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FEDERAL RESERVE BANK OF SAN FRANCISCO
WORKING PAPER SERIES

Aggregate Implications of Changing Sectoral Trends
Andrew Foerster
Federal Reserve Bank of San Francisco
Andreas Hornstein
Federal Reserve Bank of Richmond
Pierre-Daniel Sarte
Federal Reserve Bank of Richmond
Mark Watson
Princeton University and NBER
January 2022
Working Paper 2019-16
https://www.frbsf.org/economic-research/publications/working-papers/2019/16/

Suggested citation:
Foerster, Andrew, Andreas Hornstein, Pierre-Daniel Sarte, and Mark Watson. 2022.
“Aggregate Implications of Changing Sectoral Trends,” Federal Reserve Bank of San
Francisco Working Paper 2019-16. https://doi.org/10.24148/wp2019-16
The views in this paper are solely the responsibility of the authors and should not be interpreted
as reflecting the views of the Federal Reserve Bank of San Francisco or the Board of Governors
of the Federal Reserve System.

Aggregate Implications of Changing Sectoral Trends∗
Andrew T. Foerster
Federal Reserve Bank of San Francisco

Andreas Hornstein
Federal Reserve Bank of Richmond

Pierre-Daniel G. Sarte
Federal Reserve Bank of Richmond

Mark W. Watson
Princeton University and NBER

January 20, 2022

Abstract
We find disparate trend variations in TFP and labor growth across major U.S. production sectors
and study their implications for the post-war secular decline in GDP growth. We describe how capital
accumulation and the network structure of U.S. production interact to amplify the effects of sectoral trend
growth rates in TFP and labor on trend GDP growth. We derive expressions that conveniently summarize
this long-run amplification effect by way of sectoral multipliers. These multipliers are quantitatively large
and for some sectors exceed three times their value added shares. We estimate that sector-specific factors
have historically accounted for approximately 3/4 of long-run changes in GDP growth, leaving common
or aggregate factors to explain only 1/4 of those changes. Trend GDP growth fell by nearly 3 percentage
points over the post-war period with the Construction sector alone contributing roughly 1 percentage
point of that decline between 1950 and 1980. Idiosyncratic changes to trend growth in the Durable Goods
sector then contributed an almost 2 percentage point decline in trend GDP growth between 2000 and
2018. Remarkably, no sector has contributed any steady significant increase to the trend growth rate of
GDP in the past 70 years.

Keywords: trend growth, sectoral linkages, investment network, growth accounting
JEL Codes: C32, E23, O41

∗

The views expressed herein are those of the authors and do not necessarily represent the views of the Federal Reserve Banks
of San Francisco, Richmond, or the Federal Reserve System. We thank Jon Samuels of the Bureau of Economic Analysis for
advice on the data, as well as Bill Dupor, Ivan Petrella, the editor, and four anonymous referees for thoughtful discussions
and suggestions. We also thank John Fernald, Xavier Gabaix, Chad Jones, and participants at various university seminars
and conferences for their comments. Daniel Ober-Reynolds, Eric LaRose, James Lee, and Lily Seitelman provided outstanding
research assistance. Andrew Foerster: andrew.foerster@sf.frb.org, Andreas Hornstein: andreas.hornstein@rich.frb.org, PierreDaniel Sarte: pierre.sarte@rich.frb.org, Mark Watson: mwatson@princeton.edu.

Introduction

Following the so-called Great Recession of 2008-2009, U.S. GDP recovered only very gradually resulting
in a low average growth rate in the ensuing decade. Fernald, Hall, Stock, and Watson (2017) found that
this weak recovery stemmed mainly from slow growth in total factor productivity (TFP) and a fall in
labor input, and note that these adverse forces preceded the Great Recession. Antolin-Diaz, Drechsel, and
Petrella (2017) likewise document a slowdown in output growth that predates the Great Recession.1 This
paper studies what has in fact been a steady decline in trend GDP growth over the entire post-war period,
1950 − 2018. We explore the implications of TFP and labor input in accounting for this secular decline but
we do so at a disaggregated sectoral level. We document disparate trend variations in TFP and labor growth
across sectors and estimate the extent to which these trends are driven by idiosyncratic rather than common
factors. We then study the implications of our empirical findings for trend growth within a multi-sector
framework with linkages that mimic those of the U.S. economy including, crucially, in the production of
investment goods.
We first document that common trend factors play a relatively small role in explaining sectoral trends
in labor and TFP growth. For example, in Durable Goods, only 3 percent of the overall trend variation
in labor and TFP growth is explained by their respective common trend factors. These findings, therefore,
highlight the quantitative importance of idiosyncratic forces not only for business cycle fluctuations (see
Gabaix (2011), Foerster, Sarte, and Watson (2011), and Atalay (2017)), but also for variations in trends.
There are, however, exceptions in that in some service sectors, the trend variation in labor is explained to
a greater degree by the common trend factor. Common trends explain a higher fraction of aggregate trend
variation in labor and TFP growth because aggregation reduces the importance of sector-specific trends.
We estimate that approximately 1/3 of the variation in the trend growth rate of aggregate TFP is common
across sectors while roughly 2/3 is common for labor. One cannot, however, infer from these findings the
role that common and sectoral growth trends in labor and TFP play in the overall trend growth rate of
GDP. The reason is that capital accumulation and the network structure of U.S. production play a key role
in translating those trends to the aggregate economy.
To explore the historical implications of changing sectoral trends for the long-run evolution of GDP
growth, we derive balanced growth accounting equations in a dynamic multi-sector framework where sectors
use not only materials but also investment goods produced in other sectors. We then use these new growth
accounting equations to assess the aggregate effects of observed sectoral changes in the trend growth rates of
labor and TFP. Our analysis, therefore, generalizes the work of Greenwood, Hercowitz, and Krusell (1997) on
investment-specific technical change (ISTC) to an environment with multiple investment and intermediate
1
Cette, Fernald, and Mojon (2016) suggest that a slowdown in productivity growth that began prior to the Great Recession
reflects in part the fading gains from the Information Technology (IT) revolution. This view is consistent with the long lags
associated with the productivity effects of IT adoption found by Basu et al. (2004), and the collapse of the dot-com boom in
the early 2000s. Decker, Haltiwanger, Jarmin, and Miranda (2016) point to a decline in business dynamism that began in the
1980s as an additional force underlying slowing economic activity.

goods sectors that are interconnected in production.2 At the same time, our focus on estimating common
and idiosyncratic sources of sectoral trends, and what their long-run aggregate implications are, differs from
the literature building on Greenwood et al. (1997). Specifically, Fisher (2006), Justiniano, Primiceri, and
Tambalotti (2010, 2011) and Basu, Fernald, Fisher, and Kimball (2013) are primarily concerned with the
business cycle implications of sectoral shocks and, in particular, investment-specific shocks. More recently,
vom Lehn and Winberry (2022) show that the input-output network of investment goods is critical in
accounting for shifts in the cyclicality and relative volatilites of aggregate time series since the 1980s.3
We show that capital accumulation together with the network structure of U.S. production markedly
amplifies the aggregate long-run growth effects of sectoral changes in the trend growth rates of TFP and
labor. This amplification mechanism can be conveniently summarized in the form of sectoral multipliers that
reflect the knock-on effects of production linkages. Given observed U.S. production linkages, the influence
of individual sectors on GDP growth may be as large as 3 times their share in the economy including in
Durable Goods, Construction, and Professional and Business Services.
Combining our empirical findings with the amplification effects of sectoral multipliers, we find that
sector-specific trends have accounted for roughly 3/4 of the trend variation in GDP growth over the postwar period, leaving aggregate or common factors to explain only 1/4 of those changes. The secular decline
in U.S. GDP growth since 1950, therefore, is a phenomenon largely driven by idiosyncratic rather than
aggregate forces. These findings arise in part because of the knock-on effects of production linkages but
also because sector-specific changes in TFP and labor have been historically large in some sectors. Thus,
U.S. trend GDP growth fell by nearly 3 percentage points between 1950 and 2018. The combination of
large trend TFP growth variations in the Construction sector along with its large sectoral multiplier means
that it contributed roughly 1 percentage point of that decline between 1950 and 1980. The Durable Goods
sector, after contributing significantly to an economic expansion in the 1990’s, then contributed another 2
percentage point decline in trend GDP growth between 2000 and 2018. While Professional and Business
Services stands out as having the second largest sectoral multiplier, smaller trend TFP growth variations
in that sector imply that its contributions to the secular decline in GDP growth have been more muted to
this point. Remarkably, no sector has contributed any steady significant increase to the trend growth rate
of GDP over the post-war period.
Our paper also falls within the literature on equilibrium models with sectoral production networks first
developed by Long and Plosser (1983) and later Horvath (1998, 2000) and Dupor (1999). Since then, a large
2
Ngai and Pissarides (2007) provide a seminal study of balanced growth in a multi-sector environment. They consider
both multiple intermediates and multiple capital-producing sectors but not at the same time. Importantly, their analysis
abstracts from pairwise linkages in both intermediates and capital-producing sectors that play a key role in this paper. Ngai
and Samaniego (2009) generalizes the model in Greenwood et al. (1997) to three sectors which allows for an input-output
network in intermediate goods in carrying out growth accounting. Duarte and Restuccia (2020) include input-output linkages
across sectors in a multi-sector environment abstracting from capital and study the implications of cross-country productivity
differences in non-traditional service sectors.
3
Basu et al. (2013) also construct a multi-sector extension of the Greenwood et al. (1997) environment, but they work with
an aggregate capital stock and an aggregate labor endowment with each factor being perfectly mobile across sectors. In contrast
to this paper, the authors study short-run responses to TFP shocks.

body of work has explored important features of those models for generating aggregate fluctuations from
idiosyncratic shocks. We maintain the original assumptions of competitive input and product markets as
well as constant-returns-to-scale technologies. Even absent non-log-linearities in production emphasized by
Baqaee and Farhi (2019), for example, and beyond the role of idiosyncratic shocks in explaining aggregate
cyclical variations, the analysis reveals that sector-specific changes also dominate long-run variations in U.S.
GDP growth.4
This paper is organized as follows. Section 2 gives an overview of the behavior of trend GDP growth
over the past 70 years. Section 3 provides an empirical description of the trend growth rates of TFP and
labor growth by industry and estimates the contributions of sector-specific and common factors to these
trends. Section 4 develops the implications of these changes at the sector level in the context of a dynamic
multi-sector model with production linkages in materials and investment. This model serves as the balanced
growth accounting framework that we use to determine the aggregate implications of changes in the sectoral
trend growth rates of labor and TFP. Section 5 presents our quantitative findings. Section 6 discusses
salient implications of our work and outlines directions for future research. Section 7 concludes. A detailed
online Technical Appendix contains a comprehensive description of the data, statistical methods, economic
model, discussions of departures from our benchmark assumptions, and includes additional figures and tables
referenced in the text.

The Long-Run Decline in U.S. GDP Growth

Figure 1 shows the behavior of U.S. GDP growth over the post-WWII period. Here, annual GDP growth is
measured as the share-weighted value added growth from 16 sectors comprising the private U.S. economy;
details are provided in the next section.
Panel A shows aggregate private-sector growth rates computed by chain-weighting the sectors, and by
using three alternative sets of fixed sectoral shares computed as averages over the entire sample (1950−2018),
over the first fifteen years of the sample (1950 − 1964), and over the final fifteen years (2004 − 2018). Panel A
shows large variation in GDP growth rates – the standard deviation is 2.5 percent over the period 1950−2018
– but much of this variation is relatively short-lived and is associated with business cycles and other relatively
transitory phenomena. Moreover, to the extent that sectoral shares have changed slowly over time, these
share shifts have little effect in Panel A. In other words, changes in aggregate growth largely stem from
changes within sectors rather than between them. Our interest, however, is in longer-run variation.
Panel B, therefore, plots centered 11−year moving averages of the annual growth rates. Here too there
is variability. In the 1950s and early 1960s average annual growth exceeded 4 percent. Average growth fell
to 3 percent in the 1970s, rebounded to nearly 4 percent in the 1990s, but plummeted to less than 2 percent
4

See Gabaix (2011), Foerster, Sarte, and Watson (2011) and Atalay (2017) for assessments of the importance of idiosyncratic
shocks in driving business cycle fluctuations. We introduce explicit dynamic considerations into the work of Acemoglu et al.
(2012), Baqaee and Farhi (2019), and Miranda-Pinto (2019) combined with an empirical model that parses out common and
idiosyncratic components of sectoral trend input growth.

