Endowment Structures, Industrial Dynamics, and Economic Growth

Transcription

1
Jiandong Jua; , Justin Yifu Linb , Yong Wangcy
a
Shanghai University of Finance and Economics and Tsinghua University
b
c
Peking University
Hong Hong University of Science and Technology
Received Date; Received in Revised Form Date; Accepted Date
2
3
Abstract
4
Motivated by four stylized facts about industry dynamics established in this paper, we propose a
5
theory of endowment-driven structural change by developing a tractable growth model with in…nite
6
industries. We show that, while the aggregate economy follows the Kaldor facts, the composition
7
of the underlying industries changes endogenously as the economy grows. Each industry exhibits
8
a hump-shaped life cycle: As capital accumulates and reaches a certain threshold, a new industry
9
appears, prospers, and then declines, to be gradually replaced by a more capital-intensive industry,
10
which appears, booms, and eventually declines in the same manner, ad in…nitum. Analytical solu-
11
tions are obtained to fully characterize the whole process of the repetitive and perpetual structural
12
changes of industries beneath the constant aggregate growth path.
13
Keywords: Structural Change, Industrial Dynamics, Economic Growth, Capital Accumulation
14
JEL classi…cation: O11, O14, O33, O41
Corresponding author: [email protected]
y
We are grateful to the associated editor and the referee for their highly insightful comments and suggestions. For useful comments, we would also like to thank Philippe Aghion, Jinhui Bai, Eric Bond, Gene
Grossman, Veronica Guerrieri, Chang-Tai Hsieh, Chad Jones, Sam Kortum, Andrei Levchenko, Francis T.
Lui, Erzo Luttmer, Rodolfo Manuelli, Rachel Ngai, Paul Romer, Roberto Samaneigo, Danyang Xie, Dennis
Yang, Kei-Mu Yi, as well as seminar and conference participants at the CUHK, Dallas Fed, Georgetown University, George Washington University, HKUST, World Bank, Yale University, Econometric Society World
Congress (2010), SED (2010), Midwest Macro Meetings (2011), NBER-CCER annual conference (2011) and
Shanghai Macro Workshop (2012). Xinding Yu and Zhe Lu provided very able assistance. Yong Wang
acknowledges the …nancial support from GRF(grant #644112). All the remaining errors are our own.
1
1.
1
Introduction
2
Sustainable economic growth relies on the healthy development of underlying industries.
3
Enormous research progress has been made on industrial dynamics, yet many important
4
aspects still remain imperfectly understood within the context of economic growth, espe-
5
cially at the high-digit disaggregated industry level. Consider, for example, the automobile
6
industry and the apparel industry. How di¤erent are the evolution patterns of two industries
7
along the growth path of the whole economy? Which industry should we expect to expand
8
or decline earlier than the other and why? How long, if at all, does a leading industry main-
9
tain its predominant position? What fundamental forces drive these dynamics? What is
10
the relationship between individual industrial dynamics and aggregate GDP growth? These
11
questions are interesting to economists, policy makers, and private investors.
12
The goal of this paper is to shed light on these issues by studying the dynamics of all
13
high-digit industries simultaneously within a growth framework. We establish four stylized
14
facts about industrial dynamics using the NBER-CES data set of the US manufacturing
15
sector, which covers 473 industries at the 6-digit NAICS level from 1958 to 2005. First, there
16
exists tremendous cross-industry heterogeneity both in capital intensities and productivities.
17
Second, the value-added share (or the employment share) of an industry typically exhibits
18
a hump-shaped life cycle, that is, an industry …rst expands, reaches a peak, and eventually
19
declines. Third, a more capital-intensive industry reaches its peak later. Fourth, the further
20
away an industry’s factor intensity deviates from the factor endowment of the economy,
21
the smaller is the employment share (or output share) of this industry in the aggregate
22
economy, which we may call the congruence fact. Similar patterns are also found in the
23
UNIDO data set, which covers 166 countries from 1963 to 2009 at the two-digit level (23
24
sectors). In fact, documentation and analyses of a subset of the above-mentioned patterns
25
of industrial dynamics can be dated at least back to the 1960s. For example, Chenery and
26
Taylor (1968) show that the major products in the manufacturing sector gradually shift from
1
2
the labor-intensive ones to more capital-intensive ones as an economy develops.
2
Motivated and guided by these four stylized facts, we develop a tractable endogenous
3
growth model with multiple industries to explain these observed patterns of industrial dy-
4
namics. In light of Fact 3 and Fact 4, together with theoretical arguments for the importance
5
of capital accumulation in the growth literature, we revisit the role of the endowment struc-
6
ture, measured by the aggregate capital-labor ratio, in both determining the production
7
structure and driving the industrial dynamics.1 More speci…cally, our model is constructed
8
to explain the second, third and fourth stylized facts simultaneously by taking the …rst fact
9
as given.
10
To understand how endowment structure may a¤ect di¤erent industries, one may appeal
11
to the Rybczynski Theorem, which states that in a static model with two goods and two
12
factors, the capital-intensive sector expands while the labor-intensive sector shrinks as the
13
economy becomes more capital abundant. This result holds for both an autarky and an open
14
economy. The intuition is that the rental wage ratio declines as capital becomes more abun-
15
dant, so this change in the endowment structure disproportionately favors the production of
16
the capital-intensive good, holding other things constant.
17
This logic is straightforward and well known, but it is not obvious whether capital accu-
18
mulation itself can simultaneously explain Facts 2 through 4 in a growth model with a large
19
number of industries. Recall that the Rybczynski Theorem says almost nothing about what
20
happens when there are more than two goods (Feenstra (2003)). However, for our purpose,
21
a model with many (ideally in…nite) industries is indispensable, not only because it is much
22
more realistic at a high-digit industry level, but also because the multilateral interactions
23
among di¤erent industries could be potentially important for industrial dynamics.
1
In his Marshall Lectures, Lin (2009) proposes that many development issues such as growth, inequality,
and industrial policies can all be better understood by analyzing the congruence of the industrial structure
with the comparative advantages determined by the endowment structure and its change. There, the endowment structure is more formally de…ned as the composition of the production factors (including labor, human
capital, physical capital, land and other natural resources) as well as the soft and hard infrastructures.
3
1
More important technical challenges for such a formal model come from the dynamic part
2
of the problem in the presence of many industries, each of which evolves nonlinearly. More
3
precisely, …rst, the model predictions should be not only consistent with the aforementioned
4
stylized facts at the disaggregated industry level, but also consistent with the Kaldor facts
5
at the aggregate level.2 Second, to keep track of the life-cycle dynamics of each industry
6
along the whole path of aggregate growth, we must fully characterize transitional dynamics,
7
which is well-recognized to be di¢ cult even for a two-sector model, but now we have in…nite
8
industries with an in…nite time horizon.3 Third, it turns out that endogenous structural
9
change in the underlying industries eventually forces us to characterize a Hamiltonian system
10
with endogenously switching state equations, as will be clearer in the model.
11
All these technical challenges create important obstacles for theoretical research on in-
12
dustrial dynamics. Fortunately, a desirable feature of our model is precisely its tractability:
13
We obtain closed-form solutions to fully characterize the whole process of the hump-shaped
14
industrial dynamics for each of the in…nite industries along the aggregate growth path. The
15
model predictions are qualitatively consistent with all the stylized facts about the industrial
16
dynamics at the micro industry level and the Kaldor facts at the aggregate level.
17
We …rst develop a static model with in…nite industries (or goods, interchangeably) and
18
two factors (labor and capital). With a general CES production function for the …nal com-
19
modity, we obtain a version of the Generalized Rybczynski Theorem: For any given endow-
20
ment of capital and labor, there exists a cuto¤ industry such that, when the capital en-
21
dowment increases, the output will increase in every industry that is more capital intensive
22
than this cuto¤ industry, while the output in all the industries that are less capital intensive
23
than this cuto¤ industry will decrease. Moreover, the cuto¤ industry moves toward the more
24
capital-intensive direction as the capital endowment increases. As a special case, when the
2
Kaldor facts refer to the relative constancy of the growth rate of total output, the capital-output ratio,
the real interest rate, and the share of labor income in GDP.
3
See King and Rebelo (1993), Mulligan and Sala-i-Martin (1993), Bond, Trask and Wang (2003), and
Mehlum (2005).
4
1
CES substitution elasticity is in…nity, generically only two industries are active in equilib-
2
rium and the capital-labor ratios of the active industries are the closest to the capital-labor
3
ratio of the economy. This result is consistent with the congruence fact. The model implies
4
that the structures of underlying industries are endogenously di¤erent at di¤erent stages of
5
economic development.
6
Then the model is extended to a dynamic environment where capital accumulates en-
7
dogenously. The dynamic decision is decomposed into two steps. First, the social planner
8
optimizes the intertemporal allocation of capital for the production of consumption goods,
9
which determines the evolution of the endowment structure. Then, at each time point the
10
resource allocation across di¤erent industries is determined by the capital and labor endow-
11
ments in the same way as the static model. Endogenous changes in the industry composition
12
of an economy translate into di¤erent functional forms of the endogenous aggregate produc-
13
tion function and the capital accumulation function; therefore, we must solve a Hamiltonian
14
system with endogenously switching state equations because of the endogenous structural
15
change. When the CES substitution elasticity is in…nity (linear case), we show that there
16
always exists a unique aggregate growth path with a constant growth rate, along which the
17
underlying industries shift over time by following a hump-shaped pattern with the more
18
capital-intensive ones reaching a peak later (Facts 2 and 3). We analytically characterize the
19
life-cycle dynamics of each industry and also show that the model predictions are empirically
20
consistent with the data.
21
Our paper is most closely related to the growth literature on structural change, which
22
studies the resource allocation across di¤erent sectors as an economy grows (see Herrendorf,
23
Rogerson and Valentinyi (2013) for a recent survey). This literature mainly tries to match
24
the Kuznets facts, namely, the agriculture share in GDP has a secular decline, the industry
25
(manufacturing) share demonstrates a hump shape, and the service share increases. How-
26
ever, such sectors are too aggregated to address questions such as those raised in the …rst
5
1
paragraph. Insu¢ cient e¤ort is devoted to reconciling Kaldor facts with the aforementioned
2
stylized facts about industrial dynamics at much more disaggregated levels.
3
The empirical contribution of this paper to the literature is that we formally establish
4
four stylized facts about industry dynamics. In addition, our paper makes two theoretical
5
contributions. First, we develop a highly tractable growth model with in…nite industries to
6
fully characterize the industrial dynamics, which are qualitatively consistent with the four
7
motivating facts. Second, we show how capital accumulation serves as a new and important
8
mechanism that drives the structural change. The existing literature mainly discusses two
9
mechanisms of structural change in autarky. One is the preference-driven mechanism, in
10
which the demand for di¤erent goods shift asymmetrically as income increases due to the
11
non-homothetic preferences.4 The second mechanism is that unbalanced productivity growth
12
rates across sectors drive resource reallocation.5
13
Unlike these two mechanisms, we propose that improvement of endowment structure
14
(capital accumulation) itself is a new and fundamental mechanism that drives industrial
15
dynamics, which we refer to as endowment-driven structural change. To highlight the theo-
16
retical su¢ ciency and distinction of this new mechanism, we assume a homothetic preference
17
to shut down the preference-driven mechanism. Conceptually, it is not obvious how to order
18
industries according to their income demand elasticities at a highly disaggregated level. We
19
also assume in the benchmark model that productivity is constant over time in all the in-
20
dustries to shut down the productivity-driven mechanism. Instead, our model, as motivated
21
by Fact 1, assumes that industries di¤er in their capital intensities, deviating from the stan-
22
dard assumption that di¤erent sectors have equal capital intensity, including models without
23
capital.6
4
For instance, the Stone-Geary function is used in Laitner (2000), Kongsamut, Rebelo, and Xie (2001),
Caselli and Cole (2001), Gollin, Parente, and Rogerson (2007). Hierarchic utility functions are adopted in
Mastuyama (2002), Foellmi and Zweimuller (2008), Buera and Kaboski (2012a), among others.
5
See, e.g., Ngai and Pissarides (2007), Hansen and Prescott (2002), Duarte and Restuccia (2010), Yang
and Zhu (2013), Uy, Yi and Zhang (2013).
6
Ngai and Samaneigo (2011) study how R&D di¤ers across industries and contributes to industry-speci…c
6
1
An important exception is Acemoglu and Guerrieri (2008), who study structural change
2
in a two-sector growth model with di¤erent capital intensities, but their model does not aim
3
to explain or generate the repetitive hump-shaped industrial dynamics because the life cycle
4
of each sector is truncated. Instead, their analytical focus is on the asymptotic aggregate
5
growth rate in the long run, by which time one industry dominates the economy in terms of
6
employment share and structural change virtually ends. In contrast, we have in…nite sectors
7
so the structural change is endless and this setting allows us to analyze the complete life-cycle
8
dynamics of every industry at the disaggregated levels during the whole growth process.
9
Ngai and Pissarides (2007) study structural change in a growth model with an arbitrary
10
but …nite number of sectors, which potentially allows for the life-cycle analysis of disaggre-
11
gated industries. However, they do not treat capital accumulation as a major force to drive
12
structural change, even in their appendix, where they introduce di¤erent capital intensities
13
across di¤erent sectors. Nor do they attempt to keep track of the life cycles of each industry
14
to explain their dynamics along the growth path. Moreover, structural change will be over
15
in the long run since there are …nite sectors in their model.
16
Our paper is also closely related to the strand of growth literature that studies the life
17
cycle dynamics of industries, …rms, establishments or products. The key mechanisms that
18
drive the life cycle dynamics are di¤erent in di¤erent models. For example, some highlight
19
the role of innovation and creative destruction (see Stokey (1988), Grossman and Helpman
20
(1991), Aghion and Howitt (1992), Jovanovic and MacDonald (1994)); some highlight the role
21
of speci…c intangible capital such as organizational capital (see Atkenson and Kehoe (2005))
22
or technology-speci…c or industry-speci…c human capital (see Chari and Hopenhayn (1992),
23
Rossi-Hansberg and Wright (2007)); some focus on productivity change and destruction
24
shocks (see Hopenhayn (1992), Luttmer (2007), Samaniego (2010)), still others highlight
TFP growth. Acemoglu (2007) argues that technology progress is endogenously biased toward utilizing
the more abundant production factors, which indicates that endowment structure is also fundamentally
important even in accounting for TFP growth itself.
7
1
demand shift due to consumers’heterogeneous preferences together with product awareness
2
(Perla (2013)) or non-homothetic preferences (Mastuyama (2002)). Our paper di¤ers from
3
and complements these approaches by focusing on the role of endowment structure via the
4
endogenous relative factor prices. For simplicity and clarity, we abstract away complications
5
in other dimensions: There are no industry-speci…c or technology-speci…c factors, everything
6
is deterministic, and all the technologies already exist and are freely available.
7
The paper is organized as follows: Section 2 documents and summarizes the empirical
8
facts about industrial dynamics. Section 3 presents the static model. The dynamic model
9
is analyzed in Section 4. Section 5 considers two extensions: an alternative dynamic setting
10
and a more general CES aggregate production function. Section 6 concludes. Technical
11
proofs and some other extensions are in the Appendix.7
12
2.
Empirical Patterns of Industrial Dynamics
13
This section documents several important facts of industrial dynamics to motivate and
14
guide our theoretical explorations. Ideally, we need a su¢ ciently long time series of pro-
15
duction data for each industry at a su¢ ciently high-digit industry level. Due to limited
16
availability of quality data, here we will focus on the manufacturing sector.
17
2.1. Evidence from US Data
18
We use the NBER-CES Manufacturing Industry Data for the US, which adopts the 6-
19
digit NAICS codes and covers 473 industries within the manufacturing sector from 1958 to
20
2005.8
21
We …nd that there exists tremendous cross-industry heterogeneity in both physical cap7
International trade could be another important force to drive structural change (see Ventura (1997),
Matsuyama (2009) and Uy, Yi and Zhang (2013)). We show that our main results remain valid in a small
open economy, both with and without international capital ‡ows. It is relegated to the appendix due to
space limits.
8
The data set can be downloaded at http://www.nber.org/nberces.
8
1
ital intensities and productivities.9 For example, among all industries in 1958, the highest
2
capital-labor ratio is 522; 510 US dollars per worker, which is 986 times larger than the lowest
3
one; the highest capital expenditure share (one minus the ratio of labor expenditure to the
4
value added) is 0.866 while the lowest share is 0.132. In 2005, the highest capital-labor ratio
5
is still about 148 times higher than the lowest one. The highest labor productivity is 50.94
6
(in the unit of thousand dollars/worker) in 1958, which is 35 times larger than that in the
7
least-e¢ cient industry. In 2005 the top-bottom cross-industry labor productivity di¤erence
8
is 92-fold. (See Table A1 in the Appendix for detailed descriptive statistics).
9
In addition, the capital intensity (measured by the capital-labor ratio or the capital
10
expenditure share) increases over time within each industry (see Table A1). This may be
11
because each industry consists of many di¤erent products or techniques of di¤erent capital
12
intensities and the production activity within each industry gradually switches from labor-
13
intensive products/procedures/tasks to capital-intensive ones. Moreover, it is likely that the
14
rank of capital intensity for two highly disaggregated industries can be reversed over time.
