln(Y ) = εa +βi ln(V i ) +X X βik

{z

|

X

i

i

|

Cobb-Douglas

}

k

2

ln(V i ) ln(V k ) with βik = βki ,
(2.2)
{z

}

second-order terms

where V denotes one of five different inputs indexed by i and k, namely, energy (e), production
worker, non-production worker, capital, and material. Here, we allow for a flexible substitution
structure among the five inputs. In contrast, conventional macroeconomic models use either two
inputs (labor and capital) or three inputs (labor, capital, and material) with implicitly imposed
restrictions on the substitution pattern among these inputs.
The simplest method to recover the parameters in Equation (2.2) is to regress log output on log
inputs and treat the residual as the unobserved productivity. This approach has two key problems forour purpose. First, as we have already emphasized, it is extremely challenging to estimate all twenty
parameters in this specification even with the rich variation available in the panel data. Second, as
widely documented in the productivity estimation literature (e.g., Hall 1988; Evans 1992; Fernald
2014), flexible inputs are likely to be correlated with productivity, which generates inconsistentestimates of the parameters. For example, with the industry-level data, productive industries arelikely to use more inputs relative to other industries.
To avoid these two concerns, we exploit a firm’s first-order condition. Consider a firm’s first-ordercondition with respect to an input V i :
Pi
P
|{z}

real input price"#= τ i βi +X

βik ln(V k )

k

|

{z

Y
,
Vi
}

(2.3)

marginal product of input

where P i is the nominal price of the input V i , and τ i is the wedge or gap between the real input
price P i /P and the marginal product of input V i . The input-specific wedge τ i allows Equation (2.3)
to be consistent with a large class of models that deviates from the frictionless economy. Without
frictions, the marginal product of input equals the real input price, and τ i could be treated as aclassical measurement error or the ex-post productivity shock as in Gandhi et al. (2020). Once
researchers allow frictions, such as input adjustment cost (Hall 2004), imperfect competition in the
output (Rotemberg and Woodford 1999) and input (Berger et al. 2019) markets, and financial friction(Jermann and Quadrini 2012; Arellano et al. 2019; Bigio and La’O 2020), such frictions generate awedge between the marginal product of input and the real input price. This wedge is part of the6labor wedge, which is the difference between the marginal product of labor and the marginal rate of
substitution (see, e.g., Chari et al. 2007; Karabarbounis 2014; Bils et al. 2018). Note that assuming
βik = 0 for all i = 1, ..., 5 recovers the conventional first-order condition under the Cobb-Douglasiproduction function: PP = τ i βi VYi .
To avoid the extra structural assumptions required to measure real output and the output priceindex (see, e.g., Hottman et al. 2016), we rewrite Equation (2.3) as follows:"#

si = τ i βi +X

βik ln(V k ) ,

(2.4)

k
i

i

where si = PP VY is the input expenditure share out of total sales for the input V i . The right-handside is the wedge τ i multiplied by an output elasticity with respect to the input V i , which is aunit-free measure of the marginal product of input. The special cases of Equation (2.4) are used
to calibrate parameters or inform price markup cyclicality in the previous macroeconomics models.
Under the frictionless economy with Cobb-Douglas technology, an exponent of each input in the
production function equals the corresponding input share: si = βi . This restriction is often used
to recover the Cobb-Douglas production function parameters from the observed income shares. Inmodels of imperfect competition with the Cobb-Douglas production function, the wedge is interpreted
as the inverse of the price-cost markup, and the inverse of input share identifies the markup up to a1
constant: si = βi markup. This restriction allows researchers to inform on markup behavior with the

input share.
To input the data into Equation (2.4), we log-linearize the equation around the steady state
and allow the input share, wedge, and all inputs to vary across industry and time, which are indexed
by industry j and time t, respectively:ŝijt =Xi
δik V̂jtk + τ̂jt
,

(2.5)

k

where δik ≡ βik

 i
τ̄
s̄i

, x̂ denotes the log-deviation from the steady state value, and x̄ denotes thesteady state value for any variable x. The log-linearization eases the estimation by making theequation linear in parameters and is consistent with the DSGE analysis in Section 3.1.
We estimate Equation (2.5) in the panel data by double-demeaning across industry and time,
which eliminates both the industry-specific and time-specific components. The empirical counterpartof Equation (2.5) is:ŝˆijt =Xiδek V̂ˆjtk + τ̂ˆjt
,

(2.6)

k
1 PJ

ˆjt = ln xjt − J
where x̂h P
T1

j=1 ln xjt − T

t=1



P
ln xjt − J1 Jj=1 ln xjt

ifor any variable x. Technically,

the double-demeaning is identical to allowing the industry and time fixed effects; this method7eliminates both the industry-specific and time-specific components, including the aggregate trend.
Note that although we demean the variables and remove the aggregate components from all variables,
our parameters of interest, δik , match the aggregate parameters. Our benchmark estimation does
not weight observations, but weighting different industries to improve the match with the aggregate
moments does not change the main empirical results, as shown in Appendix B.6.
Since the input share sijt and all five inputs Vjtk are observed in the data, Equation (2.6) can be
i as a residual.
estimated by regressing the input share on all five inputs and treating the wedge τ̂ˆjt

However, there are two major problems in estimating Equation (2.5) directly for every input i. First,
i may contain industry-time-varying components, such as an adjustment cost,
the wedge term τjt

that are correlated with inputs and generate a confounding relationship. For example, consider
any positive aggregate shock that raises inputs in the production. If an industry j faces higher
input adjustment costs relative to other industries, then industry j is likely to utilize fewer inputs
relative to other industries.9 Second, since the input share si contains the input V i , there is aipositive mechanical correlation between the input share si ≡ V i × PPY and the input V i when V i
has a measurement error. See, for example, Berman et al. (2015) for the formal derivation of such amechanical correlation.
To ease the first estimation concern, we choose an energy input as a choice variable and focus
on estimating the energy efficiency (output elasticity with respect to energy). As a result of using
energy input, there are fewer components in the wedge that can be correlated with V k , particularly
related to the adjustment cost. The energy input and the intermediates in general are known tohave smaller adjustment costs relative to other inputs and are typically assumed away (e.g., Basu
1995; Bils et al. 2018). In addition, other potential concerns related to the monopsony power orheterogeneous input quality are mitigated when we focus on the energy shares.10
In addition to using the energy input, we utilize lagged double-demeaned input prices as
instrumental variables to avoid the mechanical correlation and to relax concerns related to theremaining wedge term. As previously discussed, the input share and the input usage generate apositive mechanical correlation when the variables are measured with error. Unless a researcher has
unusually detailed micro-level data, the input variables in any data have measurement error problems.
For example, it is difficult to allow for bulk discounts or quality differences in material inputs or to
9Note that this problem resembles the issue in estimating Equation (2.2), which arises from the correlation between
productivity and inputs.
10
Regarding the monopsony power, there are fewer concerns on how firms exercise market power in the markets forenergy inputs, and such friction would not appear as a wedge term. Previous studies have documented such frictions inlabor input (e.g., Berger et al. 2019). In contrast, energy production in the U.S. has faced heavy regulation and otherrestrictions by Congress, such as tax preferences, spending subsidies, and environmental regulations. As a result, it isvery unlikely that firms exert market power in their energy inputs. Regarding the heterogeneous input quality, energy
input is likely to have homogeneous quality across industries relative to other inputs. Since input prices partiallyreflect the quality of input, higher input prices and input shares might reflect a higher quality of inputs, which willappear as a wedge in Equation (2.5). See, for example, De Loecker et al. (2016) for the structural treatment of input
quality differences in the IO literature.

8control education, experience, and specific skills of the labor input. Capital input is known to havea large measurement error even at the firm level (Collard-Wexler and De Loecker 2020), and theperpetual inventory method in the NBER-CES data requires an assumption on initial capital stock.
Instrumenting inputs with the lagged input prices, which does not involve input usage, solves themechanical correlation problem that arises from these measurement errors.
e , even after eliminating time- and industry-specificRegarding the remaining energy wedge τ̂ˆjt

terms, there might be industry-time-varying wedge components that are correlated with inputs. By
e is not correlatedusing the lagged input prices as instruments, we assume that the idiosyncratic τ̂ˆjt
e is seriallywith the idiosyncratic component of the predetermined input prices. If the idiosyncratic τ̂ˆjt

uncorrelated, then the lagged double-demeaned input prices satisfy the exogeneity assumption since
they do not affect the current energy wedge. At the same time, the instrumental variables satisfy therelevance condition if they are autocorrelated and are correlated with input usage; Appendix A showsthat these instruments are highly correlated with V̂ˆ i for all i. In Appendix B.1, we also use two- andjt

e with idiosyncratic
three-year lagged double-demeaned input prices and allow the correlation of τ̂ˆjt

lagged input prices up to one year and find that the main empirical results remain robust.
In Appendices B.2 and B.4, we additionally conduct robustness exercises by putting specifice based on previous macroeconomic models and find that controlling such elementsinterpretations in τ̂ˆjt
e has limited effects on the estimation results. For example, consider a multi-industry businessin τ̂ˆjt

cycle model with heterogeneous price rigidity across industries (e.g., Nakamura and Steinsson 2010).
In this case, the energy wedge is price markup and can vary across time and industry. Moreover, if
industries that face larger price rigidity alter their future markups (inverse wedge) and affect their
input prices by changing the input usage relative to other industries, the exogeneity assumptionof instruments could be violated. To address such a concern, we control the industry-time-varying
measures of market power following previous studies, such as the price-cost markups (De Loecker
et al. 2020) and the Lerner index (Gutierrez and Philippon 2017). We also include the measuresof price rigidity and the inventory-to-sales ratio, which are known to be closely related to pricemarkup, and measures of financial friction, adjustment costs, and fixed costs in production. Our main
empirical results do not change with these alternative specifications, likely because double-demeaningat the detailed industry-time-level already eliminates most of the variations in the energy wedge thatoriginates from the predictions of previous models.
Equation (2.6) clearly illustrates which variation in the data identifies the substitution pattern
among energy and other inputs. Suppose that the coefficient of labor in Equation (2.6) is positive;the energy share increases with an increase in labor input, holding other inputs constant. Under
the translog technology, such an increase in the share of energy is interpreted to be a result of anincrease in energy efficiency that arises from an increase in labor input. In this case, the energy andlabor inputs are complements, and the coefficient δek captures the strength of the complementarity.

9eNote that a large magnitude of δek translates to a much smaller magnitude of βek = δek τ̄s̄ e due to thesmall energy share s̄e in the data.
Since our empirical strategy is heavily motivated by the productivity estimation literature onindustrial organizations, it is useful to compare our estimation technique with those in the literature.
One key difference in our approach is that we rely on the representative production function, as in a
typical DSGE model. Since the goal of our exercise is to discuss the aggregate parameters, we do notthink that this assumption is particularly worrisome. Besides, in Section 3.1, we show that a model
counterpart of the industry-level energy share (2.5) can be derived by aggregating the corresponding
firm-level first-order condition with an assumption of the firm-level translog production function.
Similarly, Appendix C.2 illustrates the representative translog production function by explicitly
aggregating the firm-level translog production functions. Within the structure of our model, theinterpretations of the production parameters are the same and the estimation assumptions are similar
across different levels of aggregations.
Taking the industry-level production function as a given, our use of the first-order condition
follows Gandhi et al. (2020), who estimate the first-order condition to allow a flexible substitution
pattern in a production function. One advantage of using the industry-level data is the availabilityof the entity-level input price measures, which are rarely available in more micro-level data. We
use input price deflators as instrumental variables to alleviate the mechanical correlation problems
and the endogeneity concerns. In addition, we use them to measure the quantity of inputs at theentity-level and therefore avoid input price bias in using the product- or firm-level data (see, e.g.,
De Loecker and Goldberg 2014). Double-demeaning, instrumenting, and controlling variables toaddress the potential endogeneity of the energy wedge in estimating the first-order condition issimilar to the methods that address the endogeneity of the total factor productivity in estimating
the production function. Our use of instrumental variables is similar to the method of Doraszelski
and Jaumandreu (2013), who use the first-order conditions and the lagged input prices to address
the issue of simultaneity. The double-demeaning with instrumental variable approach resembles the
dynamic panel data method (e.g., Arellano and Bond 1991; Blundell and Bond 1998), and controlling
the potential energy wedge in our robustness exercise is similar to the control function approach (e.g.,
Olley and Pakes 1996; Levinsohn and Petrin 2003; Ackerberg et al. 2015). Note that for our DSGE
analyses, we separately estimate the key complementarity parameter with the Bayesian method,which aligns with the estimate from this industry-level analysis.
Estimation Results Table 1 presents the estimated parameters in Equation (2.6). Column (1) isbased on the baseline instrumental variable estimation with all five inputs. The most economicallyand statistically significant estimate is the coefficient in front of production workers, which measuresthe complementarity between production workers and energy input. The estimated parameter shows10

Table 1: Estimation of Equation (2.6)
Dependent Variable: Energy ShareIV (Lagged Input Prices)
OLS
(1)

(2)

(3)

(4)

(5)

(6)

Energy-0.557**(0.230)-0.399**(0.161)-0.603***(0.229)0.847***(0.029)0.842***(0.029)0.822***(0.032)

Production worker

1.694***(0.446)

1.411***(0.312)

1.775***(0.457)-0.328***(0.029)-0.326***(0.030)-0.467***(0.032)

Non-production worker-0.593*(0.325)-0.454*(0.264)-0.560*(0.335)-0.119***(0.022)-0.124***(0.023)-0.224***(0.025)Material

0.167(0.218)0.321**(0.152)-0.411***(0.025)-0.421***(0.026)

Capital

0.507(0.436)
6.04
.3
23220

J-test(p-value)
Observations

0.774**(0.339)
5.8
.45
23220

6.76
.34
23220

-0.031(0.021)23220-0.243***(0.027)

23220

23220

Note. Columns (1), (2) and (3) show the regression result with instrumental variables, and (4), (5), and (6) show theordinary least square (OLS) result. For the instrumental variable analyses, we use both one-year lagged and two-year
lagged input prices. For the implementation, we use the GMM specification with the weighting matrix that accounts
for the arbitrary correlation among observations within sectors. Five different inputs are used in this regression,
specifically, energy, material excluding energy, non-production workers, production workers, and capital. All inputs,
lagged input prices, and the energy input share are logged and double-demeaned across industries and time. The
standard errors in parentheses are clustered at the NAICS industry level. J-test and the p-value refer to Hansen’s
J-statistics and p-value for overidentifying restriction, respectively. *, **, and *** indicate significance at the 0.1, 0.05,
and 0.01 levels, respectively.

that an increase in production workers of one percent leads to an increase in the energy share of
approximately 1.7 percent. The coefficient in front of energy is negative, which shows that theenergy input becomes less efficient as an economy uses more energy, thereby capturing the lawof diminishing returns for energy. Non-production workers are a substitute, but the coefficient ismarginally statistically significant. The J-test of Hansen (1982) does not statistically reject thevalidity of our instruments at the conventional level.
Since the coefficients of capital and material are not statistically significant in the baseline
specification of column (1), we exclude each input and re-do the analysis in columns (2) and (3). We
find that including these inputs separately leads to positive and statistically significant estimates,
indicating that material and capital may be a complement to the energy input, like production

11

workers. Given that the industry-level material and capital are highly correlated once they are
double-demeaned, it is not surprising to observe that putting both variables together does not
generate statistically significant estimates at the conventional level. Columns (4)-(6) show the OLS
results. As expected, due to the mechanical correlation that arises from the measurement errors, thecoefficient of energy input is positive, and it is difficult to make a meaningful inference.
Our results suggest a need for a more flexible production function with respect to energy
input for macroeconomic models. Many of the coefficients in Equation (2.6) are economically andstatistically different from zero, formally rejecting the Cobb-Douglas production function that requires
the energy input share to be invariant with respect to any factor input. Appendix B.5 considers a
CES production function, but the empirical results do not support this specification. In addition,
for the robustness checks, Appendix B revisits the main empirical results by considering various
other specifications, such as adjusting for the fixed costs in production, controlling more variables,
allowing the regression weights, using a smaller number of inputs, and using KLEMS data. The key
complementary between labor and energy, δel > 0, is both economically and statistically significant at
the conventional level in all the specifications; it ranges from 1.1 to 2.2 depending on the specifications,which are largely consistent with the Bayesian estimation results in Section 3.2.