Figure 1: U.S. GDP Growth Rates 1950-2018
(percentage points at an annual rate)
(a) GDP Growth

(b) GDP Growth
11-Year Centered Moving Average

5
4

5
3
2
0
1
-5
1950 1960 1970 1980 1990 2000 2010 2020

0
1950 1960 1970 1980 1990 2000 2010 2020

(d) Cyclically Adjusted GDP Growth
11-Year Centered Moving Average
5

3
2
2
0
1

Constant Mean Weights (Full Sample)
Chain Weights
Constant Mean Weights (First 15 Years)
Constant Mean Weights (Last 15 Years)

0
1950 1960 1970 1980 1990 2000 2010 2020

-2
1950 1960 1970 1980 1990 2000 2010 2020

Notes: Growth rates are share-weighted value added growth rates from 16 sectors making up the private U.S. economy.
Cyclical adjustment uses a regression on leads and lags of the first-difference in the unemployment rate.

in the 2000s (See Table 1). At these lower frequencies, the effects of slowly shifting shares over the sample
become more visible, but still play a relatively minor role.
Panels C and D refine these calculations by eliminating the cyclical variation using an Okun’s law
regression as in Fernald et al. (2017). Thus, panel C plots the residuals from a regression of GDP growth
rates onto a short distributed lead and lag of changes in the unemployment rate (∆ut+1 , ∆ut , ∆ut−1 ). This
cyclical adjustment eliminates much of the cyclical variability evident in panel A. In addition, the 11−year
moving average in Panel D now produces a more focused picture of the trend variation in the growth rate of
5

Table 1: Average GDP Growth Rates

Dates
1950-2018
1950-1966
1967-1983
1984-2000
2001-2018

Const Mean Weights
Full Sample
Growth
Cyc-adj
rates
rates
3.3
3.2
4.5
3.1
3.9
1.9

4.2
3.6
3.4
1.8

Time-Varying
Weights
Growth Cyc-adj
rates
rates
3.3
3.2
4.3
3.0
3.9
2.0

4.0
3.5
3.4
2.0

Const Mean Weights
First 15 Years
Growth
Cyc-adj
rates
rates
3.2
3.1
4.3
2.8
4.0
1.7

Const Mean Weights
Last 15 Years
Growth
Cyc-adj
rates
rates
3.4
3.4

4.0
3.4
3.4
1.7

4.6
3.4
3.8
2.0

4.3
3.9
3.4
2.0

Notes: The values shown are averages of the series plotted in Figure 1 over the periods shown.

private GDP. Again, time-varying share weights have discernible but relatively small effect on the aggregate
growth rate or its 11-year moving average.
The numbers reported in Table 1 frame the key question of this paper: why did the average growth rate
of GDP fall from 4 percent per year in the 1950s to just over 3 percent in the 1980s and 1990s, and then
further decline precipitously in the 2000s? As the different columns of the table make clear, this question
arises regardless of the shares used in constructing GDP. We look to inputs – specifically TFP and labor at
the sectoral level – for the answer. That is, interpreting long-run variations of the data as a time-varying
balanced growth path, changes in trend GDP growth are in part determined by changes in the trend growth
rates of those sectoral inputs. However, as the analysis in Section 4 makes clear, not all sectoral inputs are
created equal. Sectors not only differ in their size, or their value-added share in GDP, but also in the share
of materials or capital that they provide to other sectors.
Before investigating these input-output interactions, we begin by briefly describing the sectoral data and
how sectoral value-added as well as labor and TFP inputs have evolved over the post-WWII period. In
much of our analysis, we construct aggregates using constant weights computed using full-sample averages.
As Figure 1 and Table 1 suggest, results using these constant shares are robust to alternative weighting
schemes.

An Empirical Description of Trend Growth in TFP and Labor

As a first step, we estimate an empirical model of TFP and labor growth for different sectors of the U.S.
economy. Our paper applies as a benchmark the insights of Hulten (1978) on the interpretation of aggregate
total factor productivity (TFP) changes as a weighted average of sector-specific value-added TFP changes.
In particular, under constant-returns-to-scale and perfect competition in product and input markets, the

sectors’ weights are the ratios of their valued added to GDP.5
We calculate standard TFP growth rates at the sectoral level, construct trend growth rates using a
‘low-pass’ filter, and estimate a statistical model to decompose these trend growth rates into common and
sector-specific components.

3.1

Data

Sectoral TFP growth rates are calculated using KLEMS data from of the Bureau of Economic Analysis
(BEA) and the Bureau of Labor Statistics (BLS) Integrated Industry-Level Production Accounts (ILPA).
These data are attractive for our purposes because they provide a unified approach to the construction
of gross output, the primary inputs capital and labor, as well as intermediate inputs (‘materials’) for a
large number of industries. The KLEMS data are based on U.S. National Income and Product Accounts
(NIPA) and consistently integrate industry data with input-output tables and Fixed Asset tables. The
ILPA KLEMS data build on seminal work studying sectoral productivity accounting by Jorgenson and his
collaborators and first summarized in Jorgenson et al. (1987).
Table 2 lists the 16 sectors we consider along with the growth rates of value added, labor, and TFP
for each sector. Section 7 of the Technical Appendix provides a detailed discussion of the data and the
construction of labor and TFP from quantity and price indices available in KLEMS. For each sector, the
table shows average cyclically adjusted growth rates of value added, labor, and value added TFP over
1950 − 2018, and it also shows their average shares in aggregate value added and labor input. The aggregate
growth rates in the bottom row are the value-weighted averages of the sectoral growth rates with average
value added and labor shares used as fixed weights.
Clearly sectors grow at different rates and this disparity is hidden in studies that only consider aggregates.
Average real value added growth rates range from 1.4 percent in Mining to 4.9 percent in Information,
bracketing the aggregate value added growth rate of 3.3 percent. With the exception of the Durable Goods
sector, most sectors with growth rates that exceed the aggregate growth rate provide services. Similarly,
labor input growth rates range from −1.3 percent in Agriculture to 3.5 percent in Professional and Business
Services (PBS), bracketing the average aggregate growth rate of 1.6 percent. Again, most sectors with labor
input growth rates that exceed the aggregate growth rate provide services. Finally, TFP growth rates range
from −0.4 percent in Utilities to 3.1 percent in Agriculture, bracketing the average aggregate TFP growth
rate of 0.8 percent. Sectoral TFP growth rates are less aligned with either value added or labor input growth
rates. There are four sectors with TFP declines, namely Utilities, Construction, FIRE (x-Housing), and
Education and Health, as well as a number of sectors with stagnant TFP levels. Negative TFP growth rates
are a counter-intuitive but known feature of disaggregated industry data. These are in part attributed to
5

In the absence of constant-returns-to-scale, perfect competition, or frictionless factor mobility, Basu and Fernald (1997,
2001) and Baqaee and Farhi (2018) show that aggregate TFP changes also incorporate reallocation effects. These effects reflect
the movement of inputs between low and high return sectors in the absence of equalization of marginal rates of transformation
and substitution.

Table 2: 16 Sector Decomposition of the U.S. Private Economy
(1950-2018)

Sectors

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16

Agriculture
Mining
Utilities
Construction
Durable Goods
Nondurable Goods
Wholesale Trade
Retail Trade
Trans. & Ware.
Information
FIRE (x-Housing)
PBS
Educ. & Health
Arts, Ent., & Food Svc.
Other Services (x-Gov)
Housing
Aggregate

Average growth rate
Cyclically adjusted data
(Percentage points at an
annual rate)

Average share
(Percentage points)

Value
Added
2.41
1.38
2.09
1.69
3.65
2.27
4.61
3.13
2.58
4.93
3.88
4.45
3.43
2.48
1.99
3.45
3.32

Value
Added
2.69
2.11
2.37
4.99
13.32
9.20
7.15
8.18
4.16
4.97
9.97
8.79
6.22
3.74
2.94
9.20
100

Labor

TFP

-1.29
0.37
1.00
1.76
0.54
0.14
1.67
1.19
0.91
1.35
2.77
3.51
3.34
1.79
0.52
0.86
1.55

3.12
0.39
-0.42
-0.23
2.10
0.83
1.81
1.07
1.27
1.04
-0.03
0.36
-0.29
0.36
1.04
0.24
0.82

Labor
3.23
1.55
1.04
7.62
15.5
8.80
6.63
9.61
5.03
3.74
7.53
11.25
9.35
4.56
4.37
0.20
100

Notes: The values shown are average annual growth rates for the 16 sectors. The row labelled
“Aggregate” is the constant share-weighted average of the 16 sectors.

measurement issues with respect to output, though land and the regulatory environment are also a factor
in sectors such as Construction.6
To a first approximation, the contributions of the different sectors to aggregate outcomes are given by
the nominal value added and labor input shares in the last two columns of Table 2. In those columns,
two notable contributors to value added and TFP are Durable Goods and FIRE (x-Housing). The two
largest contributors to labor payments are Durable Goods and Professional and Business Services. Over
time, the shares of goods-producing sectors has declined while the shares of services-producing sectors has
6

See for example Herkenhoff, Ohanian, and Prescott (2018).

increased. However, despite these changes, aggregating sectoral outputs and inputs using constant mean
shares, as opposed to time-varying shares, has little effect on the measurement of aggregate outputs and
inputs (Figure 1).

3.2

Empirical Framework

The empirical analysis used to characterize the long-run properties of the data proceeds in 3 steps. First,
we carry out a cyclical adjustment of sectoral TFP and labor raw growth rates to eliminate some of their
cyclical variability. Second, we make use of methods discussed in Müller and Watson (2020) to extract
smooth trends capturing the long-run evolution of the data. Finally, we carry out a factor analysis that
quantifies the relative importance of common and sector-specific factors in driving these smooth trend
components.
3.2.1

Cyclical Adjustment

Let ∆e
xi,t denote the growth rate (100 × the first difference of the logarithm) of annual measurements of
labor or TFP in sector i at date t. These sectoral growth rates are volatile and, in many sectors, much of
the variability is associated with the business cycle. Our interest is in trend (i.e., low-frequency) variation,
which is more easily measured after cyclically adjusting the raw growth rates. Thus, as with the cyclically
adjusted measure of GDP shown in Figure 1, we follow Fernald et al. (2017) and cyclically adjust these
growth rates using the change in the unemployment rate, ∆ut , as a measure of cyclical resource utilization.
That is, we estimate
∆e
xi,t = µi + βi (L)∆ut + ei,t ,
where βi (L) = βi,1 L + βi,0 + βi,−1 L−1 and the leads and lags of ∆ut capture much of the business-cycle
variability in the data. Throughout the remainder of the paper, we use ∆xi,t = ∆e
xi,t − β̂i (L)∆ut , where
b
βi (L) denotes the OLS estimator, and where xi,t represents the implied cyclically adjusted value of sectoral
TFP (denoted zi,t ) or labor input (denoted `i,t ) growth rates.
3.2.2

Extracting Low-Frequency Trends

We begin by extracting low-frequency trends from the data using a framework presented in Müller and
Watson (2008). That framework is useful because on the one hand, it yields smooth trends that capture the
long-run evolution of the growth rate of GDP and the associated growth rates of sectoral labor and TFP,
and on the other hand, it simultaneously provides a convenient framework for statistical analysis. We give
an overview below of the approach here. Müller and Watson (2020) provides a detailed Handbook discussion
of statistical analysis using this framework.7
7

The methods used are closely related to well-known spectral analysis methods using low-frequency Fourier transforms of
the data. See Müller and Watson (2020) for a detailed discussion and references.

Figure 2: Trend Rate of Growth of GDP
(percentage points at an annual rate)
6

Growth Rate (Cyc. Adjusted)
11-year centered moving average
Low-frequency trend

-1
1950

1960

1970

1980

1990

2000

2010

2020

Notes: The low-frequency trend captures variability for periodicities longer than 17 years.

To extract low frequency trends in the growth rates of GDP, TFP and labor input, generically denoted
by ∆xt , we regress these series onto a constant and a set of low-frequency periodic functions. In particular,
√
let Ψj (s) = 2 cos(jsπ) denote a cosine function on s ∈ [0, 1] with period 2/j. The fitted values from the
OLS regression of ∆xt onto a constant and Ψj ((t − 1/2)/T ) for j = 1, ..., q and t = 1, ..., T capture the
low-frequency variability in the sample corresponding to periodicities longer than 2T /q. Moreover, let Ψ(s)
denote the vector of regressors [Ψ1 (s), ..., Ψq (s)]0 with periods 2 through 2/q, ΨT the T × q matrix with tth
row Ψ((t − 1/2)/T )0 and Ψ0T = [1T , ΨT ] where 1T is a T × 1 vector of ones. The specific form used for
0
the cosine weights implies that the columns of Ψ0T are orthogonal with T −1 Ψ00
T ΨT = Iq+1 . Thus, the OLS

0 −1 Ψ00 ∆x
coefficients from the regression of ∆xt onto Ψ0T , that is Ψ00
1:T , amount to q + 1 weighted
T ΨT
T

averages of the data, T −1 Ψ00
T ∆x1:T , which we partition as (x, X) where x is the sample mean of ∆xt . In our
application, T = 69 so that with q = 8, the regression captures long-run variation with periodicities longer
than 17.25 (= 2 × 69/8) years. These are the low-frequency growth rate trends analyzed in this paper.8
Figure 2 plots the growth rates of (cyclically-adjusted) GDP, its centered 11-year moving average, and
8
Calculations presented in Müller and Watson (2008) show that these low-frequency projections approximate a low-pass filter
for periods longer than 2T /q. That said, there is some leakage from higher frequencies and this makes the cyclical adjustment
discussed above useful.

Figure 3: Labor Growth Rates and Trends by Sector
(percentage points at an annual rate)
Agriculture

Mining

-10

Utilities

5
0

0
-5
1960 1980 2000 2020

Nondurable Goods

Wholesale Trade
4

-2

-4

-2
1960 1980 2000 2020

Trans. & Ware.

Information

-2
1960 1980 2000 2020

4
5

0
0

-5

1960 1980 2000 2020

Arts, Ent. & Food Svc.