15
These issues have long plagued empirical trade economists when testing the Heckscher-Ohlin
16
trade theory because the standard classi…cation of industries is not based on their capital
17
intensities and also capital intensities of products in the same industry may change over
18
time. Schott (2003) proposes to solve these problems by rede…ning the industries according
19
to their capital-labor ratios. Anticipating similar problems for our model, we follow the
20
same practice. We …rst rank all the 22275 observations (473 industries for 48 years, and 429
21
observations are dropped due to missing values in capital-labor ratio) by the capital-labor
22
ratios in an increasing order, and then equally divide all these observations into 99 bins
23
(newly-de…ned industries). Within each newly-de…ned industry, we have 225 observations.
24
The …rst bin, called “industry 1”, is most labor-intensive by construction. Likewise, the most
9
Note that the TFP growth rate obtained from the NBER-CES data set is industry-speci…c and hence the
level of TFP is not directly comparable across industries. So we mainly report and use labor productivity
for our cross-section regression analyses. However, the same pattern is still found robust when we use the
TFP-measured productivity after normalizing the TFP values for all the industries in year 2002 to 100.
9
1
capital-intensive bin is “industry 99”. In this way, the rank of capital intensity among all
2
these newly-de…ned industries is clear and time-invariant.10 For the purpose of illustration,
3
Figure 1 plots the time series of the HP-…ltered employment shares of six typical industries
4
from 1958 to 2005.11
[Insert Figure 1 here]
5
6
Except for a few anomalies, a discernible pattern emerges. The employment share de-
7
creases over time in the most labor-intensive industries, exhibits a hump shape in the “mid-
8
dle”capital-intensive industries, and increases over time in the most capital-intensive indus-
9
tries. If longer time series become available, we would expect to see the rising stage of the
10
most labor-intensive industries before 1958 and perhaps the eventual decline of the capital-
11
intensive industries after 2005. Similar patterns are also observed for the value-added share
12
or when capital intensity is measured alternatively by the share of capital expenditure.12 We
13
refer to this non-monotonic development pattern of an industry as the hump-shaped pattern
14
of industrial dynamics. Moreover, we can also see that a more capital-intensive industry
15
reaches its peak later, which we refer to as the timing fact. More formally, we regress the
16
peak time of an industry’s share (either employment share or value-added share) on its cap-
17
ital intensity. The results are in Table 1. We indeed see that the peak time of an industry
18
is positively correlated with its capital intensity.
[Insert Table 1 here]
19
10
A more detailed description for the method of industry reclassi…cation is in the Appendix 8.
The labor productivity, capital-labor ratio and capital expenditure share at in the newly de…ned “industry n”are calculated by using the aggregates of capital, labor, labor expenditure, and value added for all the
observations at t in “industry n”. When generating Figure 1, if there are missing values in the constructed
time series for certain years, we replace the missing values by the simple average of the observations before
and after. For how employment shares change over time in all the 99 industries within the manufacturing
sector, please refer to Figure A1 in the Appendix.
11
12
Alternatively, when the observations are ranked according to the capital expenditure share, 75 observations with non-positive capital expenditure shares are dropped. So we end up with a sample of 22200
observations, which are equally divided into 100 bins, with 222 observations in each bin.
To further understand what determines the industrial structures and their dynamics, we
1
2
10
run the following regression:
LSit =
3
0
+
1
Kit =Lit Kt =Lt
+
Kt =Lt
2 Tit
+ industry_dummy + "it ;
(1)
4
where LSit is the ratio of newly de…ned industry i’s employment to the total manufacturing
5
employment at year t and
6
between industry i’s capital-labor ratio and the aggregate capital-labor ratio at year t. Tit
7
is the labor productivity of the newly-de…ned industry i at year t. The results are reported
8
in Table 2. We can see that
9
employment share becomes smaller if the di¤erence between the industry’s capital intensity
10
and the aggregate capital-labor ratio is larger, which suggests that if an industry’s capital
11
intensity and the economy’s endowment structure are more congruent, that industry is rel-
12
atively larger. As shown in Table 2, this result is robust if capital intensity is alternatively
13
measured by the capital expenditure share or if the employment share is replaced by the
14
value-added share. We call this the congruence fact. In addition, an industry’s employment
15
share (or valued-added share) is signi…cantly and positively correlated with the level of labor
16
productivity.
Kit =Lit Kt =Lt
Kt =Lt
1
is the absolute value of a normalized di¤erence
is negative and signi…cant, indicating that an industry’s
17
18
How robust are these patterns to the alternative ways in which an industry is de…ned?13
19
To address this concern, we return to the original de…nition of industries with 6-digit NAICS
20
codes. We are particularly interested in the robustness of the hump-shaped pattern and the
21
timing fact. So we run the following regression by including a quadratic term for time:
yit =
22
13
0
+
1t
+
2
2t
+
3
kit t +
4 Tit
+
5 Di
+
6 GDP GRt
+ "it ;
(2)
As one may expect, a potential limitation of Schott’s method in the dynamic context is that the newlyde…ned “industry 1” by construction has more data points for the earliest years and “industry 99” has
disproportionately more data points for the latest years. However, it is not necessarily biased for most of
the “middle industries.” Anyway, the robustness of all the patterns is worth checking using the traditional
de…nition of industries, which we do below.
11
1
where yit is the output or value-added share of industry i in the total manufacturing sector
2
at year t; kit and Tit are the capital expenditure share and the labor productivity of industry
3
i at year t, respectively; Di is the industry dummy; GDP GRt is the GDP growth rate; and
4
"it is the error term. If the hump-shaped dynamic pattern is statistically valid, we should
5
expect the coe¢ cient for the quadratic term,
6
after controlling for the labor productivity and the industry …xed e¤ect, we know from (2)
7
that industry i reaches its peak at tmax
i
8
9
2,
1+ 3
2
2
to be negative and signi…cant. In addition,
ki;t
. That is,
If the timing fact is statistically valid, we should expect
3
should be positive when
2
3
2
@yit
@t
> 0 if and only if t < tmax
.
i
to be positive, or equivalently,
is negative. Moreover, the peak time tmax
must be positive
i
10
when
11
two patterns are indeed statistically signi…cant, so the results are robust.
1
is positive. Table 3 summarizes all the regression results, which con…rm that these
12
13
To check the robustness of the congruence fact, we use the original NAICS classi…cation
14
and run regression (1) again. The results are as follows (with standard errors in parentheses):
LSit = 2:233
(0:016)
0:071
(0:019)
Kit =Lit Kt =Lt
Kt =Lt
4:33
10 4 Tit + industry_dummy:
(1:04 10
4)
15
The coe¢ cient
16
the congruence fact.14 The employment share now is negatively correlated with the labor
17
productivity.
1
is again negative and signi…cant at the 99% signi…cant level, con…rming
18
Finally, we examine the correlation between the industrial capital-labor ratios and the
19
labor productivities. We obtain the following regression results (with standard errors in
14
When the value-added share is used as the dependent variable, or when the capital expenditure share is
used in the congruence term, the congruence fact cannot be found for the data with the NAICS classi…cation.
20
1
12
parentheses):
Tit = 24:184 + 0:0415(Kit =Lit ):
(0:004)
(3)
2
It shows that industrial productivities and capital intensities are positively correlated at the
3
99% signi…cant level. This positive correlation remains signi…cant and robust when capital
4
intensities are measured by capital expenditure shares and/or when controlling for industrial
5
output (or value added, employment) and year dummies.
6
2.2. Evidence from Cross-Country Data
7
We now turn to the evidence from the cross-country data, and we are particularly in-
8
terested in the patterns for developing countries. Haraguchi and Rezonja (2010) explore the
9
UNIDO data set of manufacturing development covering 135 countries from 1963 to 2006.
10
Their data set adopts the 2-digit ISIC codes and entails 18 manufacturing sectors. They
11
group all the observations together in their regressions. One of their most robust …ndings is
12
that the logarithm of the value-added share of an industry in the total GDP can be best …t-
13
ted by including a quadratic term of the logarithm of real GDP per capita in the regressions;
14
the coe¢ cient is both negative and signi…cant after controlling for various country-speci…c
15
factors (such as population and natural resource abundance) and country dummies. This is
16
consistent with the pattern of the hump-shaped industrial dynamics we documented earlier
17
for the US data. Following Chenery and Taylor (1968), Haraguchi and Rezonja (2010) group
18
industries into three separate panels: “early sectors”, “middle sectors” and “late sectors”.
19
They …nd that industries in “early sectors” reach their peaks at lower levels of GDP per
20
capita (“earlier”) than those in “middle sectors”, which in turn reach their peaks “earlier”
21
than those in “late sectors”. In other words, a more capital-intensive industry reaches a
22
peak at a higher level of GDP per capita (“later”), which is consistent with our timing fact.
23
The congruence fact is also empirically found by Lin (2009) in the cross-country empirical
24
analyses.
25
1
13
To further investigate the cross-country empirical evidence for industry dynamics, especially for developing countries, we extend regression (2) to a cross-country regression:
yitc =
2
0
+
1t
+
2
2t
+
3
kitc t +
4 Titc
+
5 Dic
+
6 GDP GRtc
+ "itc
(4)
3
where subscript c represents a country index. The UNIDO manufacturing data set we are
4
using covers 166 countries from 1963 to 2009 at the two-digit level (23 sectors). The results
5
are summarized in Table 4.
6
7
The …ndings are consistent with the results in Table 3, which suggests that the industry
8
dynamics patterns observed in the US are actually quite general and also true for other
9
countries.
10
Next we check the robustness of the empirical patterns for the subsample that only
11
consists of developing countries (following the World Bank’s de…nition). The results are
12
reported in Table 5.
13
14
As can be seen again,
1
is positive,
2
is negative and
3
is positive. Moreover, these
15
three coe¢ cients are signi…cant in almost all the cases. Therefore, we conclude that the
16
hump-shaped industry pattern and the timing fact are robust for developing countries as
17
well.
18
2.3. Summary of Empirical Facts
19
20
21
22
Combining all the empirical results, we summarize the four facts for industrial dynamics
in the manufacturing sector.
Fact 1 (cross-industry heterogeneity): There exists tremendous cross-industry heterogeneity in capital-labor ratios, capital expenditure shares and labor productivities.
14
Fact 2 (hump-shaped dynamics): An industry typically exhibits a hump-shaped
23
1
dynamic pattern: its value-added share …rst increases, reaches a peak, and then declines.
Fact 3 (timing fact): The more capital intensive an industry is, the later its value-
2
3
added share reaches its peak.
4
Fact 4 (congruence fact): The further an industry’s capital-labor ratio deviates from
5
the economy’s aggregate capital-labor ratio, the smaller is the industry’s employment share.
6
In this paper, we take Fact 1 as exogenously given and develop a growth model with
7
multiple industries of di¤erent capital intensities to simultaneously explain Facts 2, 3 and 4.
8
3.
9
3.1. Set-up
Static Model
10
Consider an economy with a unit mass of identical households and in…nite industries.
11
Each household is endowed with L units of labor and E units of physical capital, which
12
can be easily extended to incorporate intangible capital as well. A representative household
13
consumes a composite …nal commodity C; which is produced by combining all the interme-
14
15
diate goods cn , where n 2 f0; 1; 2; :::g. Each intermediate good should be interpreted as an
industry, although we will use “good”and “industry”interchangeably throughout the paper.
For simplicity, assume the production function of the …nal commodity is
16
C=
17
1
X
(5)
n cn ;
n=0
represents the marginal productivity of good n in the …nal good production.15 We
18
where
19
require cn
20
is CRRA:
21
U=
15
n
0 for any n: The …nal commodity serves as the numeraire. The utility function
C1
1
1
; where
2 (0; 1]:
(6)
It is not unusual in the growth literature to assume perfect substitutability for the output across di¤erent
production activities; see Hansen and Prescott (2002). We will relax this assumption and study the general
CES function in Section 5.
15
All the technologies exhibit constant returns to scale. In particular, good 0 is produced
22
1
with labor only. One unit of labor produces one unit of good 0. To produce any good n
2
both labor and capital are required and the production functions are Leontief:16
Fn (k; l) = minf
3
k
; lg;
an
1,
(7)
4
where an measures the capital intensity of good n. All the markets are perfectly competitive.
5
Let pn denote the price of good n. Let r denote the rental price of capital and w denote the
6
wage rate. The zero pro…t condition for a …rm implies that p0 = w and pn = w + an r for
7
n
1.
Without loss of generality, the industries are ordered such that an is increasing in n.
Since the data suggest that a more capital-intensive technology is generally more productive
(refer to regression result (3)), we assume that
n
is increasing in n: To obtain analytical
solutions, we assume
n
a
=
1>
n
; an = an ;
(8)
> 1:
(9)
must be imposed to rule out the trivial case that only the most capital-intensive good
8
a>
9
is produced in the static equilibrium, and we strengthen the assumption further to a
10
1>
to simplify the analysis as good 0 requires no capital.17
The household problem is to maximize (6) subject to the following budget constraint:
11
C = wL + rE:
12
16
(10)
Leontief functions are also used in Luttmer (2007) and Buera and Kaboski (2012a, 2012b). See Jones
(2005) for more discussions of functional forms. It can be easily shown that our key qualitative results
will remain valid when the production function is Cobb-Douglas, but that will enormously increase the
nonlinearity of the problem in the multiple-sector environment, making it much harder to obtain closed-form
solutions, especially for the dynamic analysis.
17
If = 1, the equilibrium would be trivial because only good 0 is produced in this linear case. Later, we
will allow for = 1 when (5) is replaced by a general CES function in Section 5.
16
13
3.2. Market Equilibrium
1
We …rst establish that at most two goods are simultaneously produced in the equilibrium
2
and that these two goods have to be adjacent in the capital intensities (see Appendix 1 for
3
the proof). The intuition is the following. Suppose goods n and n + 1 are produced for some
4
n
5
must be equal to their price ratio: M RTn+1;n =
1, then the marginal rate of transformation (MRT) between the two intermediate goods
8
9
pn+1
pn
=
w+an+1 r
,
w+an r
which yields
r
1
= n
:
w
a (a
)
6
7
=
Obviously, M RTj;j+1 >
1
aj
1 (a
)
(11)
pj
pj+1
whenever
r
w
>
1
aj (a
)
; and M RTj;j
1
>
pj
pj 1
whenever
r
w
<
for any j = 1; 2; ::. Therefore, when (11) holds, good n+1 must be strictly preferred
to good n + 2; because the MRT is larger than their price ratio. This means that cj = 0 for
10
all j
n + 2. Using the same logic, we can also verify that cj = 0 for all 1
11
addition, condition (9) ensures that good 0 is not produced. Similarly, when goods 0 and 1
12
are produced in an equilibrium, good 1 is strictly preferred to any good n
j
n
1. In
2:
The market clearing conditions for labor and capital are given respectively by
cn + cn+1 = L;
(12)
cn an + cn+1 an+1 = E:
(13)
13
The market equilibrium can be illustrated in Figure 2, where the horizontal and vertical
14
axes are labor and capital, respectively. Point O is the origin and Point W = (L; E) denotes
15
the endowment of the economy. When an L < E < an+1 L; as shown in the current case,
16
only goods n and n + 1 are produced. The factor market clearing conditions, (12) and (13),
17
determine the equilibrium allocation of labor and capital in industries n and n + 1; which
19
are represented respectively by vector OA and vector OB in the parallelogram OAW B.
!
!
Oan = (1; an ) cn and Oan+1 = (1; an+1 ) cn+1 are the vectors of factors used in producing cn
20
and cn+1 in the equilibrium. If the capital increases so the endowment point moves from W
18
17
21
to W 0 , the new equilibrium becomes parallelogram OA0 W 0 B 0 so that cn decreases but cn+1
1
increases. When E = an L, only good n is produced. Similarly, if E = an+1 L; only good
2
n + 1 is produced.18
[Insert Figure 2 Here]
3
More precisely, the equilibrium output of each good cn , the relative factor prices
4
5
r
,
w
and
the corresponding aggregate output C are summarized in Table 6.
Table 6: Static Trade Equilibrium
0
an L
E < aL
E
a
c0 = L
c1 =
cn =
Ei
a
, E0;1 =
r
w
a
C =L+(
a
1) Ea
(C
1
L)
Ei an Li
an+1 an
cj = 0 for 8j 6= n; n + 1
1
=
1
Lan+1 E
an+1 an
cn+1 =
cj = 0 for 8j 6= 0; 1
r
w
E < an+1 L for n
C=
=
n+1
1
an (a
n
an+1 an
h
, En;n+1 = C
E+
n
)
n
(a )
L
a 1
(a )
L
a 1
i
an+1 an
n+1
n
:
The static equilibrium is summarized verbally in the following proposition.
6
7
Proposition 1 Generically, there exist only two industries whose capital intensities are the
8
most adjacent to the aggregate capital-labor ratio,
9
exhibits a hump shape: the output …rst remains zero, then increases and reaches its peak and
10
then declines, and …nally returns to zero and is fully replaced by the industry with the next
11
higher capital intensity.
E
.
L
As
E
L
increases, each industry n (n
1)
12
The equilibrium outcome, as summarized in the above proposition and Table 6, is logi-
13
cally consistent with Facts 2 through 4 documented in Section 2. Table 6 also shows that
18
This graph may appear similar to the Lerner diagram in the H-O trade models with multiple diversi…cation cones (see Leamer (1987)). However, the mechanism in our autarky model is di¤erent from the
international specialization mechanism in the trade literature.
