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Aggregate Implications of Changing
Sectoral Trends
WP 19-11R
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
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
August 27, 2019
Working Paper No. 19-11R
Abstract
We find disparate trend variation in TFP and labor growth across major U.S. production sectors
over the post-WWII period. When aggregated, these sector-specific trends imply secular declines in the
growth rate of aggregate labor and TFP. We embed this sectoral trend variation into a dynamic multisector framework in which materials and capital used in each sector are produced by other sectors. The
presence of capital induces important network effects from production linkages that amplify the consequences of changing sectoral trends on GDP growth. Thus, in some sectors, changes in TFP and labor
growth lead to changes in GDP growth that may be as large as three times these sectors’ share in the
economy. We find that trend GDP growth has declined by more than 2 percentage points since 1950, and
that this decline has been primarily shaped by sector-specific rather than aggregate factors. Sustained
contractions in growth specific to Construction, Nondurable Goods, and Professional and Business and
Services make up close to sixty percent of the estimated trend decrease in GDP growth. In addition, the
slow process of capital accumulation means that structural changes have endogenously persistent effects.
We estimate that trend GDP growth will continue to decline for the next 10 years absent persistent increases in TFP and labor growth.
Keywords: trend growth, multi-sector model, production linkages
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 for a helpful discussion. We also thank John Fernald, Chad Jones,
and participants at various university seminars and conferences for their comments and suggestions. Daniel Ober-Reynolds
and Eric LaRose provided outstanding research assistance. Andrew Foerster: andrew.foerster@sf.frb.org, Andreas Hornstein:
andreas.hornstein@rich.frb.org, Pierre-Daniel Sarte: pierre.sarte@rich.frb.org, Mark Watson: mwatson@princeton.edu.
1
1
Introduction
The U.S. economy is currently on track for the longest expansion on record in the aftermath of the Great
Recession. However, it has also become evident that output has been growing conspicuously slowly during
this expansion. Fernald et al. (2017) find that slow growth in total factor productivity (TFP) and a fall in
labor force participation are the main culprits behind this weak recovery. Importantly, the authors also find
that these adverse forces are mostly unrelated to the financial crisis associated with the Great Recession.
Cette et al. (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.1 This view is consistent
with the long lags associated with the productivity effects of IT adoption found by Basu and Fernald (2001),
and the collapse of the dot-com boom in the early 2000s. Moreover, Decker et al. (2016) point to a decline
in business dynamism that began in the 1980s as an additional force underlying slowing economic activity.2
This paper highlights the steady decline in trend GDP growth over the post-war period, 1950 − 2016.
Building on Fernald et al. (2017), we explore the roles played by TFP and labor inputs in explaining this
secular decline, but we do so at a disaggregated sectoral level. We estimate an empirical model where, in
each industry, TFP growth and labor growth have unobserved persistent and transitory components, and
where each component can itself stem from either aggregate or idiosyncratic forces. The estimates reveal
that trends in TFP and labor growth have steadily decreased across a majority of U.S. sectors since 1950.
Interestingly, more than 2/3 of the secular decline in aggregate TFP growth results from the combination of
sector-specific disturbances, thus leaving only an ancillary role for the aggregate component. Therefore, if
technical progress in general purpose technologies has helped drive trends in sectoral TFP growth, this thrust
has not been shared widely enough across sectors to generate comovement in TFP growth. This finding aligns
with the observation in Decker et al. (2016) that over the last 30 years, manufacturing has gradually shifted
from producing “general purpose” technologies to producing “special purpose” technologies. Similarly to
TFP growth, trend labor growth has generally been dominated by sector-specific factors, especially after
1980 and the latter part of the post-war period. The decline in trend labor growth is especially large in
the Durable Goods sector, though even in that sector trend growth always remains positive. In general, we
find that secular changes in TFP and labor growth have been mostly driven by sector-specific rather than
common components.
We define the process of structural change in different sectors as concurrently determined by the observed
low frequency behavior of TFP and labor growth in those sectors. We then embed those changes into a
dynamic multi-sector framework in which materials and capital used by different sectors in the economy are
1
From a measurement standpoint, Byrne et al. (2016) also argue that the slowdown in TFP growth that preceded the last
recession is not likely the result of mismeasurement of IT related goods and services. Aghion et al. (2017) find that the process
of creative destruction does lead growth to be understated when inflation is imputed from surviving products. However, this
missing growth did not accelerate much after 2005, and was roughly constant before then.
2
Fernald and Jones (2014) more generally make the case that diminishing marginal returns to the discovery of ideas ultimately
curbs economic expansion. Gordon (2014) points to additional headwinds that have contributed to a general slowdown in growth,
while Gordon and Sayed (2019) argue that a similar slowdown has taken place in the ten largest European economies.
2
produced by other sectors. We use balanced-growth accounting to determine the aggregate effects of sectoral
changes in trend rates of growth in labor and TFP. This paper, therefore, falls partially within the literature
on equilibrium multi-sector models first developed by Long and Plosser (1983), and later Horvath (1998,
2000), and Dupor (1999). Since then, a large body of work including Gabaix (2011), Foerster et al. (2011),
Acemoglu et al. (2012), di Giovanni et al. (2014), Atalay (2017), Baqaee and Farhi (2017b), Miranda-Pinto
(2019), and others have worked out important features of those models for generating aggregate fluctuations
from idiosyncratic shocks.3 In contrast to this literature, the focus herein centers on the implications of
production linkages for secular dynamics and the determination of both sectoral and aggregate trend growth
rates.4 While recent work has suggested a somewhat muted role for aggregate shocks in explaining cyclical
variations in GDP growth, we now find that sector-specific disturbances also explain most of the trend
variations in U.S. GDP growth.
Our paper returns to the original multi-sector model of Long and Plosser (1983) and maintains the original assumptions of competitive input and product markets as well as constant-returns-to-scale technologies.
However, we explicitly allow different industries to produce investment goods for other industries. Unlike
Horvath (1998) or Dupor (1999), capital is not constrained to be sector-specific and is allowed to depreciate
only partially within the period. We assume unit elastic preferences and technologies that allow us to derive
analytical expressions for the model’s sectoral and aggregate balanced growth paths. These expressions
highlight how changes in trend TFP or labor growth in different sectors affect value added growth in every
other sector and, therefore, GDP growth. The implied elasticities reflect induced changes in capital trend
growth rates across sectors. Thus, our analysis in part extends Greenwood et al. (1997) to a multi-sector
environment.5
The fact that changes in TFP or labor growth in a sector affect value added growth in every other
sector hinges critically on the presence of capital. This feature of the environment leads to quantitatively
important multiplier effects from sectoral linkages to GDP growth. The size of this multiplier for a sector
depends on its importance as a supplier of capital or materials to other sectors. The density of production
linkages more generally determines the degree to which the sectoral network propagates structural changes
in a sector to the rest of the economy. The U.S. Capital Flow tables produced by the Bureau of Economic
Analysis (BEA) indicate that the Construction and Durable Goods sectors produce roughly 80 percent of
the capital used in almost every industry. The strength of these linkages results in GDP growth multipliers
for those sectors that are almost 3 times their share in the economy. Professional and Business Services and
3
An additional dimension of this work explores the implications of frictions for aggregate fluctuations in these models,
including Bigio and La‘O (2016), Baqaee (2018), Grassi (2018), and Baqaee and Farhi (2017a). Other recent work has also
investigated the implications of production linkages for higher order moments, for instance Acemoglu et al. (2017), and Atalay
et al. (2018)
4
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. More importantly, the analysis
abstracts from pairwise linkages in both intermediates and capital-producing sectors that play a key role in this paper.
5
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.
3
Wholesale Trade are also associated with relatively large GDP growth multipliers because of their central
role as suppliers of materials. We find that changing sectoral trends in the last 6 decades, translated through
the economy’s production network, have on net lowered trend GDP growth by roughly 2.2 percentage points.
Construction more than any other sector stands out by a considerable margin for its contribution to the
trend decline in GDP growth since 1950, accounting for close to 1/3 of this decline. Structural changes in
Professional and Business Services and Nondurable Goods together account for another 25 percent.
This paper is organized as follows. Section 2 gives an overview of the behavior of trend GDP growth over
the past 60 years. Section 3 provides an empirical description of TFP and labor growth by industry that
allows for persistent and transitory components, where each component itself may be driven by idiosyncratic
or aggregate forces. Section 4 develops the implications of these structural 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 concludes and discusses possible directions for future research.
2
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, GDP is measured
annually as the share-weighted value added 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 using time-varying shares (i.e., chainweighting) and using average shares (fixed-weights), with virtually no difference between the two calculations.
Panel A shows large variation in GDP growth rates – the standard deviation is 2.5 percent over the period
1950 − 2016 – but much of this variation is relatively short-lived and is associated with business cycles and
other relatively transitory phenomena. Our interest here is in longer-run variation.
Panel B, therefore, plots centered 15−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. This fell to 3 percent
in the 1970s, rebounded to nearly 4 percent in the 1990s, but plummeted to less than 2 percent in the 2000s
(See Table 1).
Panels C and D refine these calculations by eliminating the cyclical variation using an Okun’s law
regression in GDP growth rates as in Fernald et al. (2017).6 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 15−year moving average in Panel D now produces a more focused picture of the trend
variation in the growth rate of private GDP.
6
Compared to other measures of cyclical slack or resource utilization, Fernald et al. (2017) point out that the civilian
unemployment rate has two key advantages. First, it has been measured using essentially the same survey instrument since
1948. Second, changes in the unemployment rate have nearly a mean of zero over long periods.
4
Figure 1: US GDP Growth Rates 1950-2016
(Percentage points at an annual rate)
A: GDP Growth
10
B: GDP Growth, 15-Year Centered Moving Average
5
4
5
3
2
0
1
Constant Mean Weights
Time-Varying Weights
-5
1950
1960
1970
1980
1990
2000
2010
0
1950
2020
C: Cyclically Adjusted GDP Growth
6
5
1960
1970
1980
1990
2000
2010
2020
D: Cyclically-Adjusted GDP Growth,
15-Year Centered Moving Average
5
4
4
3
3
2
2
1
1
0
-1
1950
1960
1970
1980
1990
2000
2010
0
1950
2020
1960
1970
1980
1990
2000
2010
2020
Notes: Growth rates are share-weighted value added 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.
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? We look to inputs – specifically TFP and labor at the sectoral
level – for the answer. That is, interpreting the data as variations around a balanced growth path, changes
in GDP growth are primarily determined by changes in the growth rates of those two inputs. However,
5
Table 1: 15-Year Averages of GDP Growth Rates
Dates
Growth rates
1950 − 2016
3.3
Cyclically-adjusted
growth rates
3.2
1950 − 1965
1966 − 1982
1983 − 1999
2000 − 2016
4.3
3.1
3.9
1.8
4.1
3.7
3.3
1.9
Notes: The values shown are averages of the series plotted in
Figure 1, panels (A) and (C), over the periods shown.
as the analysis in Section 4 makes clear, not all sectoral inputs are created equal. Some sectors have a
large value-added share in GDP and also provide a large share of materials or capital to other sectors. Put
another way, input variation across sectors is a particularly important driver of low frequency movements
in aggregate GDP growth.
Before investigating these input-output interactions, we begin by describing the sectoral data, both how
these data are measured and how sectoral value-added as well as labor and TFP inputs have evolved over
the post-WWII period.
3
An Empirical Description of Trend Growth in TFP and Labor
We begin by estimating an empirical model of TFP and labor growth for different sectors of the U.S.
economy. As a benchmark, our paper applies the insights of Hulten (1978) on the interpretation of aggregate
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.
7
We calculate standard TFP growth rates at the sectoral level
following Jorgenson et al. (2017) among others, and estimate permanent and transitory components in these
growth rates.
3.1
Data
Sectoral TFP growth rates are calculated using the KLEMS dataset constructed by Jorgenson and his
collaborators, as well as its recently updated version in the form of the BEA’s Integrated Industry-Level
Production Accounts (ILPA). These datasets are attractive for our purposes because they provide a unified
7
In the absence of constant-returns-to-scale or perfect competition, 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 returns to scale sectors stemming from changes in relative sectoral TFP.
6
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 KLEMS dataset contains quantity and price indices for inputs and outputs across 65
industries. The growth rate of any one industry’s aggregate is defined as a Divisia index given by the
value-share weighted average of its disaggregated component growth rates. Labor input is differentiated
by gender, age, education, and labor status. Labor input growth is then defined as a weighted average of
growth in annual hours worked across all labor types using labor compensation shares of each type as weights.
Similarly, intermediate input growth reflects a weighted average of the growth rate of all intermediate inputs
averaged using payments to those inputs as weights. Finally, capital input growth reflects a weighted average
of growth rates across 53 capital types using payments to each type of capital as weights. Capital payments
are based on implicit rental rates consistent with a user-cost-of-capital approach. Total payments to capital
are the residuals after deducting payments to labor and intermediate inputs from the value of production.
Put another way, there are no economic profits.
An industry’s TFP growth rate is defined in terms of its Solow residual, specifically output growth less
the revenue-share weighted average of input growth rates. This calculation is consistent with the canonical
theoretical framework we adopt in Section 4 where all markets operate under perfect competition and
production is constant-returns-to-scale. For earlier versions of Jorgenson’s KLEMS data up to 1990, Basu
and Fernald (1997, 2001) compute total payments to capital as the sum of rental rates implied by the
user-cost-of-capital and find small industry profits on average that amount at most to three percent of gross
output. In the presence of close to zero profits, elasticities to scale and markups are equivalent. Basu and
Fernald (2001) estimate returns-to-scale across industries and find few significant deviations from constant
returns or, alternatively, little evidence of markups. More recently, an active debate has emerged on the
extent to which the competitive environment has changed in the U.S. over the last two decades. On the
one hand, Barkai (2017), also applying the user-cost-of-capital framework but using post 1990 data, finds
substantial profit shares over that period. Moreover, De Loecker and Eeckhout (2017), estimating industry
production functions from corporate balance sheets, present evidence of rising markups and returns to
scale since the 1980s. On the other hand, Karabarbounis and Neiman (2018) argue that the user-costof-capital framework, to the extent that it implies high profit shares starting in the 1990s, also implies
unreasonably high profit shares in the 1950s. In addition, Traina (2018) argues that the evidence on rising
markups from corporate balance sheets depends crucially on the measurement of variable costs and weights
in aggregation.8 In this paper, we maintain the assumptions of competitive markets and constant-returnsto-scale as a benchmark from which to study the aggregate implications of sectoral changes in labor and
TFP inputs.