0
-5
1960 1980 2000 2020

-10

-5

1960 1980 2000 2020

Housing

1960 1980 2000 2020

Other Services (x-Gov.)

PBS
10

6
2

Educ. & Health

1960 1980 2000 2020

FIRE (x-Housing)

1960 1980 2000 2020

Retail Trade

1960 1980 2000 2020

-2

1960 1980 2000 2020

-5

1960 1980 2000 2020

Durable Goods
10

-5

-20
1960 1980 2000 2020

Construction

1960 1980 2000 2020

Notes: Each panel shows the cylically adjusted growth rate of labor for each sector in black, along with its lowfrequency trend in blue.

its trend computed as the fitted values from the low-frequency regression we have just described.9 The
low-frequency trend smooths out the higher-frequency variation in the 11-year moving average. While the
aggregate importance of sectoral shocks is known for business cycles – generally cycles with periods ranging
from 2 to 8 years – our interest here is on the role of sectoral shocks for the aggregate trend variations shown
in Figure 2. Thus, we will focus on cycles longer than 17 years as captured by the Ψ-weighted averages of
the data.
Figures 3 and 4 plot the cyclically adjusted growth rates of labor and TFP for each of the 16 sectors
9

An 11-year moving average is a crude low-pass filter with more than half of its spectral gain associated with periods longer
than 17 years.

Figure 4: TFP Growth Rates and Trends by Sector
(percentage points at annual rate)
Agriculture

Mining

Utilities

Construction

20
0
-20
-20
1960 1980 2000 2020

-20

1960 1980 2000 2020

Durable Goods

-10
1960 1980 2000 2020

Nondurable Goods

1960 1980 2000 2020

Wholesale Trade

Retail Trade
10
5
0

-10

1960 1980 2000 2020

Trans. & Ware.

-5

-10

-10
1960 1980 2000 2020

Information

1960 1980 2000 2020

FIRE (x-Housing)

PBS

5
0
-10

-5

Educ. & Health
5

-5

-10
1960 1980 2000 2020

1960 1980 2000 2020

Arts, Ent. & Food Svc.

Other Services (x-Gov.)

1960 1980 2000 2020

0
0

-2

-4

-5
1960 1980 2000 2020

Housing
2

5
0
-5

1960 1980 2000 2020

-10

1960 1980 2000 2020

Notes: See notes for Figure 3.

along with their low frequency trends. The disparity in experiences across different sectors stands out.
In particular, the trends show large variations across sectors and through time. For example, labor input
was contracting at nearly 4 percent per year in Agriculture in the 1950s but stabilized near the end of the
sample. In contrast, labor input in the Durables and Nondurable Goods sectors was increasing in the 1950s
but has been contracting since the mid-1980s. At the same time, the trend growth rate of labor in several
service sectors exhibit large ups and downs over the sample. Similar disparities are apparent in the sectoral
growth rates of TFP. Trend TFP growth in Construction, for example, was around 5 percent in the 1950s,
declined over the next couple of decades, and flattened out thereafter. In contrast, TFP trend growth in
Durable Goods increased somewhat steadily from the 1950s to 2000 but has since collapsed by more than 5

percentage points. In Sections 4 and 5, we quantify the aggregate implications of these sectoral variations
in labor and TFP inputs.
3.2.3

Decomposition of Trend Growth Rates into Common and Sector-Specific Factors

To fix notation, let gt denote the trend growth rate constructed from our data on TFP or labor input,
∆xt . That is, gt is the fitted value from the OLS regression of ∆xt onto a constant and the q periodic
functions Ψj ((t − 1/2)/T ), and is the low-frequency trend plotted in Figures 2-4. We saw earlier that
because the regressors are mutually orthogonal, the OLS regression coefficients are (x, X) where x is the
sample mean of ∆xt and X is the q × 1 vector of OLS regression coefficients from the regression of ∆xt onto
Ψj ((t − 1/2)/T ) for j = 1, ..., q. Importantly, because the regressors are deterministic, the stochastic process
for gt is completely characterized by the probability distribution of the (q + 1) random variables (x, X), and
variation in gt over the sample is determined by the q × 1 vector X.10
A key implication of these results is that the original sample of T observations on ∆xt contains only
q pieces of independent information on the long-run properties of ∆x. In our context, the T = 69 annual
observations contain only q = 8 observations describing the long-run variation for periods longer than 17
years. This makes precise the intuition that a statistical analysis of long-run growth is inherently a ‘small
sample’ problem. Conveniently, however, this small sample problem involves variables that are averages of
the T observations – the elements of X – and that are, therefore, (approximately) normally distributed and
readily analyzed using standard statistical methods.
Examination of the trends plotted in Figures 3 and 4 suggests that some of the trend variation may be
common across sectors while some are sector-specific. In addition, in some sectors, trend variation in labor
appears to be correlated with trend variation in TFP (and interestingly this correlation generally appears
to be negative). We now outline an empirical model that captures these features.
Let ∆ ln `i,t denote the rate of growth of labor input in sector i in period t, and let ∆ ln zi,t denote the
rate of growth of TFP. Consider the factor model
"

∆ ln `i,t
∆ ln zi,t

"
=

λ`i

λzi

ft`
ftz

"
+

u`i,t
uzi,t

#
,

(1)

where ft = (ft` ftz )0 are unobserved common factors, λi = (λ`i λzi )0 are factor loadings, and ui,t = (u`i,t uzit )0
are sector-specific disturbances. Denote the trend growth rates in (∆ ln `i,t , ∆ ln zi,t , ft` , ftz , u`i,t , uzi,t ) by re`
z
` , gz , g` , gz , g`
spectively (gi,t
i,t f,t f,t u,i,t , gu,i,t ). Let Xi denote the q × 1 vector of OLS coefficients associated with

Ψj ((t − 1/2)/T ), j = 1, ..., q, in the regression of ∆ ln `i,t on a constant and these periodic functions, and
similarly for Xzi , F` , Fz , U`i and Uzi . Pre-multiplying each element in equation (1) by T −1 ψt0 , where T −1 ψt0
10

By construction, the low frequency trends are highly serially correlated, and this needs to be accounted for in the statistical
analysis. As it turns out, this is relatively straightforward given the framework described above. We highlight a few key features
of this framework in Section 1 of the Technical Appendix and refer the reader to Müller and Watson (2020) and references
therein for more detail.

is the tth row of ΨT , and summing, yields a factor decomposition of the trends and cosine transforms of the
form (abstracting from the constant),
"

X`i
Xzi

"
=

λ`i Iq

λzi Iq

F`
Fz

"
+

U`i
Uzi

#
,

(2)

which characterizes the low-frequency variation in the data. We estimate a version of (2) and use it to
` , g z ), and sector-specific components, (g `
z
describe the common components, (gf,t
u,i,t , gu,i,t ), of the trend
f,t

growth rates in sectoral labor input and TFP.11
The model is estimated using Bayes methods. While large-sample Bayes and Frequentist methods often
coincide, the analysis of long-run trends is predicated on a small sample: in our application, the variation in
each trend is characterized by only q = 8 observations. Hence, large-sample frequentist results are irrelevant
for our ‘small-sample’ empirical problem, and Bayes analysis will in general depend on the specifics of the
chosen priors. Section 1 of the Technical Appendix contains details of the estimation method and empirical
results for the low-frequency factor model.12
The priors we use are relatively uninformative except for the factor loadings. Let λ` = (λ`1 , ..., λ`16 )0
and note that the scale of λ` and F` are not separately identified. Thus, we normalize s0` λ` = 1, where s`
denotes the vector of average sectoral labor shares shown in Table 2. This imposes a normalization where
P
P
the growth of aggregate labor, say ∆ ln `t = i=1 s`,i ∆ ln `i,t , satisfies ∆ ln `t = ft` + i s`,i u`i,t . That is, a
one unit change in f ` corresponds to a unit change in the long-run growth rate of aggregate labor.
The prior for λ` is λ` ∼ N (1, P` ) where 1 is a vector of 1s and P` = η 2 (I16 − s` (s0` s` )−1 s0` ) which enforces
the constraint that s0` λ` = 1. The parameter η governs how aggressively the estimates of λ`i are shrunk
toward their mean of unity. Our benchmark model uses η = 1, so the prior puts approximately 2/3 of its
weight on values of λ`i between 0 and 2. Smaller values of η tighten the constraint, making negative factor
loadings less likely, while larger values of η loosen it. To gauge the robustness of our conclusions to the
choice of η, we will also show results with η = 1/2 and η = 2 in Section 5. We use an analogous prior for λz .

3.3

Estimated Sectoral and Aggregate Trend Growth Rates in Labor and TFP

For our purposes, the key results are summarized in a table and three figures. The table summarizes salient
features of the stochastic process describing the long-run evolution of the sectoral growth rates. The figures
11
See Müller et al. (2020) for a related application studying long run growth and long horizons forecasts for per-capita
GDP values of a panel of 113 countries. In equation (1), the common factors affect all sectors without leads and lags, an
unrealistic assumption made for expositional purposes. The low-frequency model in (2) allows for lags in (1) that are short
relative to the sample size. That said, lags of a decade or longer, such as those associated with general purpose technologies
(e.g., semiconductors) will confound common and sector-specific sources of variation in the growth rates. See, for example,
Basu et al. (2004)’s discussion of the diffusion of information and communications technology (ICT) and its delayed effects on
productivity growth in ICT-using industries in the US and the UK.
12
Frequentist methods for small-sample problems such as these are discussed, for example, in Müller and Watson (2008, 2016,
2018). As practical matter, these methods apply only to univariate and bivariate settings. Our application here involves 32
time series.

Table 3: Changes in Trend Value of Labor and TFP Growth Rates

Sector
Agriculture

Mining

Utilities

Construction

Durable Goods

Nondurable Goods

Wholesale Trade

Retail Trade

Trans. & Ware.

Information

FIRE (x-Housing)

PBS

Educ. & Health

Arts, Ent., & Food Svc.

Other Services (x-Gov)

Housing
Aggregate

λ`
2.01
(1.24, 2.71)
0.73
(-0.17, 1.64)
1.13
(0.41, 1.82)
1.55
(0.95, 2.08)
0.40
(-0.23, 1.03)
0.59
(-0.20, 1.38)
1.09
(0.62, 1.49)
0.80
(0.26, 1.29)
-0.04
(-0.75, 0.72)
1.34
(0.69, 2.01)
1.92
(1.34, 2.48)
1.87
(1.48, 2.29)
0.59
(-0.06, 1.05)
1.19
(0.69, 1.75)
0.68
(-0.10, 1.48)
0.82
(-0.13, 1.74)
1.0

λz
0.59
(-0.59, 1.64)
1.10
(0.10, 2.09)
1.36
(0.36, 2.35)
1.26
(0.21, 2.66)
1.31
(0.44, 2.17)
1.22
(0.36, 2.13)
0.88
(0.06, 1.74)
1.14
(0.17, 2.82)
0.88
(-0.02, 1.79)
0.77
(-0.18, 1.81)
0.35
(-0.42, 1.34)
0.90
(-0.01, 1.80)
1.36
(0.24, 2.49)
0.37
(-0.39, 1.31)
0.74
(-0.10, 1.63)
0.75
(0.08, 1.50)
1.0

R`2
0.21
(0.06, 0.44)
0.01
(0.00, 0.07)
0.24
(0.04, 0.58)
0.33
(0.10, 0.61)
0.03
(0.00, 0.18)
0.06
(0.01, 0.29)
0.53
(0.17, 0.81)
0.26
(0.04, 0.60)
0.05
(0.00, 0.23)
0.22
(0.04, 0.51)
0.76
(0.35, 0.92)
0.64
(0.31, 0.87)
0.16
(0.01, 0.56)
0.37
(0.11, 0.67)
0.06
(0.01, 0.23)
0.01
(0.00, 0.04)
0.67
(0.48, 0.82)

Rz2
0.02
(0.00, 0.13)
0.01
(0.00, 0.04)
0.05
(0.00, 0.29)
0.02
(0.00, 0.19)
0.03
(0.00, 0.15)
0.04
(0.00, 0.23)
0.04
(0.00, 0.20)
0.05
(0.00, 0.85)
0.06
(0.00, 0.28)
0.03
(0.00, 0.19)
0.08
(0.01, 0.40)
0.06
(0.00, 0.39)
0.10
(0.01, 0.54)
0.05
(0.00, 0.26)
0.02
(0.00, 0.10)
0.10
(0.01, 0.44)
0.30
(0.10, 0.58)

corr(`, z)
-0.32
(-0.52, -0.15)
-0.35
(-0.63, -0.06)
0.22
(-0.06, 0.58)
-0.25
(-0.55, -0.04)
-0.35
(-0.63, -0.05)
-0.36
(-0.65, -0.06)
0.20
(-0.06, 0.53)
0.06
(-0.25, 0.62)
0.06
(-0.25, 0.36)
-0.25
(-0.56, -0.00)
0.01
(-0.41, 0.40)
-0.92
(-0.98, -0.67)
-0.63
(-0.88, -0.26)
-0.18
(-0.51, 0.02)
-0.07
(-0.35, 0.17)
0.07
(-0.21, 0.40)
-0.29
(-0.72, -0.13)

Notes: The estimates are posterior medians with 68 percent credible intervals shown parentheses. The entries under cor(`, z)
`
z
`
z
are the correlations between gu,i,t
and gu,i,t
for the rows corresponding to sectors, and correlations between gf,t
and gf,t
for

the row labeled Aggregate.

summarize the historical evolution of the long-run growth rates over the sample period.
Table 3 reports the posterior medians for λ along with 68 percent credible intervals.13 Also reported
13

Throughout the paper, we report 68 equal-tail percent credible intervals. Section 1 of the Technical Appendix also reports

Figure 5: Aggregate Trend Growth Rates in Labor and TFP: Common and Sector-Specific Components
(percentage points at annual rate)
(a) Aggregate Labor

(b) Labor: Common

-1

-2

Data (demeaned)
Low frequency trend

-2
1960

1980

2000

2020

1960

(d) Aggregate TFP

1980

2000

2020

1960

(e) TFP: Common
2

-1

-2

1980

2000

2020

1960

1980

2000

2020

(f) TFP: Sector-Specific

1960

1980

2020

1960

1980

2000

2020

Notes: Panels (a) and (d) show the growth rates (deviated from their sample mean) and the low-frequency trend.
The other panels show the low-frequency trend and its decomposition into common and sector-specific components.
The red lines denote the posterior median and the shaded areas are (pointwise) equal-tail 68% credible intervals.
` , g z ),
is the fraction of the trend variability in each sector explained by the common trend factors, (gf,t
f,t

which is denoted by R`2 and Rz2 in the table. Finally, the table also reports the correlation between the
`
z ), in each sector and the correlation between the common
sector-specific labor and TFP trends, (gu,i,t
, gu,i,t
` , g z ).
trends (gf,t
f,t

Looking first at the median values of the factor loadings, Agriculture, FIRE (x-Housing), and PBS have
the largest factor loadings for labor, and Transportation and Warehousing, and Durable and Nondurable
selected 90 percent credible intervals, which in some cases are markedly wider. We remind the reader that these long-run
empirical results use only q = 8 independent observations on labor input and TFP for each of the 16 sectors.