18
14
the aggregate production function (C as a function of L and E) has di¤erent forms when the
1
endowment structures are di¤erent, re‡ecting the endogenous structural change in the un-
2
derlying industries. Accordingly, the coe¢ cient right before E in the endogenous aggregate
3
production function is the rental price of capital, and the coe¢ cient before L is the wage
4
rate. So the relative factor price is
5
stair-shaped fashion as E increases. This discontinuity results from the Leontief assumption.
6
Observe that the capital income share in the total output is given by
rE
=
rE + wL
7
r
w
=
1
an (a
)
when E 2 [an L; an+1 L), and it declines in a
1
E
a 1
n (a
a
1
E + a 1 )L
a 1
(14)
8
when E 2 [an L; an+1 L) for any n
9
with capital within each diversi…cation cone and then suddenly drops to
1. So the capital income share monotonically increases
(
1)
a 1
as the economy
10
enters a di¤erent diversi…cation cone, but the capital income share always stays within the
11
interval [ a
12
income share is fairly stable over time.19
1 ( 1)a
;
]
1 (a 1)
for any n
1. This is consistent with the Kaldor fact that the capital
Table 6 also shows that the equilibrium industrial output and the industrial employment
13
14
are not a¤ected by . The intuition is straightforward: an increase in
15
productivity of good n + 1 versus good n. But their relative price
16
thus these two forces exactly cancel out each other. However, this result is no longer valid
17
when the substitution elasticity between di¤erent goods is less than in…nity, which we will
18
show in Section 5.
19
4.
pn+1
pn
raises the relative
is also equal to ;
Dynamic Model
20
Now we will develop a dynamic model to fully characterize the industrial dynamics along
21
the growth path of the aggregate economy, where the capital changes endogenously over
22
time.
19
See Barro and Sala-i-Martin (2003) for more discussion on the robustness of the Kaldor facts.
23
19
4.1. Theoretical Model
4.1.1.
Environment
1
2
There are two sectors in the economy: a sector producing capital goods and a sector
3
producing consumption goods. Capital goods and consumption goods are distinct in nature
4
and not substitutable. Moreover, they are produced with di¤erent technologies. Capital
5
goods are produced using an AK technology: One unit of capital good produces A units
6
of new capital goods, where A captures the e¤ect of learning by doing. It also highlights
7
the feature that the technology progress is investment-speci…c, so it occurs in the capital
8
(investment) goods sector rather than in the consumption goods sector (see Greenwood,
9
Herkowitz, and Krusell (1997)).20 Let K(t) denote the capital stock available at the beginning
10
of time t, so the output ‡ow coming out of the capital good sector is AK(t), which is then
11
split between two di¤erent usages:
(15)
AK(t) = X(t) + E(t);
12
13
where X(t) denotes capital investment and E(t) denotes the ‡ow of capital used to produce
14
consumption goods at t. E(t) fully depreciates, so capital in the whole economy accumulates
15
as follows:
K(t) = X(t)
16
where
(16)
K(t);
is the depreciation rate in the capital goods sector. Substituting (15) into the above
equation and de…ning
=A
, we obtain
K(t) = K(t)
E(t):
17
At time t, capital E(t) and labor L (assumed to be constant) produce all the intermedi-
18
ate goods fcn (t)g1
n=0 with technologies speci…ed by (7), which are ultimately combined to
20
Notice that this dynamic setting di¤ers from the most standard setting. For the sake of expositional
convenience, detailed comparisons and justi…cations are postponed to Subsection 4.1.3 and Section 5.
20
19
1
2
produce the …nal consumption good C(t) according to (5). Based on Table 6, de…ne
8
>
( 1)
<
E+L
if
0 E < aL
a
;
F (E; L)
n
n
>
(a )
n
n+1
: n+1
E + a 1 L if a L E < a L for n 1
an+1 an
(17)
which is the endogenous aggregate production function derived in Section 3. Therefore,
(18)
C(t) = F (E(t); L) = r(t)E(t) + w(t)L;
3
4
where r(t) and w(t) are the rental price for capital and the wage rate at time t, respectively.
5
With some abuse of notation, let E(C(t)) denote the total amount of capital goods needed
6
to produce …nal consumption goods C(t), so F (E(C(t); L)
7
C and all the intermediate goods fcn g1
n=0 are non-storable.
8
9
10
C(t). Final consumption goods
By the second welfare theorem, we can characterize the competitive equilibrium by resorting to the following social planner problem:
Z 1
C(t)1
1 t
max
e dt
C(t) 0
1
(19)
subject to
K(t) = K(t)
(20)
E(C(t));
K(0) = K0 is given;
11
where
12
positive consumption growth and, to exclude the explosive solution, we also assume
13
is the time discount rate. Following conventions, we assume
> 0 to ensure
(1
) < . Putting them together, we impose
0<
14
<
:
(21)
The social planner decides the intertemporal consumption ‡ow C(t) and makes optimal
investment decisions X(t), which in turn determine the evolution of the endowment structure
K(t)
L
and the optimal amount of capital allocated for consumption goods production E(t).
21
Note that, at any given time t, once E(t) is determined, the optimization problem for the
whole consumption goods sector is exactly the same as the static problem in Section 3. From
the bottom row of Table 6, we know that E(C) is a strictly increasing, continuous, piece-wise
i
linear function of C. It is not di¤erentiable at C =
L, for any i = 0; 1; ::. Therefore, the
above dynamic problem may involve changes in the functional form of the state equation:
(20) can be explicitly rewritten as
K=
15
8
>
>
>
>
<
when
K;
C<L
K E0;1 (C); when
>
>
>
>
: K En;n+1 (C); when
L
n
L
C<
n+1
L; for n
where En;n+1 (C) is de…ned in the bottom row of Table 6 for any n
4.1.2.
;
C< L
1
0.
Equilibrium Characterization
1
2
We can verify that the objective function is strictly increasing, di¤erentiable and strictly
3
concave while the constraint set forms a continuous convex-valued correspondence, hence
4
the equilibrium must exist and also be unique.
Let t0 denote the last time point when aggregate consumption equals L (that is, only
good 0 is produced), and tn denote the …rst time point when C =
is produced) for n
n
L (that is, only good n
1: As can be veri…ed later, aggregate consumption C is monotonically
increasing over time in equilibrium, hence the problem can also be written as
max
C(t)
Z
0
t0
C(t)1
1
1
e
t
dt +
1 Z
X
n=0
tn+1
tn
C(t)1
1
1
e
t
dt
22
subject to
K=
8
>
>
>
>
<
when
K
K E0;1 (C); when
>
>
>
>
: K En;n+1 (C); when tn
t
where tn is to be endogenously determined for any n
0.
0
t
t0
t0
t
t1
tn+1 ; for n
;
1
K(0) = K0 is given;
5
1
2
3
Table 6 indicates that goods 0 and 1 are produced during the time period [t0 ; t1 ] and
a
tn+1 for n
1; goods n and n + 1
h
i n+1 n
n
(a )
are produced. Correspondingly, E(C) = En;n+1 (C)
C
L an+1 a n . If K0 is
a 1
E(C) = E0;1 (C)
1
L). When tn
(C
t
4
su¢ ciently small (this is more precisely shown below), then there exists a time period [0; t0 ]
5
in which only good 0 is produced and all the working capital is saved for the future, so E = 0
6
when 0
7
goods h and h + 1 for some h
t0 . If K0 is large, on the other hand, the economy may start by producing
t
1, so t0 = t1 =
= th = 0 in equilibrium.
To solve the above dynamic problem, following Kamien and Schwartz (1991), we set the
discounted-value Hamiltonian in the interval tn
tn+1 , and use subscripts “n; n + 1”to
t
denote all the variables during this time interval:
Hn;n+1 =
+
where
n;n+1
C(t)1
1
n+1
e
t
n+1
n+1
L
n;n+1 (
L C(t)
+
n;n+1
C(t)) +
n+1
n;n+1
and
0 and C(t)
n
is the co-state variable,
two constraints
1
L
[ K(t)
En;n+1 (C(t))]
n
n;n+1 (C(t)
n
n;n+1
n
L)
(22)
are the Lagrangian multipliers for the
0, respectively. The …rst-order condition
23
and Kuhn-Tucker conditions are
@Hn;n+1
= C(t)
@C
n+1
n+1
L
n;n+1 (
C(t)) = 0;
n
n;n+1 (C(t)
8
L) = 0;
an+1
an
n+1
n
n;n+1
n+1
n;n+1
0;
n+1
n
n;n+1
0; C(t)
L
n+1
n;n+1
+
n
n;n+1
= 0;
(23)
C(t) > 0;
n
L
0:
We also have
0
n;n+1 (t)
1
2
n
t
e
@Hn;n+1
=
@K
=
In particular, when C(t) 2 (
C(t)
3
e
t
=
n
an+1
n;n+1
n+1
n;n+1
L;
n+1
(24)
:
L),
n+1
n;n+1
=
n
n;n+1
= 0; and equation (23) becomes
an
(25)
n:
4
The left-hand side is the marginal utility gain from increasing one unit of aggregate con-
5
sumption, while the right-hand side is the marginal utility loss due to the decrease in capital
6
because of that additional unit of consumption, which by the Chain’s Rule can be decom-
7
posed into two multiplicative terms: the marginal utility of capital
8
capital requirement for each additional unit of aggregate consumption
9
Taking the logarithm on both sides of equation (25) and di¤erentiating with respect to t; we
10
n;n+1
and the marginal
an+1 an
n+1
n
(see Table 6).
obtain the consumption growth rate from the regular Euler equation:
C(t)
=
C(t)
gc
11
(26)
;
12
for tn
13
consumption ‡ow C(t) must be continuous and su¢ ciently smooth (without kinks); hence
14
from (26) we obtain:
15
t
tn+1 for any n
C(t) = C(t0 )egc (t
t0 )
0. The strictly concave utility function implies that the optimal
for any t
t0 > 0:
(27)
16
Following Kamien and Schwartz (1991), we have two additional necessary conditions at
17
t = tn+1 :
Hn;n+1 (tn+1 ) = Hn+1;n+2 (tn+1 );
n;n+1 (tn+1 )
18
1
n+1;n+2 (tn+1 ):
(28)
(29)
Substituting equations (28) and (29) into (22), we can verify that K (tn+1 ) = K + (tn+1 ). In
other words, K(t) is also continuous. Observe that
C(t0 )egc (tn
2
3
=
24
t0 )
= C(tn ) =
n
L when t0 > 0;
(30)
which implies
n
log C(tL0 ) +
t0
, when t0 > 0.
(31)
4
tn =
5
De…ne mn
6
n and good n + 1 are produced (that is, the duration of the diversi…cation cone for good n
7
and good n + 1). We must have
tn , which measures the length of the time period during which both good
tn+1
mn = m
8
9
gc
log
:
gc
(32)
The comparative statics for equation (32) is summarized in the following proposition.
10
Proposition 2 The full life span of each industry n
11
industrial upgrading (measured by frequency
12
but increases with the aggregate growth rate gc . More precisely, the industrial upgrading is
13
faster when technological e¢ ciency
14
1
1
)
2m
1 is equal to 2m. The speed of
decreases with the productivity parameter
increases, or the intertemporal elasticity of substitution
increases, or the time discount rate
decreases.
The intuition for the proposition is the following. Suppose good n and good n + 1 are
15
16
produced. When the productivity parameter
17
(
18
of good n+1 (
n
is larger, the marginal productivity of good n
) becomes bigger, making it pay to stay at good n longer; but the marginal productivity
n+1
) also becomes bigger, making it optimal to leave good n and move to good
19
25
n + 1 more quickly. It turns out that the …rst e¤ect dominates the second e¤ect because (9)
1
implies that, by climbing up the industrial ladder, the productivity gain
is su¢ ciently small
2
relative to the additional capital cost re‡ected by the cost parameter a. Thus the net e¤ect
3
is that industrial upgrading slows down. On the other hand, industrial upgrading is faster
4
when the consumption growth rate gc increases because larger consumption is supported by
5
more capital-intensive industries, as implied by Table 6.
6
When the household is more impatient (larger ), it will consume more and save less
7
and hence capital accumulation becomes slower and thus the endowment-driven industrial
8
upgrading also becomes slower. When the production of the capital good becomes more
9
e¢ cient ( ), capital can be accumulated faster, so the upgrading speed is increased. When
1
), the household is
10
the aggregate consumption is more substitutable across time (larger
11
more willing to substitute current consumption for future consumption, which also boosts
12
saving and then causes quicker industrial upgrading.
13
We are now ready to derive the industrial dynamics for the entire time period. The
14
industrial dynamics depend on the initial capital stock, K(0): We show in the Appendix
15
that there exists a series of increasing constants, #0 ; #1 ;
16
K(0)
17
#0 (Appendix 3 fully characterizes this case); if #n < K(0)
18
by producing goods n and n + 1 for any n
19
for any K(0) < #n : That is, irrespective of the level of initial capital stock, the economy
20
always starts to produce good n + 1 whenever its capital stock reaches #n :
; #n ; #n+1 ;
; such that if 0 <
#0 ; the economy will start by producing good 0 only until the capital stock reaches
#n+1 ; the economy will start
0: Furthermore, we can show that K(tn )
To be more concrete, consider the case when #0 < K(0)
#n
#1 , where the threshold values
#0 and #1 can be explicitly solved (see Appendix 2). That is, the economy will start by
producing goods 0 and 1: Using equation (27) and Table 6, we know that when t 2 [0; t1 ],
E(t) =
a
1
(C(t)
L) =
a
1
(C(0)e
t
L):
26
Correspondingly,
a
K = K(t)
1
t
(C(0)e
L):
Solving this …rst-order di¤erential equation with the condition K(0) = K0 , we obtain
aC(0)
1
K(t) =
"
aL
t
+ K0 +
+
(
1)
e
aC(0)
1
+
aL
(
1)
#
e t;
which yields
#1
21
a L
1
K(t1 ) =
"
aL
+
+ K0 +
(
1)
When t 2 [tn ; tn+1 ], for any n
1
2
(a
)
an+1
L n+1
a 1
t
C(0)e
+
aL
(
1)
#
L
C(0)
:
1; the transition equation of capital stock (20) becomes
n
K = K(t)
aC(0)
1
an
n
when t 2 [tn ; tn+1 ], for any n
1:
Solving the above di¤erential equation, we obtain:
K(t) =
n
+
t
ne
+
ne
t
when t 2 [tn ; tn+1 ], for any n
(33)
1
where
n
=
n
=
=
an
n+1
n
an+1
an
n+1
n
n
an+1
L
C(0)
n
n
(
#n +
(a
)L
;
(a 1)
C(0)
;
n+1
(a
n
a )L
1
"
1
+
(a
(a
)
1)
#)
:
3
Again the endogenous change in the functional form of the capital accumulation path (33)
4
re‡ects the structural changes that underlie the aggregate economic growth. Note that
5
f#n g1
n=2 are all constants, which can be sequentially pinned down: #n
1
computed from equation (33) with K(tn 1 ) known.
27
K(tn ) can be
For each individual industry, using equation (27) and Table 6, we obtain
cn (t) =
c1 (t) =
8
>
>
>
>
<
>
>
>
>
:
8
>
>
>
>
<
t
C(0)e
>
>
>
>
:
8
>
< L
c0 (t) =
>
:
n
L
n 1
t
C(0)e
n+1
n
+
1
when t 2 [tn 1 ; tn ]
L
;
1
when t 2 [tn ; tn+1 ]
t
L
1
t
C(0)e
2
2
otherwise
0;
C(0)e
; for all n
+
when t 2 [0; t1 ]
;
L
;
1
when t 2 [t1 ; t2 ] ;
otherwise
0;
t
C(0)e
1
0;
L
; when t 2 [0; t1 ]
;
otherwise
2
where C(0) can be uniquely determined by the transversality condition (see Appendix 2) and
3
the endogenous time points tn are given by (31) for any n
4
above mathematical equations fully characterize the industrial dynamics for each industry
5
over the whole life cycle while aggregate consumption growth is still given by (26). If the
6
initial capital stock is su¢ ciently small such that K0 < #0 as characterized in Appendix 3,
7
then the economy will …rst have a constant output level equal to L (Malthusian regime) until
8
the capital stock K(t) = #0 , which occurs at t0 > 0, after which the aggregate consumption
9
growth rate permanently changes to
1. Recall t0 = 0 in this case. The
(Solow regime). All these mathematical results can
10
be read as follows:
11
Proposition 3 There exists a unique and strictly increasing sequence of endogenous thresh-
12
old values for capital stock, f#i g1
i=0 , which are independent of the initial capital stock K(0).
13
The economy starts to produce good n when its capital stock K(t) reaches #n
14
K(t) evolves following equation (33), while total consumption C(t) remains constant at L
1
for any n
1.
15
until t0 , after which it grows exponentially at the constant rate
28
: The output of each indus-
1
try follows a hump-shaped pattern: When capital stock K(t) reaches #n 1 ; industry n enters
2
the market and booms until capital stock K(t) reaches #n ; its output then declines and …nally
3
exits from the market at the time when K(t) reaches #n+1 :
The industrial dynamics characterized in Proposition 3 are depicted in Figure 3.