2.3

The Returns to Scale Cyclicality

This section formally defines the returns to scale of the translog production function and accesses its
cyclicality. Conceptually, returns to scale measures how much the percentage of output increases
when all inputs increase by one percent. More precisely, by deriving the local elasticity of scale(Hanoch 1975; Epifani and Gancia 2006) for the translog production function F ({V i }; εa ) in Equation(2.2), the industry-time-specific returns to scale is expressed as follows:"
rtsjt =

X

#

βi +

X

i

where rtsjt ≡βik ln(Vjtk ) ,

(2.7)

k

i };εa )∂ log[F ({λVjtt ]|λ=1 denotes the returns to scale, and λ reflects the proportional∂ log(λ)

changes in all inputs. Under the conventional Cobb-Douglas production function, the returns to scaleP
does not depend on inputs: rtsjt = i βi . Under the translog production function, however, thereturns to scale changes with the input usage. The degree of change depends on the parameter βik ,which governs the substitution pattern among inputs. If inputs are complements (substitutes), an
increase in one input raises (lowers) both the efficiency of other inputs and the returns to scale. The
returns to scale in Equation (2.7) nests the constant returns to scale (rts = 1) embedded in the CES
and the standard Cobb-Douglas production functions.
To examine the returns to scale with the estimated parameters, we follow the previous studies
on returns to scale (Hall 1990; Basu and Fernald 1997) and additionally assume that the wedge does12

i = τ . This assumption still allows the common components of
not differ across different inputs: τjt
jt

the wedge across different inputs, such as the price markup and fixed cost of production, and it isconsistent with the standard DSGE models presented in Section 3. Despite its consistency with theprevious work and the DSGE model, one potential concern about this assumption is the presence ofan input-specific adjustment cost that is likely to differ across different inputs. In Appendix B.3, we
explicitly integrate the input-specific adjustment cost and show that the returns to scale cyclicality
results are robust to this concern.
With the common wedge assumption, we can rewrite the returns to scale by combining Equation(2.7) with the sum of Equation (2.4) across all input shares:−1
rtsjt = τjt
× salljt ,where sall
jt ≡

P

(2.8)

i
i sjt denotes the sum of all five input expenditure shares of total sales for industry j

at time t. With an additional assumption that the wedge is the inverse price markup, one can link
the returns to scale, price markup, and profit share, as in the previous studies of Hall (1990) andBasu and Fernald (1997): rtsjt = Markupjt × (1 − Profit Sharejt ).
We log-linearize Equation (2.8) to assess the cyclicality of the returns to scale:c jt = −τ̂jt + ŝall
rts
jt ,

(2.9)

where ŝall
jt is observed in the data and τ̂jt can be recovered as a residual of Equation (2.6) withthe estimated parameters reported in Column (1) of Table 1; we use double-demeaned variables torecover the log-linearized variables. Due to the demeaning, our estimation strategy only identifies
the wedge up to a constant and does not shed light on the level of returns to scale. However, we canstill analyze the association between the returns to scale and the value added and infer the cyclicalityof the returns to scale.
Figure 1 presents empirical evidence that the returns to scale is procyclical, and it arises fromthe procyclicality of the inverse wedge. Figure 1a shows a strong positive relationship between the
returns to scale and the value added. We interpret this result as empirical evidence that supports thenotion of the procyclical returns to scale; when industries experience larger (smaller) value added,
they preserve larger (smaller) returns to scale. Correlating each component of the returns to scale
and the value added reveals the importance of the inverse wedge in generating the procyclical returns
to scale. The scatter plot of the inverse wedge and value added in Figure 1b closely replicates thescatter plot of the returns to scale and value added in Figure 1a, whereas the scatter plot of thesum of input shares (1 - profit share) and value added shows the negative relationship and does notcontribute to the positive relationship plotted in Figure 1a.
The economic intuition behind Figure 1 is straightforward once we specify a structural assumption

13

Figure 1: Returns to scale and its components(a) Returns to Scale(b) Inverse Wedge(c) Sum of Input SharesNote. Figure 1a assesses the variability in the returns to scale, and Figures 1c and 1b show the variability in the twocomponents of the returns to scale, the inverse wedge and the sum of input shares. The y-axis is returns to scale, andit decomposes into the inverse wedge and the sum of input shares; the x-axis is value added. The returns to scale isrecovered based on Equation (2.9). All the variables are in double-demeaned logged values, which shows how a 1%change in value added affects approximately x% of the returns to scale, inverse wedge, and sum of input shares. The
slopes of the linear lines in Figures 1a, 1c, and 1b are 1, 1.05, and -.05, respectively.

that the wedge is the inverse price markup, which is consistent with the previous literature. Under thisassumption, the strong input complementarity between energy and other inputs, such as production
workers, leads to a larger increase in the marginal product of energy in expansion. As a result, thedifference between the marginal product of energy and the real energy price—the price markup or
the inverse marginal cost—becomes larger. The resulting decrease in the real marginal cost makes
the production more efficient and increases the returns to scale. The rise in profit share decreases
the economic efficiency in the expansion, but this effect is not strong enough to overturn the returns
to scale procyclicality. Our results imply that there is a tight connection between the returns toscale cyclicality and price markup cyclicality, which we explore carefully in a standard medium-scale
DSGE model in Section 3.

14

3

Macroeconomic Implications

Motivated by our empirical results, we explore the macroeconomic implications of the complementarityinduced procyclical returns to scale. We integrate the flexible translog production function intoa standard medium-scale DSGE model (Smets and Wouters 2007). We re-estimate the model byusing a Bayesian method with aggregate time-series data and confirm the procyclical returns toscale that arises from the complementarity between labor and energy. By comparing our benchmark
model with the standard models of Cobb-Douglas production functions, we find that the model withthe procyclical returns to scale (i) generates procyclical price markups, (ii) matches the different
cyclicality of input shares, and (iii) decreases nearly half of the contribution of price and wage markup
shocks to output fluctuation.

3.1

Model

This section describes how our model extends the DSGE model of Smets and Wouters (2007), which
is nested in our model. Our discussion in this section focuses on illustrating the key differences fromthe standard model and the relationship with the empirical framework in Section 2. In particular, wecharacterize the translog production function with an energy input, the corresponding changes in
the first-order conditions, and the modeling of the energy market. The other structure of the model
follows Smets and Wouters (2007) closely. There are households, labor unions, final good producers,intermediate goods producers, the government and central bank, and global energy consumers andsuppliers in the model. The model features sticky prices, sticky wages, utilization adjustment costs,
investment adjustment costs, and consumption habits. The details of the model are relegated toOnline Appendix A.
Production function Consider the following translog production function for intermediate goodi at time t:yt (i) = exp(εat )[kts (i)]βk [lt (i)]βl [et (i)]βe|{z

Cobb-Douglas}lt (i)¯lβel log(et /ē) |{zet (i)ēβel log(lt /l̄)second-order terms

−
}

υ
|{z}

fixed costs(3.1)

with βk + βl + βe = 1,
where kts (i) is capital services used in production, lt (i) is labor, and et (i) is energy. lt and et are
aggregate labor and energy, respectively, which individual firms take as a given when maximizing
their profit, and ē and ¯l are steady-state values of et and lt , respectively. Aggregate productivity
exp(εat ) follows an exogenous process, and υ is the fixed costs in production. The first part of the
production function has a conventional Cobb-Douglas form with constant returns to scale, and the15second part captures the second-order terms. All variables in Equation (3.1) are expressed withlowercase letters, which denote the detrended variables around the balanced growth path.11
We extend the standard two-factor Cobb-Douglas function with labor and capital to integrate
the core of the empirical findings. Given the economically and statistically significant estimate, whichindicates complementarity between labor and energy, we introduce an energy input et (i) and allow
the translog substitution structure between labor and energy via the substitution parameter βel in thesecond-order terms. It is straightforward to see that our production function nests the three-factor
Cobb-Douglas with βel = 0 and the two-factor Cobb-Douglas used in Smets and Wouters (2007) with
βel = 0 and βe = 0. We utilize these two special cases of our general translog production functions toemphasize how our new production function changes the traditional business cycle results.
By comparing the model’s production function (3.1) with the empirical production function(2.2), we make two major changes to parsimoniously adapt the translog structure to the businesscycle framework. First, we propose the three-factor translog production function with labor, capital,and energy; we neither additionally include material input with the input-output structure nor divide
labor input into production worker and non-production worker. Although adding more inputs is apotentially interesting margin to explore, this is not necessary for the procyclical returns to scale
results and could massively complicate the already complex analyses of the medium-scale DSGE model.
Instead of adding more inputs to the model, in Appendix B.7, we redo our empirical analyses with asmaller number of factors by excluding materials and aggregating non-production and production
workers into a single labor input. In all cases, we observe strong complementarity between labor andenergy.12 Because the empirical estimates are neither significant nor robust for the parameters otherthan the labor-energy complementarity, we focus on βel and the corresponding translog structure in
the model.13 This minimal adjustment from the conventional framework highlights the role of thetranslog structure while avoiding overcomplications in the theoretical investigation.
Second, we normalize the production function. We introduce the inputs into second-orderterms as a deviation from the steady-state values. This formulation carefully follows the previous
studies that integrate the generalized production function into business cycle models to make theproduction function parameters dimensionless or unit-free (see, e.g., Cantore and Levine 2012;(i)+υKoh and Santaeulàlia-Llopis 2017). By expressing the production function (3.1) as log ytȳ+υ
,

it is straightforward to see that the production function parameters are dimensionless with ournormalization. Appendix C explicitly shows that the production parameters depend on units ofK s (i)Et (i)
Specifically, yt (i) = Yγt (i), lt (i) = Lt (i), and kts (i) = γt t , where γ denotes the steady-state grosst , et (i) =γtgrowth rate.
12
In Appendix B.8, we consider the KLEMS data with three inputs. The strong complementarity between laborand energy remains robust.
13
For example, the substitution parameter between energy and capital is not statistically significant based on
the three-factor specification (Appendix B.7). The parameter for the energy square term is negative based on the
NBER-CES data but positive based on the KLEMS data (Appendix B.8).
1116

variables without making such a normalization. The normalization also makes the second-order term
disappear at the steady-state and makes our production function directly comparable to the standard
Cobb-Douglas specifications without changing the long-run balanced growth path.
When compared to the empirical specification (2.2), there are other minor changes in thenormalized translog production function (3.1) that closely follows the specifications in Smets andWouters (2007). We allow individual firm i’s translog production function in the economy andintroduce the second-order terms related to the returns to scale, lt and et , as an aggregate externality
for individual firms so that they do not choose their own returns to scale.14 In addition, we incorporate
the fixed costs in production υ as in the previous study. Given that our main empirical analyses
in Section 2 do not allow these costs, Appendix B.4 revisits the empirical analyses and finds thatthe input complementarity and procyclical returns to scale and markup results are robust to theexistence of the fixed costs. Finally, we include the labor-augmenting deterministic growth rate in theeconomy so that we can compare the model’s outcome to the time-series data with trend. Appendix
C presents the other general properties of the normalized translog production function (3.1).
First-order conditions Given real wage wt , the real price of capital service rtk , and the real priceof energy pet , firm i solves its cost-minimization problem subject to the translog production function(3.1). The first-order conditions with respect to energy, labor, and capital are given by
pet = mct (i) y (i)yt (i) + υ tβe + βel ˆlt,yt (i)et (i)(3.2)

yt (i) + υyt (i)(βl + βel êt ),yt (i)
lt (i)(3.3)

yt (i) + υ yt (i)βk s ,
yt (i)
kt (i)(3.4)

wt = mct (i)rtk = mct (i)where mct (i) is the real marginal cost of production or the inverse price markup that arises frommonopolistic competition, and x̂ is the log deviation from the steady state value x̄ for any variable x.
As in our empirical analyses (Equation (2.3)), Equation (3.2) shows that the real energy price equals
the marginal product of energy multiplied by the wedge that includes the real marginal costs andthe term related to the fixed cost. Although we evaluate Equation (3.2) at the aggregate level inthis model, evaluating the same equation at the industry-level recovers the industry-time-varying
first-order condition used in Section 2.15
14This specification is similar to the externality assumption in the increasing returns to scale literature (e.g., Baxter
and King 1991) or the redistributive shock introduced in Rios-Rull and Santaeulalia-Llopis (2010).
15
Specifically, consider a continuum of firms, which is indexed by i and operates in industry j. yt (i, j) is given

17Compared to the model with a Cobb-Douglas specification with an energy input, the only
extensions with the translog are βel ˆlt and βel êt in the first-order conditions (3.2) and (3.3), up to
log-linearization. Clearly, βel has a first-order effect on factor demand. Assuming the complementarity
between labor and energy (βel > 0), an increase in labor or energy input in expansion raises themarginal product of the other input more than the standard case. In contrast, there is a smaller
increase in the marginal product of input if labor and energy are substitutes (βel < 0). Note thatchanging βel has no direct first-order effect in the production function (3.1). Our production function
is identical to the Cobb-Douglas specification with log-linearization, as explicitly shown in Appendix
C. This feature of the model is desirable given that our empirical framework in Section 2 relies solely
on the first-order conditions to identify the complementarities between labor and energy inputs,
which is consistent with our model. Within the structure of the standard model, we reassess ourempirical results by re-estimating the key parameter βel with the aggregate data.
Energy market Given that we introduce an energy input into the production function, we need to
specify the energy market. We introduce a global energy market into our model where the real energy
price pet is determined subject to energy demand and supply shocks. We impose a parsimonious
structure on the energy market to focus on the general production function and minimize deviations
from the benchmark Smets and Wouters (2007) model.
Denote the global energy demand, excluding the US industrial usage et , as edt . The energy
market clearing condition is given by
et + edt = est ,(3.5)

where est is the global energy supply. The global energy supply is determined by the energy price andexogenous disturbances:
est
=
ēs

 e κspt
exp(εes
t ),
p̄e

(3.6)

where κs denotes the price elasticity of est , and the exogenous supply disturbances εest follow the
es
s
s
2
s
AR(1) process of εes
t = ρes εt−1 + ηt , where ηt ∼ (0, σes ). We assume that φe fraction of et is produced

domestically. Thus, the US net energy import is given by et − φe est , which implies that the grossby f (kts (i, j), lt (i, j), et (i, j); lt (j), et (j), εat ), where lt (j) and et (j) are aggregated labor and energy at the industry
level, respectively, and f (kts (i), lt (i), et (i); lt , et , εat ) denotes the production function yt (i) in Equation (3.1). Each firm(i, j) takes lt (j) and et (j) as agiven. Under this condition, the firm-level cost minimization problem implies thatset (i, j) = τt (i, j) βe + βel ˆlt (j) , where set (i, j) and τt (i, j) are the energy share and the wedge at the firm-industrytime-level, respectively. When firms in the same industry j are exposed to the same
 industry-level realization of
Calvo shocks, these firms become symmetric, and we obtain set (j) = τt (j) βe + βel ˆlt (j) , which features industry-level
labor inthe marginal
 product of energy. Furthermore, cross-sectional aggregation implies that, up to log-linearization,
set = τt βe + βel ˆlt , which is the same aggregate equation that we can derive from Equation (3.2).

18

domestic product (GDP) is ytGDP = yt − (et − φe est ).
The global energy demand, excluding US industrial usage, depends on the energy price pet ,GDPexogenous disturbances to the demand εedt , US GDP yt−1 , and real interest rates Et−1 [Rt−1 /Πt ],where Rt and Πt represent gross nominal interest rates and gross inflation, respectively:
edt
=
ēd

GDP
yt−1
ȳ GDP!ρey 

Et−1 [Rt−1 /Πt ]R̄/Π̄ρerr  e −κdpt
exp(εed
t ),
p̄e

(3.7)

where κd denotes the price elasticity of the global energy demand. The exogenous demand disturbances
ed
ed
d
d
2
εed
t follow the AR(1) process of εt = ρed εt−1 + ηt , where ηt ∼ (0, σed ). Clearly, global economic

activity positively affects energy demand (Balke and Brown, 2018; Kilian, 2009). We include lagged
US GDP and real interest rates on the right-hand side as proxies for global economic activity.
Furthermore, real interest rates capture the states of financial markets that might affect energy prices,
as discussed in Basak and Pavlova (2016) and Kilian (2014).
The remaining parts of the model are identical to those in the Smets and Wouters (2007) model.
See Online Appendix A for the full structure of the model.

3.2

Bayesian Estimation

This section discusses the Bayesian estimation by illustrating the construction of the likelihood
function and by presenting the prior and the posterior. For the objective comparisons across models,
we apply the Bayesian techniques, prior, and data identical to those in Smets and Wouters (2007)
to both our benchmark model with the three-factor translog function and the model with thethree-factor Cobb-Douglas function (βel = 0). We also bring in the energy data and make relevant
prior assumptions to estimate the new parameters introduced with the energy market in both models.
The corresponding estimation results not only confirm the input complementarity (βel > 0) we findin Section 2 but also reveal more rigid prices and wages and less persistent markup shocks. Thesechanges are important in understanding the sources of the business cycle, which we analyze in Section
3.3.3.
We use a Kalman filter to compute the likelihood of a state-space system. For the state equation,
we employ the algorithm suggested in Sims (2002). Our observation equation consists of nine variables,s ), the logarithm
including the growth rate of the global energy supply ∆ log(Ets ) ≡ log(Ets ) − log(Et−1of the real energy price log pet , and the seven major macroeconomic variables used in Smets andWouters (2007, Equation (15)):19  
  
GDP
γ̄
0
ŷtGDP − ŷt−1     dlCON SU M P T IONt   γ̄   ĉt − ĉt−1   0            dlIN V EST M EN Tt   γ̄   ît − ît−1   0         
  
γ̄
ŵ
−
ŵ
dlW
AGE
   
0

t
t−1 
t

     
 =  ¯l  + 
 + 0,ˆltlHOU RSt        
  



 0



π̄
π̂
dlP
RICE
t
t
          r̂t
F EDF U N DSt
  r̄   0   
  

 0
 lEN ERGY P RICE  p̄e  
e
p̂
t 
t
  
  

dlEN ERGYtêst − êst−1γ̄νtdlGDPt(3.8)

where ĉt , ît , π̂t , and r̂t denote the log deviations of (detrended) consumption, investment, gross priceinflation, and gross nominal risk-free return, respectively.
We obtain the global energy quantity data {Ets } from BP Energy (2020). The real energy
price is computed by the ratio of the producer price index for total energy to the GDP deflator.
Because it is an index, we demean the logarithm of the energy price in Equation (3.8). For the othermacroeconomic variables, we use the dataset constructed by Smets and Wouters (2007). Our sample
period is from 1966:q1 to 2004:q4, which is identical to that in Smets and Wouters (2007). Because
Ets data are available only at an annual frequency, we interpolate the annual series to construct aquarterly measure and introduce a measurement error νt in the observation equation. The standard
deviation of νt is denoted by σν . In total, we have nine observable variables, nine structural shocks(global energy supply and demand shocks and seven macroeconomic shocks in Smets and Wouters(2007)), and one measurement error. See Online Appendix B for further details of the data.
Table 2 presents the prior and the posterior of the parameters that arise from the integration of
the translog structure and the energy market; we assume the same prior in Smets and Wouters (2007)
for the common parameters used in the other two models. Among the new parameters, se = ēēscannot be separately identified from the other parameters that characterize the dynamics of theEtenergy market. Thus, we set se at the average value of E
s in the data during our sample period,twhich is 0.08. Based on previous studies on the energy shares of value added, we assume that βe has
a Gamma distribution with a 5% mean and 2% standard deviation.16
For the prior of the substitution parameter βel , we utilize our empirical estimate. By loglinearizing Equation (3.2) and taking cross-sectional average, we can relate βel to δel in Equation

16
The energy shares of value added are assumed to be 10% in Backus and Crucini (2000), 5.17% in Dhawan andJeske (2008), 4.3% in Finn (2000), and 4% in Rotemberg and Woodford (1996).