8
Similarly, Rossi-Hansberg et al. (2018) show while sales concentration has unambiguously risen at the national level since
the 1980s, concentration has steadily declined at the Core-Based Statistical Area, county, and ZIP code levels over the same
period. While these facts can seem conflicting, the authors present evidence that large firms have become bigger through the
opening of more establishments or stores in new local markets, but this process has lowered concentration in those markets.
7
Our calculations rely on the 2017 version of the Jorgenson KLEMS dataset which covers the period
1946 − 2014, and the ILPA KLEMS dataset which covers the period 1987 − 2016.9 For ease of presentation,
and in order to consider the role of structural change in individual sectors separately, we carry out the
empirical analysis using private industries at the two-digit level. In particular, we aggregate the 65 industries
included in the two KLEMS datasets into 16 two-digit private industries following the procedure in Hulten
(1978).10 Another advantage of the aggregation into two-digit industries is that any differences between
the two KLEMS datasets are attenuated and we feel comfortable splicing the levels of the two datasets in
1987.11 That is, we use the growth rates calculated using the Jorgenson KLEMS data before 1987 and using
ILPA data after that date.
Finally, we note that the KLEMS data rely on U.S. NIPA measures for fixed assets. Specifically, these
measures are based in part on estimates of capital goods prices. To the degree that these prices perennially
understate quality growth in capital goods, then KLEMS data understate productivity improvements in the
investment goods sector.12 If capital accumulation is an important driver of growth, our results would then
provide a lower bound for the growth contributions of the capital goods producing sectors.
Table 2 lists the 16 sectors we consider. For each sector, the table shows average cyclically adjusted
growth rates of value added, labor, and TFP over 1950 − 2016, 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 1.1 percent in Mining to 4.7 percent in Wholesale Trade,
bracketing the aggregate value added growth rate of 3.2 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 a negative 1.4 percent in Agriculture to 3.3 percent in Professional
and Business Services, bracketing the average aggregate growth rate of 1.5 percent. Again, most sectors
with labor input growth rates that exceed the aggregate growth rate provide services. Finally, TFP growth
rates range from -0.9 percent in FIRE (ex-Housing) to 3.2 percent in Agriculture, bracketing the average
aggregate TFP growth rate of 0.6 percent. Sectoral TFP growth rates are less aligned with either value
added or labor input growth rates. There are three sectors with notable TFP declines, namely Utilities,
Other Services, and Construction, as well as a number of sectors with stagnant TFP levels. Negative TFP
9
The Jorgenson dataset is downloaded from http://www.worldklems.net/data.htm and the ILPA is downloaded from
https://www.bea.gov/data/special-topics/integrated-industry-level-production-account-klems. A detailed description of the Jorgenson data can be found in Jorgenson et al. (2014), as well as Jorgenson et al. (2017), and the BEA dataset is
described in Fleck et al. (2014).
10
The procedure is described in detail in the online-only Technical Appendix and Supplementary Material to this paper,
Foerster et al. (2019).
11
While the ILPA builds on the Jorgenson KLEMS data, the two datasets are not exactly identical for the time period in
which they overlap. Since both datasets are constructed to be consistent with the BEA’s input-output tables, they mostly agree
on industry details and both cover the same 65 private industries. Nevertheless, there remain differences but these are reflected
mostly in the levels of variables and not their growth rates.
12
See Gordon (1990) or Cummins and Violante (2002).
8
Table 2: 16 Sector Decomposition of the U.S. Private Economy
(1950-2016)
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
2.63
1.06
1.87
1.67
3.36
2.29
4.65
3.16
2.43
4.58
3.93
4.36
3.32
2.44
2.02
3.51
Labor
TFP
Labor
-1.42
0.27
0.98
1.60
0.42
0.07
1.68
1.08
0.85
1.30
2.78
3.26
2.75
2.00
2.38
1.68
3.16
-0.06
-0.74
-0.10
1.88
0.79
1.13
1.07
1.35
0.86
-0.90
0.30
0.04
0.22
-0.60
0.43
Value
Added
2.70
2.15
2.42
5.04
13.48
9.31
6.92
8.44
4.19
4.90
10.03
8.59
6.01
3.70
2.98
9.15
3.24
1.49
0.62
100
100
3.25
1.66
1.05
7.64
15.59
8.84
6.50
9.86
4.91
3.68
7.41
10.88
9.28
4.67
4.50
0.25
Notes: The values shown are average annual growth rates for the 16 sectors. The row labelled
“Aggregate” is the share-weighted average of the 16 sectors.
growth rates are a counter-intuitive but well known feature of disaggregated industry data. To the degree
that they occur in service industries, they are in part attributed to measurement issues with respect to
output.
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. Two notable contributors
to value added and TFP are Durable Goods and FIRE excluding 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 increased. However,
9
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 1A).
3.2
Aggregate Balanced Growth Implications in a Canonical Model without Linkages
Before describing the secular evolution of sectoral labor growth and TFP growth in more detail, we briefly
consider the implications of the long-run averages shown in Table 2 through the lens of the standard onesector growth model. In particular, let ∆ ln z denote the average growth rate of aggregate TFP from 1950 to
P
2016, ∆ ln z = nj=1 svj ∆ ln zj , where n, svj , and ∆ ln zj denote respectively the number of sectors, constant
mean shares of sectoral value added in GDP, and the average growth rates of sectoral TFP that are shown
in Table 2. Similarly, let ∆ ln ` represent the post-war average growth rate of aggregate labor. Therefore,
P
∆ ln ` = nj=1 s`j ∆ ln `j where s`j and ∆ ln `j represent respectively average sectoral labor shares and average
sectoral labor growth rates in Table 2. Suppose that the economy admits an aggregate production function
such that at any date t, Vt = zt ktα `1−α
, where Vt is aggregate value added or GDP and kt represents
t
aggregate capital. Then, along a balanced growth path, the capital-output ratio is constant and
∆ ln V =
1
∆ ln z + ∆ ln `.
1−α
(1)
Over the period 1950 − 2016, ∆ ln z is 0.62 percent while ∆ ln ` is 1.49 percent in Table 2. Assuming a share
of aggregate labor in GDP, 1 − α, of 2/3, equation (1) then implies that GDP would have grown by around
2.42 percent on average over the same period. In other words, the predicted growth rate from equation (1)
falls short of actual average GDP growth, 3.24 percent, by more than 3/4 of a percentage point. Motivated
in part by this discrepancy, we explore and highlight below the key role of an economy’s sectoral network
in determining its aggregate growth rate.13 In particular, we show that linkages between sectors give rise
to powerful sectoral multipliers that amplify the role of idiosyncratic structural changes in the economy.
This effect also accounts for much of the inconsistency between the long-run growth rate of GDP implied
by equation (1) and that in Table 2.
3.3
Empirical Framework
Let ∆ ln x
ej,t denote the growth rate (100 × the first difference of the logarithm) of annual measurements of
labor or TFP in sector j at time 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.
13
This discrepancy is also noted in Whelan (2003), and Fernald (2014), who point to the usefulness of modeling multiple
sectors but abstract from studying the role of sectoral linkages explicitly.
10
That is, we estimate
∆ ln x
ej,t = µj + βj (L)∆ut + ej,t ,
(2)
where βj (L) = βj,1 L + βj,0 + βj,−1 L−1 and the leads and lags of ∆ut captures much of the business-cycle
variability in the the data. Throughout the remainder of the paper, we use ∆ ln xj,t = ∆ ln x
ej,t − β̂j (L)∆ut ,
where β̂j (L) denotes the OLS estimator, and where xj,t represents the implied cyclically adjusted value of
sector TFP (denoted zj,t ) or labor input (denoted `j,t ) growth rates.14
Figures 2 and 3 plot centered 15−year moving averages of the cyclically adjusted growth rates of labor
and TFP. These are shown as the thick blue lines in the figures (ignore the thin dotted red line for now).
The disparity in experiences across different sectors stands out. In particular, the moving averages show
large sector-specific variation 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 rate of growth of labor in several service sectors are shown to exhibit
large ups and downs over the sample. Looking at TFP, there are important differences across sectors as
well. In Sections 4 and 5, we quantify the aggregate implications of these sectoral variations in labor and
TFP inputs.
In the economic model presented in Section 4, we treat zj,t and `j,t as exogenous processes and study
the implied values of output and value-added that arise from realizations of these processes. We do so in a
dynamic model that features input-output and capital flow linkages between the sectors. This multi-sector
accounting exercise requires joint stochastic processes for the sectoral values of zj,t and `j,t . For this purpose,
we use a reduced-form statistical model that captures the salient cross-sectional and dynamic correlations
in the data.
Cross-correlations and autocorrelations summarized in the online Technical Appendix suggest a reducedform model with three features. First, the sectoral growth rates of labor (∆`j,t ) are somewhat correlated
across sectors; there is also (weak) cross-sector long-run correlation in the sectoral TFP growth rates (∆zj,t ).
Second, there is little evidence of long-run correlation between (∆`j,t ) and (∆zj,t ) across or within sectors.15
Finally, ∆ ln `j,t and ∆ ln zj,t exhibit substantial year-to-year variation around slowly varying levels.
These features lead us to consider independent stochastic processes for ∆ ln `j,t and ∆ ln zj,t , where
the processes have a structure that includes factors common to all sectors together with sector-specific
factors, and where these factors include both slowly-varying level terms (modeled as martingales) and terms
capturing more transitory variation (modeled as white noise). Specifically, we consider a dynamic factor
14
Other measures of variable utilization are estimated in Kimball et al. (2006). These are generally found to be stationary
so that any differences across measures of utilization will likely affect the transitory components of TFP or labor rather than
their trends.
15
Specifically, using the methods developed in Müller and Watson (2018), 32 percent of the pairwise long-run correlations for
labor and TFP are statistically significantly different from zero at the 33 percent level, while only 24 percent of the labor-TFP
cross correlations are statistically significant. The point estimates are also consistent with small correlations: the average
estimated sectoral pair-wise long-run correlation is 0.10 for labor and 0.03 for TFP; the average estimated labor-TFP long-run
cross-correlation is -0.03. See the Technical Appendix for detailed results.
11
Figure 2: Trend Growth Rate in Labor by Sector
(Percentage points at an annual rate)
Agriculture
Mining
Utilities
4
2
4
2
1
2
0
0
Construction
0
-2
0
-2
-4
-1
1960
1980
2000
2020
Durable goods
1960
2
1980
2000
2020
-2
1960
Nondurable goods
3
1980
2000
2020
Wholesale trade
1960
1980
2000
2020
Retail trade
2
4
2
2
0
1
1
0
-2
1960
3
1980
2000
2020
0
1960
Trans. & Ware.
1980
2000
2020
0
1960
Information
1980
2000
2020
FIRE (x-Housing)
4
2
1
0
4
4
3
3
2
2
1
1
0
-2
-1
1960
1980
2000
2020
Educ. & Health
1960
1980
2000
2020
1980
2000
2020
1960
Other services (x-Gov)
Arts, Ent., & Food svc.
4
2
2
2
0
0
0
2000
2020
2000
2020
0
1960
4
1980
PBS
2
0
1960
1980
Housing
4
4
2
0
1960
1980
2000
2020
1960
1980
2000
2020
1960
Cyc-Adj Data, 15 Year Centered MA
1980
2000
2020
1960
1980
2000
2020
DFM Trend
Notes: The solid line is the centered 15-year moving average of the annual rate of growth of labor in each of the
sectors shown. The dotted line is the estimated trend component (common + idiosyncratic) for sectoral growth rates
estimated from the DFM.
model (DFM) of the form,
x
x
∆ ln xj,t = λxj,τ τc,t
+ λxj,ε εxc,t + τj,t
+ εxj,t ,
(3)
x , εx , {∆τ x , εx }n
where x = z or ` and ∆τc,t
c,t
j,t j,t j=1 are i.i.d. Gaussian random variables with mean zero and
12
Figure 3: Trend Growth Rate in TFP by Sector
(Percentage points at annual rate)
Agriculture
6
Mining
5
Utilities
4
0
4
0
2
-5
0
-10
Construction
2
-1
0
-2
-2
1960
1980
2000
2020
1960
Durable goods
1980
2000
2020
1960
Nondurable goods
3
1980
2000
2020
Wholesale trade
1980
2000
2020
Retail trade
2
4
1960
2
2
1
2
0
1
1
0
0
0
1960
3
1980
2000
2020
1960
Trans. & Ware.
1980
2000
2020
1960
Information
1
2
2
1980
2000
2020
FIRE (x-Housing)
0
1980
2000
2020
Educ. & Health
2020
-1
-3
1960
2
2
2000
1
0
-2
1960
2020
-2
-1
-1
2000
-1
0
0
1980
PBS
2
1
1
1960
1980
2000
2020
1960
Arts, Ent., & Food svc.
1980
2000
2020
Other services (x-Gov)
1960
1
2
1
0
1
0
-1
0
1980
Housing
0
-2
-1
1960
1980
2000
2020
-2
1960
1980
2000
2020
-1
1960
Cyc-Adj Data, 15 Year Centered MA
1980
2000
2020
1960
1980
2000
2020
DFM Trend
Notes: See notes to Figure 2.
variable-specific variances.
The τ −terms are random walks and describe the slowly varying (or ‘local’) levels in the growth rate of
x , and some is sector-specific, through τ x . Deviations of
xj,t . Some of this variation is common, through τc,t
j,t
the data from their local levels are represented by the ε−terms, part of which is common, εxc,t , and part of
13
which is sector-specific, εxj,t .16
The sectoral model produces aggregate versions of `t and zt that also have random-walk-plus-white-noise
representations. In particular, let sj denote the (time-invariant) share weight of sector j. The aggregate
P
Pn
Pn
x
x
x
value of xt then satisfies ∆ ln xt = nj=1 sj ∆ ln xj,t = τt + εt . Here, τt = τc,t
j=1 sj λj,τ +
j=1 sj τj,t is a
random walk that represents the ‘local’ level of the aggregate growth rate and εt , defined analogously, is
white noise.