4
[Figure 3]
5
6
It is clear that our theoretical predictions for the industrial dynamics are well consistent
7
with Facts 2, 3, and 4 established in Section 2. Meanwhile, the theoretical results are also
8
consistent with the Kaldor facts that the growth rate of total consumption remains constant
9
and the capital income share is relatively stable, as shown in equation (14).
10
4.1.3.
Discussion
11
Now we brie‡y discuss how this dynamic setting deviates from the most standard envi-
12
ronment (standard setting, henceforth) and why the current alternative setting is desirable.21
13
The standard setting postulates that there is one all-purpose …nal good, which can be either
14
consumed or used for investment (think about coconut), so the production of consumption
15
goods and capital goods are not separated. Thus (16) is still valid, but (15) and (18) are
16
now replaced by
C(t) + X(t) = F (K(t); L) = r(t)K(t) + w(t)L;
17
(34)
18
where function F (K(t); L) is de…ned in (17) with E being replaced by K. Combining (34)
19
and (16) yields
K(t) = F (K(t); L)
20
21
C(t)
K(t);
Discussions in this subsection and in subsection 5.1 are inspired by the referee.
(35)
29
21
which di¤ers from (20). The social planner then solves (19) subject to (35) and that K(0) =
1
K0 is given. In contrast, our multi-sector model assumes that capital goods and consumption
2
goods are di¤erent goods, which are used for di¤erent purposes and also produced with
3
di¤erent technologies. In particular, technological progress takes place in thecapital goods
4
sector in an AK fashion, but there is no technological progress in theconsumption goods
5
sector.
6
There are four main reasons that we prefer the current setting over the standard setting.
7
First, generally speaking, due to their technological nature, capital goods (such as ma-
8
chines) usually cannot be directly used for consumption, nor can consumption goods (such
9
as shirts) be used for capital investment, so the assumptions in the current setting are more
10
realistic. Even for purely theoretical purposes, such a dichotomy has a long tradition in the
11
growth literature, dating at least back to Uzawa (1961).
Second, the current setting is more consistent with two important empirical facts than
the standard setting. Hsieh and Klenow (2007) show that the price of capital goods relative
to consumption goods is systematically higher in poor countries, but the standard setting
implies that this relative price is always equal to one, independent of the income level.
However, this fact can be explained in the current setting. To see this, note that (17) implies
the following relative price of capital goods to consumption goods:
r
E
L
8
>
<
>
:
1
if
a
n+1
n
an+1 an
0
n
if a L
E < aL
n+1
E<a
L for n
:
1
12
Since
is larger in richer countries, n is larger, and therefore r is smaller because r decreases
13
with n due to (9). Another fact which contradicts the predictions of the standard setting
14
is that this relative price keeps falling dramatically over time within the same country such
15
as the US, which is well established in the literature on investment-speci…c technological
16
progress (see Greenwood, Herkowitz, and Krusell (1997) and Cummins and Violante (2002)).
30
17
In contrast, this fact is consistent with our setting, because industrial upgrading is perpetual
1
(n keeps increasing) when the aggregate economy attains sustainable growth (gc > 0), so
2
the relative price r(t) keeps falling over time. In fact, Proposition 2 further states that
3
more e¢ cient production in capital goods (higher ) leads to faster industrial upgrading (n
4
increases faster as life spans of industries become shorter), thus r(t) decreases faster.
5
Third, the primary purpose of our model is to highlight the role of capital accumulation
6
in driving industrial dynamics, so it is conceptually clearer to explicitly distinguish capital
7
goods from non-capital goods. Also, motivated by the literature on investment-speci…c
8
technological progress, we assume thecapital goods sector makes faster technological progress
9
than theconsumption goods sector.
10
Fourth, it turns out that closed-form solutions are obtained both for the life-cycle dy-
11
namics of each industry and for the aggregate growth path in our current setting, but we lose
12
this high tractability in the standard setting. To make this point more clearly, in Subsection
13
5.1, we further explore the implications of the standard setting (with technological progress
14
in theconsumption goods sector).
15
4.2. Empirical Evidence
Propositions 2 and 3 predict that the industrial dynamics exhibit a hump-shaped pattern
and the life span of an industry (2m) decreases with the aggregate growth rate (gc ) but
increases with its productivity level (recall that the productivity level is strictly increasing
in ). To empirically test these predictions, we …rst compute the life span of industry i; di ;
based on regression (2) and Table 3 in Section 2. Note that as the right-hand side of equation
(2) is a quadratic function of time t; there are two roots of the parabola t1i and t2i ; at which
the output (or value-added) share of industry i; yit ; equals zero. Therefore, the whole life
31
span of industry i, can be measured by the distance of the two roots of the parabola:
dit =
q
(
1
2
3 kit )
+
4
2
(
0
+
4 Tit
+
5 Di
+
6 GDP GRt )
:
2
@di
< 0; so we immediately have that sign( @T
) = sign(
i;t
From Table 3, we see that
2
1
@di
) = sign(
and sign( @GDP
GRt
6)
2
@di
tions that an industry’s life span increases with the industrial labor productivity ( @T
> 0)
i;t
3
@di
< 0). For developing countries,
but decreases with the aggregate GDP growth rate ( @GDP
GRt
4
note that, based on Tables 4 and 5, the e¤ect of labor productivity on the life span of an in-
5
dustry is still consistent with the model prediction. However, the e¤ect of aggregate growth
6
rate ( 6 ) is not signi…cant, although the sign is still consistent with the model prediction in
7
most of the subsample regression results.22
8
5.
9
5.1. Alternative Dynamic Setting
16
4)
>0
< 0: This empirical evidence supports the theoretical predic-
Extensions
10
In this subsection, we explore industry dynamics and aggregate growth in an alternative
11
and more standard environment, where all capital is used to produce one …nal good, which
12
can be either consumed or saved as capital investment.
Suppose the aggregate production function becomes
13
A(t)F (K(t); L);
14
(36)
15
where function F (K(t); L) is de…ned as in (36) with E(t) being replaced by K(t), and A(t)
16
captures the exogenous Hicks-neutral technology progress.23 A(0) is given and normalized
22
This may suggest that further research is needed to better understand the interaction between aggregate
growth and industry dynamics for developing countries.
23
The labor-augmenting exogenous technological progress does not yield perpetual industry upgrading for
this case, although it generates sustainable aggregate growth. The intuition is the following: When the
labor-augmenting technological progress is micro-founded at the industry level by entering into the Leontief
K(t)
industry production function, A(t)L
for the aggregate economy will be a constant on the balanced growth
17
1
to one. Notice that, unlike Ngai and Pissarides (2007), the TFP growth is still balanced
across di¤erent sectors. Let
K = A(t)
2
3
32
When an L
1)
a
K(t) + A(t)L
K(t)
K(t) < an+1 L for some n
K = A(t)
4
(
denote the capital depreciation rate. When K(t) < aL;
n+1
n
an+1
an
(37)
C(t):
1, the capital accumulation function is
n
K(t) +
K(t)
(a
)
A(t)L
a 1
(38)
C(t):
5
The social planner solves (19) subject to (37) and (38) with K(0) taken as given. Following
6
the same method as in Section 4, we can verify that su¢ cient conditions are satis…ed to
7
ensure that the equilibrium exists and is also unique. Solving the Hamiltonian, we obtain
8
the following Euler equation when an+1 L > K(t)
n+1
A(t) an+1
C
=
C
9
11
12
13
1:
n
an
(39)
;
n+1
10
an L for any n
which is a positive constant if and only if A(t) an+1
n
an
is a su¢ ciently large positive number
independent of n. To satisfy this requirement, we assume
8
>
< B ann if t 2 [tn ; tn+1 ) for any n 1;
A(t) =
>
:
B a if t 2 [0; t1 )
(40)
a 1
where B is a positive constant and tn is de…ned in Section 4. That is, the Hicks-neutral
14
productivity A(t) is a step function, which generically keeps constant and only jumps up
15
discontinuously at the time point when a new industry just appears.24 Then (39) becomes
C
=
C
16
17
18
and
n;n+1
n;n+1
assume
1
B
a 1
=
1
B
a 1
; 8t 2 [0; 1):
1
B.
a 1
>
(41)
To ensure positive growth and also rule out explosive growth, we
1
B
a 1
> 0. Under those conditions, sustainable growth is achieved
path, so industrial upgrading also stops at some point. Please refer to Appendix 4.2 for more details.
24
Note that without technological progress F (K; L) is a piece-wise linear, continuous and concave function
of K for any given L. Thus A(t) must jump at each kink point of F (K; L) to ensure that A(t)F (K; L) has
a constant return to K. The discontinuity of A(t) could be avoided when the aggregate production function
X is of general CES form with a positive and …nite substitution elasticity, as is shown below.
33
19
together with perpetual industry upgrading from labor-intensive industries to more and more
1
capital-intensive industries step by step, because now the return to capital is sustained at a
2
constant level
3
the endogenous capital accumulation equations sequentially (see Appendix 4).
1
B.
a 1
Following a method similar to that in Section 4, we could also derive
4
However, this alternative standard setting is much less tractable than the setting in
5
Section 4. In particular, the life-cycle dynamics of each industry no longer have a neat, closed-
6
form characterization.25 Moreover, to obtain sustainable growth and perpetual industry
7
upgrading in this alternative setting, we must require the exogenous technology progress to
8
have discontinuous jumps. This is because the functional form of the aggregate production
9
function changes endogenously due to the endogenous structural change in the underlying
10
disaggregate industry compositions. Nevertheless, the key qualitative features remain valid,
11
including sustainable aggregate growth at a constant rate (41), perpetual industry upgrading
12
(equivalent to
13
of each industry, more capital-intensive industries reaching the peaks later, etc.
K(t)
L
continuously and strictly increasing to in…nity), hump-shaped dynamics
14
It is worth emphasizing that the perpetual hump-shaped industry dynamics (or structural
15
change at the disaggregate sectors) are still driven by capital accumulation, which is in turn
16
sustained by the (balanced) aggregate technology progress. So the theoretical mechanism
17
for structural change is still di¤erent from non-homothetic preference or unbalanced TFP
18
growth rates across sectors.
25
There are two main reasons that the setting in Section 4 is more tractable: (1) It gives us more linearity
as it features the linear capital production of AK technology with a constant return to capital , whereas the
aggregate production function is only piece-wise linear and concave in capital (without technology progress)
in the standard alternative setting. (2) The consumption-capital relation at the aggregate level derived
from the static model (the bottom row of Table 6) can be directly applied to the dynamic analysis in the
current setting, whereas in the more standard setting, the static model only yields the production-capital
relation at the aggregate level instead of the consumption-capital relation, so the dynamic consumption
analysis (Euler equation) becomes more complicated as its functional form is endogenously di¤erent over
time without appropriately introduced technology progress.
19
34
5.2. General CES Function
1
A salient counterfactual feature of the previous analysis is that at any time point at
2
most two industries coexist in equilibrium due to the in…nite substitution elasticity across
3
industries. In this section we show that all the industries coexist when the aggregate pro-
4
duction function is generalized to CES with …nite substitution elasticity, but the key results
5
remain valid, namely, that each industry still demonstrates a hump-shaped life-cycle pattern
6
with more capital intensive industries reaching their peaks later. However, no closed-form
7
solutions can be obtained due to the “curse of dimensionality” because no industries exit
8
any more.
9
10
11
Assume everything remains the same as in Sections 3 and 4 except that now (5) is replaced
by a more general CES function:
"1
#1
X
;
C=
n cn
(42)
n=0
12
where
< 1. We obtain the following result.
13
Proposition 4 (Generalized Rybzynski Theorem): Suppose the production function is CES
14
as de…ned in (42). All the industries will coexist in equilibrium. In addition, when capital
15
endowment E increases, output and employment in industry n (for any n
16
exhibit a hump-shaped pattern: an industry …rst expands with E and then declines with E
17
after the industrial output reaches its peak. The more capital intensive an industry, the larger
18
the capital endowment at which the industry reaches its peak.
1) will both
19
The proof is relegated to Appendix 5. Recall that the classical Rybczynski Theorem
20
is silent about what happens when the number of goods exceeds two. In a model with
21
N (N > 2) goods and two factors, as the country becomes more capital abundant, the
22
existing theory does not give clear predictions on which sectors expand and which sectors
23
shrink (Feenstra, 2003). Proposition 4 generalizes the Rybczynski Theorem by extending the
35
24
1
commodity space from two-dimensional to in…nite-dimensional, so we call this proposition
the Generalized Rybzynski Theorem.
For the dynamic analysis, we characterize the equilibrium in the same dynamic setting as
in Section 4.26 It is worth emphasizing that the hump-shaped dynamics and the timing fact
also hold in the dynamic model even if all industries have the same productivity in …nal good
production (
n
= 1 for all n) when the aggregate production function is of CES with
<1
because now each industry must always exist at any …nite price. So the only mechanism that
drives industrial upgrading is still the change in the factor endowment because of homothetic
preference and balanced productivity growth across sectors. More speci…cally, observe that
E
="
C
1
X
an
n
1
1
1+an
n=1
1+
1
X
n=1
n
1
(1 + an )
1
#1 ;
2
where denotes the endogenous rental-wage ratio, which turns out to be a strictly decreasing
3
function of E as shown in the proof of Proposition 4 (see Appendix 5). So the above function
4
implicitly determines E
5
0. This is because the marginal productivity of capital in the …nal good production must
6
be diminishing since the total labor supply is …xed while the aggregate production function
7
is homogeneous of degree one with respect to capital and labor. De…ne
8
which is the elasticity of the change in marginal capital expenditure relative to the aggregate
9
consumption change. For any speci…c time point t in the dynamic equilibrium, we must have
C(t)
C(t)
G(C), where function G(C) must satisfy G0 (C) > 0 and G00 (C)
(C)
CG00 (C)
,
G0 (C)
10
the Euler equation:
11
as in the linear case ( = 1).27 This indicates that, as long as C(t) > 0, E(t) must be strictly
12
increasing over time in equilibrium (holding L …xed, we still assume
26
=
+ (C(t))
, which would degenerate into (26) when (C) = 0, just
> 0 as before).
Due to space limits, the equilibrium in the more standard alternative dynamic setting (that is, the one
outlined in Subsection 5.1) is studied in Appendix 6. The key qualitative results turn out to be preserved.
27
If lim (C(t)) ! 1 as C(t) ! 1, then the asymptotic aggregate consumption growth rate will be zero.
36
13
Consequently, the dynamic path of the output of each individual industry must still exhibit
1
a hump-shaped pattern in the full-‡edged dynamic model, but it is no longer possible to
2
obtain closed-form solutions for the entire dynamic path.
3
6.
Conclusion
4
In this paper we explore the optimal industrial structure and the dynamics of each in-
5
dustry in a frictionless and deterministic economy with a highly tractable in…nite-industry
6
growth model. Closed-form solutions are obtained to fully characterize the endogenous
7
process of repetitive hump-shaped industrial dynamics along the sustained growth path of
8
the aggregate economy. The model generates all the features consistent with the stylized
9
facts we observe in the data. We highlight the improvement in the endowment structure,
10
which is given at any speci…c time but changeable over time (capital accumulation), as the
11
fundamental driving force of the perpetual structural change at the disaggregated industry
12
level in an economy. The model highlights the structural di¤erence at di¤erent development
13
levels by showing that the optimal industrial compositions are endogenously di¤erent as the
14
endowment structure changes over time.
15
Several directions for future research seem particularly appealing. First, various market
16
imperfections could be introduced into this …rst-best environment to see how industrial
17
dynamics can be distorted by those frictions, which may allow for discussions on welfare-
18
enhancing government policies. Second, it would be interesting to explore whether the
19
mechanism highlighted in this paper can be extended to explain the dynamics of the sub-
20
sectors in the service sector (see Buera and Kaboski (2012a, 2012b)). Third, it may be fruitful
21
to incorporate …rm heterogeneity into each industry and study the entry, exit, and growth of
22
…rms within di¤erent industries together with industrial dynamics (see Hopenhayn (1992),
23
Luttmer (2007), and Rossi-Hansberg and Wright (2007), Samaniego (2010)). We hope the
24
model developed in this paper can be useful to help us think further about all these issues.
25
37
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18
1
2
3
7.
42
Appendices (Online Publication)
7.1. Appendix 1
Here we show that in competitive equilibrium, if two di¤erent industrial goods are produced simultaneously, these two goods have to be adjacent in their capital intensities.
First set up the Lagrangian for the household problem with the multiplier
Proof:
denoted by , and we obtain the following optimality condition for consumption:
n
1
X
n
cn + c0
n=1
!
pn ; for 8n
(43)
0;
“ = ”when cn > 0:
By contradiction, suppose the statement is not true, then there can exist some good j and
good m such that 1
j
m 2 , and cj and cm are both strictly positive in the equilibrium.
Then
M RTm;j =
m j
=
am r + w
;
aj r + w
which means
m j
r
= j m j
w
a (a
1
m j
)
:
Now we show that at this relative price, good j + 1 strictly dominates good j, that is
M RTj+1;j
4
6
m j
aj aj (am
1
m j
j
j
1
m j
)
)
+1
+1
> 0;
which is equivalent to
n
5
m j
aj+1 aj (am
aj+1 r + w
=
aj r + w
an
1
n
<
1
This is true because
a
, for some integer n
m
j
2:
(44)
43
n
1
=
n
an
(
1)
a n
n
(
=
Pn
n 1
1
i=0
1)
(a
i
1
Pn
(
=
n
1 i
i=0
P
i
) ni=01 a
1)
a
=
Pn
1 i
i=0
Pn 1 a i
i=0
P
1) ni=01 i
P
) ni=01 i an 1 i
1
(
(a
<
1
a
for any n
2:
7
Therefore, it contradicts that good j is produced and consumed. It is straightforward to
1
show that it is possible to have both good 0 and good 1 under some conditions.