20

Table 2: The prior and the posterior of the new parameters
Parameter

Prior

HKL

HKL-CD

Mean

St.
Dev

FamilyPosteriorMode

Credible Set(5%, 95%)

PosteriorMode

Credible Set(5%, 95%)

βe

0.05

0.02

Gamma

0.023

(0.017, 0.033)

0.019(0.013, 0.028)

βel

0.1

0.025

Gamma

0.042

(0.031, 0.057)

-

-

ρey

1

0.8

Normal0.08(0.02, 0.21)

0.12(0.03, 0.26)

ρerr

1

0.8

Normal0.37(0.16, 0.90)

0.37(0.16, 0.91)

κd

0.1

0.08

Gamma

0.014

(0.0005, 0.167)

0.017(0.005, 0.154)

κs

0.1

0.08

Gamma

0.12

(0.04, 0.16)

0.11(0.04, 0.14)

σed

0.8

2

Inv. Gamma

0.74(0.66, 1.40)

0.77(0.70, 1.37)

σes

0.8

2

Inv. Gamma

0.82(0.56, 1.02)

0.78(0.55, 0.91)

ρed

0.9

0.05

Beta

0.9993(0.9989, 0.9994)

0.9994(0.9991, 0.9995)

ρes

0.9

0.05

Beta

0.9972(0.9961, 0.9978)

0.9983(0.9981, 0.9985)

e

0

2

Normal-0.31(-3.51, 3.03)-0.10(-3.28, 3.30)

σν

0.1

0.1

Inv. Gamma

0.05(0.04, 0.24)

0.05(0.04, 0.26)

φe0.03370.0147

Gamma

0.033

(0.016, 0.075)

0.029(0.015, 0.065)p̄

Notes: HKL denotes our benchmark model with the translog production function. HKL-CD refers to the Cobb-Douglas
specification with energy but without complementarity (βel = 0).

(2.6) within the structure of this model by using the following equation:
βel = [δel + (Φ − 1)βl ]βe,
1 − βe(3.9)

where Φ is the gross price markup at the steady state.17 When δel = 1.694, which is our benchmark
estimate in Table 1, and Φ, βl , and βe equal their prior means of 1.25, 0.65, and 0.05, respectively,
Equation (3.9) implies that βel = 0.098. When we multiply the standard error of δ̂el in Table 1 byβe
1−βe , which is evaluated at the prior mean of βe = 0.05, we obtain 0.024. Thus, we assume that βel

has a Gamma distribution with a mean of 0.1 and standard deviation of 0.025; our posterior estimate
and the corresponding theoretical results are nearly identical when we consider the alternative valuesof δel estimated in Appendix B.
For the rest of the parameters, we use the following priors. We demean log pet in our observation
equation and thus assume that p̄e has a normal distribution with a zero mean and standard deviationof two. This choice is similar to the prior of ¯l in Smets and Wouters (2007). For the standard deviationof the measurement error, we choose mean and standard deviation of 0.1. We decide the prior of the
global share of the US energy production φe given the following considerations. The US net import   ni  ni 
E
E
Etof energy, Etni , is given by Et − Ets φe . Then, φe = E− Ets = se − Ets . By subtracting the
s
t

t

t

sample average of Etni /Ets from se , we obtain the prior mean of φe , which is 0.0337. The sample
17

Equations (A.70) and (A.80) in Online Appendix A are also used in the derivation.

21

Table 3: The Calvo and the markup shock parameters in the three models
Parameter

Prior

Posterior Mode

Mean

St. Dev

FamilyHKLHKL-CD

S&W

ξp

Calvo sticky price

0.5

0.1

Beta

0.75

0.69

0.67

ξw

Calvo sticky wage

0.5

0.1

Beta

0.84

0.78

0.76

ρp

Price markup shocks:

0.5

0.2

Beta

0.83

0.91

0.89

µp

p
,
λ̂pt = ρp λ̂pt−1 + ηtp − µp ηt−1

0.5

0.2

Beta

0.69

0.77

0.73

σp

where ηtp ∼ N (0, σp2 )0.1

2Inv. Gamma

0.15

0.14

0.14

ρw

Wage markup shocks:

0.5

0.2

Beta

0.92

0.96

0.97

µw

w
w
w
λ̂w
t = ρw λ̂t−1 + ηt − µw ηt−1 ,w2where ηt ∼ N (0, σw )

0.5

0.2

Beta

0.85

0.90

0.91

0.1

2

Inv. Gamma

0.23

0.24

0.25

σw

Notes: HKL denotes our benchmark model with the translog production function. HKL-CD refers to the CobbDouglas specification with energy but without complementarity (βel = 0). S&W is the Smets and Wouters (2007)
model.

standard deviation of Etni /Ets in the data is 0.0147, which is used as a prior standard deviation.
For the shock processes, we rely on a preliminary regression analysis. Specifically, we estimatedlog Ets = c + dt + κs log pet + εest and a similar equation for log Et by using the lagged values of Romerand Romer (2004) monetary policy shocks as instrumental variables. Based on the point estimatesand standard errors, we set the prior mean and standard deviations of ρey , ρerr , κd , κs , σed , σes , ρed ,and ρes . See Online Appendix B.2 for these regression results.
The posterior is computed by using a random walk Metropolis-Hastings algorithm with a chain
length of 500,000. We consider three different models: our benchmark model (HKL), the model
without input complementarity in production (βel = 0; HKL-CD), and the Smets and Wouters(2007) model (βel = 0, βe = 0; S&W). The acceptance rates of the chains are 29%, 35%, and 23%,
respectively.
As shown in Table 2, the following parameters are newly introduced and estimated comparedto Smets and Wouters (2007). The energy share in the steady-state βe is estimated to be 2.3%.
The posterior mode of βel is 0.042. If we use Equation (3.9) to compute the value of δel that
corresponds to the posterior mode of βel , Φ, βl , and βe , we obtain 1.36, which is within the range of
one standard error of δ̂el in Table 1. The price elasticities of global energy demand and supply, κdand κs , respectively, are small at the posterior mode, which is consistent with the results in Hamilton(2009), Kilian (2009), and Kilian and Murphy (2012). Both energy supply and demand shocks arehighly persistent. Furthermore, the sizes of these shocks are estimated to be larger than the otherstructural shocks because energy prices substantially fluctuated during the sample period. Finally,
the measurement errors induced by the interpolation are estimated to be relatively small.
Among the other parameters common to the Smets and Wouters (2007) model, Table 3 reports
22the parameters that show the most noticeable changes across the three models: the parameters
associated with the Calvo stickiness and the markup shocks. The estimation results show that ourbenchmark model features larger price and wage rigidity and less persistent markup shocks comparedto the two other models with Cobb-Douglas production functions.18 More rigid prices are intuitive,
given the input complementarity (βel > 0) reported in Table 2. As will be shown explicitly in Section
3.3.1, the input complementarity generates large countercyclical fluctuations in the marginal costs. As
a result, the Bayesian estimator in our benchmark model prefers the stickier prices or the flatter price
Philips curve to match the correlation of inflation and GDP observed in the data. Given the stickier
prices, the stickier wages in our benchmark model are also intuitive. As the Bayesian estimator
matches the real wage fluctuation in the data (Equation (3.8)), it changes the wage Philips curve
parallel to the flatter price Philips curve and makes nominal wages more rigid.19 Corresponding to
the flatter Philips curves, the real variables’ responses to structural shocks are amplified, yielding
smaller residual variations previously explained by the markup shocks. Consistent with this intuition,
the contribution of price and wage markup shocks to output fluctuations is substantially smaller inthe benchmark model than in the other two models, as will be shown in Section 3.3.3.
For the rest of the parameters, the estimation results are similar across three models. See Online
Appendix B.3.

3.3

Business Cycle Implications

This section illustrates the business cycle implications of the normalized translog production function.
Despite its parsimonious structure, our benchmark model features procyclical returns to scale andprocyclical price markups that are consistent with the empirical findings in Section 2. Moreover,
our benchmark model generates countercyclical labor shares and procyclical energy shares, which
are consistent with the data, and maintains the countercyclical capital shares and procyclical profit
shares, similar to the previous studies. Finally, we show that our framework leads to a considerable
decrease in the importance of markup shocks on output fluctuations compared to the models with
Cobb-Douglas production functions.
3.3.1

Returns to Scale and Price Markups

Armed with the estimated model using aggregate data, we revisit the returns to scale and pricemarkup procyclicality results in Section 2. Given the normalized translog production function in
Equation (3.1), which is denoted by yt (i) = f (kts (i), lt (i), et (i); lt , et , εat ), the returns to scale that firm
∂ log[f (λkts (i),λlt (i),λet (i);lt ,et ,εat )]i faces is given by rtst (i) ≡|λ=1 = [1 + βel (ˆlt + êt )] yt (i)+υ . Note that∂ log(λ)

yt (i)

18

Despite the differences in the Calvo parameters across the three models, they are all broadly consistent with theempirical estimates in Nakamura and Steinsson (2008) and Barattieri et al. (2014) and the model-based estimates inJustiniano et al. (2010, 2011).
19
See Equations (A.88) and (A.92) in Online Appendix A for the price and wage Phillips curves.

23

firms take the aggregate labor lt and energy et as given when they change their scales proportionally.
Since the cross-sectional dispersion of log(yt (i)) is of the second order, the average returns to scale
across firms, up to the first order, is given by
rtst = rtstrans× rtsft ixth
i  y + υ t≡ 1 + βel ˆlt + êt ×,yt(3.10)


ˆlt + êt denotes the part of the returns to scale that arises from the translog
where rtstrans
≡
1
+
β
el
t
structure, and rtsft ix ≡ yty+υ
denotes the part of the returns to scale that emerges because of thetpresence of the fixed costs in production. Note that the rtst is conceptually the same as our empiricalmeasure of returns to scale rtsjt in Section 2.3 since both of them are based on the cross-sectional
average of the firm-level returns to scale. As a result of the adjustment in the production function,compared to Equation (2.7), Equation (3.10) has only one production parameter βel and additionallyfeatures the term relevant to the fixed costs in production rtsft ix .
Clearly, the returns to scale varies over the business cycle with the translog structure and fixed
costs. In the simplest case with Cobb-Douglas specification (βel = 0) without the fixed costs (υ = 0),
the returns to scale is time-invariant: rtst = βk + βl + βe = 1. Allowing fixed costs in production
recovers the returns to scale in Smets and Wouters (2007), rtsft ix = yty+υ
, which is countercyclical.
t
The reason is because in a recession, firms face relatively higher fixed costs and thus larger returns toscale compared to the expansion. Additionally, allowing the translog structure makes the returns toscale more procyclical if inputs are complements (βel > 0) and more countercyclical if inputs aresubstitutes (βel < 0). Depending on the degree of input complementarity, the returns to scale can beprocyclical with the presence of fixed costs.
For completeness, we define the aggregate returns to scale, which endogenizes the changes inaggregate labor and energy. Using the fact that yt is obtained by aggregating yt (i) is the same asf (kts , lt , et ; lt , et , εat ) up to the first-order, the aggregate returns to scale is given by
i  y + υ tˆRT St = 1 + 2βel lt + êt ×,ythwhere RT St ≡∂ log[f (λkts ,λlt ,λet ;λlt ,λet ,εat )]|λ=1 .
∂ log(λ)(3.11)

The aggregate returns to scale is derived under theassumption that all firms change their inputs proportionally such that the aggregate inputs changeby the same proportion. The difference between RT St and rtst (i) is that for RT St , the exponentsof the second-order term in Equation (3.1) endogenously vary with aggregate inputs when all firms
adjust their inputs. In contrast, rtst (i) treats the exponents of the second-order term as constantbecause only firm i changes its inputs. This difference yields an additional term βel (ˆlt + êt ) to theaggregate returns to scale in Equation (3.11). Note that RT St = rtst without the translog structure24

Table 4: Cyclicality of returns to scales and price markups

Correlation with log(ytGDP )HKL
HKL-CD (βel = 0)
Smets and Wouters (2007)

(1)

(2)

(3)

(4)

(5)

log(rtstrans)tlog(rtsft ix )log(rtst )log(RT St )log(Φt )

0.74

-1

0.20

0.55

0.19

(0.63, 0.85)(-1, -1)(-0.25, 0.54)(0.30, 0.74)(0.03, 0.26)

-

-1

-1

-1

0.01

-

(-1, -1)(-1, -1)(-1, -1)(-0.14, 0.18)

-

-1

-1

-1

0.00

-

(-1, -1)(-1, -1)(-1, -1)(-0.17, 0.18)

lt + êt ) and rtsft ix ≡ yty+υ
Notes: rtstrans≡ 1 + βel (ˆare the returns to scales that arise from the translog production
t
t
trans
function and fixed costs, respectively. rtst ≡ rtst
× rtsft ix is the average returns to scale that each firm faces,
RT St ≡ [1 + 2βel (ˆlt + êt )] yty+υ
istheaggregate
returns
to scale, and Φt is the gross price markup. All variablest
are log-linearized. For each of the three models, we report the correlations of each variable with ytGDP at theposterior mode and 90% credible intervals. HKL denotes our benchmark model with the translog production function,
HKL-CD refers to the Cobb-Douglas specification with energy (βel = 0), and Smets and Wouters (2007) is the
Cobb-Douglas specification without energy (βel = 0, βe = 0).

(βel = 0).
Given the structure of our model, it is straightforward to recover the price markup. The changes
in price markups can be most easily understood from the following expression of real marginal costs.
By rearranging and combining the first-order conditions (3.2), (3.3), and (3.4) and the production
function (3.1), we have:Φ̂t = −mcc t = βel (ˆlt + êt ) − βk r̂tk − βl ŵt − βe p̂et + εat ,(3.12)

where Φt is the aggregate price markup, which depends on the translog part of the returns to scale,c trans
rts= βel (ˆlt + êt ), real input prices (r̂tk , ŵt , p̂et ), and productivity shocks (εat ). The price markuptrises when firms employ more complementary inputs, face a decrease in input prices, or experience
positive productivity shocks. Equation (3.12) clarifies how traditional models link price markup
cyclicality to price and wage rigidity. Previous studies have focused on the relative rigidities of
prices and wages, which affect the cyclicality of ŵt . Consider the simple case of a Cobb-Douglas
specification with only labor input; the price markup then becomes Φ̂t = −βl ŵt + εat . In this case,conditional on any shocks other than productivity shock, price markup cyclicality is governed by
the cyclicality of real wage ŵt , which is tightly related to the relative rigidities of prices and wages.
More generally, in previous studies, the cyclicality of real input prices affects price markup cyclicality.
In our setup, we identify a new term that additionally changes the price markup, βel (ˆlt + êt ), by
adopting a more general production function that allows a flexible input substitution pattern.
Columns (1)-(4) in Table 4 show that the returns to scale is procyclical in our benchmark model(HKL), which is consistent with the results in Section 2, but it is countercyclical in the other models.