While the empirical model has a simple dynamic and cross-sectional structure, it fits the sectoral labor
and TFP data well (details are provided in the Technical Appendix) and versions of the model have proved
useful in describing sectoral data in other contexts (cf. Stock and Watson (2016)). The dynamic factor model
is estimated using Bayesian methods together with a Gaussian likelihood for the various shocks. Priors are
standard (normal priors for the λ−coefficients and inverse gamma priors for the variances). The scale of
the common factors and the λ coefficients are not separately identified; we impose a normalization where
x and εx have unit variance and the average value of λ is non-negative. The priors for the variance of
∆τc,t
c,t
the idiosyncratic terms are reasonably uninformative but we use more informative priors for λ. Details are
provided in Appendix A.
3.4
Estimated Sectoral and Aggregate Trend Growth Rates in Labor and TFP
Appendix A and The Technical Appendix contain details of the estimation method and results for the
empirical models. For our purposes, the key results are summarized in three figures and a table. Figures 2
x + τx )
and 3, introduced earlier, show the composite estimated trend component (posterior mean) (λxj,τ τc,t
j,t
as the red dotted line along with the 15−year moving averages of the cyclically adjusted growth rates.
While the estimated trends from the dynamic factor model closely track the 15−year moving averages
for most of the sectors, these trends now also allow for a decomposition into common and sector-specific
components. Table 3 shows the changes in trend growth for labor and TFP over different periods, as well
as the decomposition of these changes into various components. In the table, common and sector-specific
changes in trend growth rates add up to the aggregate change.
Figure 4 plots the aggregate values of the growth rates of labor and TFP along with their estimated
trends from the sectoral empirical model. Panels A and D show the growth rates and the estimates of τ ;
panels B and E show the 15−year moving averages of the data along with the estimate of τ and associated
68 percent credible intervals; and panels C and F decompose the estimate of τ into its common component,
Pn
Pn
x
x
x 17 As with the sectoral data, the implied
j=1 sj λj,τ × τc,t , and its sector-specific component,
j=1 sj τj,t .
16
There is an apparent tension between the DFM model, which includes the random walk τ terms and the model presented in
the next section, which requires stationarity of the growth rates. The tension is resolved by assuming the τ terms follow highly
persistent, but stationary AR(1) models with AR roots local to unity. These processes are approximated as random walks in
the DFM.
17
Share weights for labor and TFP use labor compensation and value added weights respectively. The initial values for the
common and sector-specific trend values are not separately identified - the data is only informative about their sum - so that
Panels C and F normalize the initial values in 1950 to be zero.
14
Figure 4: Aggregate Trend Growth Rate in Labor and TFP
(Percentage points at annual rate)
A: Agg. Labor: Data and DFM Trend
B: Agg. Labor: 15-Year MA and DFM Trend
C: Agg. Labor: DFM Trends (1950=0)
2.5
0.5
2
0
1.5
-0.5
1
-1
0.5
-1.5
3
2
1
0
0
Cylically-Adjusted Data
DFM Trend
-1
Data, 15-Year Centered MA
DFM Trend
-0.5
1960
1980
2000
2020
D: Agg. TFP: Data and DFM Trend
3
DFM Trend (Common+Sector-specific)
DFM Trend (Common)
DFM Trend (Sector-specific)
-2
-2.5
1960
1980
2000
2020
E: Agg. TFP: 15-Year MA and DFM Trend
1.5
1960
1980
2000
2020
F: Agg. TFP: DFM Trends (1950=0)
0.5
2
1
1
0
0
0.5
-1
-0.5
0
-2
Cylically-Adjusted Data
DFM Trend
-3
-0.5
1960
1980
2000
DFM Trend (Common+Sector-specific)
DFM Trend (Common)
DFM Trend (Sector-specific)
Data, 15-Year Centered MA
DFM Trend
2020
-1
1960
1980
2000
2020
1960
1980
2000
2020
Notes: The dotted lines in panels (A), (B), (D), and (E) are 68 percent credible intervals for the DFM trends. In
panels (C) and (F) the estimated DFM trends are normalized to 0 in 1950.
aggregate trends estimated from the dynamic factor model closely track the low-frequency movements in
the aggregate data.
Panels A and B include error bands (68 percent posterior credible intervals) computed from the dynamic
factor model. The width of these error bands (approximately 0.50 percentage points) highlights the inherent
uncertainty in estimating the level of time series from noisy observations. This uncertainty carries over to the
structural exercise in Section 4 and is amplified by uncertainty concerning the economic model postulated
15
in that exercise, its calibrated parameters, as well as the quality of the data. However, to the degree that
our sectoral trends estimates mimic the behavior of 15-year moving averages, the economic model with
parameters informed by BEA estimates traces out how these trends propagate to the rest of the economy.
Panels C and F suggest that much of the low-frequency variation in aggregate labor and TFP, as identified
by the dynamic factor model, is associated with sector-specific rather than common trends. In particular,
less than 20 percent of the trend decline in TFP growth is the result of shocks common to all sectors, and
only about 10 percent of the trend decline in labor is attributable to common shocks.
Panel C shows that aggregate trend labor growth fell by around 1.4 percentage points between 1950
and 2016. It also shows considerable variation over this period. In particular, from 1950 to 1980, the trend
growth rate of aggregate labor increases as the common component of the trend more than offsets the decline
in its sector-specific component. This early period coincides with the entrance of the Baby Boomers into the
labor force, which is consistent with the idea of a common demographic change that is distributed among
the different sectors. However, starting in 1980, the last of the Baby Boomers (those born in 1964) turn 16
years old and have become part of the labor force. The decline in the sector-specific component of trend
labor growth now begins to dominate the aggregate trend. One interpretation of this finding is that in the
latter period, idiosyncratic changes in the demand for labor absorb most of the demographic forces that
now push towards a declining trend growth rate.18 Under this interpretation, shocks to the idiosyncratic
factors are not strictly sector-specific – for example, some workers being hired at a lower rate because of
‘sector-specific’ forces in the Durable Goods sector presumably have labor opportunities in other sectors –
but rather the dominant feature of these shocks is that they appear in a specific sector.19
Panel F focuses on TFP and displays considerable swings in the trend of aggregate TFP growth between
1950 and 2016, with long stretches of rising and falling growth over different decades. It also shows that
through these swings, trend TFP growth has fallen by approximately 0.5 percentage points in the last 65
years. Furthermore, Panel F reveals that the secular behavior of aggregate TFP growth reflects to a large
degree the (weighted) sum of idiosyncratic TFP trends rather than a aggregate trend. By 2016, virtually
all of the secular decline in TFP growth is accounted for by its sector-specific component. This finding
suggests that technical progress in general purpose technologies (e.g. personal computers, information
technology, nanotechnology, etc.), to the extent that it has affected trends in sectoral TFP growth, has
not generated pronounced comovement in these trends across sectors. In particular, the rapid technical
progress associated with the 1990s IT boom was not accompanied by an increase in the common trend TFP
growth rate. The finding is also consistent with Decker et al. (2016) who document a shift away from the
production of “general purpose” technologies in manufacturing towards technologies that meet more specific
18
Aside from a U.S. population that begins to age in 1980, the participation rate of prime-age working males sees a steady
decline over the entire post-war period while that of females rises until 1999 and then begins to fall.
19
As measured in KLEMS, sector-specific labor represents an aggregate of a variety of labor types (i.e. gender, age, education,
labor status, etc.) with different sectors employing different compositions of labor types. Thus, there likely is a limit to how
substitutable different types of labor are across sectors. This limit means that declining trend labor growth in specific sectors
also reflects idiosyncratic changes in the composition of labor.
16
Table 3: Changes in Trend Value of Labor and TFP Growth Rates
1950 − 2016
1950 − 1982
1982 − 1999
1999 − 2016
Labor
TFP
Labor
TFP
Labor
TFP
Labor
TFP
Aggregate
-1.44
-0.51
-0.04
-0.47
-0.41
0.49
-0.99
-0.52
Common
Sector specific (total)
-0.11
-1.33
-0.08
-0.43
0.40
-0.44
-0.03
-0.44
-0.02
-0.39
0.12
0.36
-0.49
-0.50
-0.17
-0.35
Sector specific (by 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.
Oth. serv. (x-Gov)
Housing
0.13
-0.03
-0.01
-0.07
-0.96
-0.15
-0.07
-0.02
0.08
-0.08
-0.17
0.01
0.05
0.03
-0.08
0.00
0.00
0.01
-0.02
-0.14
0.13
-0.20
0.02
-0.03
-0.13
0.06
0.08
-0.13
-0.06
-0.01
0.01
-0.03
0.05
0.00
0.00
-0.02
-0.55
-0.08
-0.02
0.02
0.07
-0.05
-0.08
0.05
0.12
0.03
0.02
0.00
0.00
-0.04
-0.02
-0.14
0.13
-0.05
0.05
-0.04
-0.07
0.02
0.00
-0.13
-0.11
0.00
0.02
-0.07
0.05
-0.02
-0.01
0.02
-0.16
-0.13
-0.03
-0.01
0.02
0.08
-0.06
0.01
-0.05
-0.02
-0.06
0.00
0.00
0.06
0.00
-0.04
0.36
-0.07
0.01
0.04
-0.03
0.00
0.06
-0.05
0.04
0.00
-0.01
-0.01
0.03
-0.01
0.01
-0.07
-0.25
0.05
-0.01
-0.03
-0.02
-0.11
-0.03
-0.05
-0.01
0.02
-0.03
0.00
0.00
-0.01
0.00
0.04
-0.36
-0.08
-0.04
-0.03
-0.02
0.04
0.02
0.04
0.01
-0.01
-0.01
0.05
Notes: This table shows the change in the DFM trend growth rates over the period shown. For example, the first column
shows the DFM estimates of τ2016 − τ1950 . The first row shows the results for the share-weighted aggregate; the following two
rows decompose this aggregate change into the component associated with the common τ and the sector-specific τ ’s, which is
further decomposed by sector in the remaining rows.
or idiosyncratic purposes.20
Figures 5 and 6 illustrate the estimated common and sector-specific trends in the growth rates of labor
and TFP for each sector. Table 3 highlights selected values of the changes in the trends plotted in the
figures. From the beginning of the sample in 1950 until the end of the sample in 2016, the annual trend
20
As shown below, idiosyncratic trends in sectoral TFP growth can still move together in subsets of sectors, even if not across
all sectors, over different time periods. The sectoral composition of these subsets changes over time.
17
Figure 5: Sector-Specific and Common Growth Rate Trends for Labor from the DFM
Agriculture
4
2
2
0
Mining
Construction
Utilities
0
2
-1
0
0
-2
1960
1980
2000
2020
-2
1960
Durable goods
1980
2000
2020
Nondurable goods
0
-2
1960
1980
2000
2020
Wholesale trade
1960
0.5
1
0
0.5
-0.5
0
1980
2000
2020
Retail trade
0
-2
-1
-4
-2
-6
-1
1960
3
1980
2000
2020
Trans. & Ware.
1960
2000
2020
-0.5
1960
Information
2
2
1980
2
1980
2000
2020
FIRE (x-Housing)
1980
2000
2020
2000
2020
PBS
2
1
0
1960
1
0
1
-2
-2
-4
-1
1960
1980
2000
2020
1960
Educ. & Health
0
-1
0
1980
2000
2020
-1
1960
Arts, Ent., & Food svc.
2
1980
2000
2020
1960
Other services (x-Gov)
1980
Housing
1
1
1
0
0.5
0
0
-1
0
-1
1960
1980
2000
2020
-2
-2
1960
1980
2000
DFM Trend (Common)
2020
1960
1980
2000
2020
1960
1980
2000
2020
DFM Trend (Sector-specific)
Notes: The figure plots the DFM common and sector-specific trend estimates for each sector. The trends are
normalized to equal 0 in 1950.
rate of growth of aggregate labor fell from 1.92 percent to 0.48 percent, a decline of 1.44 percentage points.
Much of this decline (1 percentage point) occurred between 1999 and the end of the sample. The dynamic
factor model attributes most of the full-sample decline to sector-specific factors (1.33 percentage points) that
themselves primarily reflect labor growth declines in the Durable Goods sector (0.96 percentage points). As
shown in Figure 5, the secular decline in labor growth in Durable Goods has been large (almost 6 percentage
18
Figure 6: Sector-Specific and Common Growth Rate Trends for TFP from the DFM
Agriculture
Utilities
Mining
2
Construction
0
0
-0.5
-2
0
0
-0.5
-2
-1
-1
1960
4
1980
2000
2020
1960
Durable goods
1980
2000
2020
Nondurable goods
1
0
2
-4
1960
1980
2000
2020
Wholesale trade
1980
2000
2020
Retail trade
0.5
0
0.5
-1
1960
-0.5
0
0
-2
1960
1980
2000
2020
Trans. & Ware.
-1
1960
1980
2000
2020
1960
Information
2
1980
2000
2020
FIRE (x-Housing)
1960
1980
2000
2020
2000
2020
PBS
1
1
0
0
1
0.5
-1
0
-2
-3
1980
2000
2020
Educ. & Health
1960
2
1980
2000
2020
1960
Arts, Ent., & Food svc.
1
0
1
0
-1
0
-1
-2
-1
1960
1980
2000
2020
-2
-0.5
-1
1960
-1
0
1980
2000
2020
Other services (x-Gov)
1980
2000
DFM Trend (Common)
2020
1980
Housing
0
-0.5
-1
-2
1960
1960
1960
1980
2000
2020
1960
1980
2000
2020
DFM Trend (Sector-specific)
Notes: See notes to Figure 5.
` ,
points relative to 1950) and steady throughout the period. Overall, variations in sector-specific trends, τj,t
tend to be larger than those in common trends. This result underscores a diversity of sectoral experiences at
secular frequencies that stays otherwise hidden in analyses of long-run trends that rely solely on aggregate
data.