Now we need to show that if cn > 0 for some n
2 and cj = 0 for any j
2 and j 6= n,
then c0 = 0. From (43), the household would strictly prefer good n to good n + 1 if
n+1
<
n
2
or equivalently,
3
r
w
<
an
1
an
1
r
w
an+1
1
an+1
an
1;
; and good n is strictly preferred to good n
1 if and only if
. This means
1
4
>
pn+1
w + an+1 r
=
; for n
pn
w + an r
an
<
r
< n
w
a
1
an
1
(45)
:
By contradiction, suppose c0 > 0, then
n
w+an r
w
=
because of (43), that is
n
an
1
=
r
.
w
So we
must have
n
1
an+1
an
<
1
an
5
where the second inequality is equivalent to
6
will exist no integer n
7
is no smaller than
8
7.2. Appendix 2
n
<
1
1
1
an
an
<
a
a
1
;
. However, since a
1 > , there
2 that can satisfy the above inequality because the left-hand side
+ 1. Therefore we must have c0 = 0.
Here we solve for the initial value of total consumption C(0) when #0 < K(0)
#1 ,
and also show how to derive the threshold values for #i ; i = 0; 1; 2; :::. The transversality
44
condition is derived from
lim H(t) = 0;
t!1
so
C(t)1
lim
t!1
1
1
t
e
+
K(t)
n(t);n(t)+1
En(t);n(t)+1 (C(t))
= 0:
Note that
C(t)1
1
e
t!1
1
"
C(0)1 e
= lim
t!1
1
t
lim
= lim
t!1
= lim
t!1
(
= lim
t!1
t
(0) e
(0)
= lim K(t)e
t!1
K(t)
En(t);n(t)+1 (C(t))
)t
e
K(t)
t
+
K(t)
n(t);n(t)+1
En(t);n(t)+1 (C(t))
En(t);n(t)+1 (C(t))
"
#
(a
)
an(t)+1
L n(t)+1
a 1
n(t)
K(t)
C(0)e
e
t
K(t)e
t
n(t);n(t)+1
(1
n(t);n(t)+1
"
+
t
an(t)
n(t)
#
#)
an(t)+1 an(t)
(a
)
L
a 1
1
t
;
thus we must have lim K(t)e
t
= 0. When t 2 [0; t1 ],
t!1
E(t) =
a
1
(C(t)
L) =
a
1
(C(0)e
t
L);
Correspondingly,
K = K(t)
9
1
E(C(t)) = K(t)
a
1
(C(0)e
t
L)
Solving this …rst-order di¤erential equation with the condition K(0) = K0 , we obtain
"
#
aC(0)
aC(0)
aL
aL
1
1
K(t) =
e t+
+ K0 +
+
e t;
(46)
(
1)
(
1)
45
2
from which we obtain
K(t1 ) =
a L
1
1
aC(0)
1
When t 2 [tn ; tn+1 ] for 8n 1, we have
"
n
an+1 an C(0)e t
(a
+
K(t) =
n+1
n
(a
2
3
4
"
aL
+ K0 +
+
(
1)
aL
+
(
#
)L
+
1)
1)
n:n+1 e
t
L
C(0)
(47)
:
(48)
:
which gives
(
n
n:n+1
5
L
=
C(0)
K(tn ) +
an+1
an
L
1
"
1
(a
+
(a
Therefore, we obtain equation (33). Substituting t = tn+1 =
K(tn+1 ) =
6
#
K(tn ) +
an+1
an
L
1
"
+
log
)
1)
n+1 L
C(0)
(a
#)
(49)
into (33), we obtain
)(
(a
1)
1)
#
Using recursive induction, we get
n 1
K(tn ) =
7
(n 1)
K(t1 ) + (a
1)B
a 1
(n 2)
a
; for any n
1
2;
(50)
a
where B is a parameter de…ned as
B
L
1
2
(a
4
The transversality condition lim K(t)e
t!1
K(t1 ) + (a
)
+
t
(a
9
1)
3
5:
= 0 implies that
1)B
a
2
1
8
1
= 0;
a
so
K(t1 ) =
(a
1)B
1
a
a
:
(51)
46
10
To ensure #0 > 0, we must impose an additional assumption:
a<
1
(52)
;
2
which means that capital accumulation speed
3
capital intensity parameter a, otherwise the economy will never start by only producing
4
good 0 no matter how small K0 is. To ensure #1
(a
)
must be su¢ ciently small relative to the
K(t1 ) > 0, we need to further assume
1
>
(53)
:
(1
a
5
) + (a
)
1
This ensures B < 0. Also note that (53) guarantees that
"
K0 +
=
aC(0)
1)
(
+
a L
1)
(
aL
(
1)
#
(a
aL
+
> . According to (47), we have
(
1)
L
C(0)
1)B
1
a
.
(54)
a
6
We can verify that the right-hand side is strictly positive. Observe that the left-hand side
7
is a strictly decreasing function of C(0), therefore we can uniquely pin down the optimal
8
C (0). (54) immediately implies
> 0 and
@C (0)
@L
> 0.
Note that (51) implies that K(t1 ) does not depend on K(0), therefore (50) tells that
9
10
@C (0)
@K0
K(tn ) for all n
1 are independent from K(0):
To ensure C (0)
a
(a 1)B
L, we need K0
, which requires
1 a
K0
a
#1
1
1
1
a
2
a
4
(1
)+
(a
)
1
3
5 L:
We also need to ensure C (0) > L, which requires
"
K0 +
aL
(
1)
+
aL
(
1)
#
>
(
a L
1)
+
aL
(
(a
1)
1)B
1
a
a
;
that is
K0 > # 0
11
1
2
a
(
4
1)
1
1
(1
1
)
a
47
3
5 L:
7.3. Appendix 3
Here we prove that there exists a series of constant numbers, #0 ; #1 ;
;such that, if
#0 ; the economy will start from producing good 0 only until the capital stock
2
0 < K(0)
3
reaches #0 ; if #0 < K(0)
4
#n < K(0)
5
K(tn ) = #n whenever K(0) < #n :
#1 ; the economy will start from producing goods 0 and 1; if
#n+1 ; the economy will start from producing goods n and n + 1: Furthermore,
Now let us characterize the solution to the above dynamic problem when K0 2 (0; #0 ]
while keeping all the other assumptions unchanged. The economy must start by producing good 0 only. The discounted-value Hamiltonian with the Lagrangian multipliers is the
following
H0 =
C(t)1
1
1
K(t) +
0
0 (L
C(t) = 0 when
0
0
> 0:
@H0
=
@K
:
e
t
+
0
C(t)):
The …rst-order condition and K-T condition are
C(t)
0
0 (L
e
t
=
0
0;
C(t)) = 0;
L
and
0
=
0
They immediately imply that C (t) = L, which implies that only good 0 is produced. Since
no capital is used for production, we have K(t) = K(t). When capital stock K exceeds #0
by an in…nitesimal amount, the economy produces both good 0 and good 1. From that point
48
on, the problem is exactly the same as the one we have just solved in the main text. The
de…nition of t0 implies that it is the time point when K just equals #0 . Thus K0 e
so t0 =
log
#0
K0
= #0 ,
. Therefore
8
>
<
C (t) =
6
t0
(tj t0 )
Observe that Le
>
: Le
= C(tj ) =
when
L;
j
(t t0 )
t
; when
t0
:
t > t0
log
L, so tj = t0 +
j.
Correspondingly, the capital stock on the equilibrium path is given by
K(t) =
where
8
>
>
>
>
>
>
>
>
<
K0 e t ;
aL
1
(
aL
1)
+ #0 +
#0
K0
+
aL
(
1)
e
(t t0 )
for
t 2 [0; t0 ]
; for
t 2 [t0 ; t1 ]
for
F (t);
an+1
an
n+1
n
n
]
, tj = t0 +
1
t0 =
2
before for any n
(
"
Le
K(tn ) +
log
n
t
+
an+1
j for any j
an
L
1
(a
(a
"
)L
1)
1
#
(a
+
(a
)
1)
#)
;
t 2 [tn ; tn+1 ];
any n
+[
3
+
>
>
>
>
>
>
>
>
:
F (t)
log
(t t0 )
e
aL
1
e
(t t0 )
1
;
1, K(t0 ) = #0 ; and K(tn ) is exactly the same as
1.
Using a similar algorithm, we can fully specify the transitional dynamics when K0 > #1 .
4
We have already provided an algorithm to compute #i for i
2 in the main text by using
5
(50). An alternative way is to back out the threshold value #i from the corresponding
6
transversality conditions for any i
7
for both algorithms, and that all these threshold values are independent of K0 . In other
2. It can be veri…ed that all these values are the same
49
8
words, K0 only has a level e¤ect (i.e. it only a¤ects C(0)) but it has no speed e¤ect on
1
industrial upgrading. The main reason is that this economy is perfectly stationary, thus
2
di¤erent initial capital levels only translate into di¤erent initial aggregate consumption and
3
initial industrial structures.
4
7.4. Appendix 4
5
7.4.1.
6
7
8
Alternative Dynamic Setting
For analytical convenience, we slightly modify the industry production function as follows:
8
>
< n minf kn ; lg if n 1
a
:
(55)
Fn (k; l) =
>
:
L
if n = 0
The aggregate production function is given by
X=A
9
1
X
(56)
xn ;
n=0
where xn is output of industry n and A(t) captures the Hicks-neutral technology progress.
Note that xn is no longer equal to the quantity of indirect consumption for industry n because
only an endogenous part of the …nal output X will be consumed in the dynamic setting. At
any given level of K for any time point, we have the following result.
Table A. Static Equilibrium
0
K < aL
K
a
x0 = L
x1 =
K
a
F (K; L)
K < an+1 L for n
xn =
r
w
a
(
1)
a
K +L
n+1 K an L
an+1 an
xj = 0 for 8j 6= n; n + 1
1
=
1
n Lan+1 K
an+1 an
xn+1 =
xj = 0 for 8j 6= 0; 1
r
w
an L
F (K; L)
=
n+1
1
an (a
n
an+1 an
)
K+
n
(a )
L
a 1
50
10
Consequently, the endogenous aggregate production function is given by (36). Observe that
n
cn for any n
0 but the aggregate production function is identical to that in Table
1
xn
2
4. The associated discounted-value Hamiltonian when an L
3
by
Hn;n+1 (t) =
C(t)1
1
1
n+1
e
n+1
n+1
L
n;n+1 (a
+
t
+
A(t)
n;n+1
K(t)) +
K < an+1 L for n
n
an+1
a
n
n;n+1 (K(t)
1is given
n
K(t) +
n
(a
)
A(t)L
a 1
C(t)
an L);
which yields the following …rst-order conditions and Kuhn-Tucker conditions:
C(t)
e
t
=
n;n+1 (t);
@H
=
@K
n;n+1 (t) =
n+1
n+1
L
n;n+1 (a
n
n;n+1 (K(t)
n;n+1
A(t)
K(t)) = 0;
n+1
n;n+1
0, an+1 L
an L) = 0;
n
n;n+1
0, K(t)
n+1
n
an+1
an
+
n
n;n+1
n+1
n;n+1
;
K(t) > 0;
an L
0:
These conditions imply the Euler equation (39). To derive the capital accumulation equation,
we …rst rewrite (37) as
B(
a
K=
1)
1
K(t) + B
a
a
1
L
C(t);
for any t < t1 . The solution to the above …rst-order di¤erential equation is
K0;1 (t) =
a
(a
1)
B(
1)
BL
C(0)e
1
B
a 1
1
a 1B
t
+
1
B
a 1
(a
0;1 e
1
B
1
)t ;
51
where K0;1 (t) refers to the capital stock function when the underlying industries are 0 and
1. Using K0;1 (0) = K0 , we obtain
= K0 +
0;1
C(0)
1
B
a 1
a
(a
1
B
a 1
1)
B(
1)
BL:
Using continuity of capital stock and K0;1 (t1 ) = aL, we obtain
aL
aBL
1)
B(
(a
2
= 4K0 +
4
1
1)
+
C(0)e
1
a 1B
(57)
1
B
a 1
1
B
a 1
C(0)
1
B
a 1
t1
a
(a
1
B
a 1
1)
B(
1)
3
BL5 e( a
1
B
1
)t1
which implies that t1 is a function of C(0). Observe that (38) can be rewritten as
K=
a
1
B
1
K(t)+
2
Recall K(tn ) = an , for any n
3
equation (58), we obtain
Kn;n+1 (t) =
4
(a
an (a
)
1)
B(
an (a
)
BL C(t); when t 2 [tn ; tn+1 ) for any n
a 1
1 and K(t) must be a continuous function of t. Solving
1)
BL
for t 2 [tn ; tn+1 ), where the parameter
C(0)e
1
B
a 1
n;n+1
1
a 1B
t
+
(a
n;n+1 e
1
B
1
n;n+1
=
)t ;(59)
1
B
a 1
is uniquely pinned down by applying the
condition K(tn ) = an L to (59):
(
1.(58)
"
1
B) (a 1) an Le ( a 1 B )tn C(0)e
+
1
B
(a 1)
B(
1)
a 1
1
a 1B
(
1
B
a 1
#
)
tn
:
1
B
a 1
52
Substituting the above equation into (59) yields
Kn;n+1 (t) =
an (a
)
1)
B(
2
(a
+ e( a
1
B
1
)t 6
6(
4
1)
C(0)e
BL
1
a 1B
t
(60)
1
B
a 1
1
B
a 1
1
B
a 1
B) (a 1) an Le (
(a 1)
B(
1)
)tn
+
C(0)e
"
1
a 1B
(
1
B
a 1
1
B
a 1
1
B
a 1
#
)
tn
3
7
7:
5
5
The continuity of K(t) implies lim Kn;n+1 (t) = Kn+1;n+2 (tn+1 ). Together with (59), we
1
obtain the following recursive equation for any n
t!tn+1
n;n+1
=
an+1 (a
(a 1)
)(a
B(
1:
1)
BLe ( a
1)
1
B
1
)tn+1 +
n+1;n+2 ;
which implies
1;2
"1
a2 (a
)(a 1)BL X j ( a
=
ae
(a 1)
B(
1) j=0
1
B
1
)t2+j
#
+ lim
n!1
n;n+1 .
The transversality condition is lim H(t) = 0, which can be rewritten as
t!1
lim
t!1
C(t)1
1
1
e
t
+
n;n+1
1
B
1
a
K(t) +
an(t) (a
)
BL
a 1
C(t)
or equivalently,
lim
8
>
>
<
t!1 >
>
:
C(0)1
1
1
B
a 1
"
(a
1
1B
)(1
)
e
K(t) +
an(t) (a
a 1
#
t
)
+ (0)e ( a
BL
1
B
1
C(0)e
)t
1
a 1B
t
9
>
>
=
>
>
;
= 0;
=0
which is reduced to lim K(t)e ( a
1
B
1
)t = 0. We can also show that lim
t!1
n!1
n;n+1
53
= 0.
Plugging it into (49), we obtain
1;2
"1
X
)(a 1)
BL
aj e ( a
B(
1)
j=0
a2 (a
=
(a 1)
#
1
B
1
)t2+j :
On the other hand, combining (57), (59), and K1;2 (t1 ) = aL, we obtain
1;2
= K0 +
C(0)
1
B
a 1
a
(a
1
B
a 1
(a 1
)B
aLe ( a
(a 1)
B(
1)
1
B
1
1)
B(
1)
BL
)t1 :
So we have
(a
)(a
= [(a
1)BL
1)
"
1
X
aj e (
1
B
a 1
)tj
j=2
B(
1)] K0 +
#
[B (
(61)
1)
(a
1) ] C(0)
1
B
a 1
1
B
a 1
aBL
(a
) BaLe ( a
1
1
B
1
)t1 ;
which implicitly determines C(0) because tn is also a function of C(0) for any n
1 , as
determined by the conditions lim Kn;n+1 (t) = an+1 L for all n. Recall, for example, t1 (C(0))
t!tn+1
is determined by (57); for any n
(B
B(
=
1, tn+1 (C(0)) is recursively and implicitly determined by
) (a 1) an L
1) (a 1)
C(0)e
1
B
a 1
1
a 1B
e( a
tn+1
1
B
a 1
1
B
1
2
)(tn+1
"
4e
(
1
B
a 1
(B
(B
tn )
)
1
a 1B
#
a)
)
(tn+1 tn )
(62)
3
15 ;
2
which is obtained by using (60) and the condition lim Kn;n+1 (t) = Kn+1;n+2 (tn+1 ) = an+1 L.
1
As C(0) and all the endogenous time points tn are implicitly determined, the capital accu-
t!tn+1
54
2
mulation function (60) is also determined sequentially.
1
7.4.2.