25

Column (1) shows the cyclicality of the new returns to scale term that arises from the translog
structure rtstrans ≡ 1 + βel (ˆlt + êt ). Given the posterior mode of βel = 0.042, the correlation of

t
trans
rtst
and GDP is 0.74; in boom periods, the larger use of each input generates synergies and makes
the economy produce more. In contrast, as shown in column (2), rtsft ix features perfect or nearly

perfect countercyclical returns to scale in all three models as the fixed costs become relatively larger
in recession.20 Column (3) indicates that the total average returns to scale rtst is mildly procyclicalin our benchmark model, which shows that the procyclical effect of the translog structure dominates
the countercyclical effect of the fixed costs. The aggregate returns to scale RT St features stronger
procyclicality due to the endogenous movement in aggregate labor and energy, as appears in column(4). In the other models with Cobb-Douglas production functions, the returns to scale equals rtsft ix
and is countercyclical.
Column (5) shows that the price markup is procyclical in our benchmark model (HKL), as wec trans .
hinted in Section 2. This procyclicality works through the new part of the returns to scale, rts
In expansion, when firms utilize more labor and energy, the complementarity between these inputs
leads to an increase in the marginal product of inputs, which in turn decreases the real marginal
costs and increases the price markup. Without this mechanism, price markups are acyclical in ourmodel, as in the models with Cobb-Douglas specification, because the net effect of real input pricecyclicality on price markup cyclicality is negligible.
3.3.2

Factor and Profit Shares

Our translog production function has a novel implication on the income distribution across factors
and profits through its effects on the first-order conditions. By rewriting Equations (3.2)-(3.4) interms of factor shares, to the first-order approximation, we have:set ≡pet et1 yt + υ=(βe + βel ˆlt ),Φt ytGDP
ytGDP(3.13)

slt ≡

wt lt
1 yt + υ=(βl + βel êt ),GDP
Φt ytGDPyt(3.14)

rt kt
1 yt + υ=βk ,GDP
Φt ytGDPyt(3.15)

skt ≡20

In the Smets and Wouters (2007) model, log(ytGDP ) and log



yt +υ
yt



are perfectly negatively correlated because

ytGDP = yt . In models with energy input, although
ytGDP = yt − (et − φs est ), we still observe the nearly perfect negative

yt +υ
GDP
correlation between log(yt) and log yt . This is because the difference between yt and ytGDP , the net energysimport et − φe et , is negligible relative to yt .

26

Table 5: Cyclicality of factor and profit shares

Correlation with log(ytGDP )HKL
HKL-CD (βel = 0)
Smets and Wouters
Data

(1)

(2)

(3)

(4)

Energy Shares

Labor Shares

Capital Shares

Profit Shares

0.46

-0.41

-0.47

0.42

(0.26, 0.61)(-0.50, -0.25)(-0.55, -0.33)(0.25, 0.50)

-0.39

-0.39

-0.39

0.39

(-0.48, -0.21)(-0.48, -0.21)(-0.48, -0.21)(0.21, 0.48)

-

-0.41

-0.41

0.41

-

(-0.51, -0.22)(-0.51, -0.22)(0.22, 0.51)

0.50

-0.38

-

-

Notes: For each of the three models, we report both the correlations of the logarithm of each variable with log(ytGDP )at the posterior mode and the 90% credible intervals. We compute the empirical energy and labor shares based on thedata used for the Bayesian estimation in Section 3.2 and employ the Baxter and King (1999) filter with a periodicity ofcycles between 6 and 32 quarters. Following Gorodnichenko and Ng (2010), we apply the same filter to the model
variables and calculate the correlation coefficients by using the representation in Croux et al. (2001, Equation (8)).
HKL denotes our benchmark model with the translog production function, HKL-CD refers to the Cobb-Douglas
specification with energy (βel = 0), and Smets and Wouters (2007) is the Cobb-Douglas specification without energy(βel = 0, βe = 0).

where set , slt , and skt are energy, labor, and capital shares, respectively. Clearly, the translog structureand energy imports generate two changes in the factor share equations from those in Smets andWouters (2007). First, the translog production function induces the extra terms in the energy and labor
shares, βel ˆlt and βel êt , which constitute the cyclical components of rtstrans . Second, the imported
energy used in production allows us to consider the energy shares and makes ytGDP marginallydifferent from yt by net energy imports. The smaller inverse input wedge—the multiplication oft +υ
the price markup (Φt ) and the inverse of the fixed-cost relevant term (1/ yyGDP)—and the largertcomplementarity (βel > 0) leads to a greater factor demand and factor shares in expansions.
By combining the input share expressions (3.13)-(3.15) with the definition of the returns toscale (3.10), the profit share becomes:
sΠ
t =



rtst
yt
,
1−
Φt ytGDP

(3.16)Πtk sewhere sΠt ≡ y GDP with the aggregate profit Πt ≡ yt − wt lt − rt kt − pt et . Thus, our model generates atrelationship among the profit share, price markup, and returns to scale, consistent with the previous
studies of the returns to scale and the empirical expression (2.8). Holding everything else constant,larger price markup or smaller returns to scale leads to smaller total input shares and larger profit
shares. Note that the returns to scale cyclicality can substitute for the price markup cyclicality in
explaining the same profit share dynamics.
Our benchmark model better matches the energy input share in the data than the model with the
Cobb-Douglas production function. As shown in column (1) of Table 5, our benchmark model features27

procyclical energy shares with a correlation of 0.46, which is both qualitatively and quantitatively
consistent with the data.21 The procyclical energy shares arise from the complementarity between
labor and energy (βel > 0), where a positive βel is motivated by the empirical analyses (Section 2)
and is estimated by the Bayesian method (Section 3.2). This procyclical effect of complementarity
t +υ
dominates the countercyclical effects of the inverse price markup (1/Φt ) and the fixed costs ( yyGDP)ton energy shares. In contrast, given the Cobb-Douglas production function, the energy share is
identically countercyclical to the other factor shares. Its simple assumption on the functional form
does not allow us to generate different input share cyclicality as in the data.
At the same time, our benchmark model maintains countercyclical labor and capital shares,
which are consistent with the models with Cobb-Douglas specifications. As presented in columns(2) and (3), the cyclicality of labor and capital shares is similar across the three different models.
In particular, the labor share cyclicality is comparable to the data and to what are reported in
Karabarbounis (2014, Table 7) and Rios-Rull and Santaeulalia-Llopis (2010, Table 2). Both thelabor and capital shares are countercyclical in all models because of the fixed costs in production,
which generate a countercyclical wedge and depress the labor and capital demand in expansions.
For the labor share, although our benchmark model has an additional procyclical complementarity
term (βl + βel êt ), it is canceled out by the procyclical price markup effect (Φt ), which results incountercyclical labor shares.22 The capital share is slightly more countercyclical in our benchmark
model relative to the capital share in the two other models due to procyclical price markups.
Finally, the profit share is procyclical in all three models, as shown in column (4), but theunderlying mechanism is largely different depending on the production function specification. The
profit share procyclicality in our benchmark model arises from the procyclical price markups. Sincefirms generate more price markups in expansion as a result of the depressed marginal costs with
input complementarity, their profit shares rise at the same time. The returns to scale is procyclical inthis model and mitigates the profit procyclicality, but this effect is not strong enough to make profit
shares countercyclical. In contrast, in the models with a Cobb-Douglas production function, theprofit share procyclicality relies on the countercyclical returns to scale. Without the complementarity,
in expansion, the fixed costs depress the returns to scale, which in turn decrease all input shares andincrease the profit shares. The price markup is acyclical in these models and does not contributeto the procyclical profit. Note that the difference between yt and ytGDP is negligible and does notcontribute to the cyclicality of profit shares in any model.
21

For U S EN ERGYt , we use monthly US industrial energy usage data from U.S. Energy Information Administration(2021). We seasonally adjust this series by using X-13 ARIMA-SEATS and aggregate it to a quarterly measure.
Because this monthly measure is available from 1973, our sample for Table 5 spans from 1973:q1 to 2004:q4.
22
Note that the magnitude of procyclical variation in the complementarity term in the labor share (βl + βel êt ) issmaller than that in the energy share, which results in a procyclical energy share but a countercyclical labor share.
ˆThis is because the energy share βe is small, and with log-linearization, βe + βel ˆlt becomes ββellt , which generates aelarge procyclical component in the energy share.

28

In addition to assessing the unconditional moments, we further verify our benchmark model byassessing the conditional moments in Online Appendix B.4. Given that our theoretical mechanism iscentered around the complementarity between labor and energy, we compare the model responsesof labor and energy to monetary and fiscal policy shocks with the corresponding empirical impulse
responses, which are estimated by using the identified structural shocks in Romer and Romer (2004)
and Auerbach and Gorodnichenko (2012). We find that the model responses are largely consistent
with the empirical responses in our benchmark model. Note that this consistency holds without
directly matching the dynamics of US industrial energy usage et in the Bayesian estimation in Section
3.2.
3.3.3

Contribution of Markup Shocks

After observing the changes in the returns to scale, price markup, and factor share cyclicality,
which are induced by the translog production function, this section seeks to understand the relative
importance of different driving forces of business cycles. For this purpose, we compute forecast
variance decompositions (FEVDs) and impulse responses of the three models with different production
functions.
The translog specification substantially decreases the contribution of price and wage markup
shocks to business cycles. Table 6 presents the FEVDs of output and labor at ten quarter horizon
based on all three models. While introducing the energy input into a Cobb-Douglas technology(HKL-CD) does not meaningfully change the FEVD from that of the Smets and Wouters (2007)
model, additionally introducing the translog specification (HKL) alters the results substantially. The
most notable change is the contribution of markup shocks. Price and wage markup shocks explain
28% and 26% of output fluctuations in the Smets and Wouters (2007) model and the model withoutcomplementarity (HKL-CD), respectively. In contrast, the corresponding FEVD decreases to 15% in
our benchmark model. The results for labor are similar. The FEVD of labor with respect to markup
shocks slightly decreases from 45% to 44% when we introduce an energy input, but it becomes 31%
when we incorporate the input complementarity between labor and energy. At alternative horizons,
we still observe that markup shocks are less important determinants of output and labor in ourbenchmark model than in the two other models. As a result, the other structural shocks, such as
shocks to productivity, demand, and energy, become more important drivers of business cycles in ourbenchmark model than in the other models. See Online Appendix B.5 for additional results.
Given the substantial changes in the contribution of markup shocks, Figure 2 presents theimpulse responses of output to price and wage markup shocks of all three models to investigate
the underlying mechanism behind the FEVDs. The top (bottom) panels show the responses to a
one-standard-deviation contractionary price (wage) markup shock, and the solid, dash-dotted, anddashed lines represent the results based on our benchmark model (HKL), the model without input
29

Table 6: Forecast error variance decomposition of output and labor (10 quarters)
Output (log ytGDP )HKL

HKL-CD

Productivity (neutral)

0.36

0.34

Risk premium

0.05

0.04

Government spending

0.09

0.08

Investment-specific productivity

0.22

Monetary policy

0.08

Price markup

Labor (log lt )

S&W

HKL

HKL-CD

S&W

0.32

0.05

0.04

0.04

0.04

0.09

0.07

0.06

0.08

0.17

0.15

0.15

0.21

0.22

0.24

0.21

0.21

0.06

0.06

0.13

0.09

0.09

0.07

0.11

0.11

0.08

0.13

0.12

Wage markup

0.08

0.15

0.17

0.22

0.31

0.33

Energy demand

0.02

0.01

-

0.01

0.00

-

Energy supply

0.03

0.01

-

0.01

0.00

-

Productivity shocks

0.58

0.54

0.54

0.29

0.25

0.25

Demand shocks

0.21

0.19

0.19

0.39

0.31

0.30

Markup shocks

0.15

0.26

0.28

0.31

0.44

0.45

Energy shocks

0.06

0.01

-

0.01

0.01

-

Panel A

Panel B

Notes: Panel A decomposes the forecast error variances into the contributions of the 9 structural shocks inthe model. Panel B summarizes the FEVDs of different types of shocks. The productivity shocks include theneutral and investment-specific productivity shocks, the demand shocks include the risk premium, government
spending, and monetary policy shocks, the markups shocks include the price and wage markup shocks, andthe energy shocks include the energy demand and supply shocks. HKL denotes our benchmark model with thetranslog production function, HKL-CD refers to the Cobb-Douglas specification with energy (βel = 0), andSmets and Wouters (2007) features the Cobb-Douglas production function without energy (βel = 0, βe = 0).

complementarity (HKL-CD), and the Smets and Wouters (2007) model (S&W), respectively. Column(a) presents the impulse responses at the posterior mode of the three models. Columns (b)-(d) set
different subsets of parameters at the posterior mode of HKL to investigate why the price markup
shocks have smaller effects in our benchmark model. We fix two categories of parameters reported inTable 3, which have noticeably different posterior modes across the models: (i) Calvo price and wage
stickiness parameters and (ii) persistence of markup shocks. For all three models, column (b) uses the
Calvo parameters, column (c) uses the markup shock parameters, and column (d) uses both the Calvo
parameters and markup shock parameters in our benchmark model. In Online Appendix B.6, we
separately test whether the positive input complementarity parameter (βel > 0) can depress the rolesof markup shocks but do not find the supporting evidence. It mostly substitutes the countercyclicalvariations in price markups in the previous models, as discussed with the profit share cyclicality
result in Section 3.3.2.
Consistent with Table 6, Figure 2 column (a) clearly shows that the markup shocks have30

Figure 2: Impulse responses of output to price and wage markup shocks

0

0

0

0

-0.1

-0.1

-0.1

-0.1

-0.2

-0.2

-0.2

-0.2

-0.3

-0.3

-0.3

-0.3

-0.4

-0.4

-0.4

-0.4

-0.5

-0.5

-0.5

-0.5

-0.6

-0.6

-0.6

-0.6

-0.7

-0.7
0

4

8

12

16

20

-0.7
0

4

8

12

16

20

HKL

-0.7
0

HKL-CD

4

8

12

16

20

0

0

0

0

-0.2

-0.2

-0.2

-0.4

-0.4

-0.4

-0.4

-0.6

-0.6

-0.6

-0.6

-0.8

-0.8

-0.8

-0.8

-1

-1

-1

-1

-1.2

-1.2

-1.2

-1.2

-1.4
0

4

8

12

16

20

-1.4
0

4

8

12

16

20

4

8

12

16

20

0

4

8

12

16

20

S&W

-0.2

-1.4

0

-1.4
0

4

8

12

16

20

Notes. The two panels in column (a) show the responses of output to a one-standard-deviation contractionary priceand wage markup shock, respectively. The solid, dashdotted, and dashed lines represent the results based on ourbenchmark model (HKL), the model without input complementarity (HKL-CD), and the Smets and Wouters (2007)
model (S&W), respectively. We use the posterior modes of each model. For column (b), we replace the Calvo sticky
price and wage parameters of HKL-CD and S&W to the posterior mode of the HKL model. Column (c) is based on
the estimated price and wage markup shock processes of HKL in Table 3. Finally, column (d) uses HKL posteriormodes for both Calvo parameters and markup shock processes.

substantially smaller effects on output in our benchmark model than that in the two other models.
The peak effects of a one-standard-deviation price markup shock on output are 0.32% (HKL), 0.42%(HKL-CD) and 0.42% (S&W) in absolute value. Similarly, for wage markup shocks, the peak effects
are 0.48% (HKL), 0.62% (HKL-CD) and 0.70% (S&W). The smaller effects of both price and wage
markup shocks with the translog production function are similar for investment, consumption, andlabor, as shown in Appendix D.
By comparing Figure 2 columns (b)-(d), we find that less persistent price markup shocks aremostly responsible for the depressed role of markup shocks in our benchmark model. As shown incolumn (b), fixing the price and wage rigidity across the three models makes the impulse response
generated from HKL-CD and Smets and Wouters (2007) to deviate even more from that of ourbenchmark model, highlighting that the larger rigidity in our benchmark model does not decrease
the importance of the markup shocks. However, the impulse responses of output are nearly identicalacross the three models when we use the same markup shock parameters, as shown in column (c).
These results show that the decrease in markup shock parameters (i.e., lower persistence of markup
31

shock processes) is essential for generating the smaller contribution of markup shocks. Column (d)fixes both Calvo and markup shock parameters, and the impulse responses are analogous to those incolumn (a).
Allowing the procyclical returns to scale leads to the less persistent markup shocks throughthe flatter price and wage Philips curves. As shown in Table 4, our benchmark model features theprocyclical price markup or the countercyclical real marginal costs because of the input complementarity. The large countercyclical fluctuations in the marginal costs per se do not amplify the
dynamics of aggregate variables. However, it makes our Bayesian estimator select stickier prices andless responsive inflation to the real marginal costs to match the empirical correlation of inflation andGDP. Correspondingly, nominal wages become stickier to match the real wage cyclicality in the data.
The larger price and wage rigidity, in turn, makes real variables respond more to structural shocks,
as shown in Figure 2 column (b), and absorb previously unexplained variations that were attributed
to price and markup shocks.

4

Conclusion

This paper studies business cycles with a translog production function. Our empirical analyses
suggest that there is complementarity between labor and energy, which leads to procyclical returns toscale. Our empirical evidence is not compatible with the tightly parameterized production functions
commonly used in the literature. Thus, we introduce the normalized translog production function
into a standard medium-scale DSGE model and re-estimate the substitution parameters within the
structure of our model. Our model rationalizes procyclical returns to scale, procyclical price markups,
countercyclical labor shares, procyclical energy shares, and procyclical profit shares. Furthermore,
we document that the contribution of price and wage markup shocks to output fluctuations in ourmodel is substantially smaller than that in the standard Smets and Wouters (2007) model. The
complementarity between labor and energy and the corresponding procyclical returns to scale are
central to the theoretical mechanism behind the results.
Our work underscores the need to employ more general forms of production functions in businesscycle research. Further efforts to utilize a general functional form will extend the understanding ofbusiness cycles.