19
Similarly, the annual trend growth rate of aggregate TFP fell from 0.83 percent to 0.32 percent over the
course of the sample, a decline of 0.51 percentage points. As we saw in Figure 4, and further underscored in
Table 3, this decline was not monotonic: trend annual growth fell by half a percentage point over the period
1950 − 1982, then rebounded over the period 1982 − 1999, before falling again from 1999 to 2016. Over
the entire post-war period, less than a fifth of the decline is common to all sectors, and the largest sectorspecific declines were in Construction (0.14 percentage points, primarily in the first half of the sample)
and Nondurable Goods (a near steady decline of 0.20 percentage points over the entire sample period).
Remarkably, Figure 6 shows that in the period from 1950 − 1999, sector-specific TFP in Durables led to
an increase in aggregate TFP growth (0.49 percentage point in Table 3) that largely offset the decrease in
several other sectors including Construction, Nondurable Goods, Transportation and Warehousing, as well
as Professional and Business Services. However, since 1999, trend TFP growth in Durable Goods has fallen
from 4 to 1.1 percent per year (or 2.9 percentage points) and by itself contributed 0.36 percentage points
to the decline in aggregate TFP.21
To assess the implications of the sectoral changes highlighted in this section for the secular behavior of
GDP growth, one needs to be explicit about how secular change in one sector potentially impacts other
sectors. Put another way, one needs to account for the fact that sectors interact through various input-output
and capital flow relationships. We show in the next section that, in the presence of capital accumulation,
production linkages between sectors can significantly amplify the effects of structural change in a sector on
GDP growth.22
4
Changing Sectoral Trends and the Aggregate Economy
This section explores how the process of structural change in individual sectors, here captured by the
behavior of sectoral TFP and labor growth, shapes the behavior of GDP growth. Consistent with our TFP
calculations in Section 3, we consider a canonical multi-sector growth model with competitive product and
input markets. Each sector uses materials and capital produced in other sectors, and we allow for less than
full depreciation of capital within the period.
The empirical specification in Section 3 leads us to distinguish between persistent and transitory sectorlevel changes that can arise from either aggregate or idiosyncratic forces. We consider preferences and
technologies that are unit elastic in which case the economy evolves along a balanced growth path in the
long run. Capital accumulation, however, allows for variations in output growth off that balanced growth
path. Given linkages across sectors, structural change in an individual sector affects not only its own value
added growth but also that of all other sectors. In particular, capital induces network effects that amplify
the effects of sector-specific changes on GDP growth and that we summarize in terms of sectoral multipliers.
21
See Oliner et al. (2007) for the role of technological improvements in the IT sector as a driver of TFP growth in the Durable
Goods.
22
See Greenwood et al. (1997) for the importance of TFP growth in investment goods producing sectors as a driver of
aggregate growth.
20
We show that the approximation in equation (1) holds sector by sector in the special case where both
materials and capital are sector-specific. In contrast, the actual structure of the U.S. economy implies a
balanced growth equation for GDP that is markedly different, and considerably more nuanced, than the
simple relationship in equation (1).
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
E0
∞
X
βt
t=0
n
Y
cj,t θj
θj
j=1
,
n
X
θj = 1, θj ≥ 0,
j=0
where θj is the household’s expenditure share on final good j.23
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].
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 , used in 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
23
1−αj
, αj ∈ [0, 1].
Here, the representative household is assumed to have full information with respect to transitory and permanent changes,
as well as between idiosyncratic and common changes, as described in Section 3. In the Technical Appendix, we also consider
an imperfect information case in which the representative household cannot distinguish between permanent and transitory
components of exogenous changes to the environment. In this alternative scenario, the household faces an additional filtering
problem in which it must infer estimates of these components in deciding how much to consume and save in the face of exogenous
disturbances.
21
Capital accumulation in each sector follows
kj,t+1 = xj,t + (1 − δj )kj,t ,
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.
(5)
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. 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 or 1 in this case. 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 .
(6)
i=1
Structural change in a sector is captured by the composite variable, Aj,t , which reflects the joint behavior
of TFP and labor growth. In particular, under the maintained assumptions, sectoral value added may be
expressed alternatively as
vj,t = Aj,t
kj,t
αj
αj
,
where
∆ ln Aj,t = ∆ ln zj,t + (1 − αj )∆ ln `j,t .
(7)
As discussed in Section 3, TFP and labor growth each have persistent and transitory characteristics,
and each reflects sector-specific and common forces to different degrees. Moreover, observe in Figures 2 and
3 that trend movements in TFP and labor growth often appear unrelated for a given sector. For example,
trend TFP growth in Construction has seen a large decline since the 1950s while trend labor growth in that
sector has been close to flat and dominated by its common trend or demographics (see Figure 5). Similarly,
trend TFP growth in the Durable Goods sector is in 2016 close to where it started in 1950 (Figure 3), with
several notable up-and-down swings along the way, while trend labor growth in that sector has steadily
declined during the same period. Trend TFP growth in Professional and Business Services has gradually
declined over the post-war period while trend labor growth in that sector is essentially flat, and so on. These
observations suggest that changes in both labor supply and labor demand potentially play an important role
at secular frequencies and that neither force necessarily dominates the other. Furthermore, to the extent
22
that changes in labor demand are tied to technical progress, the relationship is not unambiguous. New
technologies can make workers more productive and increase labor demand or, on the contrary, reduce labor
demand if these new technologies are primarily labor-saving.24
In this paper, we take as given the joint behavior of {∆ ln zj,t , ∆ ln `j,t } in each sector j and interpret these
changes as concurrently describing the process of structural change in different sectors. Each component
of {∆ ln zj,t , ∆ ln `j,t } is modeled after the empirical specification in equation (3) with one small change.
Specifically, we let
x
x
x
τc,t
= (1 − ρ)gcx + ρτc,t−1
+ ηc,t
,
(8)
x
x
x
τj,t
= (1 − ρ)gjx + ρτj,t−1
+ ηj,t
,
(9)
and
x and η x ∀j are mean-zero random variables. We assume that ρ < 1 in which
where x = z or `, and ηc,t
j,t
case the economy is characterized by a balanced growth path in the long run that we describe below. For
x and τ x become those described in Section 3.
values of ρ arbitrarily close to 1, however, the processes for τc,t
j,t
Put another way, as in the empirical section, we allow the growth rates of sectoral TFP and labor to have
both transitory and persistent - though not quite permanent - components off the balanced growth path.
In the quantitative application, we consider values of ρ close to 1 and study transition paths implied by the
observed secular behavior of ∆ ln zj,t and ∆ ln `j,t .
Because observed labor growth, ∆ ln `j,t , is treated as given, we consider two polar cases that bound
counterfactual exercises studying the implications of trend labor growth changes in individual sectors. At
one extreme, counterfactual changes in trend labor growth within a sector are independent of changes in
trend labor growth in other sectors and, therefore, fully reflected as changes in the growth rate of the
aggregate labor force. At the other extreme, counterfactual changes in trend labor growth within a sector
are reallocated across all other sectors according to their respective labor shares. In the latter case, the
growth rate of the aggregate labor force is unaffected.
4.2
Model Parameters
Our choice of model parameters follows mostly Foerster et al. (2011) and is governed by the BEA InputOutput (IO) accounts and Fixed Asset Tables (FATs).
The consumption bundle shares, {θj }, 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
24
At business cycle frequencies, the behavior of labor and TFP changes across sectors also suggest an important role for
labor supply shocks. Specifically, absent supply shocks, idiosyncratic TFP shocks in the standard model imply strong negative
comovement in output growth and labor growth across sectors while, in the data, this comovement is unambiguously positive.
One way to generate positive comovement in output growth and labor growth across sectors from TFP changes is to assume
that these changes are dominated by aggregate or common forces. However, as discussed in Section 3, this does not appear to
be the case empirically.
23
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 },
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 consumption bundle parameters, {θj }, are likewise payments for consumption to
sector j as a fraction of total consumption expenditures.
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 }, 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.
The capital depreciation rates are chosen to be consistent with capital accumulation as described in the
BEA’s Fixed Asset Tables. We construct Divisia aggregates for our 16-sector aggregation from detailed real
net capital stocks and investment, and calculate capital depreciation rates such that net-stocks and investment are consistent with the capital accumulation equation in each sector. Because the implied depreciation
rates vary over time, we fix each sector’s depreciation rate at its post-2000 average as a benchmark.
For ease of presentation, we use the following notation throughout the paper: we denote the vector
of household expenditure shares by Θ = (θ1 , ..., θn ), the matrix summarizing value added shares in gross
output by sector, Γd = diag{γj }, the matrix of input-output linkages by Φ = {φij }, the capital flow matrix
by Ω = {ωij }, the matrix summarizing capital shares in value added by sector, αd = diag{αj }, and the
matrix summarizing sector-specific depreciation rates by δd = diag{δj }.
4.3
Some Benchmark Results in an Economy Without Growth
A special case of the economic environment presented above is one where αj = 0 ∀j, and {ln zj,t , ln `j,t } are
modeled as stationary processes in levels rather than growth rates, in which case ln Aj,t is also stationary
in levels around a constant mean in each sector. This special case reduces to the economy studied in Long
and Plosser (1983) - though in that paper, materials, mj,t , are used with a one-period lag - or Acemoglu
et al. (2012). Aggregate value added or GDP, Vt , is then equivalent to the aggregate consumption bundle,
24
Qn
j=1
cj,t
θj
θ j
, and
∂ ln Vt
= svj ,
∂ ln Aj,t
(10)
where svj is sector j’s value added share in GDP, and where these shares may be summarized in a vector,
sv = (sv1 , ..., svn ), given by sv = Θ(I − (I − Γd )Φ0 )−1 Γd . Consistent with Hulten (1978) or more recently
Gabaix (2011), a sector’s value added share entirely summarizes the effects of structural change in that
sector on the level of GDP. Accordingly, Acemoglu et al. (2012) refer to the object Θ(I − (I − Γd )Φ0 )−1 Γd
as the influence vector.25
When at least some sectors use capital in production, so that αj > 0 for some j, the economy becomes
dynamic and, absent shocks, converges to a steady state in levels in the long-run. Getting rid of the t
subscripts to denote variables in that steady state, and letting A = (A1 , ..., An ) represent the long-run
vector of composite exogenous sectoral states, a version of equation (10) holds in the limit as β → 1,
∂ ln V
= ηsvj ,
∂ ln Aj
(11)
where η is an adjustment factor approximately equal to the inverse of the mean labor share across sectors.
1
1−α . The value
−1
0
0 −1
0
0 −1
Θ[Γ−1
d (I −(I −Γd )Φ )−αd Ω ] /Θ[Γd (I −(I −Γd )Φ )−αd Ω ] 1,
In particular, when sectors use capital with the same intensity, αj = α ∀j, then η =
shares in this case,
sv ,
are given by
added
where
1 is a unit vector of size n. When β < 1, the influence vector also depends on sectoral depreciation rates, δj ,
and equation (11) holds as an approximation that depends on
of β, is close to
1.26
β
1−β(1−δj ) × δj
which, for standard calibrations
As underscored by Baqaee and Farhi (2017b), both equations (10) and (11) may be
interpreted in terms of macro-envelope conditions. When preferences and technology are Cobb-Douglas,
neither expressions (10) nor (11) has sectoral states, Aj , affecting value added shares, svj .
4.4
Balanced Growth
The effects of structural changes are less straightforward during transitions to the steady state in an economy
with capital. Furthermore, because the data described in Section 3 reveals important persistent components
in sectoral growth rates, we frame the effects of sectoral structural change on the aggregate economy in
terms of growth rate elasticities, ∂∆ ln Vt /∂∆ ln Aj,t .
In an economy with steady state growth, sectoral value added shares in GDP will depend on the entire
distribution of growth rates characterizing structural changes, ∆ ln Aj , j = 1, ..., n, in addition to reflecting
the sectoral network implicit in the IO and capital flow matrices. As described above, these shares are then
used to calculate overall GDP growth, ∆ ln V , by way of a Divisia index. More importantly, because of
production linkages, structural change in a sector, ∆ ln Aj , potentially helps determine value added growth
25
Observe that TFP scales value added rather than gross output in this paper. Consequently, the influence vector reflects
sectoral value added shares in GDP rather than shares of gross output in GDP or Domar weights.
26
See Appendix B.
25
in every other sector along the balanced growth path. This mechanism, therefore, amplifies the effects of
sector-specific structural change on GDP growth in a way that can be summarized in terms of a multiplier
for each sector. In some sectors, these multipliers can scale the impact of structural changes on GDP growth
by multiple times their share in the economy.
Consider a non-stochastic steady state path where all disturbances - persistent and transitory as well as
idiosyncratic and common - are set to zero. The non-stochastic steady state is defined by a balanced growth
path determined by constant growth in each sector,
∆ ln Aj,t = g j = g zj + (1 − αj ) g `j ,
(12)
g zj = λzj,τ gcz + gjz and g `j = λ`j,τ gc` + gj` .
(13)
where
In other words, sectoral structural growth in the steady state, g j , reflects steady state sectoral TFP growth,
g zj , and sectoral labor growth, g `j . The long-run growth rates of TFP and labor in each sector in turn reflect
aggregate (common) components, (λzj,τ gcz , λ`j,τ gc` ), and idiosyncratic components, (gjz , gj` ), respectively.
4.4.1
Sectoral Growth in Value Added
Let ∆ ln µvt = (∆ ln µv1,t , ...∆ ln µvn,t )0 denote the vector value added growth by sector. Then, along the
balanced growth path, ∆ ln µv is constant and given by
−1
Γd g a ,
∆ ln µv = I + αd Ω0 I − αd Γd Ω0 − (I − Γd )Φ0
|
{z
}
(14)
Ξ0
where g a = (g 1 , ..., g n )0 is the vector of sectoral structural growth rates.27 Equation (14) describes how
structural growth in a given sector, g j , affects value added growth in all other sectors, ∆ ln µvi . This
relationship involves the direct effects of sectors’ structural 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 materials
and investment, αd Ω0 Ξ0 g a . Specifically,
n
n
k=1
k=1
X
X
∂∆ ln µj
∂∆ ln µi
= 1 + αj
ωkj ξjk and
= αi
ωki ξjk ,
∂g j
∂g j
(15)
where (ξj1 , ..., ξjn ) is a column of Ξ0 equal to the Leontief inverse, (I − αd Γd Ω0 − (I − Γd )Φ0 )−1 , diagonally
weighted by the matrix of value added shares in gross output, Γd , in equation (14). Thus, along the balanced
growth path, sectoral linkages make it possible for a structural change in a given sector j, ∂g j , to affect
value added growth in every other sector, i, so long as that sector uses capital in production, αi > 0.