Goods have the smallest. Utilities, Durable Goods and Construction have the largest loadings for TFP,
while FIRE (x-Housing) and Arts, Entertainment, and Food Services have the smallest. The 68 percent
credible intervals are relatively wide and give a quantitative sense of how information about the long-run
is limited in our sample: the average width is 1.3 for λ` and 1.9 for λz . That said, for the majority of
sectors, the posterior puts relatively little weight on negative values of the factor loadings. Detailed results
for alternative priors are available in Section 1 of the Technical Appendix.
The sectoral R2 values are typically low indicating that common trend factors play a relatively muted
role in explaining overall sectoral trends. For example, in Durable Goods, only 3 percent of the overall
trend variation in labor and TFP growth is explained by their respective common trend factors. Notable
exceptions for R`2 arise in several service sectors, for example in FIRE (x-Housing) where 76 percent of the
trend variation in labor is explained by the common trend factor. Interestingly, the posterior suggests that
the sector-specific trends in labor and TFP are generally negatively correlated, rather dramatically so for
Professional and Business Services.
The final row of the table shows the results for aggregate values of labor and TFP. By construction, the
` , g z ), are also negatively correlated.
share-weighted factor loadings sum to unity. The common trends, (gf,t
f,t

The R2 values are higher for the aggregates because aggregation reduces the importance of the sector-specific
trends. The point estimates suggest that roughly 2/3 of the variation in the trend growth rate of labor is
common across sectors while roughly 1/3 is common for TFP. However, one cannot directly infer from these
findings the role that common growth trends in labor and TFP play in the overall trend growth rate of
GDP. The reason is production linkages across sectors. In particular, the effective weight that each sector
has in the aggregate economy can differ considerably from its value added share in GDP. Thus, as we show
below, sectors such as Durable Goods, Construction, and Professional and Business Services, with extensive
linkages to other sectors as input suppliers, have an outsize influence on the aggregate trend.
Figure 5 shows a historical decomposition of the trends in aggregate labor and TFP growth rates arising
` , g z }16 . Panels (a) and (d) show
from the common factors, (gf` , gfz ), and sector specific components, {gu,i
u,i i=1

the (demeaned) values of the aggregate growth rates with the associated low frequency trend. The other
panels decompose the trend into its common (panels (b) and (e)) and sector-specific components (panels
(c) and (f)). This decomposition relies on standard signal extraction formulas to compute the posterior
distribution of (F, U) given X, and the figure includes 68% (pointwise) credible intervals for the resulting
common and sector-specific trends that incorporate uncertainty about the model’s parameter values. Figure
5, panel (b), suggests that much of the increase in the trend growth rate of aggregate labor in the 1960s
and 1970s, and subsequent decline in the 1980s and 1990s (both typically associated with demographics),
are captured by the model’s common factor in labor. Sector-specific labor factors, for the most part, played
a supporting role. In contrast, while the model’s aggregate common factor played a role in the decline of
trend TFP growth the 1970s, the low frequency variation in the series since then has been associated almost
exclusively with sector-specific sources.
Figures 6 and 7 present the trend growth rates for each of the sectors (shown previously in Figure 5)

Figure 6: Labor Trends and Sector-Specific Components
(percentage points at annual rate)
Agriculture
5

Mining

Utilities

Construction

5
0
0
-2
-5

-2

-5
1960 1980 2000 2020

1960 1980 2000 2020

Durable Goods
4
2

1960 1980 2000 2020

Nondurable Goods

Wholesale Trade

Retail Trade
2
1
0

0
-2
-2

-2
1960 1980 2000 2020

1960 1980 2000 2020

Trans. & Ware.

Information

1960 1980 2000 2020

FIRE (x-Housing)
2

-1
1960 1980 2000 2020

PBS

-2
-2
-2

-4
1960 1980 2000 2020

Educ. & Health

-2
1960 1980 2000 2020

1960 1980 2000 2020

Arts, Ent. & Food Svc.

Other Services (x-Gov.)

1
0

1960 1980 2000 2020

Housing

0
-1

-5

-2

-2
1960 1980 2000 2020

-10

-2

1960 1980 2000 2020

Notes: Each panel shows the low-frequency trend for sectoral growth rate (in blue) and its sector-specific component
(in red). The red lines denote the posterior median and the shaded areas are (pointwise) equal-tail 68% credible
intervals.
`
z ) components. Consistent with the R2 values shown in
along with the estimated sector-specific (gu,i,t
, gu,i,t

Table 3, much of the variation in the trend growth rates of sectoral TFP and labor is associated with sectorspecific factors, and this is particularly true for TFP. Notable in Figures 6 and 7 is the negative correlation
between the low frequency components of labor and TFP sectoral growth.14
14

One explanation for this negative correlation relies on complementarities in preferences (see Ngai and Pissarides (2007),
Herrendorf et al. (2013)). Technological progress in a sector leads to reduced spending on that sector’s consumption goods and,
by implication, reduced employment in that sector as well.

Figure 7: TFP Trends and Sector-Specific Components
(percentage points at annual rate)
Agriculture

Mining

Utilities

-5

-2

Construction
5

0
-10

-5

Durable Goods
5

-4
1960 1980 2000 2020

1960 1980 2000 2020

Nondurable Goods

-5
1960 1980 2000 2020

Wholesale Trade

0
0

Retail Trade

-2
-2
-4

-5

1960 1980 2000 2020

Trans. & Ware.
2

FIRE (x-Housing)

Educ. & Health

0
-1

-2

1960 1980 2000 2020

Arts, Ent. & Food Svc.

Other Services (x-Gov.)

-2

PBS
2

-2
1960 1980 2000 2020

1960 1980 2000 2020

Information

-2

1960 1980 2000 2020

Housing

-2
1960 1980 2000 2020

-1
1960 1980 2000 2020

1960 1980 2000 2020

Notes: See notes to Figure 6.

Sectoral Trends and the Aggregate Economy

Given the evolution of sectoral trend growth rates for labor and TFP over the past 70 years, this section
explores their implications for long-run GDP growth. The key consideration here is that production sectors
are linked because each sector uses capital goods and materials produced in other sectors. Therefore, we
consider a multi-sector growth model that features these interactions. Consistent with our TFP calculations
in Section 3, the model also features competitive product and input markets.
We consider a structural framework with preferences and technologies that are unit elastic so that
the economy evolves along a balanced growth path in the long run. Capital accumulation interacts with

production linkages to amplify the effects of sector-specific sources of growth. In particular, changes in the
growth rate of labor or TFP in one sector affect not only its own value added growth but also that of all
other sectors. We derive closed form expressions for the long-run multipliers summarizing the aggregate
growth implications of these network effects for each sector. The magnitude of the multiplier associated with
a given sector depends on its role and importance as a supplier of capital and materials to other sectors.
We first outline a general n-sector model that we use in our quantitative analysis. After introducing the
general model, we present several special cases using n = 2 sectors to highlight key mechanisms and their
relationship to previous work. We then return to the general n-sector model.

4.1

Economic Environment

Consider an economy with n distinct sectors of production indexed by j (or i). A representative household
derives utility from these n goods according to

∞
X

t=0

n
Y
cj,t θj
θj

j=1

n
X

θj = 1, θj ≥ 0,

j=1

where θj is the household’s expenditure share on final good j.
Each sector produces a quantity, yj,t , of good j at date t, using a value added aggregate, vj,t , and a
materials aggregate, mj,t , using the technology,

yj,t =

vj,t
γj

γj

mj,t
1 − γj

(1−γj )
, γj ∈ [0, 1].

(3)

The quantity of materials aggregate, mj,t , used in sector j is produced with the technology,
mj,t =

n
Y
mij,t φij
i=1

φij

n
X

φij = 1, φij ≥ 0,

(4)

i=1

where mij,t denotes materials purchased from sector i by sector j. The notion that every sector potentially
uses materials from every other sector introduces a first source of interconnectedness in the economy. An
input-output (IO) matrix is an n × n matrix Φ with typical element φij . The columns of Φ add up to the
degree of returns to scale in materials for each sector, in this case unity. The row sums of Φ summarize the
importance of each sector as a supplier of materials to all other sectors. Thus, the rows and columns of Φ
reflect “sell to” and “buy from” shares respectively for each sector.
The value added aggregate, vj,t , from sector j is produced using capital, kj,t , and labor, `j,t , according
to

vj,t = zj,t

kj,t
αj

αj

`j,t
1 − αj

1−αj
, αj ∈ [0, 1].

(5)

Capital accumulation in each sector follows
kj,t+1 = xj,t + (1 − δj )kj,t ,

(6)

where xj,t represents investment in new capital in sector j, and δj ∈ (0, 1) is the depreciation rate specific
to that sector. Investment in each sector j is produced using the quantity, xij,t , of sector i goods by way of
the technology,
xj,t =

n
Y
xij,t ωij
ωij

i=1

n
X

ωij = 1, ωij ≥ 0.

(7)

i=1

Thus, there exists a second source of interconnectedness in this economy in that new capital goods in
every sector are potentially produced using the output of other sectors. This additional source of dynamic
linkages in the economy, mostly absent from structural multisector studies, is shown to be a key propagation
mechanism over the business cycle in vom Lehn and Winberry (2022). Similarly to the IO matrix, a capital
flow matrix is an n × n matrix Ω with typical element ωij . The columns of Ω add up to the degree of returns
to scale in investment for each sector which here is unity. The row sums of Ω indicate the importance of
each sector as a supplier of new capital to all other sectors.
The resource constraint in each sector j is given by
cj,t +

n
X

mji,t +

i=1

n
X

xji,t = yj,t .

i=1

Sectoral change is defined by changes in the composite variable, Aj,t , that reflect the joint behavior of
both TFP and labor growth. In particular, under the maintained assumptions, sectoral value added may
be alternatively expressed as

vj,t = Aj,t

kj,t
αj

αj
,

where
∆ ln Aj,t = ∆ ln zj,t + (1 − αj )∆ ln `j,t .

(8)

In this paper, we condition on the observed joint behavior of TFP and labor growth rates, {∆ ln zj,t , ∆ ln `j,t },
in each sector j and derive their implications for aggregate value added or GDP growth. In particular, we
provide general growth accounting expressions that quantify the effects of changes in trend input growth in
a given sector in light of its production linkages to all other sectors.
While we condition on observed labor growth rates, the growth accounting expressions we derive are
largely unchanged in a model where the allocation of labor is endogenous. In particular, a conventional
treatment of labor supply produces a growth expression that is isomorphic to that presented below. In that
expression, the way in which capital accumulation and the network features of production determine the
influence of different sectors on aggregate growth is unchanged, as are the effects of long-run changes in
TFP growth on GDP growth. The key difference is that with endogenous labor supply, the common and
21

idiosyncratic components of labor input now carry a structural interpretation. Specifically, the common
component is associated with broad demographics such as population growth and how these demographics
affect labor input in each sector. The idiosyncratic component reflects sector-specific factors such as those
determining the disutility cost of working in different sectors, including a sector-specific Frisch elasticity, or
sector-specific labor quality adjustments.15
For ease of presentation, we use the following notation throughout the paper: we denote the matrix
summarizing value added shares in gross output in different sectors by Γd = diag{γj }, the IO matrix by
Φ = {φij }, the capital flow matrix by Ω = {ωij }, and the matrix summarizing capital shares in value added
in different sectors by αd = diag{αj }.