32References
Ackerberg, Daniel A., Kevin Caves, and Garth Frazer, “Structural Identification of Production Functions,” Econometrica, 2015, 83 (6), 2411–2451.
Anderson, Eric, Sergio Rebelo, and Arlene Wong, “Markups Across Space and Time,” Working
paper, 2020.
Antras, Pol, “Is the U.S. aggregate production function Cobb-Douglas? New estimates of theelasticity of substitution,” Contributions to Macroeconomics, 2004, 4 (1).
Arellano, Cristina, Yan Bai, and Patrick J. Kehoe, “Financial Frictions and Fluctuations in
Volatility,” Journal of Political Economy, 2019, 127, 2049–2103.
Arellano, Manuel and Stephen Bond, “Some Tests of Specification for Panel Data: Monte Carlo
Evidence and an Application to Employment Equations,” The Review of Economic Studies, 04
1991, 58 (2), 277–297.
Atalay, Enghin, “How Important Are Sectoral Shocks,” American Economic Journal: Macroeconomics, 2017, 9 (4), 254–280.
Auerbach, Alan J. and Yuriy Gorodnichenko, “Measuring the output responses to fiscal policy,”
American Economic Journal: Economic Policy, 2012, 4 (2), 1–27.
Backus, David K. and Mario J. Crucini, “Oil prices and the terms of trade,” Journal ofInternational Economics, 2000, 50 (1), 185–213.
Balke, Nathan S. and Stephen P.A. Brown, “Oil supply shocks and the US economy: An
estimated DSGE model,” Energy Policy, 2018, 116, 357–372.
Baqaee, David Rezza and Emmanuel Farhi, “The Darwinian Returns to Scale,” Working paper,
2021.
Barattieri, Alessandro, Susanto Basu, and Peter Gottschalk, “Some evidence on the importance of sticky wages,” American Economic Journal: Macroeconomics, 2014, 6 (1), 70–101.
Bartelsman, Eric J., Randy A. Becker, and Wayne Gray, “The NBER-CES Manufacturing
Industry Database,” 2000. National Bureau of Economic Research Working Paper, Cambridge,
MA.
Basak, Suleyman and Anna Pavlova, “A model of financialization of commodities,” Journal of
Finance, 2016, 71 (4), 1511–1556.
Basu, Susanto, “Intermediate goods and business cycles,” American Economic Review, 1995, 85(3), 512–531.
and John Fernald, “Returns to Scale in U.S. Production: Estimates and Implications,” Journalof Political Economy, 1997, 105 (2), 249–283.
Baxter, Marianne and Robert G. King, “Productive externalities and business cycles,” Institutefor Empirical Macroeconomics, Federal Reserve Bank of Minneapolis, Discussion Paper 53, 1991.
and , “Measuring business cycles: approximate band-pass filters for economic time series,”
Review of Economics and Statistics, 1999, 81 (4), 575–593.

33

Benhabib, Jess and Roger E. A. Farmer, “Indeterminancy and Increasing Returns,” Journal of
Economic Theory, 1994, 63, 19–41.
and , “Indeterminancy and Sector-Specific Externalities,” Journal of Monetary Economics,
1996, 37, 421–443.
Berger, David, Kyle Herkenhoff, and Simon Mongey, “Labor Market Power,” 2019. Working
Paper.
Berman, Nicolas, Antoine Berthou, and Jérôme Héricourt, “Export Dynamics and Sales at
Home,” Journal of International Economics, 2015, 96 (2), 298–310.
Bigio, Saki and Jennifer La’O, “Distortions in Production Networks,” 2020. Working Paper.
Bils, Mark, “The Cyclical Behavior of Marginal Cost and Price,” American Economic Review, 1987,
77 (5), 838–855.
, Peter J. Klenow, and Benjamin A. Malin, “Testing for Keynesian Labor Demand,” in“NBER Macroeconomics Annual 2013,” University of Chicago Press, 2013, pp. 311–349.
, , and , “Resurrecting the Role of the Product Market Wedge in Recessions,” American
Economic Review, 2018, 108 (4-5), 1118–1146.
Blundell, Richard and Stephen Bond, “Initial conditions and moment restrictions in dynamic
panel data models,” Journal of Econometrics, 1998, 87 (1), 115–143.
BP Energy, “BP Statistical Review of World Energy,” 2020. June.
Cantore, Cristiano and Paul Levine, “Getting normalization right: Dealing with ‘dimensional
constants’ in macroeconomics,” Journal of Economic Dynamics and Control, 2012, 36 (12), 1931–1949.
, Miguel León-Ledesma, Peter McAdam, and Alpo Willman, “Shocking Stuff: Technology,
Hours, and Factor Substitution,” Journal of the European Economic Association, 2014, 12 (1),
108–128.
, Paul Levine, Joseph Pearlman, and Bo Yang, “CES technology and business cycle fluctuations,” Journal of Economic Dynamics and Control, 2015, 61, 133–151.
Chari, V. V., Patrick J. Kehoe, and Ellen R. McGrattan, “Business Cycle Accounting,”
Econometrica, 2007, 75 (3), 781–836.
, , and , “New Keynesian Models: Not Yet Useful for Policy Analysis,” American Economic
Journal: Macroeconomics, 2009, 1 (1), 242–266.
Chrinko, Robert S., “σ:The Long and Short of It,” Journal of Macroeconomics, 2008, 30 (2),671–686.
Christensen, Lauritis, Dale Jorgensen, and Lawrence Lau, “Transcendental logarithmic
Production Frontier,” Review of Economics and Statistics, 1973, 55 (1), 28–45.
, , and , “Transcendental logarithmic utility functions,” American Economic Review, 1975, 65(3), 367–383.
Collard-Wexler, Allan and Jan De Loecker, “Production Function Estimation and Capital
Measurement Error,” Working paper, 2020.
34

Croux, Christophe, Mario Forni, and Lucrezia Reichlin, “A measure of comovement foreconomic variables: Theory and empirics,” Review of Economics and Statistics, 2001, 83 (2),
232–241.
De Jong, Frits J., Dimensional Analysis for Economists, North Holland, 1967.
De Loecker, Jan and Pinelopi K. Goldberg, “Firm Performance in a Global Market,” Annual
Review of Economics, 2014, 6, 201–227.
, Jan Eeckhout, and Gabriel Unger, “The Rise of Market Power and the Macroeconomic
Implications,” Quarterly Journal of Economics, 2020, 135 (2), 561–644.
, Pinelopi K. Goldberg, Amit K. Khandelwal, and Nina Pavcnik, “Prices, Markups andTrade Reform,” Econometrica, 2016, 84 (2), 445–510.
Dhawan, Rajeev and Karsten Jeske, “Energy price shocks and the macroeconomy: the role ofconsumer durables,” Journal of Money, Credit and Banking, 2008, 40 (7), 1357–1377.
Doraszelski, Ulrich and Jordi Jaumandreu, “R&D and Productivity: Estimating Endogenous
Productivity,” Review of Economic Studies, 2013, 126 (3), 1027–1084.
Epifani, Paolo and Gino Gancia, “Increasing returns, imperfect competition, and factor prices,”
Review of Economics and Statistics, 2006, 88 (4), 583–598.
Evans, Charles L., “Productivity shocks and real business cycles,” Journal of Monetary Economics,
1992, 29 (2), 191–208.
Fernald, John, “A quarterly, utilization-adjusted series on total factor productivity,” 2014. Federal
Reserve Bank of San Francisco Working Paper 2012-19.
Finn, Mary G., “Perfect competition and the effects of energy price increases on economic activity,”
Journal of Money, Credit and Banking, 2000, 32 (3), 400–416.
Galí, Jordi, Monetary policy, inflation, and the business cycle: an introduction to the new Keynesianframework and its applications, Princeton University Press, 2015.
Gali, Jordi, Mark Gertler, and J. David Lopez-Salido, “Markups, Gaps, and The Welfare
Costs of Business Fluctuations,” Review of Economics and Statistics, 2007, 89 (1), 44–59.
Gandhi, Amit, Salvador Navarro, and David Rivers, “On the Identification of Production
Functions,” Journal of Political Economy, 2020, 128 (8), 2973–3016.
Gilchrist, Simon, Raphael Schoenle, Jae W. Sim, and Egon Zakrajsek, “Inflation Dynamics
During the Financial Crisis,” American Economic Review, 2017, 107 (3), 785–823.
Goodfriend, Marvin and Robert King, “The New Neoclassical Synthesis and the Role ofMonetary Policy,” in “NBER Macroeconomics Annual 1997, Volume 12,” MIT Press, 1997, pp. 231–
296.
Gorodnichenko, Yuriy and Serena Ng, “Estimation of DSGE models when the data are persistent,” Journal of Monetary Economics, 2010, 57 (3), 325–340.
Gutierrez, German and Thomas Philippon, “Declining Competition and Investment in the
U.S.,” 2017. NBER Working Paper No. 23583.
Hall, Robert E., “The relation between price and marginal cost in US industry,” Journal of Political
35

Economy, 1988, 96 (5), 921–947.
, “Invariance properties of Solow’s Productivity Residual,” in Peter Diamond, ed., Growth/Productivity/Unemployment: Essays to Celebrate Bob Solow’s Birthday, Cambridge, MA: MIT Press,
1990, pp. 72–112.
, “Measuring Factor Adjustment Costs,” Quarterly Journal of Economics, 2004, 119 (3), 899–927.
, “What the cyclical response of advertising reveals about markups and other macroeconomic
wedges,” Working Paper 18370, National Bureau of Economic Research, Inc. October 2013.
Hamilton, James D., “Understanding crude oil prices,” Energy Journal, 2009, 30 (2), 179–206.
Hanoch, Giora, “The elasticity of scale and the shape of average costs,” American Economic Review,
1975, 65 (3), 492–497.
Hansen, Lars Peter, “Large sample properties of generalized method of moments estimators,”
Econometrica, 1982, 50 (4), 1029–1054.
Hassler, John, Per Krusell, and Conny Olovsson, “Directed technical change as a response tonatural-resource scarcity,” Working paper, 2019.
Hottman, Colin J., Stephen J. Redding, and David E. Weinstein, “Quantifying the Sourcesof Firm Heterogeneity,” Quarterly Journal of Economics, 2016, 131 (3), 1291–1364.
Jermann, Urban and Vincenzo Quadrini, “Macroeconomic Effects of Financial Shocks,” American Economic Review, 2012, 102 (1), 238–271.
Justiniano, Alejandro, Giorgio E. Primiceri, and Andrea Tambalotti, “Investment shocks
and business cycles,” Journal of Monetary Economics, 2010, 57 (2), 132–145.
,, and , “Investment shocks and the relative price of investment,” Review of Economic
Dynamics, 2011, 14 (1), 102–121.
Kaldor, Nicholas, “A Model of Economic Growth,” Economic Journal, 1957, 67 (268), 591–624.
Karabarbounis, Loukas, “The labor wedge: MRS vs. MPN,” Review of Economic Dynamics, 2014,17 (2), 206–223.
and Brent Neiman, “The Global Decline of the Labor Share,” Quarterly Journal of Economics,
2014, 129 (1), 61–103.
Kilian, Lutz, “Not all oil price shocks are alike: Disentangling demand and supply shocks in the
crude oil market,” American Economic Review, 2009, 99 (3), 1053–1069.
, “Oil Price Shocks: Causes and Consequences,” Annual Review of Resource Economics, 2014, 6(1), 133–154.
and Daniel P. Murphy, “Why agnostic sign restrictions are not enough: understanding the
dynamics of oil market VAR models,” Journal of the European Economic Association, 2012, 10 (5),1166–1188.
Koh, Dongya and Raül Santaeulàlia-Llopis, “Countercyclical Elasticity of Substitution,” 2017.
Working Paper.
Levinsohn, James A. and Amil Petrin, “Estimating Production Functions Using Inputs to36

Control for Unobservables,” Review of Economic Studies, 2003, 70 (2), 317–340.
Nakamura, Emi and Jon Steinsson, “Five Facts about Prices: A Reevaluation of Menu Cost
Models,” Quarterly Journal of Economics, 2008, 123 (4), 1415–1464.
and , “Monetary Non-Neutrality in a Multi-Sector Menu Cost Model,” Quarterly Journal of
Economics, 2010, 125 (3), 961–1013.
Nekarda, Christopher J. and Valerie A. Ramey, “The cyclical behavior of the price-cost
markup,” Journal of Money, Credit and Banking, 2020, 52 (S2), 319–353.
Oberfield, Ezra and Devesh Raval, “Micro data and macro technology,” Econometrica, 2021, 89(2), 703–732.
Olley, Steve G. and Ariel Pakes, “The Dynamics of Productivity in the Telecommunications
Equipment Industry,” Econometrica, 1996, 64 (6), 1263–1297.
Rajan, Raghuram G. and Luigi Zingales, “Financial Dependence and Growth,” American
Economic Review, 1998, 88 (3), 559–586.
Raval, Devesh R., “The micro elasticity of substitution and non-neutral technology,” RAND
Journal of Economics, 2019, 50 (1), 147–167.
Ravn, Morten O., Stephanie Schmitt-Grohé, and Martin Uribe, “Deep Habits,” Review of
Economic Studies, 2006, 73, 195–218.
Rios-Rull, Jose-Victor and Raul Santaeulalia-Llopis, “Redistributive shocks and productivity
shocks,” Journal of Monetary Economics, 2010, 57 (8), 931–948.
Romer, Christina D. and David H. Romer, “A new measure of monetary shocks: Derivationand implications,” American Economic Review, 2004, 94 (4), 1055–1084.
Rotemberg, Julio J., “Comment on Testing for Keynesian Labor Demand,” in “NBER Macroeconomics Annual 2013,” Vol. 27 (1), University of Chicago Press, 2013, pp. 362–370.
and Michael Woodford, “Markups and the Business Cycle,” in “NBER Macroeconomics Annual
1991,” MIT Press, 1991, pp. 63–129.
and , “Imperfect competition and the effects of energy price increases on economic activity,”
Journal of Money, Credit, and Banking, 1996, 28 (4), 550–577.
and , “The Cyclical Behavior of Prices and Costs,” in John B. Taylor and Michael Woodford, eds.,
Handbook of Macroeconomics, North Holland: Elsevier Science, 1999, chapter 16, pp. 1051–1135.
Schmitt-Grohé, Stephanie, “Endogenous Business Cycles and the Dyanmics of Output, Hours,
and Consumption,” American Economic Review, 2000, 90 (5), 1136–1159.
Sims, Christopher A., “Solving linear rational expectations models,” Computational Economics,
2002, 20 (1-2), 1–20.
Smets, Frank and Rafael Wouters, “Shocks and Frictions in US Business Cycles: A Bayesian
DSGE Approach,” American Economic Review, 2007, 97 (3), 586–606.
Stroebel, Johannes and Joseph Vavra, “House Prices, Local Demand, and Retail Prices,” Journalof Political Economy, 2019, 127 (3), 1391–1436.

37

Syverson, Chad, “What determines productivity?,” Journal of Economic Literature, 2011, 49 (2),326–365.
U.S. Energy Information Administration, Monthly Energy Review February 2021, Washington,
DC: Office of Energy Statistics, U.S. Department of Energy, 2021.

38

Appendix A

NBER-CES: Summary Statistics

Table A.1 reports the basic summary statistics for the main dataset we use in the paper (NBER-CES
manufacturing database). The data covers 473 6-digit NAICS manufacturing industry in 1958-2009.
The quantity index of five inputs are measured with the energy (energy/pien), material excluding
energy ((matcost/pimat) - (energy/pien), since the units in the value and price of material andenergy are identical), production workers (prode), non-production workers (emp - prode), and capital(cap). The energy share is measured as the share of the value of shipment (energy/vship). We
use the input prices for the instrumental variables, and they are the energy price deflator (pien),material price deflator (pimat), production worker wage (prodw/prode), worker wage (pay/emp),
and investment deflator (piinv ). Note that we do not construct the price indexes separately forthe material excluding the energy input and the non-production worker, as they are not given in
the original dataset and requires more subtraction and addition, which would like to exacerbate
the measurement error problems in such variables. For example, we find that if we measure thenon-production worker wage by ourselves ((pay-prodw)/(emp-prode)), we observe that this variable
has large negative values. Table A.2 reports the first-stage regression results for the instrumental
variables used in this paper.