27
See Appendix C.
26
Otherwise, value added growth in a sector with αj = 0 is entirely determined by its own structural growth
rate,
∂∆ ln µj
∂g j
= 1. In this sense, the presence of capital accumulation plays a central role for the sectoral
growth implications of production linkages.
To gain intuition into how structural growth in different sectors help determine value added growth
in other sectors, observe that the effect of a change in sector j’s structural growth rate on sector i is
given by the (i, j) element of αd Ω0 Ξ0 . Each of these (i, j) elements in turn contains all of the elements
of the vector (ξj1 , ..., ξjn ) in the j th column of the weighted Leontief inverse, Ξ0 , as described in equation
(15). The j th column of the weighted Leontief inverse in turn will reflect the j th column of the transposed
capital flow matrix, Ω0 (i.e. the j th row of Ω or the degree to which sector j produces new capital for
other sectors), as well as the j th column of the transposed IO matrix, Φ0 , (i.e. the j th row of Φ or the
degree to which sector j produces materials for other sectors). To see this, observe that the Leontief
inverse, (I − αd Γd Ω0 − (I − Γd )Φ0 )−1 , can be alternatively expressed as the limit of (αd Γd Ω0 + (I − Γd )Φ0 ) +
(αd Γd Ω0 + (I − Γd )Φ0 )2 + (αd Γd Ω0 + (I − Γd )Φ0 )3 + ... + (αd Γd Ω0 + (I − Γd )Φ0 )n . Each column of Ω0 and
Φ0 is weighted by that column’s corresponding sector’s share of value added and materials in gross output
respectively, Γd Ω0 and (I−Γd )Φ0 . Ultimately, sectors that play a central role in producing capital or materials
for other sectors will be associated with a column of the weighted Leontief inverse, Ξ0 , whose elements are
relatively large. The individual elements of the Neumann series, (αd Γd Ω0 + (I − Γd )Φ0 )k , k = 1, .., n, describe
feedback effects in which a change in the structural growth rate of some sector, j, impacts the price and
quantities of j 0 s goods purchased by another sector, i, which in turn impacts the price and quantities of i’s
goods purchased by other sectors including j. This process then feeds back into the prices and quantities of
goods that sector j sells to sector i in the next round, and so on.
Table 4 shows the Capital Flow matrix of the U.S. economy, Ω, for the 16 sectors considered in this
paper. As the table makes clear, the production of investment goods in the U.S. is concentrated in relatively
few sectors, with many sectors not producing any capital for other sectors while Construction and Durable
Goods produce close to 80 percent of capital in almost every sector. Construction comprises residential
and non-residential structures, including infrastructure such as power plants or pipelines used to transport
crude oil for example, but also the maintenance and repair of highways, bridges, and other surface roads.
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. As a practical matter, the distinction between materials
and investment goods is not always straightforward. The BEA distinguishes between materials and capital
goods by estimating the service life of different commodities and, consistent with a time period in this paper,
commodities expected to be used in production within the year are defined as materials.28 From the Capital
28
As noted in Foerster et al. (2011), there are a couple of notable measurement issues related to the construction of the
Capital Flow table. First, the table accounts for the purchases of new capital goods but not used assets. Thus, for example, a
27
Table 4: Ω, Capital Flow Table
Agr
Min
Util
Const
Dur gds
Nd gds
Wh trd
Ret trd
T&W
Inf
FIRE (x-Hous)
PBS
Ed&H
A,E&FS
Oth serv
Housing
Agr
Min
Util
Const
0.00
0.00
0.00
0.12
0.70
0.00
0.10
0.04
0.02
0.00
0.00
0.02
0.00
0.00
0.00
0.00
0.00
0.50
0.00
0.07
0.31
0.00
0.05
0.01
0.01
0.02
0.00
0.04
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.47
0.36
0.04
0.04
0.01
0.01
0.03
0.00
0.04
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.03
0.76
0.00
0.08
0.04
0.02
0.04
0.00
0.04
0.00
0.00
0.00
0.00
Dur
gds
0.00
0.00
0.00
0.15
0.58
0.00
0.07
0.01
0.01
0.03
0.00
0.16
0.00
0.00
0.00
0.00
Nd
gds
0.00
0.00
0.00
0.14
0.61
0.00
0.08
0.02
0.01
0.02
0.00
0.12
0.00
0.00
0.00
0.00
Wh
trd
0.00
0.00
0.00
0.14
0.60
0.00
0.07
0.03
0.01
0.05
0.00
0.09
0.00
0.00
0.00
0.00
Ret
trd
0.00
0.00
0.00
0.52
0.35
0.01
0.04
0.02
0.01
0.02
0.00
0.03
0.00
0.00
0.00
0.00
T&W
Inf
0.00
0.00
0.00
0.14
0.69
0.00
0.06
0.01
0.01
0.02
0.00
0.06
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.20
0.52
0.00
0.06
0.02
0.01
0.10
0.00
0.10
0.00
0.00
0.00
0.00
FIRE
x-H
0.00
0.00
0.00
0.15
0.58
0.01
0.06
0.07
0.01
0.04
0.00
0.08
0.00
0.00
0.00
0.00
PBS
0.00
0.00
0.00
0.12
0.45
0.01
0.07
0.04
0.01
0.06
0.00
0.25
0.00
0.00
0.00
0.00
Ed&H A,E,FS Oth
sv
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.38
0.57
0.41
0.43
0.30
0.42
0.00
0.01
0.00
0.09
0.05
0.05
0.02
0.03
0.04
0.00
0.01
0.01
0.03
0.01
0.02
0.00
0.00
0.00
0.06
0.03
0.04
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
Hous
0.00
0.00
0.00
0.91
0.05
0.00
0.00
0.01
0.00
0.00
0.02
0.00
0.00
0.00
0.00
0.00
Notes: Ωij shows the share of investment goods used by sector j that originated in sector i. Entries are computed from the
1997 Capital Flow.
Flow table, we expect that columns of the Leontief inverse, Ξ0 , associated with Construction and Durable
Goods will have relatively large elements.
Table 5 shows the IO matrix or Make-Use table for the U.S. economy, Φ. Compared to the Capital Flow
table, the production of materials is considerably less concentrated in that all sectors produce materials
for all other sectors, though to varying degrees. From the Make-Use table, Professional and Business
Services, Finance and Insurance, and to a degree Nondurable Goods, all play an important role in providing
intermediate inputs to the U.S. economy. Observe that while Professional and Business Services figures
prominently in Φ, this sector is not nearly as dominant as Durable Goods or Construction are in Ω.
In contrast to the sectors that play a key role in Ω or Φ, output produced in sectors such as Agriculture,
Forestry, Fishing, and Hunting, Entertainment and Food Services, or Housing, is mostly consumed as a final
good. Therefore, columns of the Leontief inverse associated with these sectors will have elements that tend
to be small. An extreme case is housing which produces neither intermediate nor capital goods for other
sectors. But even for housing the diagonal element of Ξ0 is large since it reflects the value added share of
housing’s gross output, close to 90 percent.
Putting the information from the Capital Flow and Make-Use tables together, Table 6 shows the resulting
Leontief inverse matrix, weighted by the diagonal matrix Γd . As expected, elements of the Construction and
Durable Goods columns of the Leontief inverse are uniformly large, while those of the Agriculture, Forestry,
firm’s purchase of a used truck in a sector from another sector will not be recorded as investment in the capital flow table even
though the truck’s remaining service life may be well in excess of a year. Second, McGrattan and Schmitz (1999) note that a
non-trivial portion of maintenance and repair takes place using within sector resources, yet many of the diagonal elements of
the Capital Flow table are very small or zero. Results presented here are robust up to an adjustment that assumes that an
additional 25 percent of capital expenditures takes place within sectors.
28
Table 5: Φ, Make-Use Table
Agr
Min
Util
Const
Dur gds
Nd gds
Wh trd
Ret trd
T&W
Inf
FIRE (x-Hous)
PBS
Ed&H
A,E&FS
Oth serv
Housing
Agr
Min
Util
Const
0.39
0.01
0.01
0.01
0.04
0.26
0.10
0.00
0.05
0.00
0.07
0.04
0.00
0.00
0.00
0.00
0.00
0.27
0.02
0.04
0.15
0.10
0.05
0.00
0.08
0.02
0.05
0.22
0.00
0.01
0.00
0.00
0.00
0.32
0.02
0.04
0.02
0.11
0.03
0.00
0.17
0.02
0.08
0.16
0.00
0.02
0.01
0.00
0.00
0.02
0.00
0.00
0.38
0.13
0.09
0.15
0.04
0.02
0.03
0.13
0.00
0.01
0.01
0.00
Dur
gds
0.01
0.02
0.01
0.00
0.57
0.09
0.09
0.00
0.04
0.02
0.02
0.13
0.00
0.01
0.00
0.00
Nd
gds
0.13
0.16
0.01
0.00
0.06
0.37
0.07
0.00
0.05
0.01
0.01
0.11
0.00
0.01
0.00
0.00
Wh
trd
0.00
0.00
0.01
0.00
0.03
0.04
0.08
0.00
0.13
0.06
0.18
0.42
0.00
0.02
0.04
0.00
Ret
trd
0.00
0.00
0.01
0.00
0.03
0.05
0.05
0.01
0.14
0.06
0.27
0.33
0.02
0.01
0.02
0.00
T&W
Inf
0.00
0.00
0.01
0.01
0.07
0.21
0.07
0.01
0.27
0.02
0.12
0.19
0.00
0.01
0.01
0.00
0.00
0.00
0.00
0.00
0.12
0.04
0.04
0.00
0.03
0.37
0.07
0.25
0.00
0.06
0.02
0.00
FIRE
x-H
0.00
0.00
0.04
0.04
0.01
0.01
0.00
0.00
0.02
0.05
0.52
0.25
0.00
0.03
0.02
0.00
PBS
0.00
0.00
0.01
0.00
0.06
0.05
0.03
0.00
0.04
0.08
0.19
0.47
0.00
0.05
0.03
0.00
Ed&H A,E,FS Oth
sv
0.00
0.01
0.00
0.00
0.00
0.00
0.02
0.02
0.01
0.00
0.00
0.01
0.07
0.05
0.19
0.12
0.22
0.07
0.04
0.04
0.04
0.00
0.01
0.03
0.03
0.03
0.02
0.04
0.04
0.04
0.31
0.18
0.36
0.28
0.30
0.18
0.03
0.00
0.01
0.03
0.06
0.02
0.04
0.02
0.03
0.00
0.00
0.00
Hous
0.00
0.00
0.00
0.40
0.06
0.01
0.01
0.03
0.00
0.00
0.42
0.07
0.00
0.00
0.00
0.00
Notes: Φij shows the share of materials used by sector j that originated in sector i. Entries are computed from the 2015 BEA
Make and Use Tables.
Table 6: Ξ0 , Weighted Leontief Inverse
Agr
Min
Util
Const
Dur gds
Nd gds
Wh trd
Ret trd
T&W
Inf
FIRE (x-Hous)
PBS
Ed&H
A,E&FS
Oth serv
Housing
Agr
Min
Util
Const
0.53
0.01
0.01
0.01
0.02
0.07
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.02
0.01
0.01
0.08
1.06
0.16
0.05
0.06
0.18
0.03
0.03
0.05
0.04
0.03
0.02
0.03
0.04
0.03
0.05
0.02
0.01
0.67
0.01
0.01
0.02
0.01
0.01
0.01
0.01
0.03
0.01
0.01
0.01
0.01
0.01
0.09
0.09
0.18
0.61
0.07
0.09
0.07
0.10
0.07
0.10
0.09
0.05
0.05
0.10
0.07
0.54
Dur
gds
0.43
0.38
0.35
0.37
0.87
0.40
0.29
0.23
0.31
0.38
0.31
0.21
0.18
0.23
0.23
0.39
Nd
gds
0.14
0.07
0.07
0.07
0.08
0.50
0.04
0.04
0.09
0.06
0.05
0.04
0.05
0.08
0.04
0.06
Wh
trd
0.19
0.12
0.11
0.12
0.15
0.16
0.76
0.08
0.11
0.12
0.09
0.07
0.06
0.09
0.07
0.12
Ret
trd
0.03
0.02
0.03
0.05
0.02
0.02
0.02
0.60
0.02
0.02
0.03
0.02
0.01
0.02
0.02
0.06
T&W
Inf
0.07
0.06
0.07
0.05
0.05
0.07
0.06
0.06
0.60
0.05
0.04
0.03
0.03
0.04
0.03
0.05
0.05
0.05
0.05
0.04
0.05
0.05
0.06
0.05
0.05
0.69
0.07
0.05
0.04
0.05
0.04
0.04
FIRE
x-H
0.10
0.07
0.07
0.06
0.07
0.07
0.09
0.12
0.10
0.09
0.73
0.09
0.12
0.11
0.12
0.10
PBS
0.28
0.29
0.27
0.24
0.31
0.34
0.30
0.27
0.28
0.35
0.32
0.93
0.23
0.28
0.20
0.25
Ed&H A,E,FS Oth
sv
0.00
0.01
0.01
0.00
0.01
0.01
0.00
0.02
0.01
0.00
0.01
0.01
0.00
0.02
0.01
0.00
0.02
0.01
0.00
0.01
0.02
0.01
0.01
0.02
0.00
0.01
0.01
0.00
0.03
0.02
0.00
0.02
0.02
0.00
0.02
0.02
0.59
0.02
0.02
0.00
0.55
0.02
0.00
0.01
0.65
0.00
0.01
0.01
Hous
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.90
Notes: See text for definition of Ξ.