Labor-Augmenting Technological Progress in Alternative Dynamic Setting
Here we show that the labor-augmenting technology progress does not yield perpetual
industry upgrading from labor-intensive to capital-intensive industries in the steady state,
although it does generate sustainable aggregate growth. The intuition is as follows: The key
driving force for perpetual industry upgrading is that the increase in endowment structure
(K=L) dominates the increase in the capital intensity of di¤erent industries. Suppose the
aggregate exogenous labor-augmenting technology progress is micro-founded at the the industry level via the Leontief industry production function, the ratio of capital to e¤ective
labor, K=AL; would be constant in the steady state, and therefore the industry upgrading
must stop at certain point. To see this more clearly, suppose we introduce the following
aggregate production function
G(K(t); L)
F (K(t); A(t)L)
8
>
( 1)
<
K(t) + A(t)L
if
0 K(t) < aA(t)L
a
n
n+1
n
>
: n+1
K(t) + a(a 1 ) A(t)L if an A(t)L K(t) < an+1 A(t)L for n
a
an
;
1
where function F ( ; ) is given by Table A and A(t) captures the Harrod-neutral technology
progress. Suppose
A(t)
= gA ;
A(t)
2
where gA is a constant. A(0) is given and normalized to one.
The micro-foundation for this new endogenous aggregate production function can be as
follows:
Fn (k; A l) =
8
>
<
>
:
n
minf akn ; A lg if n
A L
1
if n = 0
:
55
The optimization problem is stated as follows.
max
C(t)
Z
1
0
C(t)1
1
1
e
t
dt
subject to
K = F (K(t); A(t)L)
K(t)
C(t);
K(0) = K0 is given.
The problem can also be written as
max
C(t)
Z
t0
0
C(t)1
1
1
e
t
dt +
1 Z
X
tn+1
tn
n=0
C(t)1
1
1
e
t
dt
subject to
K=
8
>
>
>
>
<
>
>
>
>
:
K
1) Ka
A L+(
n+1
n
an+1 an
K+
n
(a )
A
a 1
K
L
C;
when
0
t
t0
when
t0
t
t1
C; when tn
K
t
tn+1 ; for n
;
1
K(0) = K0 is given;
De…ne e
c(t)
C(t) e
; k(t)
A(t) L
K(t)
; ye(t)
A(t) L
max
e
c(t)
Z
0
1
X(t)
.
A(t) L
The above problem can be restated as
e
c(t)1
1
1
e
[
(1
)gA ]t
dt
subject to
e
k(t) = F (e
k(t); 1)
(gA + )e
k(t)
K0
e
k(0) =
is given,
A0 L
e
c(t);
56
where
F (e
k(t); 1) =
8
>
<
>
:
(
a
n+1
n
an+1
an
1) e
if
k(t) + 1
e
k(t) +
n
(a )
a 1
0
if an
e
k(t) < a
e
k(t) < an+1 for n
1
Following the similar method as in the text, we can obtain the following equilibrium. There
must exist a unique integer n
b such that
n
b+1
anb+1
3
1
n
b
anb
> ( + gA ) + [
(1
) gA ]
n
b+2
anb+2
n
b+1
anb+1
.
or
n
b+1
anb+1
n
b
anb
> + + gA
n
b+2
anb+2
n
b+1
anb+1
(63)
We conclude that in the steady state, we have
e
k
e
c
= anb+1 L;
=
n
b+1
anb+1 L:
We conclude that, in the long run, we have sustainable growth with rate gA
K(t)
C(t)
=
= gA :
C(t)
K(t)
2
3
4
However, industry upgrading will stop at industry n
b + 1, where n
b is uniquely determined by
(63). We could also characterize the life cycle dynamics along the whole transitional dynamic
path as well as the balanced growth path.
5
7.5. Appendix 5: Proof for Proposition 4
First note that for any n
1, the MRT is equal to the price ratio:
4
5
6
=
pn+1
w + an+1 r
=
;
pn
w + an r
which implies that
cn+1
=
cn
2
3
1
cn+1
cn
M RTn+1;n =
1
57
where
1
1
1
1 + an+1
1 + an
1
; 8n
r
.
w
denotes the rental wage ratio
cn = c1 (1 + a ) 1
Thus
1
n 1
1
(64)
1
1
; 8n
1 + an
(65)
1:
In particular, we have
c1
=
c0
1
(1 + a )
1
1
1
(66)
:
Combining (65) and (66), we can express the output of each good as a function of c1 and .
Using the market clearing conditions for labor and capital, we obtain
L = c1
E = c1
"
1
1+a
1
+
1
X
Thus the factor price ratio
E
=
L
8
9
10
11
1
X
an
n 1
1
a
1 + an
1+a
n 1
1
n
1
1
+
1
X
(1 + an )
1
#
;
(67)
1
1
:
(68)
1
1
:
n 1
1
1
is determined by
n=1
1
1 + an
1+a
n 1
1
n=1
n=1
7
1
X
(1 +
an
)
(69)
1
1
n=1
Recall
< a (implied by condition (9)). Here we strengthen this assumption by assuming
< a to ensure the right-hand side of (69) is well de…ned (…nite). Next we o¤er a series of
lemmas, which lead us to Proposition 4.
58
Lemma 1 shows that the rental-wage ratio
12
1
4
E
L
in-
creases.
Lemma 1.
2
3
decreases as the capital-labor ratio
and output cn for any n
strictly decreases with
0 can all be uniquely determined. In addition,
E
:
L
Proof of Lemma 1
7.5.1.
Observe that when
1
X
n 1
1
> 0,
n
(1 + a )
1
1
<
n=1
1
X
a
n 1
1
n
1
n
(1 + a )
1
1
X
<
n 1
1
(an )
1
1
n=1
n=1
=
an
1
1
1
1
n
1
X
1
;
a
n=1
where the …rst inequality holds because a > 1 and the second inequality holds because
< 1:
Hence both the denominator and the numerator of the right-hand side of (69) must be …nite.
To prove the existence and uniqueness of , …rst we show that the right-hand side of equation
(69) is a decreasing function of . Note that
1
X
a
n
n 1
1
n
(1 + a )
1
1
n=1
1
1
1
+
1
X
=
n 1
1
(1 + an )
1
1
n=1
1
X
a
n=1
1
X
n 1
1
n
1
X
1
n
(1 + a )
1
n 1
1
(1 + an )
1
1
n=1
n 1
1
(1 + an )
1
1
1
1
1
+
1
X
;
n 1
1
(1 + an )
1
1
n=1
n=1
5
and both terms on the right-hand side are positive and decreasing with . The proof for why
6
the …rst term on the right-hand side decreases with
N(
)
N
X
an
n=1
N
X
n=1
n 1
1
is the following. De…ne
1
(1 + an )
1
:
n 1
1
(1 + an )
1
1
59
First it is easy to show that
2(
2
X
1
)
1
n=1
2
)
X
7
1
n 1
1
(1+an
an
n 1
1
(1+an
1
)
1
n=1
obviously decreases with . Now for any N
N(
)=a+
N
X
j=2
(aj
n 1
1
n
1
(1 + an )
(1 + an+1 )
1
(1+an
)
1
1
, which
1
n=1
2, we can rewrite
2
N
X
6
6
j 1 6 n=j
a )6 N
6X
4
n=1
Observe that
2 1
(a2 a) 1 (1+a2 )
=a+ X
2
n 1
1
n 1
1
n 1
1
N(
(1 + an )
1
) as
3
7
7
7
7:
7
1
(1 + an ) 1 5
1
1
1
strictly increases with
1
for any n
1;
1
which immediately implies
2
N
X
6
6
6 n=j
6 N
6X
4
n=1
2
3
4
5
n 1
1
n 1
1
(1 + an )
3
7
7
7
7 must be strictly decreasing in
7
1
(1 + an ) 1 5
So we establish that
1(
1
N(
1
for any j = 2; :::; N:
) strictly decreases with . Recall that we have assumed
) is positive and …nite for any
2 (0; 1). Therefore,
1(
< a so
) is strictly decreasing in .
We have now proved that the right-hand side of equation (69) is a decreasing function of
. In addition, when
! 0, the right-hand side of (69) goes to in…nity; when
! 1, the
6
right-hand side of (69) goes to zero, therefore, (69) can uniquely determine the value of
7
the Mean Value Theorem. Once
8
cn for any other n
by
is known, c1 can be uniquely determined from (68), while
0 can be uniquely determined by (65) and (66).
9
Lemma 2 establishes that when E increases, holding L constant, the equilibrium output of
10
a higher-indexed industry increases disproportionately more (or decreases disproportionately
11
1
2
60
less) than any lower-indexed industry. So when E increases, if cn increases, then cn+h
increases for all h
Lemma 2.
1 and if cn decreases, then cn
c0n+1 (E)
cn+1 (E)
>
c0n (E)
cn (E)
cn+1
cn
d ln
dE
Proof. Note that
for any n
=
d ln
cn+1
cn
d
d
dE
h
decreases for all h
1.
0 and for any E.
and
d
dE
< 0 from Lemma 1. Now, (64) and (66)
yield
d ln cn+1
cn
1
=
d
an (a 1)
< 0; 8n
1 (1 + an+1 ) (1 + an )
1; and
d ln cc10
d
< 0;
c
3
which implies that
d ln n+1
cn
dE
> 0. Q.E.D.
4
The following lemma shows that for any given E there exists a threshold industry n such
5
that all the industries that are more capital intensive than industry n will expand when E
6
increases marginally, while industry n and all the industries that are less capital intensive
7
than n will marginally shrink. Furthermore, if the single-peaked condition is satis…ed (to
8
be speci…ed soon), the index of the threshold industry n weakly increases as E increases.
9
That is, the threshold industry itself shifts toward the more capital intensive direction as E
10
11
increases.
Lemma 3. (A)
@c0
@E
< 0. Moreover, for any given E, there must exist a unique …nite
@cn
@E
12
positive integer n (E) such that
13
will eventually decline as E becomes su¢ ciently large. (C) If the single-peaked condition is
14
satis…ed, n (E) weakly increases in E.
15
0 holds if and only if n > n (E). (B) Each industry
Proof. The factor market clearing conditions now become
1
X
cn = L
(70)
an c n = E
(71)
n=0
1
X
n=0
16
By contradiction, suppose
@c0
@E
0: Lemma 2 implies that
@cn
@E
> 0 for all n
1: Di¤erentiating
17
61
equation (70) with respect to E; the derivative of the left-hand side is positive but the
@c0
@E
< 0. Now we
1
derivative of the right-hand side is zero. That is a contradiction. Thus
2
show there must exist some N so that
3
@cn
@E
4
contradiction. Now by revoking Lemma 2, we can conclude that, for any given E, there must
5
exist a unique …nite positive integer n (E) such that
@cN
@E
0: If this is not true, then we would have
< 0 for all n: Now by di¤erentiating equation (71) with respect to E, we would reach a
@cn
@E
0 holds if and only if n > n (E).
Furthermore, we want to show that any industry n will eventually decline as E becomes
1, c0n ( ) > 0 if and only if
su¢ ciently large. First, (65) and (67) imply that for any n
G0n ( ) > 0, where
"
Gn ( )
6
h
1
h
1+a
h=1
1 + an
1
1
#
1
:
which is equivalent to
( )
1
X
h
1
ah 1 + ah
1
1
1
h=1
1+
7
1+
1
X
1
X
>
h
1
(1 + ah )
1
an
1 + an
(72)
1
h=1
8
Note that as becomes su¢ ciently small, the left-hand side of inequality (72) goes to in…nity.
9
Thus, as E becomes su¢ ciently large (so
10
11
becomes su¢ ciently small), we must have c0n ( ) >
0; which implies that c0n (E) < 0:
It remains to show that n (E) weakly increases in E. Alternatively, we want to show
12
that if c0n ( 0 ) > 0 for some
0
13
ensures that cn ( ) is single peaked in : From (65) and (66), we obtain
> 0, then c0n ( ) > 0 holds for any
<
0
: Such a property
cn = c0
n
1
ln cn = ln c0 +
,
14
c0n (
cn
)
(1 + an )
n
1
,
1
ln +
1
=
1
c00 (
c0
)
+
62
1
1
ln (1 + an )
an
:
1 (1 + an )
For any given ; c0n ( ) > 0 if and only if
1
c0
>
0
an
c0 ( ) (1
)
:
1
A su¢ cient condition for the single-peaked property to hold is that the derivative of the
2
right-hand side of the above inequality is non-negative, which is ensured if
00
3
c0 ( )c0 ( )
(c00 ( ))2
(73)
4
Inequality (73) is the single-peaked condition, which we assume to hold. Q.E.D
5
Now we combine all the previous lemmas together. For any industry n
n (E) for
6
some given E; the industry always declines when E increases. For any industry n > n (E 0 )
7
for some E 0 ; it expands as E 0 increases up to a point E 00 such that n < n (E 00 ). Industry
8
n will then decline with E after E surpasses E 00 . Therefore, the industry exhibits a hump-
9
shaped pattern as the economy becomes more capital abundant. Proposition 4 is established.
10
Q.E.D
11
For illustrative purpose, Figure A2 plots how the industrial output of the …rst six indus-
12
tries (n = 0; 1; 2::; 5) changes as the capital endowment E becomes larger. The output of
13
industry 0, c0 , becomes smaller when E increases, as predicted by Lemma 3. All the other
14
industries demonstrate a hump-shape pattern. The more capital intensive an industry, the
15
“later”it reaches its peak, as predicted by the above proposition.
63
[Insert Figure A2]
16
We can show that the hump-shaped dynamics and the timing fact also obtain even if all
1
industries have the same marginal productivity in …nal good production (
2
n). This implies that the only mechanism that drives industrial upgrading is change in the
3
factor endowment because of homothetic preference and balanced productivity growth across
4
sectors.
5
7.6. Appendix 6
6
7
n
= 1 for all
Everything is the same as in Appendix 4 except that the …nal output production function
(56) now becomes:
X=A
8
where
"
1
X
n=0
xn
#1
(74)
;
< 1. The …nal output is chosen as the numeraire. The industry production function
is still given by (55). The imperfect substitution elasticity implies that every good n
be produced in the equilibrium. First, for any n
0 will
1, the marginal rate of transformation is
equal to the price ratio:
M RTn+1;n =
9
12
pn+1
w + an+1 r 1
=
=
;
pn
w + an r
which implies that
xn+1
=
xn
10
11
1
xn+1
xn
where
1
1
1
1 + an+1
1 + an
1
; 8n
denotes the rental wage ratio
xn = x1 (1 + a ) 1
1
n 1
1 + an
(75)
1
r
.
w
Thus
1
1
; 8n
1:
(76)
13
1
64
In particular, we have
x1
=
x0
1
(1 + a )
1
1
1
(77)
:
Combining (76) and (77), we can express the output of each good as a function of x1 and .
Using the market clearing conditions for labor and capital, we obtain
L = x1
K = x1
"
1
1+a
1
+
Thus the factor price ratio
K
=
L
3
4
1
X
an
n 1
1
1
X
a
n
n 1
1
1 + an
1+a
1
1
+
1
X
1
1
#
;
(78)
1
1
:
(79)
is determined by
(1 + an )
1
1
n=1
1
1 + an
1+a
n 1
1
n=1
n=1
2
1
X
:
n 1
1
(1 + an )
(80)
1
1
n=1
Recall
< a (implied by condition (9)). Here we strengthen this assumption by assuming
5
< a to ensure the right-hand side of (80) is well de…ned (…nite). Observe that (76)-
6
(80) are mathematically isomorphic to (65)-(69) except that notations are di¤erent: K
7
and xn (8n
8
production and the individual industry production functions are also de…ned di¤erently.
9
However, it turns out these two problems are mathematically almost isomorphic to each
10
other, so the static properties stated in Lemmas 1-3 and Propostion 4 are still valid up to
11
the change of notations as mentioned above. The proofs are almost identical to those in
12
Appendix 5 and hence omitted.
0) now replace E and cn (8n
0), respectively. Observe that the aggregate
65
The major change is in the dynamic part. First of all, (76), (77) and (79) jointly imply
(1 + a ) 1
xj =
1
X
an
1
1
j
1
1
K
1+aj
n 1
1
1
1+an
1
; 8j
1;
1+a
n=1
1
(1 + a ) 1
x0 = 1
X
n 1
an 1
1
1
1+an
1+a
K
:
1
1
n=1
13
1
Plugging the above two equations into (74) yields
h
i1
1
P1
n
1
1
1 + n=1 1+an
A (1 + a )
K:
X=
1
X
1
n
1
n
a 1+an
n=1
2
where
3
K
L
4
5
6
(81)
is uniquely determined by (80) as an implicit and strictly decreasing function of
(recall Lemma 1). Substituting out
in (81) would implicitly determine X as a func-
b
tion of K and L, or written as X = A G(K;
L), where A denotes the newly introduced
b
Hicks-neutral TFP in the …nal output production. Moreover, function G(K;
L) must be
homogeneous of degree one with respect to capital and labor because each industry has a
7
constant-return-to-scale technology in this perfectly competitive neoclassical environment.
8
In addition,
9
10
11
b
@ G(K;L)
@K
> 0 and
b
K = A G(K(t);
L)
b
@ 2 G(K;L)
@K 2
K(t)
and the Euler equation is given by
h
i
b
@ A G(K; L) =@K
C(t)
=
C(t)
0. The capital accumulation function is
(82)
C(t);
;
(83)
12
which degenerates into (41) in the previous linear case when A takes the form given by (40).