39

Table A.1: Summary statistics
Variable
NAICS 6-digit Codes
Year ranges from 58 to 09
Total employment in 1000s
Total payroll in $1m
Production workers in 1000s
Production worker hours in 1m
Production worker wages in $1m
Total value of shipments in $1m
Total cost of materials in $1m
Total value added in $1m
Total capital expendture in $1m
End-of-year inventories in $1m
Cost of electric & fuels in $1m
Total real capital stock in $1m
Real capital: equipment in $1m
Real capital: structures in $1m
Deflator for VSHIP 1997=1.000
Deflator for MATCOST 1997=1.000
Deflator for INVEST 1997=1.000
Deflator for ENERGY 1997=1.000
5-factor TFP annual growth rate
5-factor TFP index 1997=1.000
4-factor TFP annual growth rate
4-factor TFP index 1997=1.000
production worker hours per worker in 1000s

Mean
327009.662
1983.534.814735.89625.42350.64
443.7964799.495
2620.972190.409
156.655
585.42997.69
2757.95
1664.517
1093.433
0.799
0.718
0.694
0.695
0.003
0.937
0.003
0.936
2.004

40

Std. Dev.
8889.717
15.00945.0531252.867
33.65166.823736.828
13196.309
9721.219
4710.187
462.108
1433.618
360.536388.034145.339
2418.355
1.552
0.357
0.309
0.402
0.066
0.257
0.066
0.257
0.119

Min.
311111
1958
0.2
2.9
0.2
0.3
1.8
19.3
8.800
9.700
0.1
1.3
0.1
4.1
1.9
2.2
0.044
0.127
0.183
0.087
-0.6420.012-0.642
0.011
1.28

Max.
3399992009559.916162.9459.9
904.1
10475.2
732728.4648048.4111187.9
17601.640084.914201.5
120110.388454.600
38874.4
47.409
2.777
1.581
2.233
1.387
13.192
1.387
13.193
3.228

N
24596
24596
24167
24167
24167
24167
24167
24167
24167
24167
24167
24162
24167
24167
24167
24167
24167
24167
24167
24167
23694
24167
23694
24167
24167

Table A.2: Regressing Inputs on Input Prices(1)
Energy-0.523∗∗∗(0.091)(2)
Material
0.158∗∗(0.071)(3)
Capital-0.166∗∗∗(0.049)(4)Workerp-0.054(0.064)(5)Workernp-0.126∗∗(0.058)energy price(t-2)-0.542∗∗∗(0.090)-0.202∗∗∗(0.068)-0.224∗∗∗(0.048)-0.292∗∗∗(0.062)-0.240∗∗∗(0.056)

material price(t-1)

0.156∗∗(0.071)-0.696∗∗∗(0.071)-0.338∗∗∗(0.048)0.085(0.057)-0.026(0.059)material price(t-2)-0.183∗∗∗(0.069)-0.033(0.068)0.015(0.047)0.081(0.055)0.111∗(0.058)

investment price(t-1)-1.899∗∗∗(0.313)-2.003∗∗∗(0.349)-1.573∗∗∗(0.216)-0.621∗∗(0.286)-2.130∗∗∗(0.294)

investment price(t-2)

2.327∗∗∗(0.317)1.601∗∗∗(0.350)1.263∗∗∗(0.216)

1.094∗∗∗(0.287)2.263∗∗∗(0.296)

production worker wage(t-1)

0.577∗∗∗(0.134)0.380∗∗∗(0.135)0.117(0.078)0.378∗∗∗(0.123)0.047(0.107)production worker wage(t-2)

0.308∗∗(0.141)0.028(0.139)0.067(0.080)0.104(0.125)-0.443∗∗∗(0.109)payroll per worker(t-1)-0.069(0.158)0.445∗∗∗(0.158)0.265∗∗∗(0.091)-0.529∗∗∗(0.149)0.219∗(0.129)payroll per worker(t-2)-0.059(0.166)0.114(0.161)0.424∗∗∗(0.092)-0.193(0.148)0.422∗∗∗(0.128)Constant

0.000(0.003)135.2923221

0.000(0.003)135.1223221

0.000(0.002)
159.3023221-0.000(0.002)
57.06
23221

-0.000(0.002)
27.53
23220

energy price(t-1)

F-statObservations

Note. Each double-demeaned input is regressed on all 10 double-demeaned lagged input prices usedin Equation (2.6). “Energy” (energy/pien) is the quantity of energy, “Material” ((matcost/pimat) (energy/pien)) is the quantity index of material that excludes the energy, “Capital” (cap) is the realcapital, “Workerp ” (prode) is the number of production workers, and “Workernp ” (emp - prode) is thenumber of non-production workers. The “energy price” is the energy deflator (pien), the “material
price” is the material deflator (pimat), the “investment price” is the investment deflator (piinv ), the“production worker wage” is the production worker wage (prodw/prode), and the “payroll per worker”
is the average wage of all workers (pay/emp).

41

Appendix B
B.1

Empirical Results: Robustness

Using Two-year and Three-year Lagged Input Prices

As a robustness exercise, we use two- and three-year lagged input prices as instrumental variables
instead of one-year and two-year lagged input prices. Table B.1 presents the results. The key
complementarity parameter between energy and production workers is still positive and statisticallysignificant.
Table B.1: Estimation of Equation (2.6)
IV (Lagged Input Prices)

OLS

(1)
-0.286(0.188)(2)-0.330**(0.163)(3)-0.351**(0.165)(4)0.843***(0.028)(5)0.838***(0.029)(6)0.816***(0.031)

production worker

1.255***(0.356)

1.341***(0.310)1.272***(0.310)-0.326***(0.029)-0.323***(0.030)-0.464***(0.032)

non-production worker-0.472*(0.246)-0.510**(0.244)-0.425*(0.221)-0.119***(0.022)-0.125***(0.022)-0.219***(0.025)material

0.403*(0.235)0.386**(0.175)-0.409***(0.025)-0.420***(0.026)

capital-0.070(0.403)22747

energyObservations

0.449*(0.239)
22747

22747

-0.032(0.021)
22747

22747

-0.247***(0.026)22747

Note. The regression specifications are identical to those in Table 1. *, **, and *** indicate significance at the 0.1,
0.05, and 0.01 levels, respectively.

B.2

Allowing Additional Control Variables

This section explicitly considers the potential omitted variables that might be correlated with our
instrumental variables in estimating Equation (2.6). For our baseline approach, we assume thatthe instrumental variables—the double-demeaned lagged input prices—are not correlated with the
double-demeaned wedge term τ . This assumption is consistent in a model that does not have anindustry-time-varying energy wedge and the non-parametric estimation strategy employed in Gandhi
et al. (2020). However, in a more general class of models, our assumption could be violated. For
example, in a dynamic model that allows the heterogeneity of output price rigidity and variation in
input prices across sectors, the industry-time-varying lagged input prices could be correlated with
the industry-time-varying contemporaneous price-cost markups, which is a part of the industry-time42

varying contemporaneous wedge τ e .
Given such a concern, we control the potential omitted variables directly and re-estimate thefirst-order conditions. First, we measure the industry-level market power and directly control it
to address such potential omitted variables. We carefully follow previous studies to measure theindustry-time-varying price-cost markups (De Loecker et al. 2020) and the Lerner index (Gutierrez
and Philippon 2017) with Compustat data. Since previous studies discuss the methodology in detail,we briefly state how we measure these two variables.
In estimating the price-cost markups, we first use two measures of inputs available in Compustat
data, specifically, the cost of goods sold (COGS) and capital, to estimate the output elasticity withrespect to the COGS. With the assumption of the firm-level Cobb-Douglas technology, elasticity is
the coefficient of the COGS input. We follow Olley and Pakes (1996) for our estimation and allow
the elasticity to vary across NACIS 2-digit industries. We leverage the firm-level first-order condition
to recover the price-cost markups, which is the estimated elasticity divided by the COGS input
share. Then, we aggregate the markups across firms within each 6-digit NAICS industry by using
the weight of a sale to recover the industry-level markups. Using a cost-based weight does not makemuch difference in the results. We measure the Lerner index as the operating income to sales ratio in
the 6-digit NAICS industries.
In addition to the two measures of market power, we consider four other control variables toaddress the industry-time-varying contemporaneous wedge term. Given that standard macroeconomic
models predict markup heterogeneity based on price rigidity, we introduce the price rigidity measurefrom Bils et al. (2013). In light of the literature that links price markup to inventory, we include
the inventory-to-sales ratio. We also measure the external financial dependent index of Rajan andZingales (1998) to control the financial friction that potentially prevents the optimal use of the energy
input.
Finally, although the adjustment costs for the energy input are typically assumed away inprevious studies, as a robustness check, we integrate the adjustment cost term explicitly. By assuming
 i
2
i
Vjt
η i Pjt iia quadratic adjustment cost Φ(Vjt ) = 2 Pjt Vjt V i− 1 and solving for the first-order conditionj,t−1(2.5) again, we have
i

ŝ =

"
X

#
δik V̂k− φ̂i + τ̂ i ,

(B.1)

k

where we have a new term in our equation, φ̂ijt ≡ η iii
Et [V̂j,t+1]−V̂jtiiV̂jt − V̂j,t−1 −with the real
1+r̄
i

η
i
i
interest rate r. Given a small r̄ and the (scaled) forecast error εj,t+1 ≡ 1+r̄(V̂j,t+1− Et [V̂j,t+1]), we


i
i
V̂
−V̂jt
i
i
have φ̂ijt = η i V̂jti − V̂j,t−1− j,t+1+ εj,t+1 ≈ −η i ∆2 V̂j,t+1+ εj,t+1 . We include the double1+r̄edifferences and double-demeaned energy input (∆2 V̂j,t+1) and estimate η e by instrumenting it withe
the corresponding lagged input price (∆2 P̂j,t−1) to address the measurement error problem arises

43

Table B.2: Estimation of Equation (2.6) with Additional Controls
IV (Lagged Input Prices)
production worker(1)
1.261***(0.359)Markup(2)
1.268***(0.369)(3)1.401***(0.397)(4)
1.517***(0.381)(5)
1.697***(0.449)(6)
1.387***(0.346)-0.158(0.113)

Lerner index-69.206**(35.152)

Frequency of price changes-0.011**(0.005)Inventory

0.497***(0.111)

RZ measure-0.006***(0.002)

Adj. cost
Observations

14591

14591

18198

23217

23220

-0.096(0.905)22275

Note. All columns show the regression result with the instrumental variables. For the instrumental variable analyses,
we use both one- and two-year lagged input prices. Five different inputs are used in this regression, specifically,energy, material excluding energy, non-production worker, production worker, and capital. The Markup, Lerner index,
Frequency of price changes, RZ measure, and Inventory are controlled directly, and the Adj. cost is instrumented withthe corresponding lagged input price. All variables are logged and double-demeaned across industries and time. The
standard errors in parentheses are clustered at the NAICS industry level. *, **, and *** indicate significance at the
0.1, 0.05, and 0.01 levels, respectively.

from the forecast error. Because Et [εj,t+1 ] = 0, εj,t+1 is orthogonal to the instruments and do not
violate the exclusion restriction.
Table B.2 shows the empirical results. Regardless of the additional measures of market power,
price rigidity, inventory, financial friction, and adjustment costs, our main empirical results of complementarity between production workers and energy input stay positive and statistically significant
at the conventional level. Columns (1) and (2) utilize the double-demeaned markup and Lernerindex. The coefficients of the measure of market power are intuitive since the market power lowers
the input share in a large class of models. Columns (3)-(6) control the measures of price rigidity,
inventory-to-sales, financial friction, and adjustment to ease the concerns related to these measures.
Consistent with the previous studies, we do not find empirical evidence on the existence of the energy

44adjustment costs.
We assess the cyclicality of returns to scale conditional on each of these six control variables. We
measure the double-demeaned returns to scale by subtracting the double-demeaned wedge term fromthe double-demeaned energy share, which is a double-demeaned version of Equation (2.9). Figure
B.1 shows that the estimated returns to scale still preserves a larger positive correlation with thevalue added in each industry.

B.3

Returns to Scale: Allowing an Input-Specific Adjustment Cost

In recovering the returns to scale in Section 2.3, we assume that τ i = τ . Although this specification isconventional in the previous studies and consistent with the DSGE model presented in Section 3, one
concern is that the input-specific adjustment cost is likely to be different across different inputs. Thissection allows the input-specific adjustment cost and shows that the returns to scale procyclicality isrobust to this concern.
Consider the firm’s cost minimization problem with the translog production function (2.2):
min C =

X Pi
P

i

ln(Y ) = ln(z) +Xβi ln(V i ) +V + Φ(V )is.t.
X X βik

i

i

2

k

i

ln(V i ) ln(V k ) with βik = βkiiwhere PP ≡ pi is the real input price and Φ(V i ) is the input-specific adjustment cost for input V i .
We consider two different functional form assumptions on the adjustment cost Φ(V i ). First,
 i
2
i
i
consider a convex adjustment cost with the following functional form, Φ(V i ) ≡ η2 PP V i VV i − 1 .
ssIn this specification, we make adjustment costs depend on the current variables and steady statevalues, similar to the specification in Rotemberg and Woodford (1999). Solving the cost-minimization
problem with respect to V i yields
"
si

ηi
1+
2



"
#
2
 i
  i #X
ViVV
− 1 + ηi−1
= mc βi +βik ln(V k ) ,
V̄ i
V̄ i
V̄ i

(B.2)

k

where mc is the Lagrangian multiplier of the production technology, which is the firm’s real marginal
cost or the inverse price markup. Rearranging and log-linearizing (B.2) leads to
"
ŝi =

#
X

δik V̂ k + mc
c − φ̂i ,

(B.3)

k

where δik ≡ βik

mc
s̄i



and φ̂i ≡ η i V̂ i . Equation (B.3) is a special case of Equation (2.5) with45

Figure B.1: Returns to Scale and Value-Added(a) Control Markup(b) Control Lerner Index(c) Control Frequency of Price Changes(d) Control RZ Measure(e) Control the Adjustment Costs(f ) Control Inventory-to-Sales

Note. The y-axis is returns to scale, and the x-axis is value added. All variables are double-demeaned across industries
within each year and across years within each industry. The slopes of the linear lines in Figures B.1a, B.1b, B.1c, B.1d,
B.1e, and B.1f are .64, .63, .91, 1.02, .87, and .86 respectively.

46τ̂ i = mc
c − φ̂i ; the input-specific wedge τ i is the linear combination of the input-specific term arisingfrom the adjustment costs and the inverse price markup in this setup.
By adding Equation (B.2) across all input shares and combining the resulting equation with thereturns to scale Equation (2.7), we have
"
 i

2
  i #VV
1 X i
ηi V i− 1 + ηi
−1
.
rts =
s 1+
i
i
mc
2 V̄
V̄
V̄ i

(B.4)

i

The log-linearization of (B.4) leads toc = ŝall − mc
rts
c + φ̂all ,where sall =i
all ≡i s and φ̂PP  s̄i 
i

φ̂i =

s̄all

(B.5)

P  s̄i  i ii s̄all η V̂ .

Equation (B.5) makes it clear that the returns to scale procyclicality results do not likely changewith the presence of the input-specific adjustment cost. The key difference between Equation (2.9)
and Equation (B.5) is φ̂all , which captures the effects of the input-specific adjustment cost. Equation(2.9) allows the common wedge τ , such as the real marginal cost mc. However, because the input V̂ iis procyclical in the data, regardless of the degree of input-specific adjustment cost η, the adjustment
cost term φ̂all tends to be procyclical. Thus, if adjustment costs exist in any input, then they would
strengthen the procyclicality of the returns to scale.
Second, we assume an alternative adjustment cost functional form that depends on lagged
2
 i
i
V
inputs, as is the case for the energy input in Appendix B.2: Φ(V i ) = η2 PPi V i V it − 1 . Undert−1
P s̄i ithis assumption, one can also derive Equation (B.5) with φ̂all
=
φ̂
with
all
t
i s̄
φ̂it = η ii ] − V̂ i
Et [V̂t+1ti
V̂ti − V̂t−1−

1 + r̄!
,
i ]−V̂ i
Et [V̂t+1
tiii ] with a small r̄,
where r is the real interest rate. Since V̂t − V̂t−1 −≈ −Et [∆2 V̂t+1
1+r̄
i
the procyclicality of returns to scale might change if −∆2 V̂t+1
is strongly countercyclical. We assessithe cyclcality of −∆2 V̂t+1
by correlating this measure with the value added for all inputs. Weifind that −∆2 V̂t+1
is either procyclical or acyclical and is not likely to weaken the returns to scaleiprocyclicality. The correlation of −∆2 V̂t+1
with Ŷt is 0.14, 0.08, 0.0001, 0.12, and 0.07 for production

worker, non-production worker, capital, material, and energy, respectively.

B.4

Fixed Costs in Production

This section extends the empirical analyses in Section 2 by allowing the fixed cost of production
along with the structural assumptions to be consistent with the DSGE model presented in Section 3.
47

In the translog production function Equation (2.2), we allow the fixed cost as in Equation (3.1):
ln(Y ) = ln(z) +Xβi ln(V i ) +X X βik
i

i

|

{z

}

Cobb-Douglas

|

k

2

ln(V i ) ln(V k ) −υ|{z}with βik = βki

fixed costs{zsecond-order terms

(B.6)

}

where υ is the fixed cost of production.
As in the DSGE model, we assume that firms face monopolistic competition and earn zero profit
at the steady state. Under these assumptions, we derive the first-order condition with respect to
input V i and rearrange the terms to obtain
"
#
X
Y +υ
k
s = mcβi +βik ln(V )
Y
i

(B.7)

k

where mc is the real marginal cost. Equation (B.7) is a special case of Equation (2.4) with
τ i = τ = mc Y Y+υ .
As in our main analyses, we choose an energy input as a choice variable (V i = E), allow thevariables to change across industry and time, and log-linearize and double-demean Equation (B.7) sothat we can implement the estimation. To recover the fixed cost of production in the data, we use
the zero-profit condition at the steady state. This condition makes firms use their profit to recover
their fixed cost of production at the steady state: υ = (Φ − 1)Ȳ , where Φ is the gross price markup
at the steady state. Replacing the fixed cost of production, log-linearizing, double-demeaning, andrearranging Equation (B.7) with V i = E leads toŝˆejt +XΦj − 1 ˆc jt
Ŷjt =
δek V̂ˆjtk + mc
Φj

(B.8)

k

e = mcc jt − Φj −1 Ŷˆjt . The steady state
where Equation (B.8) is a special case of Equation (2.6) with τ̂ˆjt
Φjindustry-specific value of price markup is recovered from the simple average of markup across years(1958-2009) within the industry, where the industry-time-varying price-cost markup is measured
based on De Loecker et al. (2020) methodology from Compustat data as described in Appendix B.2.
With the industry-specific price markup measure, Equation (B.8) can be estimated by either treating
the double-demeaned marginal cost as a residual or controlling the double-demeaned real marginal
cost directly as in Appendix B.2.
Table B.3 reports the results. Regardless of adjusting for the fixed cost of production, we stillobserve that the coefficient of the production worker, which governs the complementarity between
energy and the production worker, is positive and statistically significant. Additionally adjustingfor the price markup decreases the coefficient of the production worker but still features large
complementarity. The OLS results are largely consistent with the results in Table 1.