Fishing, and Hunting or Entertainment and Food Services columns are comparatively unimportant. Because
columns of the Leontief inverse determine the extent to which structural change in one sector affects value
added growth in every other sector, these columns also ultimately make up a network multiplier for each
sector that summarizes the importance of structural change in that sector for aggregate GDP growth.
Before turning our attention to sectoral multipliers, we make one last observation regarding the role of
29
sectoral linkages along the balanced growth path. In particular, consider the special case in which each
sector produces its own capital, Ω = I, as in Horvath (1998) or Dupor (1999), and its own materials, Φ = I.
In other words, production requires capital and materials but both are sector specific. In that case, there
are effectively no sectoral linkages and we have that
∆ ln µvj,t =
1
g z + g `j ,
1 − αj j
(16)
which is the sectoral analog to equation (1). In an environment where sectors mostly produce their own
capital and material, equation (1) will hold individually sector by sector.
4.4.2
Aggregate Balanced Growth Implications in a Canonical Model with Linkages
Given the vector of value added growth, ∆ ln µv , the Divisia aggregate index of GDP growth is ∆ ln V =
sv ∆ ln µv or
∆ ln V =
n
X
"
svj
gj +
n
X
i=1
j=1
αj ωij
n
X
#
ξki g k ,
(17)
k=1
so that, holding shares constant,
n
n
X
X
X
∂∆ ln V
ωkj ξjk +
svi αi
ωki ξjk ,
= svj + svj αj
∂g j
k=1
i6=j
(18)
k=1
where the second and third terms in this last expression capture the weighted network sectoral effects
discussed above in the context of the (diagonally weighted) Leontief inverse, Ξ0 .29 When all sectors use
little or no capital in production, αj = 0 ∀j, the effects of structural change in a given sector j, ∂g j , on
GDP growth will be well approximated by its share in the economy, svj . This case recovers a version of
of Hulten’s theorem (1978) but in growth rates. More generally, equation (18) suggests the presence of a
network multiplier effect that varies by sector and that depends not only on the importance of sectors as
suppliers of capital and materials to other sectors, ωki and ξjk , but also on the extent to which the latter
sectors use capital in their own production, αi . The effects of sectoral change, ∂g j , on GDP growth may
be summarized as a direct effect, sv I, and an additional indirect effect resulting from sectoral linkages,
sv αd Ω0 Ξ0 . Hence, we define the combined direct and indirect effects of structural change on GDP growth in
terms of network multipliers, sv (I + αd Ω0 Ξ0 ).
29
Along the balanced growth path, sectoral value added shares in GDP, sv , are functions of the model’s underlying parameters including, in the general case, the distribution of sectoral structural growth rates, g a . As shown in the Tech
Q
Qn
Θ[Γ−1
(I−(I−Γd )Φ0 )−αd Gd Ω0 ]−1
−ωki ξ`k
d
where Gd = diag 1 − (1 − δi ) n
∆i with
nical Appendix, sv =
−1
k=1
`=1 (1 + g ` )
0
0 −10
∆i =
Θ[Γd (I−(I−Γd )Φ )−αd Gd Ω
Qn
ξkj θj
j=1
k=1 (1+g k )
hQ
i
Qn
ξ
(θ −ωji )
n
1−β(1−δi )
(1+g k ) kj j
j=1
k=1
v
∂sj
β
in a sector k,
P
sum to 1, j
]
1
Qn
∂g k
∂sv
j
∂g k
. Furthermore, changes in sectoral shares induced by a change in structural growth
tend to be small for overall GDP growth, consistent with Figure 1, and the notion that since shares must
= 0.
30
Table 7: Sectoral Network 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.
Oth. serv. (x-Gov)
Housing
sv
sv αd Ω0 Ξ0
sv (I + αd Ω0 Ξ0 )
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.
Table 7 shows the direct and combined effects of structural change in the different sectors we consider
on GDP growth given Tables 4 and 5. The importance of Construction and Durable Goods as suppliers
of investment goods results in their having not only a large value added share in GDP, 5 and 13 percent
respectively, but also in their having large spillover effects on other sectors. In particular, the network
multipliers for the Construction and Durable Goods sectors come out to more than 3 times their share,
0.17 and 0.42 respectively. Considering that trend TFP growth in Construction fell by nearly 5 percentage
points between 1950 and 2016 in Figure 3, this gives us, all else equal, a roughly 0.7 percentage point effect
on trend GDP growth from TFP changes in Construction alone. In practice, the aggregate growth effects
from combined changes in TFP and labor in Construction are likely smaller since labor growth has been
moderately positive in that sector throughout the post-war. Nevertheless, the effect from Construction is
large not only because of its central role as a producer of capital, including commercial and residential
structures for all sectors, but also because capital depreciates only partially in any given year. It is apparent
from Table 7 that the effects of structural change on GDP growth are always at least as large as sectoral
shares. The sectoral network multipliers roughly double the share of Professional and Business Services, from
0.09 to 0.25, and Wholesale Trade from 0.07 to 0.15. In other sectors, such as Agriculture, Forestry, Fishing
and Hunting, Entertainment and Food Services, or Housing, the network multipliers are small or negligible
31
as suggested by the corresponding columns of the weighted Leontief inverse. Because the same network
relationships embodied in the Capital Flow table, Ω, and Make-Use table, 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 7 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 in
aggregate TFP growth. For example, consider purely idiosyncratic changes in TFP growth, ∂g zj , that leave
P
aggregate TFP growth unchanged, nj=1 svj ∂g zj = 0. Despite aggregate TFP growth not changing, these
idiosyncratic changes may nevertheless have an effect on GDP growth since the sum of sectoral multipliers
is larger than 1.
Finally, we return to Table 2 and the calculation in Section 3.2, but this time using the expression
derived for the economy with production linkages across multiple sectors, equation (17). In this expression,
we continue to use the numbers in Table 2, specifically the second and third columns for g zj and g `j , the
fourth column for svj , together with the BEA estimates of αd , Γd , Ω0 and Φ0 . Using these estimates, equation
(17) now gives a growth rate for GDP that jumps to 2.97 percent compared to 2.42 percent in the aggregate
one-sector model and 3.24 percent in the data.
Evidently, the balanced growth equations (1), (14), and (17) hold only in the steady state and ignore
important endogenous dynamics driven by capital accumulation. In other words, these equations all miss
dynamic terms that capture deviations from steady state growth. Thus, we now turn our attention to the
model solution and associated Markov decision and policy rules.
5
Quantitative Findings
Having described the balanced growth path implications of sectoral structural change, we now interpret the
secular trend in aggregate GDP growth estimated in Section 2 as part of a slow-moving transition towards
a balanced growth path. Therefore, to assess how sectoral trends in different sectors play out in other
sectors over time, as well as in terms of overall GDP growth, we first work out the transition dynamics of
the model. Given the economy’s balanced growth path, we normalize the model’s variables by appropriate
factors (these are also the factors used in Appendix C) so as to make the model’s optimality conditions and
resource constraints stationary in the detrended variables. The scaling factors used in detrending differ with
the variables under consideration but always involve the entire distribution of growth rates summarized in
g a . Detrended variables are constant along the balanced growth path and while consumption, investment,
materials, and gross output all grow at the same rate within each sector, these variables grow at different
rates across sectors. Prices in different sectors, as well as their implied price indices, also all grow at different
rates along the balanced growth path but in such a way as to generate constant shares along that path. After
linearizing the model’s dynamic equations around the steady state in detrended variables, Markov decision
and policy rules may be readily obtained using standard linear rational expectations solution toolkits,
32
including in this case King and Watson (2002).30
5.1
Historical Decompositions
In exploring the aggregate implications of structural change in individual sectors, the previous section
highlighted the importance of accounting for production linkages and their network effects. In other words,
in any counterfactual exercise, individual sectors cannot be modeled in isolation. In this section, we wish to
recover the dynamic implications of the model for sectoral value added growth, ∆ ln vj,t , and aggregate GDP
P
growth, ∆ ln Vt = nj=1 svj ∆ ln vj,t , when driven by the different sectoral processes estimated in Section 3.
In Appendix D, we show that during transitions to the balanced growth path, the vector of value added
growth rates, ∆ ln vt , evolves according to
∆ ln vt = I + αd Ω0 Ξ0 g a + αd ∆k̂t + Ât + αd Ω0 Ξ0 Ât−1 ,
(19)
where the vector k̂t = (k̂1,t , ..., k̂n,t )0 denotes the log deviations of detrended capital from its appropriate
level and Ât = (Â1,t , ..., Ân,t )0 with Âj,t = ∆ ln Aj,t − g j . The vector k̂t follows from the model’s equilibrium
Markov decision rules and is thus a function of previous endogenous states, k̂t−1 , and the various shocks
affecting the economy (i.e. persistent and transitory, as well as idiosyncratic and common). The first term
on the right-hand side of (19) captures the sectoral steady state growth paths described in equation (14).
The additional terms represent the dynamics of value added growth induced by the driving processes, Ât ,
and by endogenous adjustments to capital in different sectors, ∆k̂t . In the absence of shocks, ∆k̂t and Ât
are zero in the long run and ∆ ln vt = (I + αd Ω0 Ξ0 ) g a as in equation (14).
Equation (19) implies both contemporaneous and lagged effects of structural changes, Ât , on sectoral
value added growth, ∆ ln vt , towards balanced growth. The contemporaneous effect of a structural change in
sector j, ∂∆ ln Aj,t , on its own value added, ∆ ln vj,t , is one-for-one whereas this effect is zero on value added
growth in other sectors. It follows that in the period in which a sector experiences a structural change, the
effect of that change on GDP growth is entirely captured by that sector’s share in GDP,
∂∆ ln Vt
∂∆ ln Aj,t
= svj .
Thus, a version of Hulten’s theorem (1978) holds in growth rates on impact.
In subsequent periods, the change in GDP growth stemming from sectoral structural changes will reflect
the network effects of production linkages, sv αd Ω0 Ξ0 ∆ ln At . Along transitions, structural changes will also
affect GDP growth indirectly in-so-far as they are reflected by endogenous changes in the capital stock of
different sectors, sv αd ∆k̂t , through optimal investment decisions.
Before assessing the effects of structural changes in individual sectors, we discuss the dynamic model’s
behavior when driven by the processes estimated in Section 3. Relative to the empirical model in that
section, the structural model adds two key features: a rich network of production linkages in both materials
and capital, and endogenous dynamics by way of capital accumulation. Figure 7 illustrates the behavior of
30
For a detailed description of this sequence of steps and all their subsequent implications for this section, see the Technical
Appendix.
33
Figure 7: Sectoral Value-Added Growth Rates Data and Model
(Percentage points at an annual rate)
Agriculture
Mining
20
20
0
0
Utilities
Construction
10
20
5
0
0
-20
-20
1960
1980
2000
2020
-20
1960
Durable goods
1980
2000
2020
1960
2000
2020
Wholesale trade
Nondurable goods
10
1980
10
1980
2000
2020
Retail trade
10
10
5
5
0
0
0
0
-10
1960
10
1960
1980
2000
2020
Trans. & Ware.
1980
2000
2020
1960
Information
5
0
-5
-10
1960
1980
2000
2020
FIRE (x-Housing)
10
10
5
5
0
0
-5
-5
-10
-10
1960
1980
2000
2020
2000
2020
PBS
10
5
0
-5
1960
10
1980
2000
2020
1960
Educ. & Health
1980
2000
2020
1960
Arts, Ent., & Food svc.
10
1980
2000
2020
Other services (x-Gov)
10
0
2
0
-10
1980
2000
2020
Housing
4
0
0
1960
1980
6
5
5
1960
1960
1980
2000
Data (Cyclically Adjusted)
2020
1960
1980
2000
2020
1960
1980
2000
2020
Model-Implied Growth Rates
Notes: The figure shows the growth rate in value added for each sector. The model-implied value added is computed
from equation (19) with values of Âj,t from the DFM model.
value added growth in each sector, ∆ ln vj,t , in the data and in the model. As the figure makes clear, the
model performs remarkably well in capturing sectoral value added growth simultaneously across virtually
every sector. The fit is particularly close in some key sectors highlighted above, such as Construction and
Durable Goods, as well as most other sectors. The model misses somewhat on the dynamics of Wholesale
34
Figure 8: GDP Growth - Data and Model
A: Growth Rates
8
Cyclically Adjusted Data
Benchmark Model
Balanced Growth Approximation
6
4
2
0
-2
1950
1960
1970
1980
1990
2000
2010
2020
1990
2000
2010
2020
B: Trends
5
4
3
2
Cyclically Adjusted Data, 15-Year Centered MA
Benchmark Model
Balanced Growth Approximation
1
0
1950
1960
1970
1980
Notes: Panel (A) shows share-weighted value added growth rates from Figure 7. Panel (B) shows 15-year centered
moving averages from panel (A) (thin black line) and the model-implied values of growth rates using equation (19)
and the τ -components of Âj,t . The dashed lines are balanced growth approximations using equation (17) but with
the disturbances to TFP and labor growth in place of ḡj .
Trade in the early part of the sample.
In Figure 7, a downward trend is clearly noticeable in Construction more than in any other sector.
This feature of the data is significant in that the Construction sector was also associated with a large
35
multiplier in the previous section. This large multiplier in turn reflected the importance of that sector in
producing capital (commercial and residential structures as well as equipment) for almost every sector in the
economy. A downward trend is also noticeable in Professional and Business Services though considerably
less pronounced.
Figure 8A illustrates the behavior of GDP growth in the model and in the data. The red line depicts
the growth rate of GDP as approximated by the balanced growth formula in equation (17) but driven by
the full set of disturbances to TFP and labor growth in place of g j in that equation. In other words, the
red line shows the model’s predicted GDP growth rate implied by the dynamics embodied in the exogenous
driving processes but absent any internal (endogenous) dynamics from the model. The figure shows that
this predicted growth rate is able to move ‘locally’ with the data. However, it is also considerably more
volatile than actual GDP growth. In contrast, the blue line depicts the behavior of GDP growth in the full
model as implied by the law of motion for sectoral value added growth in equation (19) together with the
Markov decision rules for k̂t . This line is considerably smoother and closely matches the actual behavior of
GDP growth (in black). The model underpredicts GDP growth slightly in the late 1990s and overstates the
level of GDP growth somewhat in the 2000s but it matches its overall rate of decline almost exactly during
the latter period.