4.2

Balanced Growth and Sectoral Multipliers

We consider a balanced growth path (BGP) where the growth rates of TFP and labor in sector j are given
by gjz and gj` respectively. From equation (8), it follows that along that path,
∆ ln Aj,t = gja = gjz + (1 − αj ) gj` .
We now show that because of production linkages, sources of change in an individual sector, gja , help determine value added growth in every other sector along the balanced growth path. These linkages, therefore,
amplify the effects of sector-specific change on GDP growth, and this amplification can be summarized by
a multiplier for each sector. As we will see, these multipliers scale the influence of some sectors on GDP
growth by up to multiple times their share in the economy.
The sectoral multipliers are readily computed from the production linkages specified in the model. Along
the BGP, gross output in sector j, yj,t , and its uses, cj,t , mji,t and xji,t grow at the same sector-specific rate.
Thus, let gjy denote this common growth rate for yj , cj , mji , and xji . Let gjv , gjm , gjk and gjx denote the BGP
growth rates of sector j’s inputs vj , mj , kj and xj . Let g y = (g1y , ...gny )0 and define the n × 1 vectors g v , g m ,
P
etc., analogously. From (4), note that gjm = i φij giy (because mij grows at rate giy ), so that g m = Φ0 g y .
P
Similarly, from (6) and (7) gjk = gjx = i ωij giy (because xij grows at rate giy ), so that g k = Ω0 g y . Equation
(3) implies g y = Γd g v + (I − Γd )g m , with g v = g a + αd g k from (5). Collecting terms in g y then yields
g y = Γd g a + Γd αd Ω0 g y + (I − Γd )Φ0 g y , so that
g y = Ξ0 g a ,
15

(9)

See Section 4 of the Technical Appendix. The interpretation or identification of sources of labor growth will necessarily
depend on the particular model of endogenous labor supply under consideration. Because our focus is on growth accounting
(rather than counterfactuals), we take the observations on labor growth as given whatever their underlying forces. Ngai and
Pissarides (2007) explore an alternative framework where the reallocation of labor among consumption goods sectors is an
outcome of unbalanced growth among those goods while, at the same time, preserving balanced growth at the aggregate level.
Absent from their work, however, are the network considerations and the role of capital in determining network multipliers that
are central to this paper. An interesting avenue for future work, therefore, is the study of growth and structural change with
production networks.

where Ξ0 = [I − Γd αd Ω0 − (I − Γd )Φ0 ]−1 Γd is the generalized Leontief inverse.
Finally, with g v = g a + αd g k and g k = Ω0 g y , then

g v = I + αd Ω0 Ξ0 g a .

(10)

Observe that preference parameters are absent from equation (10) in that balanced growth relationships are
ultimately statements about technologies and resource constraints.
Equation (10) describes how the sources of growth in a given sector, gja , affects value added growth in all
other sectors, giv . This relationship involves the direct effects of sectors’ TFP and labor growth on their own
value added growth, Ig a , and the indirect effects that sectors have on other sectors through the economy’s
sectoral network of investment and materials, αd Ω0 Ξ0 g a . The general Leontief inverse, Ξ, is central and
summarizes the knock-on effects of sectoral changes through linkages in investment, captured in Ω, and
materials, captured in Φ.
Given the vector of sectoral value added growth rates, g v , the Divisia aggregate index of GDP growth
is g V = sv0 g v where sv = (sv1 , ..., svn ) is a vector of sectoral value added shares in GDP that are constant on
the BGP. Thus,
g V = sv0 [I + αd Ω0 Ξ0 ]g a ,

(11)

∂g V
= sv + ΞΩαd sv .
∂g a

(12)

so that, holding shares constant,

The first term in (12) shows the direct effect of g a on the growth rate of GDP and the second term captures
the network effects of g a on GDP growth induced by production linkages.16 Equation (12), therefore, defines
the vector of sectoral multipliers for each of the j sectors.
When no sector uses capital in production, αd = 0, the drivers of growth sector in j, gja , affect GDP
growth only through that sector’s share in the economy,

∂g V
∂gja

= svj . More generally, equations (10) and

(12) suggest the presence of a network multiplier effect that varies by sector and that depends not only on
the importance of sectoral interactions through the elements of Ξ, but also on the extent to which sectors
use capital produced by other sectors in their own production, that is the elements in Ω and α. From (9),
a change in input growth in sector j influences every other sector k through the network of production
linkages summarized by all non-zero jk elements (i.e., from j to k) of Ξ. Induced changes in all sectors k
in turn potentially affect investment in every other sector, i, through capital flows summarized by ωki in Ω,
(i.e, from k to i including back to j). The net effect on GDP growth is the sum of all these interactions.
Conveniently, the effects of sectoral changes, ∂g a , on GDP growth may be thought of as a direct effect, sv ,
16

In general, sectoral value added shares in GDP, sv , will also depend on the model’s underlying parameters including the
vector of sources of sectoral growth, g a . However, changes in sectoral shares induced by an exogenous change in a sector k,
∂sv
j
a , will be mostly inconsequential for overall growth, consistent with Figure 1 and the notion that since shares must sum to
∂gk
P ∂sv
1, j ∂gja = 0.
k

and an additional indirect effect resulting from sectoral linkages, ΞΩαd sv . Hence, we define the combined
direct and indirect effects of structural change on GDP growth in terms of the vector of sectoral multipliers,
sv + ΞΩαd sv .
To gain intuition, the next section discusses expressions (10) and (11) above in the context of special cases
exemplified in previous work. In particular, we provide examples of sectoral multipliers in Greenwood et al.
(1997) (henceforth GHK (1997)) and variations thereof. The Technical Appendix discusses each example
in detail as well as the case studied by Ngai and Pissarides (2007). These examples highlight the role of
capital accumulation in generating sectoral multipliers. They also underscore the fact that given a network
of intermediate goods, all sectors, even those that produce no capital goods, can have sectoral multipliers
well in excess of their share in GDP. This last feature of sectoral influence is absent in GHK (1997) which
abstracts from intermediate inputs.

4.3

Relationship to Greenwood, Hercowitz, Krusell (1997)

The one-sector environment featuring an aggregate production function in GHK (1997) also has an interpretation as a two-sector economy.17 Under that interpretation, one sector produces consumption goods (sector
1) and the other investment goods (sector 2), and each sector’s production function has the same capital
elasticity, α. For simplicity, we focus on the discussion in Section III of GHK (1997) which abstracts from
the distinction between equipment and structures. Thus, consider a two-sector economy with production
given by
α 1−α
ct = y1,t = z1,t k1,t
`1,t ,
α 1−α
xt = y2,t = z2,t k2,t
`2,t ,

kt+1 = xt + (1 − δ)kt ,
where factors are freely mobile, kt = k1,t +k2,t and `t = `1,t +`2,t , and the constant scale factors in production
(which simplify the algebra in the full model) have been dropped. Under the maintained assumptions, this
two-sector environment reduces to the one-sector framework with aggregate production described in GHK
(1997). That is, there exists a one-sector interpretation of the two-sector economy with associated resource
constraint,
ct + qt xt = z1,t ktα `t1−α ,
where qt =

z1,t
z2,t

(13)

is the relative price of investment goods, and aggregate output (in units of consumption

goods), yt = ct + qt xt , is a function of total factor endowment only, z1,t ktα `1−α
. To the extent that technical
t
progress in the investment sector, z2,t , is generally more pronounced than in the consumption sector, z1,t ,
the relative price of investment goods will decline over time as emphasized by GHK (1997).
Along the balanced growth path, all variables grow at constant but potentially different rates. Because
GHK (1997) do not consider materials, there is no distinction between gross output and value added. From
17

See Greenwood et al. (1997), Section V. A.

the market clearing conditions and the form of production technologies, it follows that sectoral output
growth rates, gjy = gjv , are given by (in terms of the notation introduced above):
gjv = gjz + (1 − α)g ` + αg k = gja + αg k , j = 1, 2.

(14)

Equation (14) makes clear that any amplification of sectoral sources of growth, gja , can only take place
through capital accumulation. In this case, it follows from the capital accumulation equation that along the
BGP, capital grows at the same rate as investment which, in sector 2 (the capital goods producing sector),
is also that of output. Thus, we have that
g2v = g k =

1
α a
g a and g1v = g1a +
g .
1−α 2
1−α 2

(15)

Note that the assumption of factor mobility across sectors has only minor implications for the characterization of the BGP. First, even with sector-specific investment, the resource constraint for investment implies
that investment and capital grow at the same rate in each sector. Second, with sector-specific labor, the
expression for output growth remains as in equation (14) with the only difference being that sector-specific
labor growth rates, gj` , now replace the aggregate labor growth rate, g ` , so that gja = gjz + (1 − α)gj` .
Aggregate GDP growth is defined as the Divisia index of sectoral value-added growth rates weighted by
their respected value added shares. Thus, from equation (15), aggregate GDP growth is
V

g =

sv1

g1a +

α a
g
1−α 2

or alternatively,
g V = sv1 g1a + sv2 g2a +

+ sv2

1
ga,
1−α 2

(16)

α a
g .
1−α 2

(17)

In this economy, sector 2 is the sole producer of capital for both sectors 1 and 2 and has both a direct and
indirect effect on the aggregate economy. The indirect effect stems from the fact that capital accumulation
amplifies the role of sectoral sources of growth. In equation (16), sector 2 contributes

α
a
1−α g2

> 0 to value

added growth in sector 1 and scales its contributions from TFP and labor to its own value added growth
by

1
1−α

> 1. Thus, in equation (17), the direct aggregate effect of an expansion in sector 2 by way of TFP

or labor growth is its share, sv2 , while its indirect aggregate effect is
sectoral multiplier,

∂g V /∂g2a ,

sv2

α
1−α .

α
1−α

> 0. It follows that sector 2’s

In contrast, because sector 1 produces goods that are only fit for

consumption, it only has a direct effect on the aggregate economy. Its sectoral multiplier, ∂g V /∂g1a , is then
simply its share in GDP, sv1 .
A straightforward application of the general framework laid out in the previous section produces the
same balanced growth path and sectoral multipliers for sectors 1 and 2 that we have just discussed. In
particular, the GHK (1997) economy is a special case with n = 2 and, since sector 2 is the only sector
producing investment goods, ω1j = 0 and ω2j = 1 for j = 1, 2. In addition, each good is produced without
intermediate inputs, γj = 1, j = 1, 2, and the sectors use the same production functions, αj = α, j = 1, 2.
25

These yield the matrices

Ω=

0 0
1 1



!
, Γd = I, αd = αI, and Ξ = (I − Ωαd )−1

1
= α
1−α


0
1 .
1−α

The associated sectoral multipliers are given by the elements of (I + ΞΩαd )sv ,
∂g V
α
∂g V
= sv1 , and
= sv2 +
.
a
∂g1
∂g2a
1−α
Actual production linkages are generally more involved than those just discussed. Importantly, even in
the context of two sectors and no materials, the simple fact that factor income shares differ across sectors
prohibits a one-sector interpretation of the economic environment with an aggregate production function. In
this case, the amplification of sources of sectoral growth on GDP growth now depends on a value-added-share
weighted average of capital elasticities.18

4.4

Strictly Positive Multipliers for Sectors That Produce No Capital

Moving beyond GHK (1997), sectoral linkages also reflect a network of materials production. Thus, we now
introduce intermediate goods into the GHK (1997) environment. Crucially, when the consumption sector
(sector 1) also produces materials for the investment goods sector (sector 2), the growth rate of capital
depends on both sectors 1 and 2. Therefore, both sectors 1 and 2 now have indirect effects on long-run GDP
growth over and above their share in the economy.
We illustrate these points via a simple network of intermediate goods. Here, sector 1 produces not only
consumption goods but also materials, m1,t , used by sector 2. Similarly, sector 2 still produces capital
goods for both sectors but also materials, m2,t , used by sector 1. Since sector 1 now produces consumption
goods and intermediate goods, we refer to sector 1 as the Nondurables sector. Thus, in terms of our general
notation, we have that γj 6= 1, ω1j = 0 and ω2j = 1 for j = 1, 2. Moreover, the relevant resource constraints
in sectors 1 and 2 are now

and

h
iγ
α1 1−α1 1
1
ct + m1,t = y1,t = z1,t k1,t
`1,t
m1−γ
2,t ,
h
iγ
α2 1−α2 2
2
xt + m2,t = y2,t = z2,t k2,t
`2,t
m1−γ
1,t ,

while the rest of the production side of the economy is as in the previous examples.
In this case, a calculation shows that the BGP growth rate of capital is given by
gk =
18

(1 − γ2 )γ1 g1a + γ2 g2a
,
∆

See Section 3 of the Technical Appendixfor exact expressions.

(18)

where ∆ = 1 − γ2 α2 − (1 − γ2 ) [γ1 α1 + (1 − γ1 )] . The growth rate of GDP is then
g V = sv1 g1a + sv2 g2a + (sv1 α1 + sv2 α2 )g k .

(19)

Two important observations emerge relative to the previous examples. First, because the Nondurable
Goods sector now produces intermediate inputs for the investment sector, the growth rate of capital goods in
equation (18) reflects sources of growth in both sectors, g1a and g2a . Hence, unlike in the previous section, both
k

sectors 1 and 2 in equation (19) have an additional indirect effect on long-run GDP growth, (sv1 α1 +sv2 α2 ) ∂g
∂g a
1

v
v
and (sv1 α1 + sv2 α2 ) ∂g
∂g a respectively, over and above their shares in the economy, s1 and s2 . Second, from
2

equation (19), the indirect effect from sector 2 on GDP growth dominates that from sector 1 if and only if
its contributions to overall capital growth,
1,

∂g k
∂g1a .