48

Table B.3: Adjusting the Fixed Cost of Production
IV (Lagged Input Prices)

OLS

energy(1)-0.849∗∗(0.344)(2)-0.449∗∗(0.196)(3)-0.861∗∗∗(0.324)(4)-0.176(0.237)(5)-0.137(0.180)(6)-0.256(0.215)(7)0.875∗∗∗(0.034)(8)0.907∗∗∗(0.031)

production worker

2.136∗∗∗(0.610)1.409∗∗∗(0.365)2.106∗∗∗(0.602)

1.139∗∗∗(0.364)1.135∗∗∗(0.289)1.248∗∗∗(0.357)-0.394∗∗∗(0.049)-0.420∗∗∗(0.050)non-production worker-0.738∗(0.411)-0.458(0.296)-0.689∗(0.398)-0.691∗∗∗(0.262)-0.653∗∗(0.259)-0.591∗∗(0.253)-0.115∗∗∗(0.032)-0.136∗∗∗(0.034)material

0.231(0.288)0.639∗∗∗(0.162)0.489∗∗∗(0.159)0.496∗∗∗(0.110)-0.191∗∗∗(0.051)-0.188∗∗∗(0.054)

capital1.113∗(0.576)0.029(0.029)0.060∗(0.034)20864-0.059(0.051)14840

1.402∗∗∗(0.395)0.071(0.325)20048-0.038(0.096)
14591

MarkupObservations

20048

20048

0.733∗∗∗(0.213)-0.024(0.073)14591-0.069(0.105)14591

Note: Columns (1)-(3) replicate the regression results in Table 1 columns (1)-(3) by redefining the left hand side
Φj −1 ˆvariable by subtracting the term relevant to the fixed cost of production ( ΦŶjt ) from the energy share. Columnsj(4)-(6) additionally control the double-demeaned price markup. *, **, and *** indicate significance at the 0.1, 0.05,
and 0.01 levels, respectively.

Figure B.2 shows that the procyclicality of returns to scale results remain robust after adjustingfor the fixed cost of production. The returns to scale expression (2.9) does not change with thepresence of the fixed cost of production except that the wedge term τ becomes the real marginal cost,
and there are changes in the estimated production function coefficients used in recovering the returns
to scale. Figure B.2a and B.2b are based on Table B.3 columns (1) and (4), respectively. Regardlessof the use of alternative specifications, the measure of returns to scale is positively correlated withthe value added.
As in our main analyses, separately correlating the components of the returns to scale, which
are the price markup and the sum of input shares, still shows the importance of price markup
procyclicality in generating the returns to scale procyclicality. Based on Figures B.2a and B.2b, thecorrelations of value added with returns to scale are 0.64 and 0.5, respectively. The correlationsof value added with the price markup are very close to these values—specifically, 0.67 and 0.56.
However, the correlation of the sum of input shares and value added is -0.28 and does not correspondto the returns to scale procyclicality.

49

Figure B.2: Returns to Scale and Value-Added(a) Adjust Fixed Cost(b) Adjust Fixed Cost and Control Markup

Note. The y-axis is returns to scale, and the x-axis is value added. All variables are double-demeaned across industries
and year. The slopes of the linear lines in Figures B.2a and B.2b are 1.21 and .59, respectively.

B.5

CES Assumption

This section considers two different methods to test the CES production functional form assumption
conditioned on including energy input in the production function. The first method tests the CES
assumption with respect to energy input only. The second method tests the CES functional form
assumption on any input with a second-order approximation of the production function.
B.5.1

CES Assumption on Energy Input

First, consider the following CES production functional form assumption on the energy input:
1
Q = c(E) + g(V i ; A) ρ

(B.9)

where g(.) is an arbitrary function of other inputs and productivity components. c is an arbitrary
constant, and A is productivity. i represents the other four inputs. Rearranging the first-ordercondition leads to the following equation, similar to Equation (2.4):log(sejt ) = log(c) − ρ log(Qjt ) + ρ log(Vjte ) + τjt(B.10)

Note that the CES assumption leads to a testable prediction: the coefficient of output must
equal the negative coefficient of energy input. Double-demeaning leads to an equation similar toEquation (2.6):
ˆ + ρV̂ˆ e + τ̂ˆŝˆejt = −ρQ̂
jt
jt
jt
50

(B.11)

Table B.4 estimates Equation (B.11) and rejects the CES assumption on the energy input.
Columns (1) and (2) conduct the regression analysis without imposing any constraint. Regardlessof the use of instrumental variables or not, the coefficient of output significantly differs from thecoefficient of energy input. An instrumental variable (IV) regression result rejects the equality of
the absolute value of the coefficients on output and energy with χ2 = 7.82 (p-value = 0.0052).
The OLS results similarly reject the functional form assumption with a near 0 p-value. Columns(3) and (4) re-estimate the coefficients while requiring that the coefficients of output and energy
are equal in absolute value. The IV result produces a near-zero coefficient, which supports the
Cobb-Douglas functional form assumption with respect to the energy input, which is rejected in our
main analysis. The OLS results show that ρ is positive, which does not align with the studies thatassume complementarity between energy and other inputs.
Table B.4: Estimation of Equation (2.6) with Additional ControlsIV(1)-0.368***(0.102)OLS(2)-0.451***(0.069)

IV
(3)
0.009
(0.007)

OLS(4)-0.497***(0.073)

Energy0.060(0.119)0.577***(0.054)-0.009(0.007)0.497***(0.073)J-test(p-value)
Observations

.1
.75
23694

24167

23221

24167

OutputNote. Columns (1) and (2) present the result based on an unconstrained regression, and columns (3) and (4) present
the result based on a constrained regression such that the coefficient of energy input is equal to the negative coefficientof output. Columns (1) and (3) show the regression result with instrumental variables, and columns (2) and (4) show
the OLS result. Output, energy, and lagged input prices are logged and double-demeaned across industries and acrosstime. The energy price deflator, total payroll, and price of investment are used to instrument the endogenous variables.
Standard errors in parentheses are clustered on the NAICS industry code. J-test refers to Hansen’s J-statistics for theoveridentifying restriction. *, **, and *** indicate significance at the 0.1, 0.05, and 0.01 levels, respectively.

B.5.2

CES Assumption on Any Input

Second, consider the CES assumption on any input in the production function that explicitlyP
integrates the energy input. We show that assuming the CES assumption requires that k δik = 0
P
in Equation (2.5). After testing the null hypothesis of k δik = 0 for columns (1), (2), and (3) inTable 1, we strongly reject the null hypothesis with χ2 of 8.51, 11.55, and 11.89, respectively; thep-values are effectively 0 for all three cases.
Without a loss of generality, we consider below the special cases of a two-factor and three-factor
51

translog production function with energy input for the exposition. It is straightforward to generalize
them to five-factor nested CES production functions.
Two-input CES production functions. We begin with a two-factor CES production function.
Suppose thaty = exp(εa ) (βl lρ + βe eρ )1/ρ ,(B.12)

where βl + βe = 1. Two inputs are complementary when ρ < 0. A positive ρ implies that two inputs
are substitutable. When ρ = 0, Equation (B.12) simplifies to a conventional Cobb-Douglas function
exp(εa )lβl eβe . In a neighborhood of this Cobb-Douglas function, we approximate Equation (B.12) tothe second order as follows.
1
1
log y = εa + βl log l + βe log e + ρβl βe (log l)2 − ρβl βe log l · log e + ρβl βe (log e)2 +O(ρ2 ). (B.13)
{z
} |2
|
2
{z
}
Cobb-Douglassecond-order terms

Thus, the CES production function in Equation (B.12) includes the second-order terms with tightly
parametrized coefficients by ρ, βl , and βe . By comparing Equation (B.13) to Equation (2.2), we
obtain βll = ρβl βe , βel = −ρβl βe , and βee = ρβl βe . As a result, βel + βee = 0, which further impliese
that δel + δee = 0 because δek = βek τ̄s̄e for all k (see discussion following Equation (2.5)). We
conclude that the above CES production function is compatible with a null hypothesis δel + δee = 0
that can be tested by using the estimated delta coefficients in Section 2.
ρρ

Derivation of Equation (B.13). Let f (ρ; l, e, εa ) be log y = εa + log(βl l ρ+βe e ) . A Taylor approximation of log y with respect to ρ around ρ = 0 implies log y = f (ρ) = f (0) + f 0 (0)ρ + O(ρ2 ). Because
it is well-known that f (0) = εa + βl log l + βe log e, we focus on f 0 (0) here. Note thatβl lρ log l+βe eρ log e
βl lρ log l+βe eρ log e
ρ + β eρ )
ρ
−
log
(β
l
− [f (ρ) − εa ]ρ +β eρelβlβl lρ +βe eρe0lf (ρ) =
=
.
ρ2
ρ

52

o0
n ρ
l+βe eρ log e
a] |
0
−
[f
(ρ)
−
ε
By L’Hospital’s rule, we obtain f 0 (0) = βl l logρρ
ρ=0 . As a result, f (0) =βl l +βe eno0
1 βl lρ log l+βe eρ log e|ρ=0 . Algebraically, we can show thatρρ
2βl l +βe e
1 [βl lρ (log l)2 + βe eρ (log e)2 ](βl lρ + βe eρ ) − (βl lρ log l + βe eρ log e)2
ρ→0 2(βl lρ + βe eρ )21
=βl (log l)2 + βe (log e)2 − (βl log l + βe log e)22
1
=βl (1 − βl )(log l)2 − 2βl βe log l · log e + βe (1 − βe )(log e)22
1= βl βe (log l − log e)2 .
2

f 0 (0) = lim

Finally, we obtain the desired result:1
log y = f (0) + f 0 (0)ρ + O(ρ2 ) = εa + βl log l + βe log e + ρβl βe (log l − log e)2 + O(ρ2 ).
2
Nested CES production functions. Next, we derive similar results for nested CES production
functions. We concentrate on nested CES functions with three inputs for exposition. First, we begin
with a case where capital is combined with a composite input of labor and energy as follows:hi1/φ
y = exp(εa ) βk k φ + (βl + βe )g(ρ; l, e)φ,(B.14)

βlwhere g(ρ; l, e) = (ξl lρ + ξe eρ )1/ρ , βk + βl + βe = 1, ξl = βl +βand ξe = 1 − ξl . Clearly, Equatione(B.14) features constant returns to scale. By repeatedly applying the above result for two-input CES
functions, we obtain
1
log y = εa + βk log k + (βl + βe ) log g(ρ) + φβk (βl + βe )(log k − log g(ρ))2 + O(φ2 ),2
1
log g(ρ) = ξl log l + ξe log e + ρξl ξe (log l − log e)2 + O(ρ2 ).
2With some algebra, we can show that(βl + βe ) log g(ρ) = βl log l + βe log e +1 ρβl βe(log l − log e)2 + O(ρ2 ),2 βl + βeφ(log k − log g(ρ))2 = φ [log k − ξl log l − ξe log e + O(ρ)]2 = φ(log k − ξl log l − ξe log e)2 + O(φρ) + O(φρ2 ).
Thus, we have the following translog approximation of Equation (B.14):
log y = εa + βk log k + βl log l + βe log e+

1 ρβl βe1(log l − log e)2 + φβk (βl + βe )(log k − ξl log l − ξe log e)2 + O(||φ, ρ||2 ),2 βl + βe253(B.15)

l βe
l βe
which implies that βek = −φβk (βl +βe )ξe , βel = −ρ ββl +β+φβk (βl +βe )ξl ξe and βee = ρ ββl +β+φβk (βl +eeeβe )ξe2 . We conclude that βek + βel + βee = 0, and therefore, δek + δel + δee = (βek + βel + βee ) τ̄s̄e = 0.
We can obtain a similar result for the case where labor is combined with a composite input1/φ
of capital and energy, y = exp(εa ) βl lφ + (βk + βe )g(ρ; k, e)φ, by switching the roles between
k and l above. Finally, we investigate the case where energy is combined with a composite input1/φ
of capital and labor: y = exp(εa ) βe eφ + (βk + βl )h(ρ; k, l)φ, where h(ρ; k, l) = (ζk k ρ + ζl lρ )1/ρ ,kand ζl = 1 − ζk . In this case, we haveζk = βkβ+βl

1log y = εa + βe log e + (βk + βl ) log h(ρ) + φβe (βk + βl )(log e − log h(ρ))2 + O(φ2 ),2
1
log h(ρ) = ζk log k + ζl log l + ρζk ζl (log k − log l)2 + O(ρ2 ),2
1 ρβk βl(βk + βl ) log h(ρ) = βk log k + βl log l +(log k − log l)2 + O(ρ2 ),2 βk + βlφ(log e − log h(ρ))2 = φ [log e − ζk log k − ζl log l + O(ρ)]2 = φ(log e − ζk log k − ζl log l)2 + O(φρ) + O(φρ2 ).
Thus, we have the following translog approximation of this nested CES production function:
log y = εa + βk log k + βl log l + βe log e+

1 ρβk βl1(log k − log l)2 + φβe (βk + βl )(log e − ζk log k − ζl log l)2 + O(||φ, ρ||2 ).
2 βk + βl
2Here, βek = −φβe (βk + βl )ζk , βel = −φβe (βk + βl )ζl and βee = φβe (βk + βl ). Again, we have
βek + βel + βee = 0.
Accordingly, for any three-input nested CES production function, we have βek + βel + βee =δek + δel + δee = 0.

B.6

Regression Weights

This section revisits the main empirical analyses reported in Table 1 by weighting different industries sothat the estimated parameter better matches the aggregate parameter. We consider both the industrytime-varying weight and industry-specific weights. As shown in Table B.5, the complementarity
between labor and energy is preserved with the weighting; the corresponding parameter ranges from1.26 to 1.51 depending on the weights.

B.7

Smaller Number of Inputs

The empirical analyses in Section 2 utilize five inputs (energy, capital, production worker, nonproduction worker, and material) available in the NBER-CES data, whereas the DSGE model
presented in Section 3 allows for only three inputs (energy, capital, and labor), which generates a54

Table B.5: Weighted Regression
IV (Lagged Input Prices)

OLS

energy(1)-0.195(0.200)(2)-0.212(0.250)(3)-0.288(0.231)(4)-0.635∗∗∗(0.214)(5)-0.454∗∗∗(0.172)(6)-0.630∗∗∗(0.209)(7)0.792∗∗∗(0.067)(8)0.885∗∗∗(0.043)production worker

1.259∗∗∗(0.419)1.302∗∗∗(0.383)

1.533∗∗∗(0.475)1.515∗∗∗(0.382)

1.157∗∗∗(0.225)1.565∗∗∗(0.358)-0.272∗∗∗(0.098)-0.334∗∗∗(0.080)non-production worker-0.496(0.382)-0.496∗(0.267)-0.670(0.444)-0.488(0.324)-0.137(0.190)-0.525(0.356)-0.081(0.071)-0.077(0.064)material-0.126(0.102)-0.114(0.099)-0.227(0.150)-0.133(0.091)-0.438∗∗∗(0.054)-0.438∗∗∗(0.053)

capital

0.011(0.242)
11.77
.04
23220

-0.033(0.072)-0.044(0.055)

23220

23220

J-test(p-value)
Observations

11.85
.07
23220

0.008
(0.271)
14.78
.02
23220

0.534
(0.372)
12.02
.03
23220

14.76
.02
23220

0.301
(0.300)
14.8
.02
23220

Note: To inform on the aggregate parameters, we weight the observations by either industry-time-specific output(“industry-time-weight”) or industry-specific output (“industry weight”). Other regression specifications are identical tothose in Table 1. *, **, and *** indicate significance at the 0.1, 0.05, and 0.01 levels, respectively.

discrepancy between the model and the data. This section revisits the empirical analyses and allows
a smaller number of inputs in the estimation, consistent with the DSGE model.
Table B.6 shows that the strong complementary between labor and energy remains regardlessof combining production and non-production workers or using a value-added production functionthat excludes material input. Column (1) combines both production worker and non-production into
all workers, column (2) excludes material input, thereby considering the value-added production
function, and column (3) considers both the combined worker input and the value-added production
function. Regardless of using different specifications, we still estimate the complementarity between
labor and energy that is within a range of the Bayesian estimate in Section 3. The OLS results also
show a similar empirical pattern as in Table 1; the estimated coefficient of energy input is positive,
and the other coefficients are negative.