13
Now consider the case when
14
15
and only if
h
i
b
@ A G(K; L) =@K = ;
< 1. The consumption growth rate in (83) is a constant if
(84)
16
where
66
is constant over time.
1
One possible way to satisfy this condition is to introduce endogenous growth mechanisms
2
as follows. Suppose that the common component of industry productivity A is an increasing
3
function of capital stock K, capturing the standard Arrow’s learning-by-doing mechanism or
4
the knowledge spillover e¤ect as in Romer (1986). Assume A(0) = 0. By integrating (84),
5
we obtain
A(K)
6
K
b
G(K;
L)
(85)
:
7
The following condition must be imposed to ensure positive growth and avoid explosive
8
growth (assume
9
we obtain
K(t)
K(t)
=
2 (0; 1]):
C(t)
C(t)
=
X(t)
X(t)
> 0. Substituting (85) into (82) and (83),
>
=
. Consequently, in the full-‡edged dynamic model,
10
the output of each individual industry must still exhibit a hump-shaped pattern with more
11
capital-intensive industries reaching their peaks later, because the counterpart to Proposition
12
4 still holds (except for the change of notations by replacing E with K) and that K(t) is
13
strictly increasing boundlessly over time. However, it is no longer possible to obtain closed-
14
form solutions for the entire dynamic path, especially for the life-cycle dynamics of each
15
industry.
16
In summary, the key qualitative dynamic features are preserved, but the major drawback
17
of this alternative approach still lies in its low tractability and the necessity of introducing
18
unconventional technological progress to ensure sustainable growth and perpetual industry
19
upgrading because of the extra complications that result from the endogenous structural
20
changes in the in…nite underlying industries.
21
7.7. Appendix 7. Open Economy
22
Our model focuses on autarky, but international trade could be another important force
23
to drive structural change (see Matsuyama (2009) and Uy, Yi and Zhang (2013)). In this
24
section, we can show that our main results remain valid in a small open economy.28
Now the price for good n in the open economy, denoted pen , is exogenously given by
1
2
3
4
5
6
67
the international market for any n = 0; 1; 2:::. Let x
en denote domestic output of good
n. Di¤erent from autarky, now the domestic demand fe
c n g1
n=0 is not necessarily equal to
domestic production fe
x n g1
n=0 . The production function for the …nal commodity is assumed
to be CES:
" 1 #1
X
e=
C
e
cn ;
n=0
2 (0; 1):
(86)
7
The domestic technologies to produce the intermediate goods are the same as before (given
8
by (7)-(9)) and these goods are all tradable. Capital and labor cannot move across borders.
9
Without loss of generality, the …nal composite good is assumed to be non-tradable.
10
7.7.1.
To simplify the analysis and also make it more instructive, suppose that the exogenous
11
12
Static Model
price pen =
n,
for 8n and that both (8) and (9) are still satis…ed. Since both labor and capital
13
are inelastically supplied by the households and the price sequence fe
p n g1
n=0 is exogenous, the
14
market equilibrium allocation can be found by solving an arti…cial social planner’s problem
15
16
17
18
19
20
21
for this small open economy in two sequentially separate steps. First, it maximizes the total
P
domestic income 1
en x
en (= wL
e + reE) for given factor endowments (E and L). Therefore,
n=0 p
in terms of the production decisions, it reduces to the same problem of the social planner
P
in autarky (maximizing welfare 1
n=0 n cn ) as in Section 3.2 and the market equilibrium is
again summarized in Table 4 (replacing cn by x
en now).
Second, the social planner maximizes the representative household’s utility (86) by choosP1
ing domestic demand fe
c n g1
ene
cn = wL+e
e
rE. We obtain
n=0 p
n=0 subject to budget constraint
28
In the Heckscher-Ohlin models, the endowment-determining trade mechanism only reinforces the mechanism we illustrate in autarky. Recent dynamic Heckscher-Ohlin models include Ventura (1997), Bond, Trask,
and Wang (2003), Bajona and Kehoe (2010), and Caliendo (2011), but there are only two sectors in these
models. Wang (2012) extends the autarky model in this paper to analyze how trade and trade policies a¤ect
industrial dynamics and economic convergence into a two-country general equilibrium trade environment.
68
22
fe
c n g1
n=0 as follows:
e
c0 =
e
cj =
8
>
<
>
:
1
1
1
j
1
e
c0 ; 8j
n+1
n
an+1
an
1
E+
L+(
n
(a )
L
a 1
1) Ea
if an L
if
E < an+1 L for n
0
1
;
(87)
E < aL
0:
In other words, the domestic demand for each intermediate good is strictly positive but generically only two goods are produced domestically for any given endowment structure. The
P1
P
net export sequence is given by fe
xn e
cn g1
en x
en = 1
ene
cn .
n=0 . Trade is balanced:
n=0 p
n=0 p
e
e = reE + wL
C
q
8 n+1 n
n (a
>
< an+1 an E+ a 1
q
=
L+(
1) E
>
a
:
q
)
L
if an L
if
1
where q is the aggregate price index de…ned as
2
7.7.2.
Dynamic Model
E < an+1 L for n
0
P1
n=0
1
(88)
E < aL
1
pen
1
1
. Obviously, q = 1
1
3
We assume that trade is balanced all the time, so the state equation (20) still applies
4
for this small open economy in the social planner’s dynamic optimization problem. Observe
5
that (88) di¤ers from the endogenous aggregate production functions in Table 4 only in that
6
7
e is
the exogenous aggregate good price q is no longer unity, but aggregate consumption C
still a linear function of E. Therefore, the aggregate consumption growth rate is still given
8
by (26). The industrial output dynamics fe
xn (t)g1
n=0 are still characterized by Propositions
9
2 and 3 (only up to an adjustment of the relative price level). Di¤erent from Section 4, now
10
fe
cn (t)g1
n=0 are always positive for any t 2 [0; 1) and can be obtained explicitly by revoking
11
e in precisely the same way as in Section 4.
(87) and tracking the time path of E(t) via C(t)
.
12
The dynamic trade ‡ows fe
xn (t)
69
e
cn (t)g1
n=0 are also determined for t 2 [0; 1).
1
The model tractability and the equivalence results both depend on the assumption im-
2
posed on the exogenous prices for the intermediate goods. For the more general case, it can
3
be shown that the key equilibrium features will be qualitatively similar as long as the exoge-
4
nous price ratio of any two neighboring goods,
5
low relative to their capital intensity ratio
6
assumption of “no factor intensity reversal” in the Heckscher-Ohlin trade literature. More
7
generally, a two-country or multiple-country general equilibrium trade model is needed to
8
endogenize the prices of the tradables. However, the dynamic analysis becomes much more
9
complicated, so we leave it for separate treatment.29
10
7.7.3.
pn+1
,
pn
an+1
,
an
is larger than unity and also su¢ ciently
which is essentially similar to the standard
International Capital Flow
11
Our main result (industries will move from labor-intensive ones to capital-intensive ones
12
following a hump-shaped path) is qualitatively robust even when allowing for international
13
capital ‡ow in a small open economy. This is because the output decisions of all the industries
14
are still guided by the equilibrium rental wage ratio, not just by the rental price of capital
15
alone. In this new environment, even if the rental price of capital is exogenously given by
16
the world market (typically assumed to be time-invariant in the literature), the wage-rental
17
ratio still increases over time because the marginal productivity of labor increases with the
18
upgrading of industries (recall
19
endogenous aggregate production functions shown in Table 4. Domestically-owned capital
20
keeps accumulating while labor supply does not increase. Empirically, the rise of the wage-
21
rental ratio is a basic growth fact, which is also implied by the Neoclassical growth models
22
(Barro and Sala-i-Martin (2003)). Thus …rms’ optimal choice is still to switch gradually
23
from labor-intensive industries to more capital-intensive ones in response to the change in
29
pn+1
pn
> 1 and production function (7)), as hinted by the
An extension to two-country general equilibrium dynamic trade is explicitly explored in Wang (2012),
where the model predictions are consistent with the stylized facts established in Section 2.
70
24
1
the relative factor price.
7.8. Appendix 8: Industry Re-classi…cation
2
Here is how we rede…ne new industries based on Schott (2003) in Subsection 2.1. Let kit
3
denote the capital-labor ratio of NAICS industry i at year t: Ranking all 22275 observations
4
of kit from low to high, we obtain a sequence kihh th
5
the lowest and the highest capital-labor ratios, respectively. Superscript h denotes the rank
6
of the capital-labor ratio. All the observations whose superscript satis…es 1
7
grouped together and de…ned as “industry 1”. Those observations satisfying 226
8
are de…ned as “industry 2”, and so on. In this way, we obtain 99 newly-de…ned industries with
9
225 observations in each. Each superscript h uniquely corresponds to an NAICS industry
10
code and year (ih , th ) pair thus we can …nd the corresponding observation of employment,
11
denoted xih th , and obtain a sequence fxih th g22275
h=1 . Note that xih th is not necessarily monotonic
12
in h.
13
22275
,
h=1
For a newly-de…ned industry n (where n = 1;
represent
where ki11 t1 and ki22275
22275 t22275
h
225 are
h
450
; 99), we have 225 observations for
14
capital-labor ratios but only 48 years, which means that there must be multiple observations
15
for certain years. To construct the employment time series fe
xnt g2005
t=1958 for the newly-de…ned
16
industry n, we …rst need to …nd all the corresponding observations that belong to this
17
18
19
20
21
22
industry, that is, fxih th g225n
h=225(n 1)+1 . We de…ne the total employment of the newly-de…ned
P
industry n at year t as x
ent
; 99 and t =
225(n 1)+1 h 225n and th =t xih th , for any n = 1;
1958; 1959; :::; 2005. Correspondingly, the employment share of the newly-de…ned industry n
is measured by the ratio of the employment in industry n to total manufacturing employment
99
X
at year t, namely, x
ent =
x
ent . The same method is used to construct the time series for
n=1
value added (share) for each newly-de…ned industry.
Peak time of employment share
Independent variable
(1)
Capital expenditure share
Constant
Observations
R-squared
(2)
0.535***
(0.070)
Capital-labor ratio
1958.309***
(3.371)
33
0.651
Peak time of value-added share
(3)
(4)
0.553***
(0.070)
123.298***
(16.550)
1915.588***
(8.876)
40
0.594
1955.004***
(3.625)
34
0.661
Table 1: Peak Time of Industries in US: 1958-2005
Note: We drop industries monotonically declining (increasing) during entire time period, and a few
industries which are not single peaked. ***indicates significance at 1% level.
Data Source: NBER-CES Manufacturing Database
97.863***
(9.841)
1924.355***
(5.525)
43
0.707
Table 2: Congruence Fact of Industry Dynamics in US: 1958-2005
Employment share×1000
(1)
Coherence term 1
-3.035***
(0.266)
Coherence term 2
T
Constant
Industry dummies
Year dummies
Observations
R-squared
(2)
0.022***
(0.006)
14.196***
(0.991)
Yes
Yes
4,515
0.133
Value-added share×1000
(3)
(4)
-3.603***
(0.262)
-33.228***
(1.562)
0.018***
(0.005)
18.595***
(1.030)
Yes
Yes
4,542
0.201
0.100***
(0.006)
14.191***
(0.975)
Yes
Yes
4,515
0.296
Note: Coherence term 1 is the absolute value of a normalized difference between sector i's
capital-labor ratio and the aggregate capital-labor ratio in the manufacturing sectors at year t.
Coherence term 2 is the absolute value of a normalized difference between sector i's capital
expenditure share and the average capital expenditure share in all manufacturing sectors at year
t. ***indicates significance at 1% level. In addition, the result of a Hausman test shows that the
fixed effect model is statistically preferred to the random effect model.
-23.594***
(1.509)
0.134***
(0.005)
14.877***
(0.995)
Yes
Yes
4,542
0.381
Table 3: Hump-shaped Pattern and Timing Fact of Industrial Dynamics in US: 1958-2005
Independent
variable
t
t^2
t×k
T
GDPGR
Constant
Industry
dummies
Observations
R-squared
Full sample
(1)
(2)
Value-added
Output
share×1000
share×1000
1.581***
1.197***
(0.223)
(0.263)
-4.07e-04***
-3.08e-04***
(5.61e-05)
(6.62e-05)
0.001***
0.001***
(7.80e-05)
(9.21e-05)
0.006***
0.004***
(1.47e-04)
(1.73e-04)
-0.710*
-0.475
(0.415)
(0.489)
-1,534***
-1,163***
(220.705)
(260.383)
Subsample
(3)
(4)
Value-added
Output
share×1000
share×1000
1.596***
1.205***
(0.223)
(0.263)
-4.11e-04***
-3.10e-04***
(5.63e-05)
(6.64e-05)
0.001***
0.001***
(7.82e-05)
(9.23e-05)
0.006***
0.004***
(1.47e-04)
(1.74e-04)
-0.688*
-0.463
(0.416)
(0.491)
-1,549***
-1,171***
(221.372)
(261.206)
Yes
Yes
Yes
Yes
20,889
0.862
20,889
0.884
20,790
0.862
20,790
0.884
Note: t, k, T and GDPGR represent Year, Capital expenditure share, Labor productivity and GDP growth rate,
respectively. ***, ** and *denote significance at the 1, 5, and 10 percent level, respectively. Standard errors are in
parentheses. Column 3 and Column 4 report the results when we drop all the eleven industries that have data for only
twelve years.
Table 4: Hump-shaped Pattern and Timing Fact of Industrial Dynamics Cross Countries: 1963-2009
t
t^2
t×k
T
GDPGR
Constant
Country×Industry
dummies
Observations
R-squared
Full sample
(1)
(2)
Value-added
Output
share×1000
share×1000
9.750**
9.432**
(4.587)
(4.507)
-0.002**
-0.002**
(0.001)
(0.001)
4.01e-05***
2.54e-05*
(1.37e-05)
(1.34e-05)
3.16e-09***
2.28e-10*
(1.28e-10)
(1.25e-10)
-3.726
-1.585
(3.874)
(3.811)
-9,519**
-9,206**
(4,556)
(4,477)
Subsample
(3)
(4)
Value-added
Output
share×1000
share×1000
10.314**
12.267**
(5.112)
(5.652)
-0.003**
-0.003**
(0.001)
(0.001)
2.42e-04***
1.51e-04**
(6.85e-05)
(7.61e-05)
2.87e-09***
2.42e-10*
(1.19e-10)
(1.32e-10)
0.831
4.515
(5.247)
(5.809)
-10,050**
-12,009**
(5,077)
(5,614)
Yes
Yes
Yes
Yes
47,328
0.820
47,179
0.847
27,236
0.774
27,447
0.796
Note: t, k, T and GDPGR represent Year, Capital expenditure share of an industry, Labor productivity of an industry,
and GDP growth rate for each specific country, respectively. ***, ** and *denote significance at the 1, 5, and 10
percent level, respectively. Standard errors are in parentheses. Regressions (3) and (4) are run for the subsample which
includes only those industries whose values are observed for over 40 years.
Data Source: UNIDO (1963-2009).
Table 5: Hump-shaped Pattern and Timing Fact of Industrial Dynamics in Developing Countries:
1963-2009
t
t^2
t×k
T
GDPGR
Constant
Country×Industry
dummies
India
ValueOutput
added
share×1
share×1
000
000
144.706
292.409
***
***
(42.073) (98.763)
0.036**
0.074**
*
*
(0.011)
(0.025)
A
B
Valueadded
share×10
00
104.543*
**
(28.761)
157.847*
*
(61.281)
0.026***
0.040***
(0.007)
(0.015)
Output
share×10
00
0.011**
0.003
0.013***
0.003
(0.005)
(0.012)
(0.002)
(0.004)
0.001**
*
(5.96e05)
-19.639
(32.614)
143,406
***
(41,945)
0.002**
*
(1.4e04)
-48.864
(76.559)
290,248
***
(98,461)
4.74e04***
(2.66e05)
-22.213
(21.245)
103,595*
**
(28,669)
7.15e04***
(5.63e05)
-42.336
(45.211)
156,722*
*
(61,085)
Yes
Yes
Yes
Yes
Valueadded
share×10
00
13.926*
(8.302)
-0.004*
(0.002)
1.42e04**
(7.10e05)
1.81e04***
(6.60e06)
-3.200
(6.314)
C
Output
share×10
00
25.969**
*
(6.958)
0.007***
(8,246)
(0.002)
1.05e04*
(6.07e05)
8.93e05***
(5.64e06)
1.424
(5.361)
25,474**
*
(6,910)
Yes
Yes
-13,433
Valueadded
share×1
000
Output
share×1
000
(8,440)
27.975*
**
(7.227)
0.007**
*
(0.002)
1.05e04*
(6.16e05)
8.93e05***
(5.84e06)
1.081
(6.137)
27,459*
**
(7,177)
Yes
Yes
14.016*
(8.498)
-0.004*
(0.002)
1.38e04*
(7.10e05)
1.73e04***
(6.73e06)
-4.597
(7.137)
-13,527
Observations
467
467
924
914
12,073
12,531
11,632
12,090
R-squared
0.920
0.762
0.903
0.746
0.836
0.900
0.824
0.885
Note: t, k, T and GDPGR represent Year, Capital expenditure share, Labor productivity and GDP growth rate,
respectively. ***, ** and *denote significance at the 1, 5, and 10 percent level, respectively. Standard errors are in
parentheses. All the regressions are run for the various subsamples with industry values observed for more than 40
years. The first column is for India, one of the BRICS countries. BRICS is the acronym for the five largest emerging
economies, i.e., Brazil, Russia, India, China and South Africa. Since the data set records no industry values for more
than 40 years in Brazil, Russia and China, Column A is for only India and South Africa together. Column B is for all the
developing countries (following the classification by the World Bank) excluding India and South Africa. Other
developing countries include Philippines, Cyprus, Panama, Hungary, Kuwait, Costa Rica, Fiji, Sri Lanka, Turkey,
Jordan, Pakistan, Tanzania, Malaysia, Singapore, Trinidad, Ecuador, Tunisia, Iran, Uruguay, Syria, Kenya, Malta,
Malawi, Egypt, Columbia and Garner. Column C is for the subsample of developing countries whose average annual
real GDP growth rates are more than 2%. Average annual real GDP growth rate is calculated by the authors based on
the data from Penn World Table 8.0.