Figure 8B illustrates the implications of the dynamic multi-sector model for trend GDP growth. Specifically, it illustrates the model’s solution when driven only by the disturbances associated with the persistent
x and ∆τ x , x = `, z. The red line represents the trend growth
components of labor and TFP growth, ∆τc,t
j,t
rate implied by the balanced growth approximation (17) when driven only by these shocks. The blue line in
Figure 8B is the model-implied trend growth rate of GDP taking into account the model’s internal dynamics. The black line depicts a 15−year centered moving average of actual (cyclically adjusted) GDP growth.
The full model indicates that trend GDP growth rate has steadily declined by about 2.2 percent over the
post-war period. In the next sections, therefore, we unpack this slow decline in GDP growth in terms of the
history experienced by individual sectors through counterfactual exercises.
5.2
Implications of Changing Sectoral Trends for GDP Growth
The multi-sector balanced growth model of Section 4 provides an accounting framework for decomposing
the changes in the trend growth rate of GDP shown in Figure 8B into sources associated with common and
sector-specific shocks. More precisely, using equation (19) and the definition of At , the model-based trend
in GDP growth rates can be written as its balanced growth value, say µ∆V , and a distributed-lag of past
trend shocks:
n h
h
i X
i
z
`
z
`
∆ ln Vt = µ∆V + βc,z (L)∆τc,t
+ βc,` (L)∆τc,t
+
βj,z (L)∆τj,t
+ βj,` (L)∆τj,t
,
(20)
j=1
where the distributed lag coefficients, β(L), are functions of the model parameters and the factor loadings
that link the common shocks to the sectors. Figure 9 plots the historical contribution of each sector to
36
Figure 9: Model-Based Historical Decomposition of Trend GDP Growth Rates:
Contributions from Sector-Specific Trends
Agriculture
Mining
0.1
0.04
Utilities
0
0
Construction
-0.2
-0.02
0
0.02
-0.4
0
-0.04
-0.1
1960
1980
2000
2020
1960
1980
2000
2020
Nondurable goods
Durable goods
0.1
0
0.2
0
-0.6
1960
1980
2000
2020
Wholesale trade
1960
0
0
-0.05
2000
2020
Retail trade
0.05
0.05
1980
-0.2
-0.2
-0.4
-0.4
1960
0
1980
2000
2020
Trans. & Ware.
-0.05
1960
2000
2020
-0.1
1960
Information
0.05
-0.05
1980
0
1980
2000
2020
FIRE (x-Housing)
1980
2000
2020
2000
2020
PBS
0
-0.1
0
-0.05
-0.1
1960
-0.2
-0.05
-0.3
-0.1
-0.15
-0.1
1960
1980
2000
2020
-0.4
1960
Educ. & Health
1980
2000
2020
Arts, Ent., & Food svc.
0.1
1960
0.05
1980
2000
2020
1960
Other services (x-Gov)
1980
Housing
0
0.02
0
0
-0.1
1960
1980
2000
2020
-0.05
0
0.01
-0.1
-0.05
1960
1980
2000
2020
1960
1980
2000
2020
1960
1980
2000
2020
z
`
Notes: These figures show the effect of the sector-specific trend shocks (∆τj,t
and ∆τj,t
) on the growth rate of GDP
using the model-based dynamic multipliers. The trend growth rates are normalized to 0 in 1950.
z + β (L)∆τ ` ; Figure 13 plots the contribution of the common shocks,
∆ ln Vt , that is ζj,t = βj,z (L)∆τj,t
j,`
j,t
z + β (L)∆τ ` (in each figure, pre-sample values of ∆τ are set to zero, so initial values of
ζc,t = βc,z (L)∆τc,t
c,`
c,t
ζ are equal to zero.)
The sectoral results shown in Figure 9 suggest that, over the entire post-war period, the Construction
sector is the largest single source of decline in trend GDP growth at approximately 0.6 percentage points.
37
That is, more than 25 percent of the trend decline in GDP growth in the last 60 years is associated with this
sector alone. Other sectors that contributed materially to the fall in trend GDP growth in the last 6 decades
include Professional and Business Services and Nondurable Goods with both sectors contributing more than
0.3 percentage point loss. The Durable Goods sector contributed large decade-long up-and-down swings in
trend GDP growth between 1950 and the end of the sample. The most-recent upswing in Durable Goods,
contributing by itself almost 0.5 percentage points to trend GDP growth between the late 1980s and 2000,
partly reflects technical progress in the production of semiconductors that was much publicized during that
period. The subsequent decrease in Durable Goods accounts for nearly half (0.7 percentage points) of the
1.5 percentage point post-1999 decline in the growth rate of GDP seen in Figure 8B. Finally, some sectors,
such as Information and Wholesale Trade, have somewhat offset the longer run decrease in GDP growth.
Figure 10 looks more closely at the aggregate effects of structural change in the construction sector. Panel
A shows the composite effect of construction-specific changes in trend values of labor and TFP that was
` ) and TFP
shown previously in Figure 9. Panels B and C show the separate effects from labor (βj,` (L)∆τj,t
z ). Evidently, decreases in the rate of growth of TFP in the construction sector explain most of
(βj,z (L)∆τj,t
its deleterious effect on trend GDP growth. Of course, one interpretation of these historical decompositions
is as counterfactual paths of GDP growth setting all other shocks to zero. In this interpretation, Panel A
of Figure 10 shows the path of the trend growth rate of GDP under the counterfactual absence of changes
z = ∆τ ` = τ z = ∆τ ` = 0 for j 6=
in common and sector-specific shocks other than construction (∆τc,t
c,t
j,t
j,t
Construction). This interpretation of Figure 10 raises the question of how to account for the reduction in
` in the construction sector. Did it disappear, as implicitly assumed in Panels A
labor associated with ∆τj,t
and C, or did it move to other sectors? Panel D shows the counterfactual historical path of trend growth rates
in GDP after allocating all of the sectoral losses of labor in Construction to the other sectors in proportion
to their labor shares, so there is no change in the aggregate supply of labor.31 Because the behavior of labor
growth is not the dominant force driving secular changes in the Construction sector, panels A and D show
similar historical paths.
Figures 11 and 12 repeat this exercise for two other important sectors: Nondurable Goods and Professional and Business Services. Taken together, Figures 11 and 12 underscore the importance of sector-specific
structural change for trend GDP growth. Remarkably, the Construction, Nondurable Goods, and Professional and Business Services sectors account for around 60 percent of the 2.2 percentage point decrease in
trend GDP growth since 1950.
In contrast, Figure 13 illustrates the implications of the common or aggregate trends in labor and TFP
growth estimated in Section 4. Figure 13 suggests that these common trends have not made a material
contribution to the aggregate trend decline in GDP growth since 1950. Thus, sector-specific rather than
aggregate factors shaped the major part of the secular behavior of GDP growth over the period 1950 − 2016.
31
Recall also our discussion of the limits on cross-sectoral substitution of KLEMS sectoral labor measures in footnote 19.
38
Figure 10: Model-Based Historical Decomposition of Trend Growth Rates:
Contributions from Construction
A: TFP and Labor
0.2
0
0
-0.2
-0.2
-0.4
-0.4
-0.6
-0.6
-0.8
-0.8
-1
1950
1960
1970
1980
1990
2000
2010
-1
1950
2020
C: Labor
0.1
B: TFP
0.2
0.2
0.05
1960
1970
1980
1990
2000
2010
2020
D: TFP and Labor with Labor Reallocation
0
0
-0.2
-0.05
-0.4
-0.1
-0.6
-0.15
-0.8
-0.2
-0.25
1950
1960
1970
1980
1990
2000
2010
-1
1950
2020
1960
1970
1980
1990
2000
2010
2020
z
`
Notes: Panels (A)-(C) show the effect of the sector-specific trend shocks (∆τj,t
and ∆τj,t
) on the growth rate of GDP
`
using the model-based dynamic multipliers. Panel (D) shows these effects after reallocating ∆τj,t
to the other sectors.
Blue dots indicate 68 percent credible intervals conditional on balanced growth model parameters. The trend growth
rates are normalized to 0 in 1950.
5.3
Implications for Future Trend Growth
Given the trends in labor growth and TFP growth estimated in this paper, and their implications for the
aggregate U.S. economy through sectoral multipliers, a natural question is: what do those trends imply
going forward? As indicated by equation (19), disturbances to trend labor growth and TFP growth are
39
Figure 11: Model-Based Historical Decomposition of Trend Growth Rates:
Contributions from Nondurable Goods
A: TFP and Labor
0.2
B: TFP
0.1
0.1
0
0
-0.1
-0.1
-0.2
-0.2
-0.3
-0.3
-0.4
-0.4
-0.5
-0.6
1950
1960
1970
1980
1990
2000
2010
-0.5
1950
2020
C: Labor
0.05
0.2
1960
1970
1980
1990
2000
2010
2020
D: TFP and Labor with Labor Reallocation
0.1
0
0
-0.05
-0.1
-0.1
-0.2
-0.15
-0.3
-0.2
1950
1960
1970
1980
1990
2000
2010
-0.4
1950
2020
1960
1970
1980
1990
2000
2010
2020
Notes: See notes to Figure 10.
endogenously propagated over time through investment decisions and thus have persistent effects. The
random walk dynamics of the trend components of labor growth and TFP growth imply that forecasts of
these components are equal to their value in 2016, the last year of the data.
Figure 14 shows predicted trend growth rates for GDP from the dynamic multi-sector model as past
disturbances play out through the model’s internal dynamics. The figure indicates that absent the realization
of positive and persistent disturbances to TFP and labor growth, trend GDP growth will continue to fall by
40
Figure 12: Model-Based Historical Decomposition of Trend Growth Rates:
Contributions from Professional and Business Services
A: TFP and Labor
0.2
0
0
-0.2
-0.2
-0.4
-0.4
-0.6
-0.6
-0.8
1950
1960
1970
1980
1990
2000
2010
-0.8
1950
2020
C: Labor
0.3
0.2
0.2
0
0.1
-0.2
0
-0.4
-0.1
-0.6
-0.2
1950
1960
1970
1980
1990
2000
B: TFP
0.2
2010
-0.8
1950
2020
1960
1970
1980
1990
2000
2010
2020
D: TFP and Labor with Labor Reallocation
1960
1970
1980
1990
2000
2010
2020
Notes: See notes to Figure 10.
around 0.60 percentage points over the next 10 years. In this scenario, both trend labor and TFP growth
remain constant, but the internal dynamics of capital accumulation imply continued slowing of trend GDP
growth. The 68 percent credible bands indicate there is substantial uncertainty about the exact level of
trend GDP going forward but, absent positive shocks, the model predicts a slowdown ahead nonetheless.
Table 8 shows the contributions of the common and sector-specific trend components in generating the
decline in trend GDP growth of around 0.65 percentage points going forward. While there is a role for
41
Figure 13: Model-Based Historical Decomposition of Trend Growth Rates:
Contributions from Common Trends
1.5
1
0.5
0
-0.5
-1
1950
1960
1970
1980
1990
2000
2010
2020
z
`
Notes: This figure show the effect of the common trend shocks (∆τc,t
and ∆τc,t
) on the growth rate of GDP using
the model-based dynamic multipliers. Blue dots indicate 68 percent credible intervals conditional on balanced growth
model parameters. The trend growth rates are normalized to 0 in 1950.
common trends, they play a smaller role than the sector-specific trends, particularly in TFP growth. The
decomposition into each individual sector shows that a decline in Durable Goods is the key driver of lower
trend GDP growth in the future. The fall in large part reflects the propagation of the collapse associated
with the Durable Goods sector that began in 2000 (see Figure 9). This effect is mitigated somewhat by
a turn-around in Construction. However, the marked slowdown in Durable Goods affects other sectors
42
Figure 14: In-Sample and Predicted Out-of-Sample Trend Growth Rate of GDP
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
1950
1960
1970
1980
1990
2000
2010
2020
2030
2040
Notes: Blue dots indicate 68 percent credible intervals conditional on balanced growth model parameters.
through production linkages, and through capital accumulation implies a pervasive impact that is sustained
10 years into the future.
6
Concluding Remarks
In this paper, we estimate trends in TFP and labor growth across major U.S. production sectors and
explore the role they have played in shaping the secular behavior of GDP growth. We find that trends in
43
Table 8: Model-based Forecasts of Changes in Trend GDP Growth
2016 − 2021
2021 − 2026
VA
Labor
TFP
VA
Labor
TFP
Aggregate
-0.49
-0.15
-0.34
-0.16
-0.03
-0.13
Common
Sector specific (total)
-0.16
-0.33
-0.05
-0.10
-0.11
-0.23
-0.03
-0.13
0.00
-0.03
-0.04
-0.10
Sector specific (by 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.
Oth. serv. (x-Gov)
Housing
-0.00
-0.02
-0.00
0.02
-0.31
0.00
-0.01
0.00
-0.00
-0.00
-0.00
-0.00
-0.00
-0.00
0.00
-0.01
-0.00
-0.01
-0.00
-0.01
-0.08
0.00
-0.00
0.00
-0.00
-0.00
0.00
-0.00
0.01
-0.00
0.00
0.00
0.00
-0.01
0.00
0.02
-0.22
0.00
-0.00
0.00
-0.00
0.00
-0.00
-0.00
-0.01
0.00
0.00
-0.01
-0.00
-0.01
-0.00
0.01
-0.12
0.00
-0.00
-0.00
-0.00
0.00
-0.00
-0.00
-0.00
0.00
0.00
-0.00
-0.00
-0.00
-0.00
-0.00
-0.03
-0.00
-0.00
0.00
-0.00
-0.00
0.00
-0.00
0.00
-0.00
0.00
0.00
-0.00
-0.00
0.00
0.01
-0.09
0.00
-0.00
-0.00
-0.00
0.00
-0.00
-0.00
-0.01
0.00
0.00
-0.00
TFP and labor growth have generally decreased across a majority of sectors since 1950. More than 3/4
of the secular decline in aggregate TFP growth results from the combination of sector-specific rather than
aggregate disturbances. Similarly, trend labor growth has also been dominated by sector-specific factors,
especially after 1980 and the latter part of the post-war period.