∂g k
∂g2a ,

are larger than the corresponding contributions from sector

From equation (18), this condition holds if and only if
γ2 > (1 − γ2 )γ1 .

This will not be true, for example, in economies where the value added share in gross output of the capital sector, γ2 , is relatively small. In that case, the main input into the production of capital goods are
intermediate inputs from the non-durables sector. That sector, therefore, ends up having more influence on
aggregate growth.
Substituting (18) into (19) yields the sectoral multipliers:19
∂g V
∂g V
(sv1 α1 γ1 (1 − γ2 ) + sv2 α2 γ1 (1 − γ2 ))
sv1 α1 γ2 + sv2 α2 γ2
v
v
and
.
=
s
+
=
s
+
1
2
∂g1a
∆
∂g2a
∆
Generally, the main lesson from these examples is that network production linkages and capital accumulation are the key components that lead to sectoral multipliers along the balanced growth path. Furthermore,
the implied amplification of idiosyncratic sources of growth on GDP growth can arise in any sector, including
those producing only Nondurable Goods.
Finally, we note a caveat to our results: they all pertain to a closed economy. Cavallo and Landry
(2010) argue that imports are also a source of equipment capital accumulation, and more generally Basu
et al. (2013) argue for including trade when studying investment-specific technical change with production
networks. In Section 3.5 of the Technical Appendix, we introduce traded investment goods into the GHK
(1997) framework along the lines of Basu et al. (2013). In that case, the amplification effects of production
networks and capital accumulation also reflect variations in the terms of trade, scaled by the share of foreign
investment goods in total investment. In this example, the quantitative implications of traded capital goods
19

Equivalently, these multipliers can be computed from the general formula (12) using the matrices for this model:

0 0
α1 0
γ1 0
0 1
Ω=
, αd =
, Γd =
, and Φ =
.
1 1
0 α2
0 γ2
1 0

Figure 8: Investment Network

Agriculture
Housing
Educ. & Health

Construction

Durable Goods
FIRE (x-Housing)

Mining
Retail Trade
Wholesale Trade

PBS

Trans. & Ware.
Other Services (x-Gov)

Nondurable Goods
Information

Arts, Ent., & Food Svc.
Utilities

Notes: This figure shows the investment network as a graph, where capital flows are represented
by edges between nodes representing sectors. A sector with a larger node indicates that other
sectors spend a larger share of their capital expenditures on average from that sector. A wider
edge between two nodes reflects larger bi-directional capital flows relative to all other capital
flows. See the capital flow table in Section 8 of the Technical Appendix.

remain limited though they have increased over time.

Quantitative Findings

This section puts together the empirical findings from Section 3 and model insights from Section 4. It shows
that sector-specific trends have played a dominant role in driving the trend rate of growth in GDP over the
postwar period. We estimate that this aggregate trend rate of growth has fallen by almost 3 percentage
points between 1950 and today.

5.1

Model Parameters

We first outline the construction of model parameters, a procedure that follows mostly Foerster et al. (2011)
and is governed by the BEA input-output and capital flow accounts.20
In our benchmark economy, value-added shares in gross output {γj }, capital shares in value added,
{αj }, and material bundle shares, {φij }, are obtained from the 2015 BEA make and use tables. The make
table tracks the value of production of commodities by sector, while the use table measures the value
of commodities used by each sector. We combine the make and use tables to yield, for each sector, a
table whose rows show the value of a sector’s production going to other sectors (materials) and households
(consumption), and whose columns show payments to other sectors (materials) as well as labor and capital.
Thus, a column sum represents total payments from a given sector to all other sectors, while a row sum
gives the importance of a sector as a supplier to other sectors. We then calculate material bundle shares,
{φij }, which constitute the IO matrix, as the fraction of all material payments from sector j that goes to
sector i. Similarly, value-added shares in gross output, {γj }, are calculated as payments to capital and labor
as a fraction of total expenditures by sector j, while capital shares in value added, {αj }, are payments to
capital as a fraction of total payments to labor and capital.
The parameters that determine the production of investment goods, {ωij }, are chosen similarly in accordance with the BEA capital flow table from 1997, the most recent year in which this flow table is available.
The capital flow table shows the flow of new investment in equipment, software, and structures towards
sectors that purchase or lease it. By matching commodity codes to sectors, we obtain a table that has
entries showing the value of investment purchased by each sector from every other sector. A column sum
represents total payments from a given sector for investment goods to all other sectors, while a row sum
shows the importance of a sector as a supplier of investment goods to other sectors. Hence, the investment
bundle shares, {ωij }, which constitute the capital flow matrix are estimated as the fraction of payments for
investment goods from sector j to sector i, expressed as a fraction of total investment expenditures made
by sector j.
Conditional on these parameters, equation (10) gives sectoral value added growth along the balanced
growth path. In constructing aggregate GDP growth from these sectoral value added growth rates, we rely
on the full sample mean value added shares from the KLEMS data that were used in our empirical analysis.
Recall also that in Figure 1, we explored using different definitions of value added shares in calculating GDP
growth. While this did not lead to meaningful differences in aggregate growth, to the extent that these
shares are changing over time as do input-output relationships, the model might nevertheless yield more
material differences in the implied sectoral multipliers. Thus, in the Technical Appendix, we show that our
benchmark sectoral multipliers are robust to versions of the model informed by mean value added shares
for the first and last 15 years, and the 1960 and 1997 make and use tables.
20
Section 8 of the Technical Appendix contains a more detailed description of the procedure, and for the 16 sectors considered
in this paper it displays the capital flow matrix, Table A10, the IO matrix, Table A11, and the associated generalized Leontief
inverse, Table A12.

5.2

Production Linkages in the U.S. Economy

The production of investment goods in the U.S. turns out to be concentrated in relatively few sectors. Construction and Durable Goods produce close to 80 percent of the capital in almost every sector. Put another
way, as shown in Figure 8, we can think of the Construction and Durable Goods sectors as investment
hubs in the production network. Construction comprises residential and non-residential structures including infrastructure. The bulk of capital produced by the Durable Goods sector resides in motor vehicles,
machinery, and computer and electronic products. Other sectors recorded as producing capital goods for
the U.S. economy include Wholesale Trade, Retail Trade, and Professional and Business Services. In the
Professional and Business Services sector, the notion of capital produced for other sectors is overwhelmingly
composed of computer system designs and related services. The BEA distinguishes between materials and
capital goods by estimating the service life of different commodities and, consistent with the annual time
period used in this paper, commodities expected to be used in production within the year are defined as
materials. As a practical matter, however, the distinction between materials and investment goods is not
always straightforward. We address measurement issues separately in Section 6.21
Compared to the capital flow matrix, the production of intermediate goods is somewhat less concentrated
as all sectors produce materials for all other sectors to varying degrees. However, from the IO matrix,
Professional and Business Services and FIRE stand out as suppliers of materials. These two sectors are the
largest suppliers of intermediate goods in the U.S. production network, making up roughly 20 percent of
materials expenditures across sectors. A key difference, however, is that intermediate inputs produced by
FIRE are used extensively in Housing which is consumed mostly as a final good. In contrast, Professional and
Business Services is the largest supplier of intermediate goods to Durable Goods (other than those Durable
Goods purchases from itself). After FIRE and Professional and Business Services, the next largest suppliers
of materials are Durable Goods and Nondurable Goods which make up around 11 percent of materials
expenditures across sectors on average, or 1/2 of those spent on Professional and Business Services.
In contrast to the sectors that play a key role in the U.S. production network, output produced in sectors
such as Agriculture, Forestry, Fishing, Housing, and Hunting, Entertainment and Food Services is mostly
consumed as final goods.
21

Major revisions of the National Income Accounts (NIAs) broadened the concept of capital by including expenditures on
software in 1999, and expenditures on R&D and entertainment, literary, and artistic originals in 2013. These investments now
come under the heading of intellectual property products (IPPs). While the KLEMS data include a broad measure of IPP
capital, the 1997 capital flow tables include software investment but not all IPP investment categories. The share of missing
investment is less than 10 percent in 1960 and about 15 percent today. As also noted in Foerster et al. (2011), capital flow tables
do not account for an industry’s purchases of used capital goods and likely miss a portion of maintenance and repair using
within sector resources. Results presented here are robust to adjustments that assume up to an additional 25 percent of capital
expenditures within sectors. Finally, we abstract from land and inventories in the production structure. Including inventories
and land is conceptually straightforward but data quality at the sector level remain an issue. Fernald (2014) includes land and
inventories into a GHK (1997) growth accounting framework and finds only a marginal effect.

Table 4: Sectoral Multipliers

Sector
Agriculture
Mining
Utilities
Construction
Durable Goods
Nondurable Goods
Wholesale Trade
Retail Trade
Trans. & Ware.
Information
FIRE (x-Housing)
PBS
Educ. & Health
Arts, Ent., & Food Svc.
Other Services (x-Gov)
Housing

ΞΩαd sv

(I + ΞΩαd )sv

0.03
0.02
0.02
0.05
0.13
0.09
0.07
0.08
0.04
0.05
0.10
0.09
0.06
0.04
0.03
0.09

0.01
0.03
0.01
0.12
0.28
0.03
0.08
0.02
0.03
0.03
0.03
0.16
0.00
0.01
0.01
0.00

0.03
0.05
0.03
0.17
0.42
0.13
0.15
0.11
0.07
0.08
0.14
0.25
0.06
0.04
0.04
0.09

Notes: This table decomposes each sector’s total multiplier (column 3) into a direct effect
(column 1) and an indirect effect (column 2). The sums do not necessarily add up because of
rounding.

5.3

Sectoral Multipliers

Table 4 shows the direct and combined effects of sectoral sources of growth on GDP growth. The importance
of Durable Goods, Professional and Business Services, and Construction, means not only that their value
added share in GDP is large, 13, 9 and 5 percent respectively, but also that they have large spillover effects
on other sectors. In particular, Durable Goods and Professional and Business Services have the two largest
sectoral multipliers, 0.42 and 0.25 respectively, while Construction’s multiplier exceeds three times its value
added share in GDP at 0.17 given its central role in the investment network. Considering that trend TFP
growth in Construction fell by almost 5 percentage points between 1950 and 1980 in Figure 4, this gives us,
all else equal, a roughly 0.85 percentage point decline in trend GDP growth from that sector alone during
that period. Similarly, the over 6 percentage point collapse in the trend growth rate of TFP in Durable
Goods since 2000 would have on its own contributed roughly a 2.5 percentage point decline in trend GDP
growth.
It is also apparent from Table 4 that the effects of sectoral change on GDP growth are always at
least as large as sectors’ value added shares in GDP. Sectoral network multipliers almost triple the share of
Professional and Business Services, from 0.09 to 0.25, and doubles that of Wholesale Trade, from 0.07 to 0.15.
In other sectors, such as Agriculture, Forestry, Fishing and Hunting, or Housing, the network multipliers are
31

Figure 9: Trend Growth Rate in GDP: Data and Model
(percentage points at annual rate)
6

1
Data
Trend (data)
Trend (model, total)
Trend (model, direct effect)

-1
1950

1960

1970

1980

1990

2000

2010

2020

Notes: The figure shows the cyclically adjusted GDP growth rate (thin black line) along with its low-frequency trend
(thick black line). Also shown are the model-implied trend using the low-frequency trends of labor and TFP growth
(solid blue line), and the trend implied by only the direct effects of labor and TFP based solely on value-added shares
(dashed blue line).

smaller since these sectors produce mainly final consumption goods. Because the same network relationships
embodied in the capital flow matrix, Ω, and the IO matrix, Φ, determine the importance that sectors have
in the economy both as a share of value added and through their spillover effects, sectors with relatively
larger shares in GDP will also tend to be associated with large network multipliers.
A key implication of Table 4 is that the effects of sectoral change on GDP growth arise in part through
a composition effect. Therefore, secular changes in GDP growth can take place without observable changes
u,z
in aggregate TFP growth. For example, consider purely idiosyncratic changes in TFP growth, ∂gu,j
, that
Pn
v
z
leave aggregate TFP growth unchanged, j=1 sj ∂gu,j = 0. In other words, the direct effect of sectoral TFP

growth in this case is zero. Despite aggregate TFP growth not changing, these idiosyncratic changes will
nevertheless have an (indirect) effect on GDP growth since the sum of sectoral multipliers is larger than 1.