B.8

Using the KLEMS Data

One potential concern in using the NBER-CES database is that it only covers the manufacturingsectors, which does not represent the entire US economy. Thus, in introducing the complementarity
parameter into the DSGE model to understand the macroeconomic implications, we use the estimatedparameter from the NBER-CES database as a prior and re-estimate the parameter with the aggregate
data.
As an additional robustness exercise, this section analyzes the KLEMS database. Unlike the55Table B.6: Estimation with a Smaller Number of Inputs
IV (Lagged Input Prices)energyall workers(1)-0.347(0.231)(2)-0.854∗∗(0.392)

1.347∗∗∗(0.395)OLS(3)-0.929∗∗(0.437)(4)0.852∗∗∗(0.029)1.858∗∗∗(0.639)-0.458∗∗∗(0.033)(5)
0.789∗∗∗(0.034)-0.692∗∗∗(0.032)

production worker

1.848∗∗∗(0.626)-0.458∗∗∗(0.035)

non-production worker-0.450(0.355)-0.228∗∗∗(0.028)material

0.257(0.227)

capital-0.190(0.370)value-added share
Observations

23221

(6)
0.794∗∗∗(0.034)-0.400∗∗∗(0.027)0.854∗(0.497)
X
23220

0.376
(0.348)X
23221-0.024(0.021)24167-0.203∗∗∗(0.024)
X
24166-0.187∗∗∗(0.024)
X
24167

Note: The “all workers” add the production worker and non-production worker. When material input is excludedin the regression, we use the value added for the energy share instead of the value of shipment. The energy shareis defined as the energy expenditure over the value added, which subtracts material expenditure from the value of
shipment. In using the instruments, only the input prices that correspond to included inputs are used. *, **, and ***indicate significance at the 0.1, 0.05, and 0.01 levels, respectively.

56

NBER-CES database, the KLEMS database covers the entire sector of the US economy. It includes
key inputs that we use in this paper, such as energy, labor, capital, and material inputs, as wellas its price indexes. It is balanced industry panel data that cover 60 industries over 26 years from1987 to 2012. This database is available on the Bureau of Economic Analysis (BEA) website andis used by influential papers, such as Bils et al. (2018). Although the data cover aggregate sectors
over a short period relative to the NBER-CES database, to the best of our knowledge, it is the only
industry-level panel data that have all the relevant information for our estimation technique andcover the entire US economy.
Table B.7: Using the KLEMS Data
IV (Lagged Input Prices)

OLS

energy(1)1.369∗∗∗(0.307)(2)
1.245∗∗∗(0.318)(3)
1.214∗∗∗(0.232)(4)
1.439∗∗∗(0.301)(5)
1.182∗∗∗(0.278)(6)
1.314∗∗∗(0.218)(7)0.906∗∗∗(0.111)(8)
0.898∗∗∗(0.090)

all workers

1.578∗∗(0.658)1.674∗∗∗(0.618)1.489∗∗∗(0.506)1.574∗∗(0.667)1.253∗∗(0.525)

1.164∗∗(0.521)-0.090(0.130)-0.210∗(0.110)material-0.064(0.292)0.048(0.278)-0.071(0.269)0.112(0.244)-0.181∗∗(0.069)-0.181∗∗∗(0.060)capital

0.165(0.302)
X

0.043(0.074)X

0.025(0.066)X

0.215(0.276)
X

1020

1020

1020

Industry-time weight
Industry weight
Observations

0.254(0.306)
X
1020

0.338
(0.277)
X
1020

X
1020

1140

X
1140

Note: The “all workers” is the labor input available in the KLEMS data. To be consistent with the exercise that uses
the NBER-CES data, we define the energy share after subtracting the service input from the output. To inform on theaggregate parameters, we weight the observations by either industry-time-specific output (“industry-time-weight”) or
industry-specific output (“industry weight”). As in Table 1, both the one- and two-year lagged input prices are used as
instrumental variables. *, **, and *** indicate significance at the 0.1, 0.05, and 0.01 levels, respectively.

We re-estimate the parameters based on Equation (2.6) with the KLEMS data and replicate
Table 1. Table B.7 presents the results. Given that the motivation of this exercise is to informon the aggregate representative parameters, we weight the sample either by industry-time-specificoutput (columns 1-3 and 7) or industry-specific output (columns 4-6 and 8). Regardless of the useof different specifications, we still find a strong complementarity between labor and energy similar
to our main empirical analysis in Table 1. In addition, the complementarity parameter between
labor and energy is in the range with the Bayesian estimation results in Section 3.2. One difference
from Table 1 is that the energy input no longer features the strong diminishing returns, potentiallybecause industries outside of the manufacturing sectors do not utilize the energy input the same asin the manufacturing sectors. The OLS results show a similar empirical pattern.

57

Appendix C
C.1

Translog Production Function in the DSGE Model

Properties of the Translog Production Function

In this section, we discuss in more detail the properties of the translog production function that we
propose. For the clarity of exposition, we abstract away fixed cost (i.e., v = 0) from Equation (3.1):
yt (i) = exp(εat )[kts (i)]βk [lt (i)]βl [et (i)]βe ×"lt (i)¯lβel log(et /ē) et (i)ēβel log(lt /l) #
.

(C.1)

Equation (C.1) can be broken down into the short-run and long-run components, in which theshort-run component is expressed as the deviation from the long-run component (i.e., the steadystate):

yt (i) = f

SR

 s


kt (i) lt (i) et (i), ¯ ,; Ωt · f LR k̄, ¯l, ē; Ω ,
ē
k̄
l

(C.2)

where
f

SR



kt (i) lt (i) et (i)
, ¯ ,
; Ωt
ē
k̄
l



≡ exp(εat )
=yt (i)ȳ s βk lt (i) βl +βel log(et /ē) et (i) βe +βel log(lt /l̄)kt (i)
¯l
ē
k¯s



(C.3)

,

β β
f LR (k¯s , ¯l, ē; Ω) ≡ k¯s k ¯l l (ē)βe (= ȳ)withΩt ≡ [βk , βl + βel log(et /ē), βe + βel log(lt /¯l)]0 ,Ω ≡ [βk , βl , βe ]0 .

(C.4)

A similar decomposition can be found in Cantore et al. (2015) and Koh and Santaeulàlia-Llopis(2017) in the context of the CES production function in an attempt to resolve dimensionality issues
Cantore and Levine (2012).23 Here, lt and et indicate the cross-sectional average of lt (i) and et (i),respectively, which individual firms take as a given.
In the above expression, Ωt is a vector of endogenous parameters that govern potentially time23

As discussed in De Jong (1967), Cantore and Levine (2012) and Cantore et al. (2014), such a distinction
between short- and long-run production functions (or normalization) was implemented to address what is called
the dimensionality issue: under CES technology, factor share parameters no longer directly measure the “share” butdepend on the underlying “dimensions”. This makes such parameters depend on the choice of units, which creates
measurement problems in calibration and estimation. Expressing the production function in deviation from thesteady-state eliminates the dimensional parameters and resolves the issue. As discussed below, our normalization also
makes the translog parameters under our production function no longer depend on the choice of units.

58

varying short-run returns to scale of the economy. We write “endogenous” parameters because thetime-varying components are endogenously determined in equilibrium, although individual firms takethese values as a given.24 Ω is a vector of strictly exogenous parameters that govern the time-invariant
and constant long-run returns to scale of the economy. If we evaluate Ωt at the steady-state or on
the long-run horizon, we have Ω = Ω.
The proposed translog production function has the following properties.
βel log(et /ē) βel log(lt /l) lt (i)et (i)
consists
1. The bracketed term on the right-hand side of (C.1),ēl̄
of the variables normalized by their steady state values. This makes the production function
collapse into the conventional Cobb-Douglas at the steady state.
There are three advantages of the normalization. First, it resolves the dimensionality issue
discussed in De Jong (1967), Cantore and Levine (2012) and Cantore et al. (2014); that is, thenormalization makes βel independent of the choice of units. To illustrate this point, note that under the normalization, the labor and energy input shares are given by wyttlt = λ [βl + βel log (et /ē)]


pe e
and t t = λ βe + βel log lt /¯l , respectively, where λ is the wedge between the marginal prodyt

uct of inputs and real input prices. From this expression, we can see that the parameter βel
is dimensionless because the inputs that appear on the right-hand sides are normalized bytheir steady state counterparts (i.e., eēt and ll̄t ), which do not depend on the choice of units.
In contrast, if we do not normalize the production function, then the labor and energy input
shares become wyttlt = λ [βl + βel log et ] andpet etyt = λ [βe + βel log lt ], respectively. In this case,since the input shares are dimensionless, βel depends on the unit of inputs (et and lt ).
Second, it makes the model compatible with the balanced-growth path.25 In this way, theinput complementarity that we introduce (and the resulting procyclical returns to scale) is a
short-run characteristic, which does not affect the long-run growth of the economy.
Finally, the normalization facilitates a comparison with the model without complementarityinduced procyclical returns to scale because the steady-state is identical across the two models.
2. At the aggregate level, the short-run component f SR has the translog expression:

log

yt
ȳ



= εat + βk log

 s
 
 
e 
e 
kt
lt
lt
t
t
+
β
log
+
β
log
+
2β
log
log
e
l
el
s
¯
¯
¯
ē
ē
k
l
l

3. The complementarity between energy and labor is reflected by a single parameter βel . If βel > 0,
our model features complementarity-induced procyclical returns to scale in the short-run,
24

Because the time-varying components in Ωt are endogenously determined in equilibrium, the terminology“parameter” can be somewhat misleading. Still, we call it an endogenous parameter because it characterizes the returns
to scale of the economy that individual firms take as exogenous. The long-run value of Ωt , Ω, is a vector of strictly
exogenous parameters in the sense that it consists only of deep structural parameters.
25
Our medium-scale DSGE model features the balanced-growth path.

59

provided that the dynamics of log(et /ē) and log(lt /¯l) are procyclical.
4. Despite procyclical returns to scale, the production function becomes scale-free up to thefirst order because log-linearizing the production function yields a form exactly identical to
the log-linearized Cobb-Douglas production function. To see this, consider the first-orderapproximation of the production function at the aggregate level:ŷt = εat + βk k̂ts + βl ˆlt + βe êt ,where x̂t ≡ log xx̄t for an arbitrary variable x.
Therefore, the procyclicality of returns to scale does not generate any additional fluctuationof output by itself and behaves exactly the same as the conventional Cobb-Douglas up to thefirst order. All interesting dynamics arise through the first-order condition of the firm. This
scale-free characteristic up to the first order is one feature that distinguishes our model from theconventional increasing returns to scale model. In the increasing returns to scale model, boththe production function and the first-order conditions are affected by the increasing returns up
to the first order, while only the first-order conditions are affected by the procyclical returns toscale in our model.
5. The short- and long-run returns to scale at the individual firm level are given by
Short-Run Returns to Scale (rtst ) = βk + βl + βe + βel log(et /ē) + log(lt /¯l)

(C.5)

Long-Run Returns to Scale (rts) = βk + βl + βe
Since individual firms take the cross-sectional average variables et and lt as given and do notinternalize their changes, each individual firm takes the returns to scale as a given.
This assumption guarantees that firms’ optimizing behavior is well-characterized by the firstorder conditions. If an individual firm can internalize the change in returns to scale of theeconomy, then by choosing a larger amount of labor and energy inputs, each firm can maketheir returns to scale arbitrarily large. This would induce firms to choose an infinite amount oflabor and energy inputs. By assuming that individual firms do not internalize the change inreturns to scale, this issue no longer arises.26

26This assumption makes our model similar to the internal increasing returns to scale (IRS) model. In contrast to
an external IRS model in which individual firms take the production externalities as a given and therefore effectively
face constant returns to scale, firms in the internal IRS model take the returns to scale parameter (which is larger thanone) as a given and therefore face increasing returns to scale. In our model, individual firms do not internalize thechange in returns to scale, and firms take as a given the returns to scale evaluated at the aggregate inputs.

60

C.2

Aggregation of the Firm-level Production Function

Suppose that Xt is an aggregate quantity of firm-level variable Xt (i) such that Xt =R

Xt (i)di. We

denote the logarithm of Xt and Xt (i) by xt and xt (i), respectively.
R
Lemma. xt = xt (i) di up to the first order.
Proof. We closely follow the discussion in Galí (2015, Appendix 3.4).
R
R
Let Ei {xt (i)} be xt (i) di. Similarly, we define vari {xt (i)} as (xt (i) − Ei {xt (i)})2 di.
From the definition of Xt ,ZZ
Xt (i)
1=
di = exp(xt (i) − xt ) diX
ZtZ1
≈ 1 + (xt (i) − xt ) di +(xt (i) − xt )2 di,2where the last line is based on the second-order Taylor expansion. Thus, we have1
xt ≈ Ei {xt (i)} +2

Z(xt (i) − xt )2 di

up to the second order.
It remains to show that At ≡R

(xt (i) − xt )2 di is of the second-order. The above equation

implies that Ei {xt (i)} − xt ≈ − 12 At up to the second order. Thus,Z(xt (i) − Ei {xt (i)} + Ei {xt (i)} − xt )2 diZZ2= (xt (i) − Ei {xt (i)}) di + 2 (xt (i) − Ei {xt (i)})(Ei {xt (i)} − xt ) di + (Ei {xt (i)} − xt )2

2
Z1≈ vari {xt (i)} − At (xt (i) − Et {xt (i)}) di +
At
2
2

1
= vari {xt (i)} +
At
2

At =

up to the second order. This result implies thatp
At ≈ 2 ± 2 1 − vari {xt (i)}up to the second order. Becausep2

t (i)}− [vari {x8t (i)}] up to the second
1 − vari {xt (i)} ≈ 1 − vari {x261order,p
At ≈ 2 ± 2 1 − vari {xt (i)}vari {xt (i)} [vari {xt (i)}]2≈2±2 1−−
2
8!
.

We know that At > 0. Thus, we obtain
vari {xt (i)} [vari {xt (i)}]2
At ≈ 2 − 2 1 −
−
2
8

!

≈ vari {xt (i)}up to the second order. Because vari {xt (i)} is of the second order, the proof is complete.
Consider the firm-level production function (3.1). This implies thatlog(yt (i) + υ) = εat + βk log(kts (i)) + βl log(lt (i)) + βe log(et (i)) + βel êt ˆlt (i) + βel êt (i)ˆlt .
For the Kimball aggregator, it is well-known that yt ≈R

yt (i) di up to the first order (Smets andWouters, 2007). Thus, by repeatedly applying the above lemma, we haveZ
log(yt + υ) ≈ log

≈log(yt (i) + υ) di
Z

a
s
¯
≈ εt + βk log kt + βl log lt + βe log et + βel êtlog(lt (i)) di − log l
Z

+ βel
log et (i) di − log ē ˆlt
 
e 
lt
t ˆ
a
s
≈ εt + βk log kt + βl log lt + βe log et + βel êt log ¯ + βel log
ltēl= εa + βk log k s + βl log lt + βe log et + 2βel ˆlt êt ,tyt (i) di + υZtwhere all the approximation is up to the first order. By taking the exponential, we recover theaggregate translog production function that mirrors the firm-level production function up to the firstorder.
yt ≈ exp(εat )[kts ]βk [lt ]βl [et ]βe βel log(et /ē)  lt
et βel log(lt /l̄)
− υ.
¯l
ē

62

Appendix D

Additional Impulse Responses

Figure D.1: Impulse responses to price and wage markup shocks

0

0

-0.05

0

0

-0.2

-0.05

-0.4

-0.1

-0.6

-0.15

-0.8

-0.2

-0.05

-0.1
-0.1
-0.15
-0.2

-0.15

-0.25

-0.2

-0.3
-0.25
-0.35
-0.3

-0.4
-0.45

-0.35
0

4

8

12

16

20

-1
0

4

8

12

16

20

HKL

0.1

0

0

-0.1

-0.25
0

HKL (CD)

4

8

12

16

20

0

4

8

12

16

20

0

4

8

12

16

20

S&W

0

0.2
0

-0.1

-0.1

-0.2
-0.2

-0.2
-0.3

-0.2

-0.4

-0.3
-0.4
-0.4

-0.6

-0.3

-0.5
-0.5
-0.8
-0.6

-0.6

-0.4
-1

-0.7

-0.7
-0.8

-0.8
0

4

8

12

16

20

-1.2
0

4

8

12

16

20

-0.5
0

4

8

12

16

20

Notes. The top panels show the responses of output, consumption, investment, and labor to a one-standard-deviation
contractionary price markup shock. The solid, dashdotted, and dashed lines represent the results based on ourbenchmark model (HKL), the model without input complementarity (HKL-CD), and the Smets and Wouters (2007)
model (S&W), respectively. Similar responses to a one-standard-deviation contractionary wage markup shock areillustrated in the bottom panels.

Figure D.1 shows the impulse responses of major macroeconomic variables in response to
price and wage markup shocks at the posterior mode of each of the three models. The top panels
illustrate the responses of output, consumption, investment, and labor to a one-standard-deviation
contractionary price markup shock. The solid, dashdotted, and dashed lines represent the results
based on our benchmark model (HKL), the model without input complementarity (HKL-CD), and the
Smets and Wouters (2007) model (S&W), respectively. Similar responses to a one-standard-deviation
contractionary wage markup shock are illustrated in the bottom panels.
Clearly, HKL features the smallest responses of major aggregate variables to price and wage
markup shocks. For instance, the peak effects on output of a one-standard-deviation price markup
shock is 0.32%, 0.42%, and 0.42% at quarterly frequencies based on HKL, HKL-CD, and S&W,
respectively. For wage markup shocks, the peak effects on output are 0.48%, 0.62%, and 0.70% at
quarterly frequencies. The results for consumption, investment, and labor are similar.

63