We embed these findings into a dynamic multi-sector framework in which materials and capital used
by different sectors are produced by other sectors. The presence of capital, in particular, allows changes
in TFP or labor growth in a given sector to affect value added growth in every other sector. This feature
leads to quantitatively important sectoral multiplier effects on GDP growth that reflect the importance of
different sectors as suppliers of capital or materials to other sectors. The strength of these linkages result
44
in GDP growth multipliers that for some sectors can be as large 3 times their value added share.
Ultimately, sector-specific rather than aggregate factors in TFP and labor growth explain the major part
of low frequency variations in U.S. GDP growth. Changing sectoral trends in the last 6 decades, translated
through the economy’s production network, have on net lowered trend GDP growth by around 2.2 percentage
points. The Construction sector, more than any other sector, stands out for its contribution to the trend
decline in GDP growth over the post-war period, accounting for 30 percent of this decline. Moreover, the
process of capital accumulation means that these structural changes have endogenously persistent effects.
Thus, absent the realization of predominantly positive and persistent disturbances to TFP and labor growth,
we estimate that trend GDP growth will continue to fall over the next 10 years.
45
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49
Appendix A: Estimation of the Multivariate Unobserved Components
Model
x + λx εx + τ x + εx , where x = z or ` and j = 1, ..., n. The τ
The DFM has the form: ∆ ln xj,t = λxj,τ τc,t
j,ε c,t
j,t
j,t
components evolve as randomwalks and the ε components
as white noise. More specifically, we assume that
x , εx , {∆τ x , εx }n
the 2(n + 1) × 1 vector ζt = ∆τc,t
c,t
j,t j,t j=1 is i.i.d. with ζ ∼ N (0, Σζ ), where Σζ is a diagonal
matrix. The factor loadings and scale of the common factors are not separately identified; thus we normalize
Pn
Pn
x
x
x ) = var(εx ) = 1,
var(∆τc,t
c,t
j=1 λj,τ ≥ 0, and
j=1 λj,ε ≥ 0 to identify the scale and sign of the factors.
We estimate the model using Bayesian methods. We use independent priors for all the parameters. The
x ) and var(εx )
priors for the factor loadings are λxj,τ ∼ N (0, 1) and λxj,ε ∼ N (0, 16). The priors for var(∆τj,t
j,t
are inverse Gamma priors with shape and scale parameters that correspond to Tprior observations with
2
x ) and T
2
2
s2prior = ωprior
/Tprior . We use Tprior = 2 and s2prior = 0.252 for var(∆τj,t
prior = 2 and sprior = 3.00
for var(εxj,t ).
The posterior is approximated using Gibbs-MCMC methods.
Let X denote the data,
ζ denote the values
x ), var(εx
for ζt for t = 1, ..., T , and θ denote the parameter values λxj,τ , λxj,ε , var(∆τj,t
j,t for j = 1, ...n. The
first Gibbs step draws ζ|(X, θ) using standard formulae related to the Kalman smoother. The second Gibbs
step draws θ from the posterior of θ|(X, ζ) that is also known in closed form from familiar linear regression
formulae. The estimates reported in the paper rely on one million MCMC draws after 5k initial draws.
The Technical Appendix contains additional details.
Appendix B: The Influence Vector in a Model with Capital
The competitive equilibrium of the economy described in the main text is defined as a set of gross output
x
prices, pyj,t , value added prices, pvj,t , materials prices, pm
j,t , and investment goods prices, pj,t , for all sectors j
and dates t, and allocations, (cj,t , Ct , xij,t , mij,t , xj,t , mj,t , vj,t , yj,t , kj,t+1 , `j,t ), such that
i) the representative household maximizes its utility,
ii) the representative firm in each sector maximizes profits
iii) all markets clear.
Absent frictions, equilibrium allocations may be found by solving a planner’s problem whose corresponding decentralization and definitions of prices are straightforward.
Suppose that sectoral structural changes embodied in {zj,t , `j,t } are stationary in levels, with nonstochastic steady state denoted by {zj , `j }. Then, in the absence of shocks, the economy converges to
a steady state in the long-run indexed by A = (A1 , ..., An ), where ln Aj = ln zj + (1 − αj ) ln `j , j = 1, ..., n.
In the steady state, first-order conditions imply a set of relationships between the prices of materials,
investment, and value added,
Θ ln py = 0,
50
ln pv = Γ−1
I − (I − Γd )Φ0 ln py ,
d
ln px = Ω0 ln py ,
where pv = (pv1 , ..., pvn )0 , px = (px1 , ..., pxn )0 , and py = (py1 , ..., pyn )0 .
From the definition of value added, we have that
ln vj = ln Aj + αj ln
kj
αj
,
and from the Euler equation governing the optimal choice of capital in each sector,
kj
=
αj
pvj vj
pxj
!
β
1 − β(1 − δj )
.
Combining these expressions gives
ln vj = ln Aj + αj ln
pvj vj
−
αj ln pxj
+ αj ln
β
1 − β(1 − δj )
,
or in matrix form,
(I − αd ) ln (pv . × v) = ln A + ln pv − αd ln px + αd ln ∆d ,
n
o
where (pv . × v) represents the vector of nominal value added, pvj vj , αd = diag(αj ), and ∆d =
β
. Substituting for investment and value added prices on the right-hand-side of this last
diag 1−β(1−δ
j)
expression, we obtain
0
0 −1
ln py = Γ−1
I
−
(I
−
Γ
)Φ
−
α
Ω
[(I − αd ) ln (pv . × v) − ln A − αd ln ∆d ] .
d
d
d
(B.1)
From the resource constraints in each sector j, and the optimal allocation of materials and investment
in the economy, it follows that
n
n
X
pvj vj
βδj
pvi vi X
= θj C +
φji (1 − γj )
+
ωji
αi pvi vi ,
γj
γi
1 − β(1 − δj )
i=1
i=1
or in matrix form,
−1
v
0
v
v
Γ−1
d (p . × v) = Θ C + Φ(I − Γd )Γd (p . × v) + Ω∆d δd αd (p . × v) ,
so that
−1 0
(pv . × v)
Θ.
= [I − Φ(I − Γd )] Γ−1
− Ω∆d δd αd
d
C
51
(B.2)
Aggregate GDP (in units of the final consumption bundle) is then given by
V = 10 (pv . × v) = 10 ψC,
where ψ = [I − Φ(I − Γd )] Γ−1
d − Ω∆d δd αd
−1
Θ0 .
Substituting the expression for value added in (B.2) into the equation for gross output prices, (B.1), and
using the fact that the ideal price index for the final consumption bundle implies Θ ln py = 0, we obtain
ln C =
0
0
Θ Γ−1
d [I − (I − Γd )Φ ] − αd Ω
[ln A + αd ln ∆d − (I − αd ) ln ψ]
,
0
0 −1 (I − α )1
Θ Γ−1
d
d [I − (I − Γd )Φ ] − αd Ω
0
0
where note that Θ Γ−1
d [I − (I − Γd )Φ ] − αd Ω
It follows that
−1
−1
(I − αd )1 = 1.
n
−1 0 o
∂ ln V
∂ ln C
=
= [I − Φ(I − Γd )] Γ−1
−
Ωα
Θ .
d
d
∂ ln Aj
∂ ln Aj
j
At the same time, the vector of value added shares, sv , is given by
−1 0
Θ
[I − Φ(I − Γd )] Γ−1
(pv . × v)
d − Ω∆d δd αd
=
−1 .
−1
0
V
1 [I − Φ(I − Γd )] Γd − Ω∆d δd αd
Θ0
Therefore, in the limit case where β → 1, ∆d δd → I, and
∂ ln V
= ηsvj ,
∂ ln Aj
where η = 10 [I − Φ(I − Γd )] Γ−1
d − Ωαd
−1
Θ0 is approximately the inverse of the mean labor share across
−1 0
10 ([I−Φ(I−Γd )]Γ−1
Θ
d −Ωαd )
sectors. To see this, observe that η can also be expressed as η = 0
. Thus,
−1
−1
1 (I−αd )([I−Φ(I−Γd )]Γd −Ωαd ) Θ0
1
.
when αj = α ∀j, η = 1−α
Appendix C: Balanced Growth with Sectoral Linkages in Materials and
Investment
We begin by normalizing some of the model’s key variables with respect to sector-specific factors, µj,t , to
be determined below, so as to yield a system of equations that is stationary in the normalized variables.
If all growth rates are constant, the resource constraint in any individual sector implies that all variables
in that constraint must grow at the same rate along a balanced growth path. Thus, define yej,t = yj,t /µj,t ,
52
e
cj,t = cj,t /µj,t , m
e ji,t = mji,t /µj,t , and x
eji,t = xji,t /µj,t . Then, the economy’s resource constraint becomes
e
cj,t +
n
X
m
e ji,t +
i=1
n
X
x
eji,t = yej,t .
i=1
Similarly, the production of investment goods may be re-written as
x
ej,t =
n
Y
x
eij,t ωij
ωij
j=1
where x
ej,t = xj,t /
ωij
j=1 µi,t ,
Qn
,
while the capital accumulation equation becomes
n
Y
µi,t−1 ωij
e
kj,t+1 = x
ej,t + (1 − δj )e
kj,t
µi,t
j=1
,
Q
ω
where e
kj,t+1 = kj,t+1 / nj=1 µi,tij .
The expression for value added may be written as
Q
ωij
e
kj,t nj=1 µi,t−1
vj,t = Aj,t
so that, defining vej,t = vj,t /Aj,t
Q
! αj
,
αj
αj
ωij
n
,
µ
j=1 i,t−1
it becomes
e
kj,t
αj
vej,t =
! αj
.
The composite bundle of materials used in sector j may be expressed as
m
e j,t =
n
Y
m
e ij,t φij
φij
j=1
with m
e j,t = mj,t /
φij
j=1 µi,t .
Qn
,
Therefore, gross output in terms of detrended variables is given by
yej,t µj,t =
vej,t Aj,t
αj ωij
j=1 µi,t−1
Qn
!γj
m
e j,t
φij
j=1 µi,t
Qn
1 − γj
γj
!1−γj
,
or alternatively,
yej,t =
vej,t
γj
γj
m
e j,t
1 − γj
1−γj
γ
n
Aj,tj Y
µj,t
53
j=1
γ α ω
(1−γj )φij
j j ij
µi,t−1
µi,t
.
We can now use the expression in square brackets to solve for the normalizing factors, µj,t , as a function of
the model’s underlying parameters.
First, re-write the term in square brackets as
Aj,tj
γ
γj αj ωij
n
Y
µi,t−1
µj,t
γ αj ωij
µj
j=1 i,t
n
Y
(1−γj )φij γj αj ωij
µi,t
,
µi,t
j=1
where this last expression involves the growth rate of µi,t . Then, along a balanced growth path where all
γj
Qn
Aj,t
(1−γj )φij γj αj ωij
µi,t
must be constant. Thus, we choose
variables grow at constant rates, the term µj,t
j=1 µi,t
µj,t such that
γ
n
Aj,tj Y
µj,t
γ αj ωij +(1−γj )φij
µi,tj
= 1.
j=1
This choice is without loss of generality since any other constant in place of 1 will be differenced out along
the balanced growth path.
Taking logs of both sides of the above expression, we have
γj ln Aj,t − ln µj,t +
n
X
(γj αj ωij + (1 − γj )φij ) ln µi,t = 0,
i=1
or in vector form,
Γd ln At − ln µt + Γd αd Ω0 ln µt + (I − Γd )Φ0 ln µt = 0,
which gives us
ln µt = Ξ0 ln At
(C.1)
where
Ξ0 = I − Γd αd Ω0 − (I − Γd )Φ0
−1
Γd
is the Leontief inverse, (I − Γd αd Ω0 − (I − Γd )Φ0 )−1 , diagonally weighted by value added shares in gross
output, Γd .
Going back to equation (12) in the main text, and writing the vector of productivity growth rates as
∆ ln At = g a , it follows that
∆ ln µt = Ξ0 g a = Ξ0 (g z + (I − αd )g ` ) .
Thus, let µvj,t denote the normalizing factor for value added in sector j defined above,
µvj,t = Aj,t
n
Y
j=1
54
αj
ω
ij
µi,t−1
.
Then, using equation (C.1), we have that
ln µvt = ln At + αd Ω0 Ξ0 ln At−1 ,
or, along a steady state growth path,
−1
∆ ln µvt = I + αd Ω0 I − αd Γd Ω0 − (I − Γd )Φ0
Γd g a .
|
{z
}
(C.2)
Ξ0
Appendix D: The Dynamics of Value Added Growth
We wish to recover the implications of the model for sectoral value added growth, ∆ ln vj,t , and aggregate
GDP growth, ∆ ln Vt .
Observe that
∆ ln vej,t = αj ∆ ln e
kj,t ,
and that ∆ ln e
kj,t = ∆k̂j,t , where ∆k̂j,t follows directly from the Markov decision rules. Then, by definition
of vej,t ,
∆ ln vj,t = ∆ ln vej,t + ∆ ln Aj,t +
= αj ∆k̂j,t + Âj,t + g j +
n
X
i=1
n
X
αj ωij ∆ ln µi,t−1
αj ωij ∆ ln µi,t−1 ,
i=1
or, in vector form,
∆ ln vt = αd ∆k̂t + Ât + g a + αd Ω0 ∆ ln µt−1 .
The sectoral detrending factors, µj,t , solve
∆ ln µt = Ξ0 ∆ ln At ,
where
Ξ0 = I − Γd αd Ω0 − (I − Γd )Φ0
−1
Γd .
Then,
∆ ln vt = αd ∆k̂t + Ât + g a + αd Ω0 Ξ0 Ât−1 + g a
55
or
∆ ln vt = I + αd Ω0 Ξ0 g a + αd ∆k̂t + Ât + αd Ω0 Ξ0 Ât−1 ,
with ∆ ln Vt = sv ∆ ln vt . When all shocks are set to zero, so that ∆k̂t = Ât = Ât−1 = 0, we recover the
sectoral balanced growth paths for sectoral value added,
∆ ln vt = I + αd Ω0 Ξ0 g a .
56
@FedFRASER