5.4

Historical Decomposition of the Trend Growth Rate of GDP

The various sectoral multiplier calculations we have just carried out depend on the balanced growth equations
(10) and (11). These equations hold only in steady state and ignore endogenous transitional dynamics
that are potentially important in explaining variations over the business cycle. However, because our
empirical focus is on variations in growth rates with periodicities longer than 17 years, we abstract from
these transitional dynamics and apply the formulas (10) and (11) directly to the trend growth rates of TFP
z and g ` , as an approximation.22 In addition, we then explore how our
and labor extracted in Section 3, gi,t
i,t
z , λ` g ` ), and sector-specific, (g z , g `
estimates of common, (λzi gi,t
i i,t
u,i,t u,i,t ), trend input growth have historically

contributed to the trend growth rates of sectoral value added and GDP. Thus, we compute the trend growth
rates of sectoral value added as,

z
z
`
`
+ gu,t
+ (I − αd ) λ` gf,t
+ gu,t
,
gtv = I + αd Ω0 Ξ0 λz gf,t
v , ..., g v ), and (λz , λ` ) are vectors containing the factor loadings from (1). GDP trend growth
where gtv = (g1,t
n,t

is then
gtV = sv0 gtv .
Figure 9 depicts the annual growth rate of GDP and its trend in black (previously shown in Figure 2)
together with the corresponding trend growth rate computed from the balanced growth multipliers (in solid
blue) and its contribution from the direct effect using sectors’ value added shares only (in dashed blue),

sv0 I gtz + (I − αd )gt` . In all, trend GDP growth fell by nearly 3 percentage point over the postwar period.
Importantly, the sizable gap between the trend with direct effects only and the full model trend implies that
the indirect effects stemming from network production linkages constitute a significant component of trend
GDP growth. There is a notable discrepancy between model and data in the 1970s, when the balanced
growth multipliers suggest a larger decline in trend GDP growth rates than in the data. In that period,
periodicities longer that 17 years may not be adequate to capture the required adjustment to capital implied
by the model.
Figure 10 decomposes the trend growth rate of GDP implied by the model into its components derived
from common factors and sector-specific factors. The model indicates that sector-specific or unique factors
in trend labor and TFP growth (panel (b)) have historically accounted for roughly 3/4 of the long-run
changes in GDP growth. Conversely, only about 1/4 of the variation in trend GDP growth since 1950 has
come from common sources of input growth (panel (a)). This is despite common factors explaining roughly
2/3 of the variation in the trend growth rate of aggregate labor noted in Section 3. To understand this
finding, recall that some sectors that have large sectoral multipliers such as Durable Goods or Construction
22

Our results then reflect the long-run amplifying effects of production linkages by way of capital accumulation. Despite
abstracting from the transition dynamics, Section 9 of the Technical Appendix shows that the model’s implied trend sectoral
capital growth rates match their counterparts in the data well, with the exception of Mining which carries a small sectoral
multiplier.

Figure 10: Decomposition of the Trend Growth Rate in GDP
(percentage points at annual rate)
(a) Common Factor

(b) Sector-Specific

2
1

1
1.5

-1

-2

-3

0.5

-3
1960

1980

2000

2020

0
1960

1980

2000

2020

0.5

Notes: Panels (a) and (b) show the (demeaned) model-implied trend GDP growth (black line), and its decomposition into changes due to the common factor and sector-specific factors (red lines). The overall trend in black
is sv0 [I + αd Ω0 Ξ0 ] (gtz + (I − αd )gt` ). The posterior median estimates of common and sector specific components,

z
`
along with their 68 percent credible intervals in red, are respectively sv0 [I + αd Ω0 Ξ0 ] λz gf,t
+ (I − αd )λ` gf,t
and
v0
0 0
z
`
s [I + αd Ω Ξ ] (gu,t + (I − αd )gu,t ). Panel (c) shows the posterior distribution for the fraction of the variance in
trend GDP growth attributed to the common factor.

(Table 3) also have large variations in trend input growth that are almost entirely driven by idiosyncratic
factors (Figures 6 and 7).
Panel (c) plots the posterior density for Rf2 , the fraction of the variance in trend GDP growth explained
by common sources. The median of the posterior for Rf2 is 0.26, the mode is less than 0.20, and 70 percent
of the posterior mass is associated with values of Rf2 that are less than 0.40. Thus, these results suggest
that most of the long-run evolution of GDP growth has historically stemmed from sector-specific factors.
The results reported thus far use the benchmark priors. Recall from Section 3 that these priors were
relatively uninformative except for the factor loadings. In particular, the prior for λ` was λ` ∼ N (1, P` ),
where P` = η 2 (I16 − s` (s0` s` )−1 s0` ) and an analogous prior was used for λz . These priors enforced the
normalization that s0` λ` = s0v λz = 1, so that unit changes in ft` and ftz lead to unit changes in the long-run
growth rate of aggregate labor and TFP. The parameter η then governed how aggressively the estimates
of λ`i or λzi are shrunk toward their mean of unity. The benchmark results use η = 1. Smaller values of η
shrink the estimates closer to 1 while larger values of η allows them to deviate from 1 more than the baseline
model. Thus, we now explore the robustness of our findings to alternative priors, η = 1/2 and η = 2. In
addition, we also run the model using q = 6 which captures long-run variations with periodicities longer
34

Figure 11: Robustness to Changes in Statistical Model
(a) Model: q = 8, η = 1.0, Rf2 = 0.28 (0.12, 0.52)
Common
2

-2

-2
1960

1980

2000

Density, Rf2 (Common)

Sector-Specific
2

1960

2020

1980

2000

2020

0
0

0.5

(b) Model: q = 8, η = 0.5, Rf2 = 0.26 (0.12, 0.46)
Common

Density, Rf2 (Common)

Sector-Specific
2

2
0

1
-2

-2
1960

1980

2000

1960

2020

1980

2000

2020

0
0

0.5

Sector-Specific

Common
2

-2

3
2
1

1960

1980

2000

2020

1960

1980

2000

2020

0
0

0.5

(d) Model: q = 6, η = 1.0, Rf2 = 0.37 (0.19, 0.60)
Common

Density, Rf2 (Common)

Sector-Specific

1
-2
-2
1960

1980

2000

2020

-4
1960

1980

2000

2020

0
0

0.5

Notes: See notes for Figure 10. Each vertical panel of the Figure shows results for a different specification of the
prior distribution, η, or long-run periodicities, q.

than 2 × 69/6 = 23 years.
Figure 11 summarizes the findings from these robustness exercises by reproducing Figure 10 for each of
these alternative models It is clear from the figure that across all cases, contributions from common sources
of trend input growth to the long-run evolution of GDP growth remain limited. Median estimates of Rf2

range from 0.28 to 0.37 with posterior distributions that place the bulk of their mass between 0 and 0.5.
We thus conclude that the result that sector-specific forces are the primary driver of trend GDP growth are
robust to changes in the priors for the factor loadings and to increasing the periodicity used that defines
long-run trends.
Given that sector-specific (rather than common) trends have played a dominant role in driving trend
GDP growth over the postwar period, Figure 12 gives the historical trend contributions to aggregate GDP
growth from the sector-specific components for each sector. Two sectors stand out, Construction and Durable
Goods. Recall that U.S. trend GDP growth fell by approximately 3 percentage points between 1950 and
2018. Comparing the beginning and the end of the sample, Figure 12 indicates that Durable Goods alone
contributed around 1 percentage point of that decline and Construction 0.75 percentage points. However,
there are also important differences in the timing and variation of those sectoral contributions. Construction
contributed roughly a 1 percentage point decline in trend GDP growth between 1950 and 1980 and was
essentially flat thereafter. In contrast, Durable Goods played a key role in raising trend GDP growth
in the 1980’s and 1990’s before contributing an almost 2 percentage point decline in trend GDP growth
after 2000. Nondurable goods also notably contributed to the post-war decline in trend GDP growth at
roughly 0.5 percentage points over the entire sample period, though offset somewhat by Mining after 1980.
Strikingly, many other sectors show relatively flat contributions to aggregate trend growth over 1950 to
2018, between −0.1 and 0.1 percentage points. Perhaps even more surprising, no sector has contributed any
steady significant increase to the trend growth rate of GDP over that period.

Discussion and Implications for Future Research

The findings we have just described result from two key notions explored above. One is largely empirical and
relates to the size of variations in trend TFP and labor growth in each sector. The other is more theoretical
and relates to the size of a sector’s multiplier given its place in the production network. The paper then brings
together two related, though so far mostly distinct, literatures. One addresses investment-specific technical
change, explored by Greenwood et al. (1997) and others, and the other studies the effects of production
networks, underscored for example by Acemoglu et al. (2012) and Baqaee and Farhi (2019). We extend
Greenwood et al. (1997) by considering the full set of materials and investment linkages that characterize U.S.
sectoral production. At the same time, we introduce their emphasis on dynamics and capital accumulation
into the static production network environments of Acemoglu et al. (2012). The analysis then provides
new results and more general insights into each of these literatures separately. Importantly, these results
combined with our empirical trend analysis provide the basis for a more complete and accurate picture of
the drivers of the secular decline in GDP growth.

Figure 12: Sector-Specific Contributions to the Trend Growth Rate of GDP
(percentage points at annual rate)
Agriculture

Mining

Utilities

0.1

0.5

0.1

Construction
1
0.5
0

-0.1

-0.5
1960 1980 2000 2020

Durable Goods

-0.5

1960 1980 2000 2020

Nondurable Goods

Wholesale Trade

0.5

2
1

0
0
-1

0.2

-0.2

-0.2
-0.4
1960 1980 2000 2020

1960 1980 2000 2020

Trans. & Ware.

Information

1960 1980 2000 2020

FIRE (x-Housing)

0.2

PBS
0.2
0
-0.2

-0.2

Retail Trade

0.2

-0.4

-0.5
1960 1980 2000 2020

1960 1980 2000 2020

Arts, Ent. & Food Svc.

Other Services (x-Gov.)

0.1

1960 1980 2000 2020

Educ. & Health

-0.1
1960 1980 2000 2020

-0.1

-0.1
1960 1980 2000 2020

Housing

-0.1
1960 1980 2000 2020

1960 1980 2000 2020

Notes: Each panel shows the implications of sector-specific trends for the trend growth rate of GDP using the modelbased multipliers. The solid lines denote the posterior median and the shaded areas are (pointwise) equal-tail 68%
credible intervals.

6.1

Production Networks and Investment-Specific Technical Change

The literature on investment-specific technical change (ISTC) has relied for the most part on a particular
set of key simplifying assumptions. One is a direct relationship between the relative price of investment
goods and the relative productivity of the capital producing sector. Another is that there is no distinction
between producer prices and final demand prices. Finally, abstracting from sectoral linkages in materials
means that only the capital producing sector can have a sectoral multiplier that exceeds its value added
share. In practice, however, as illustrated in the two-sector example of Section 4, even sectors that produce

only services will have a multiplier effect when those services are used by capital-producing sectors.
Sectoral linkages break down the one-to-one relationship between relative productivity and relative price.
In Greenwood et al. (1997), this would mean that investment-specific productivity, 1/qt , is no longer the
relative price of investment goods, qt , as in equation (13) above.23 In Section 2 of the Technical Appendix, we
show a more general mapping where productivity growth in any one sector potentially contributes to changes
in producer prices in all other sectors. Analogous to equation (10), this more complex mapping reflects the
influence of production linkages, Ω and Φ, through the general Leontief inverse, Ξ0 . The Technical Appendix
then further shows that the model’s quantitative implications for producer prices generally matches well
their data counterparts across sectors.24 That said, as in most previous work, we continue to abstract from
the distinction between producer prices and final demand prices. To make that distinction, including for
investment prices, one needs to model the allocation of the cost components from intermediation industries,
which include Retail Trade, Wholesale Trade, and Transportation, to final goods. For now, we leave exploring
these relationships to future work.25
With only two sectors and no intermediate inputs, the last key limitation of Greenwood et al. (1997)
is that the effects of TFP increases in a sector that produces mainly intermediate goods or services cannot
be easily traced. In particular, these would show up partly as a decline in the relative price of investment
goods when those services are purchased by capital-producing sectors. Therefore, without a more structural
description of the sectoral production network, any effects on growth risk being attributed to capital sectors
rather than the original service sector. For example, while Professional and Business Services (PBS) play a
notably less prominent role in the investment network than does Construction in Figure 8, Table 4 shows
that PBS nevertheless has a larger overall sectoral multiplier than Construction, 0.25 versus 0.17. Moreover,
while the ratio of Construction’s sectoral multiplier to its value added share exceeds that of PBS, both are
around three. The reason is that while PBS’ role in the investment network is small, it is a key supplier of
materials including to Durable Goods. Figure 12 shows that the contributions from PBS to trend variations
in GDP growth are smaller than those from Construction. However, this finding arises not because PBS is
less influential in the overall production network but because variations in trend TFP growth in PBS have
been historically less important (recall Figure 4). Evidently, as an empirical matter, this can change going
forward in a way that could not be captured in a starker model.
23

It should be noted though, that even in the absence of sectoral linkages, unequal capital income shares across sectors alone
introduce a wedge between relative prices and relative productivities. See Hornstein and Krusell (1996).
y
24
Let py denote the n × 1 vector of prices for sectoral output and g p the associated long-run growth rate. With multiple
y
sectors and the full set of production linkages, g p = (1Θ − I) Ξ0 g a . See Section 9 of the Technical Appendix for a comparison of
the model’s implied sectoral trend growth rates in producer prices and the data. Closer to the ISTC literature, Basu et al. (2013)
in a similar framework, but imposing that capital and labor are homogeneous and mobile across sectors, obtain an analogous
relationship that relates the relative price of capital to a weighted average of all sectoral productivities off the balanced growth
path.
25
The construction of input-output tables and gross product by industry separates intermediation industries’ contributions
to final demand and their contributions to the direct provision of goods. Therefore, commodity transactions are valued at
producers’ prices that exclude final purchasers’ payments for trade services and transportation costs to obtain the commodities.
A useful survey on the treatment of intermediation industries is Yuskavage (2007).

6.2

Measurement

While explicitly modeling the production network helps address shortcomings implied by a starker sectoral
setup, one challenge with more detailed multisector models is that output is more easily measured in some
sectors, for example Durable Goods, than others, such as PBS. In the case of PBS, this matters for at least
two reasons. One is that PBS is a large supplier of intermediate inputs to other sectors. The other is that,
after the Information sector, PBS is the second largest producer of Intellectual Property Products (IPPs).
Measurement error in PBS then potentially arises in mainly two ways. First, service price deflators that
account for quality changes in IPP industries are notoriously difficult to obtain. It is possible, therefore, that
our benchmark results incorrectly attribute sources of productivity growth across sectors by understating
output in PBS. Second, the distinction between materials and investment goods is sometimes ambiguous,
and goods can be misclassified. Over time the BEA has in several instances come to recognize expenditures
on goods as investment rather than payments for intermediate inputs. This is the case for example in the
comprehensive revisions to the National Income Accounts (NIAs) in 2013 regarding expenditures on R&D
and Entertainment Originals. Our analysis relies on capital flow tables from 1997 to determine the sources
of investment goods in different sectors. While these tables already include software as an investment good,
they do not line up exactly with the broader IPP definition used in the construction of capital stocks
in KLEMS. Our capital requirements matrix, Ω, therefore, likely does not capture all of the investment
contributions from PBS, in particular those of its IPP industries.
In the environment we study, the mismeasurement of output growth in one sector, say PBS, will generally
affect measured TFP in other sectors. Thus, suppose that sectoral output growth, g y , is measured with
some error, , so that measured sectoral output growth is g y,m = g y + . Then, given production linkages,
measured sectoral TFP, g z,m , is given by,
g z,m = g z + Ξ0

−1

Therefore, measurement error in any sector’s output growth rate, , is generally reflected in all sectors’
measured TFP growth, g z,m , through Ξ0 .26 In particular, to the degree that output growth in PBS is
under measured (j < 0 in PBS), so is its TFP growth rate while TFP growth in other sectors tends to
be overstated (because the off-diagonal elements of Ξ0 −1 are generally negative). This last expression then
allows us to carry out counterfactuals exploring the implications of measurement error in sectoral gross
output growth. In particular, removing the measurement error changes GDP growth by
− sv0 (I + αd Ω0 Ξ0 )Ξ0

−1

= −sv0 Ξ0

−1

− sv0 αd Ω0 .

(20)

In other words, correcting for downward bias in the measurement of PBS output (j < 0), the first term
on the right-hand-side of the above expression, −sv0 Ξ0 −1 , increases the contributions to GDP growth from
26

See Section 6 of the Technical Appendix for derivations.

Figure 13: Mismeasurement and Misclassification in Professional and Business Services
Construction

Durable Goods

0.8
Baseline
Mismeasured
Mismeasured and
Misclassified

0.6

PBS

0.6

1.5

0.5
0.4

0.4

0.2

0.5

0.3
0.2
0.1
-0.2

-0.5

-0.4

-1

-0.6

0
-0.1
-0.2

-1.5
1960

1980

2000

2020

1960

1980

2000

2020

1960

1980

2000

2020

Notes: The red solid lines depict the baseline contributions to trend GDP Growth from the different sectors. The
black dashed lines reflect the effects of higher productivity in PBS, implied by the price indices of its IPP industries
in the National Income Accounts, on those contributions. The solid blue lines illustrate sectoral contributions to
trend GDP growth when, in addition, the capital requirement matrix allows for investment contributions from IPP
industries within PBS.

PBS and lowers the contributions from other sectors (TFP growth is now higher in PBS and lower in other
sectors). The second term, −sv0 αd Ω0 , generally increases all other sectors’ contributions to GDP growth to
the extent that PBS sells some investment goods to these sectors. The net effect of correcting for understated
output growth in PBS, therefore, is an increase in its contributions to GDP growth, and either an increase
or decrease in the contributions from other sectors.
To explore the role of possible output mismeasurement in PBS, we consider the possibility that price
growth in its two IPP related subsectors, namely Computer System Design (BEA Industry Code 5415), and
Miscellaneous Professional, Scientific, and Technical Services (BEA Industry Code 5412OP), is overstated
in KLEMS. Alternatively, gross output growth in those sectors would be understated. In particular, in
a manner comparable to Byrne et al. (2016), we modify observed price measures in the two IPP related
subsectors of PBS to be more closely aligned with price measures of IPPs (that cover similar commodities
as BEA Industry Codes 5415 and 5412OP) in the National Income Accounts (NIA). The NIA price indices
indicate less rapid price growth and, therefore, imply higher productivity (see Section 6 of the Technical
Appendix). By using closely related NIA price indices, we interpret this exercise as a reasonable first pass at
correcting for suspected bias in the KLEMS prices, or at least providing a sense of robustness with respect
to measurement. In this case, the adjustment produces a price index for PBS that increases at a rate that
is one percentage point lower than KLEMS prices.
The dashed black line in Figure 13 shows, relative to the contributions to GDP growth originally shown
40

in Figure 12, the effects of higher productivity in PBS implied by the more rapidly declining prices of its
IPPs in the National Income Accounts. As explained above, higher measured productivity growth in PBS
affects all sectors, including Construction and Durable Goods highlighted here. The contributions from PBS
to trend GDP growth is noticeably higher both because of the direct effect of higher measured TFP in that
sector, through the corresponding element of −sv0 Ξ0 −1 in equation (20), and because the production of other
capital goods in PBS benefits from its more productive IPP sectors (and thus lower prices), captured by the
corresponding element of −sv0 αd Ω0 in (20). In contrast, the quantitative contributions from Construction
and Durable Goods to the trend growth rate of GDP do not change appreciably relative to their baseline.
On the one hand, measured TFP growth is now smaller in those sectors (i.e., the corresponding elements of
−sv Ξ0 −1 are negative). On the other hand, those sectors also benefit from employing more productive IPP
sectors in PBS in producing their own output (−sv0 αd Ω0 > 0).
The other key potential source of mismeasurement in multisector models is the misclassification of goods.
Specifically, while the 1997 capital flow tables include software as investment, they do not line up exactly
with the broader definition of IPPs used in KLEMS for capital. Thus, they likely miss contributions from
PBS to investment stemming from its IPP industries. To explore the implications of this misclassification
problem, we separate out the two sectors producing IPPs within PBS, BEA Industry Codes 5415 and
5412OP defined above, from other PBS industries producing more clearly defined intermediate inputs.27
We then construct a modified capital requirement matrix, Ω, that accounts for the possible omission of IPP
contributions from PBS to the production of new capital. In particular, as an upper bound for possible
mismeasurement in Ω, we reclassify 50 percent of the value of IPPs produced by industries 5415 and 5412OP
in PBS as final investment demand for IPP. This reclassification implies a new capital requirement matrix,
Ω, that results in a sectoral multiplier for PBS of 0.36 as compared to our baseline of 0.25.
The solid blue line in Figure 13 shows the combined effects of the new capital requirement matrix
with those correcting for possible bias in KLEMS prices of Intellectual Property Products in PBS. The
partial reclassification of Computer System Designs and Miscellaneous Professional, Scientific and Technical
Services in PBS from materials to capital raises its sectoral multiplier and lowers those of Construction and
Durable Goods. The net effect is that contributions from PBS to trend GDP growth are now higher overall
than those of Construction. While the reapportioning of 50 percent of the production value of PBS’s main
IPPs may be an upper bound on missing contributions from IPP capital in Ω, the exercise nevertheless
underscores the importance of classifying goods appropriately. Moreover, this section also highlights the
importance of continuing efforts to address challenges associated with the measurement of IPP indices.

6.3

Production Networks and Capital Accumulation

In seminal work, Hulten (1978) showed that when different sectors employ inputs produced in multiple other
sectors, aggregate TFP is a weighted average of sectoral TFP with weights given by the ratio of sectoral
27
These are Legal Services, Management of Companies and Enterprises, Administrative and Support Services, and Waste
Management and Remediation Services.

gross output to GDP, or Domar weights. This result hinges in part on interpreting TFP as scaling gross
output. When TFP is instead interpreted as scaling value added as we do here, the relevant weights become
value added shares in GDP.28 Building on Long and Plosser (1983), a number of papers over the last decade
have studied the different ways in which sectoral productivity changes influence aggregate value added.
Acemoglu et al. (2012) note that in a static multi-sector environment abstracting from the production
of capital, the same sectoral value added shares also capture the effects of sectoral productivity changes on
GDP. They interpret this observation, therefore, in terms of Hulten (1978)’s work on aggregation and refer
to the vector of value-added shares as the influence vector. They show that when some sectors serve as hubs
in the production network, the distribution of these shares is such that sectoral shocks do not generally
cancel out in aggregation. These insights are used in Gabaix (2011) to highlight the importance of shocks
to large firms for aggregate variations. Baqaee and Farhi (2019) then explore in a similar environment the
role of non-linearities in production for generating GDP effects from sectoral shocks that go beyond what
they refer to as Hulten’s theorem.
Our work recognizes that a key aspect of an economy’s production network arises through sectoral
linkages in the production of investment goods in addition to those in materials. The presence of capital,
in particular, means that the effects of sectoral changes on GDP reflect the interactions between sectoral
linkages and the dynamics of capital accumulation. These features then amplify the aggregate effects of
disturbances in different sectors beyond their value added shares. Moreover, they do so under otherwise
standard neoclassical assumptions, log-linear technologies and competitive input and product markets. Importantly, unlike the aforementioned papers on production networks, sectoral multipliers here apply to the
effects of changes in sectoral input growth on GDP growth, ∂∆ ln V /∂∆ ln A, rather than levels, ∂ ln V /∂ ln A.
This complements the empirical macroeconomics literature’s emphasis on the characterization and behavior
of growth rates, including at different frequencies, rather than levels. In this case, the new formulas we
derive make it possible to explore empirically the secular decline in GDP growth as highlighted in Figure
9.29
Beyond our focus on long-run growth, our work highlights the importance of capital accumulation within
the production network. Because the investment network plays a key role in amplifying the aggregate
effects of sectoral changes, it is reasonable to conjecture that other features related to investment or other
sources of dynamics could also play a role. While we allow for sectoral technologies that differ in their
input shares, features such as time-to-build, investment adjustment costs, or the cost of holding investment
28

These results are evidently related. When sectoral TFP, zj , is measured as scaling value added, zej,t = zj,tj becomes the

relevant scalar for sectoral gross output, where γj is j’s value added share in gross output,
y
pj yj

pv
j vj
y
pj yj

. In Hulten (1978),

∂ ln Zt
∂ ln z
ej,t

= Dj ,

where Dj is sector j’s Domar weight or ratio of gross output to GDP, V . It immediately follows from the definition of zej
t
that ∂∂lnlnzZj,t
= γj Dj , where γj Dj is then simply sector j’s value added share in GDP, svj .
29
Section 5 of the Technical Appendix shows that the findings and insights in Acemoglu et al. (2012) and subsequent work
remain nested in a static version of our economic environment without capital and where the focus is on levels rather than
growth rates. However, it also shows that this ‘levels’ result changes somewhat in the steady state of a dynamic economy with
capital. In particular, the effect of a productivity change in a sector on the level of GDP is given by its value added share (or
Domar weight) scaled by the inverse of the average labor income share.

goods in inventories likely differ across sectors. Aside from affecting the long-run amplification mechanisms
highlighted here, these features likely also help shape how sectoral disturbances play out at business cycle
or medium run frequencies. Therefore, more accurately modeling the technologies used in different sectors,
and how these technologies affect dynamics at different frequencies, is an important next step.

Concluding Remarks

In this paper, we study how trends in TFP and labor growth across major U.S production sectors have
helped shape the secular behavior of GDP growth. We find that sectoral trends in TFP and labor growth
have generally decreased across a majority of sectors since 1950. Common trends in sectoral TFP growth
contributed around 1/3 of the secular decline in aggregate TFP growth. Common trends in sectoral labor
growth contributed about 2/3 of the secular decline in aggregate labor growth.
We embed these findings into a dynamic multi-sector framework in which materials and capital used by
different sectors are produced by other sectors. These production linkages along with capital accumulation
mean that changes in the growth rate of labor or TFP in one sector affect not only its own value added
growth but also that of all other sectors. In particular, capital induces network effects that amplify the
repercussions of sector-specific sources of growth on the aggregate economy and that we summarize in terms
of sectoral multipliers. Quantitatively, these multipliers scale up the influence of some sectors by multiple
times their value added share in the economy.
Ultimately, we find that sector-specific factors in TFP and labor growth historically explain 3/4 of low
frequency variations in U.S. GDP growth, leaving common or aggregate factors to explain only 1/4 of these
variations. Changing sectoral trends in the last 7 decades, translated through the economy’s production
network, have on net lowered trend GDP growth by close to 3 percentage points. The Construction and
Durable Goods sectors, more than any other sector, stand out for their contribution to the trend decline in
GDP growth over the post-war period, though other sectors with large multipliers, such as Professional and
Business Services, could also have an outsize influence on the aggregate economy going forward. Remarkably,
no sector has contributed any steady or significant increase to the trend growth rate of GDP in the last 70
years.

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