Data Source: UNIDO (1963-2009)
SER01
SER19
.07
.06
.05
.04
.03
.02
.01
SER39
.014
.013
.013
.012
.012
.011
.011
.010
.010
.009
.009
.008
.008
.007
.007
.006
.006
60
65
70
75
80
85
90
95
00
05
.005
60
65
70
75
SER59
80
85
90
95
00
05
60
65
70
75
SER79
.016
80
85
90
95
00
05
90
95
00
05
SER99
.020
.028
.014
.024
.016
.012
.020
.010
.016
.012
.008
.012
.006
.008
.008
.004
.004
.004
.002
60
65
70
75
80
85
90
95
00
05
.000
60
65
70
75
80
85
90
95
00
05
60
65
70
75
80
85
Figure 1. HP-Filtered Employment Shares of Six Typical “Industries” (Newly
Defined) in US from 1958 to 2005
Note: The horizontal axis is the year (from 1958 to 2005), and the vertical axis is the employment
share of “Industry 1 (19, 39, 59, 79 and 99)”. An industry with a higher index is more capitalintensive by construction. If there is a missing value in the constructed time series for a certain
year, that missing value is replaced by the simple average of the observations immediately before
and after that year. The HP filter parameter λ is set to 1000.
Figure 2. How Endowment Structure Determines Equilibrium Industries
cn
*
L
0
t1
*
c0 :
t2
*
c1 :
t3
*
c2 :
t4
*
c3 :
Figure 3. How Industries Evolve over Time when K 0 ∈ (ϑ0 ,ϑ1 ) .
Note: The horizontal axis is time and the vertical axis is the consumption of good n. The
purple line represents good 0; the green line represents good 1, the yellow line represents good 2,
and the blue line represents good 3.
t
Extra Tables and Figures for Online Publication
Table A1: Descriptive Statistics for Main Variables
1958
1959
mean
32.41
33.64
max
506.10
502.20
min
0.60
0.60
sd
47.30
47.84
mean
40.83
40.26
max
522.51
527.34
min
0.53
0.73
sd
49.78
49.53
Capital Expenditure
Share
mean max min
sd
0.47
0.87 0.13 0.12
0.49
0.88 0.02 0.12
1960
33.83
544.20
0.60
47.92
41.30
533.15
0.79
49.60
0.48
0.88
0.00
0.13
10.84
62.29
1.50
5.30
1961
33.02
498.30
0.60
46.36
43.76
542.03
0.82
51.67
0.48
0.89
0.01
0.13
11.13
64.91
1.56
5.56
1962
33.87
497.20
0.50
47.01
44.05
560.17
0.89
52.26
0.49
0.89
0.01
0.13
11.85
72.35
1.66
6.03
1963
33.64
495.80
0.60
45.56
44.88
555.52
0.87
52.61
0.50
0.90
0.06
0.12
12.63
77.06
1.67
6.52
1964
34.25
527.80
0.70
46.22
45.57
566.21
0.86
53.13
0.51
0.90
0.09
0.12
13.30
81.34
1.73
6.95
1965
35.92
559.90
0.70
47.98
46.24
574.63
0.91
54.79
0.52
0.91
0.03
0.12
13.99
89.17
1.78
7.43
1966
37.96
554.10
0.70
49.89
46.90
609.33
0.96
56.57
0.52
0.91
0.02
0.12
14.71
93.91
1.88
7.81
1967
38.41
528.60
0.70
50.04
48.94
647.69
1.03
57.69
0.52
0.91
0.07
0.12
15.12
93.59
2.11
8.01
1968
38.79
528.60
0.70
50.78
50.93
672.95
1.12
59.49
0.53
0.92
0.08
0.12
16.26
96.95
2.26
8.81
1969
39.73
533.30
0.80
50.64
51.56
676.38
1.21
60.02
0.52
0.92
0.07
0.12
16.94
98.46
2.34
8.93
1970
37.92
521.90
0.80
47.46
54.98
706.90
1.29
61.95
0.52
0.92
0.07
0.12
17.55
102.04
2.40
9.42
1971
36.09
478.20
1.00
44.33
58.70
746.92
1.41
65.04
0.53
0.92
0.07
0.12
19.06
112.01
2.66
10.05
1972
37.09
465.20
1.70
44.24
59.90
758.83
1.43
70.07
0.54
0.92
0.09
0.11
20.85
120.60
2.67
10.95
1973
38.85
498.30
1.50
46.52
60.05
762.15
1.49
71.16
0.55
0.92
0.05
0.11
22.85
138.85
2.86
12.26
1974
38.58
514.00
1.60
46.37
61.70
818.13
1.67
71.98
0.57
0.92
0.12
0.12
26.62
191.09
3.15
16.81
1975
35.31
447.90
1.50
42.52
68.37
854.93
1.83
75.82
0.56
0.92
0.20
0.12
27.69
193.87
3.31
16.92
1976
36.36
448.60
1.60
43.51
69.01
889.04
1.80
78.75
0.57
0.92
0.17
0.11
30.66
227.44
3.46
18.30
1977
38.08
438.60
1.60
44.85
69.41
868.07
1.87
81.42
0.57
0.93
0.23
0.11
33.30
238.16
4.01
19.91
1978
39.57
440.40
1.70
46.15
69.65
909.67
1.91
84.00
0.57
0.93
0.26
0.11
35.89
247.69
4.60
21.24
1979
40.60
447.90
1.50
47.77
71.45
958.42
1.91
88.95
0.58
0.93
0.06
0.11
39.96
334.06
4.36
26.98
1980
39.59
400.20
1.60
46.72
75.34
962.61
1.88
91.05
0.57
0.93
0.21
0.11
42.76
298.57
4.67
27.29
1981
38.82
387.90
1.60
46.27
78.41
1043.47
1.99
94.12
0.58
0.93
0.28
0.11
46.86
328.67
5.41
29.74
1982
36.48
319.90
1.70
43.71
85.18
1111.60
2.17
101.16
0.56
0.94
0.20
0.11
48.13
304.36
5.80
28.99
1983
35.67
328.40
1.60
42.86
88.39
1144.69
2.15
106.53
0.58
0.94
0.18
0.11
53.00
309.25
5.83
31.87
1984
36.55
345.10
1.60
44.21
88.47
1195.06
2.36
109.76
0.59
0.93
0.00
0.11
57.61
340.83
6.57
35.08
1985
35.74
371.50
1.60
43.99
92.52
1239.11
2.59
114.60
0.58
0.93
0.04
0.11
59.37
343.95
6.75
36.83
1986
34.80
379.70
1.50
43.55
97.39
1315.15
2.74
123.90
0.58
0.93
0.07
0.11
63.10
388.46
7.54
40.96
1987
35.97
384.60
1.40
45.60
97.47
1226.61
2.56
124.71
0.60
0.94
0.08
0.11
70.02
477.87
8.06
49.15
1988
36.41
394.20
1.40
45.73
95.93
1222.48
2.61
119.44
0.61
0.96
0.05
0.12
76.25
583.44
8.29
59.67
1989
36.18
402.60
1.50
45.75
96.50
1237.15
2.74
118.25
0.61
0.95
0.03
0.12
79.24
625.83
8.63
64.02
1990
35.67
403.90
1.40
45.29
98.62
1238.72
2.79
116.50
0.61
0.95
0.25
0.12
81.44
742.03
9.25
68.88
1991
34.05
388.50
1.40
42.82
102.76 1272.94
2.87
117.68
0.60
0.95
0.01
0.12
83.09
825.64
9.39
70.75
1992
34.38
425.60
1.30
43.82
103.22 1337.73
2.89
121.21
0.62
0.95
0.05
0.11
88.81
976.45
10.29 75.01
1993
34.33
443.00
1.30
43.51
104.90 1374.95
2.76
125.16
0.62
0.95
0.07
0.11
92.95
755.15
10.31 71.66
1994
34.55
472.10
1.20
44.23
107.00 1471.50
2.84
130.82
0.64
0.95
0.09
0.11 100.31 859.09
10.99 78.67
Employment
K/L
Labor Productivity
mean
9.85
10.76
max
50.94
60.97
min
1.44
1.46
sd
4.51
5.16
1995
35.30
499.80
1.20
45.51
107.36 1431.76
2.81
131.88
0.64
0.95
0.10
0.11 105.75 1107.67 11.08 91.03
1996
35.13
499.60
1.20
45.49
109.29 1361.83
3.03
132.43
0.64
0.95
0.03
0.11 107.45 1143.53 11.65 89.30
1997
35.53
522.70
1.30
46.23
113.23 1574.08
3.58
148.52
0.65
0.95
0.41
0.10 113.99 1095.71 21.70 94.11
1998
35.83
534.70
1.20
47.00
120.77 1577.91
5.33
157.44
0.65
0.96
0.37
0.11 117.29 1364.74 22.04 98.68
1999
35.28
550.90
0.90
46.96
132.30 1369.66
8.32
166.41
0.64
0.97
0.38
0.11 120.94 1941.35 25.98 121.10
2000
35.21
554.90
0.90
47.54
142.19 1621.59 10.91
182.51
0.64
0.97
0.27
0.11 124.70 2186.72 28.10 131.63
2001
33.50
528.60
0.90
45.82
154.64 1975.08 14.23
195.20
0.63
0.98
0.20
0.12 123.01 2831.86 27.01 152.47
2002
31.05
488.30
0.90
42.43
169.16 1598.52 10.11
197.67
0.65
0.97
0.22
0.11 134.10 2002.93 26.39 130.16
2003
29.33
478.00
0.80
40.33
180.39 2289.47 11.72
217.31
0.65
0.97
0.22
0.11 142.74 2005.16 25.06 138.05
2004
28.32
448.50
0.80
39.32
190.17 2333.50 12.61
224.30
0.66
0.97
0.21
0.11 160.45 2222.26 28.44 174.89
2005
27.84
446.00
0.80
38.86
193.72 2224.50 15.06
225.65
0.67
0.97
0.30
0.11 175.53 3138.27 34.13 225.10
All
35.52
559.90
0.50
45.62
88.09
124.23
0.58
0.98
0.00
0.13
2333.50
0.53
60.76 3138.27
1.44
86.69
Note: All the variables are from NBER-CES Manufacturing Industry Data set for the U.S., which covers 473 industries.
Employment is in the unit of 1000 employees. K/L is the ratio of real capital stock to employment and in the unit of
thousand US dollars per worker. Capital expenditure share is one minus the ratio of labor expenditure to the value
added. Labor productivity is measured as value added per worker.
SER02
SER01
. 07
. 06
SER03
. 05
. 020
. 04
. 016
. 03
. 012
. 02
. 008
. 01
. 004
SER04
SER05
SER07
SER06
. 03
. 01
. 00
60
65
70
75
80
85
90
95
00
. 030
. 030
. 028
. 035
. 030
. 025
. 025
. 024
. 030
. 025
. 025
. 020
. 020
. 020
. 025
. 020
. 020
. 016
. 020
. 015
. 015
. 015
. 015
. 010
. 010
. 010
. 010
. 005
. 005
. 005
. 005
. 004
. 000
60
05
65
70
75
SER11
80
85
90
95
00
05
. 000
60
65
70
75
SER12
80
85
90
95
00
05
. 030
. 020
. 020
. 020
. 025
. 016
. 016
. 016
. 020
. 012
. 012
. 012
. 015
. 008
. 008
. 008
. 010
. 004
. 004
. 004
. 005
. 000
. 000
. 000
. 000
75
80
85
90
95
00
60
05
65
70
75
80
85
90
95
00
05
60
65
70
75
SER22
SER21
75
80
85
. 014
80
85
90
95
00
05
90
95
00
05
65
70
75
. 016
. 014
. 014
. 012
. 012
. 010
. 010
. 008
85
90
95
00
05
. 000
60
65
70
75
80
85
90
95
00
. 020
. 016
. 016
. 012
. 012
. 008
. 008
. 004
65
70
75
80
85
90
95
00
60
65
70
75
80
85
90
95
00
60
05
65
70
75
. 016
. 016
. 014
. 014
. 012
. 012
. 010
. 010
. 008
. 008
. 006
. 006
. 004
. 004
. 002
. 002
. 000
65
70
75
80
85
85
90
95
00
. 006
. 006
. 004
05
60
65
70
75
65
70
75
90
95
00
85
90
95
00
05
80
85
90
95
00
05
65
70
75
80
85
90
95
00
SER41
00
60
65
70
75
80
85
90
95
00
65
70
75
80
85
90
95
00
05
. 016
. 016
. 015
. 011
. 014
. 010
65
70
75
80
85
90
95
00
. 006
. 005
. 004
. 004
. 002
. 009
60
05
65
70
75
SER51
80
85
90
95
00
05
60
65
70
75
. 016
. 014
. 014
80
85
90
95
00
05
. 016
. 010
. 004
. 004
. 000
65
70
75
80
85
90
95
00
05
65
70
75
80
85
90
95
00
60
65
70
75
80
85
90
95
00
. 014
. 016
. 012
. 014
65
70
75
80
85
90
95
00
90
95
00
05
65
70
75
80
85
90
95
00
. 012
. 010
. 006
. 004
65
70
75
80
85
90
95
00
05
60
65
70
75
SER81
. 028
. 014
. 024
. 012
. 020
. 010
. 016
. 008
. 012
. 006
. 008
. 004
. 002
60
65
70
75
80
85
90
95
00
70
75
80
85
90
95
00
. 030
. 024
. 025
. 020
. 020
. 016
. 015
. 012
. 010
. 008
80
85
90
95
00
. 005
. 004
. 000
. 000
60
05
65
70
75
65
70
75
85
90
95
00
80
85
90
95
00
05
80
85
90
95
00
. 020
. 016
. 012
. 008
. 004
. 004
. 004
. 000
. 000
. 000
60
65
70
75
80
85
90
95
00
05
65
70
75
60
65
70
75
80
85
90
95
00
05
65
70
75
80
85
90
95
00
-.005
95
00
05
90
95
00
05
65
70
75
80
85
90
95
00
05
00
05
70
75
80
85
90
95
00
65
70
75
80
85
90
95
00
. 006
. 004
. 000
. 004
. 004
. 002
60
65
70
75
80
85
90
95
00
60
05
65
70
75
80
85
90
95
00
05
60
65
70
75
SER56
80
85
90
95
00
80
85
90
95
00
05
65
70
75
80
85
90
95
00
05
90
95
00
05
90
95
00
05
90
95
00
05
90
95
00
05
90
95
00
05
90
95
00
05
90
95
00
05
. 004
. 000
65
70
75
80
85
90
95
00
60
05
65
70
75
80
85
SER40
SER39
. 013
. 016
. 012
. 014
. 012
. 010
. 010
. 008
. 008
. 006
. 007
. 004
. 005
65
70
75
80
85
90
95
00
05
. 002
60
65
70
75
80
85
90
95
00
60
05
65
70
75
SER49
80
85
SER50
. 020
. 016
. 018
. 014
. 012
. 014
. 012
. 010
. 006
. 014
. 010
85
90
95
00
05
75
80
85
90
95
00
05
. 002
60
65
70
75
80
85
90
95
00
05
60
65
70
75
SER59
80
85
SER60
. 014
. 016
. 018
. 012
. 014
. 016
. 012
. 014
. 010
. 012
. 008
. 006
. 004
. 000
60
65
70
75
80
85
90
95
00
05
65
70
75
SER66
80
85
90
95
00
. 002
60
05
. 008
. 004
. 002
60
. 010
. 006
. 004
. 002
80
70
. 008
. 006
. 002
75
65
. 010
. 010
. 004
70
. 004
. 012
. 008
65
. 004
. 006
SER58
. 016
65
70
75
SER67
80
85
90
95
00
05
. 006
60
65
70
75
. 024
. 024
. 014
. 012
. 020
. 020
. 012
. 010
. 016
. 016
. 010
. 008
. 012
. 012
80
85
90
95
00
05
60
65
70
75
SER69
SER68
. 014
. 008
. 006
. 008
. 008
. 006
. 004
. 004
. 004
. 002
60
65
70
75
80
85
90
95
00
. 000
60
05
65
70
75
SER75
80
85
SER70
. 014
. 024
. 020
. 012
. 016
. 012
80
85
90
95
00
05
. 008
. 006
65
70
75
SER76
80
85
90
95
00
60
05
65
70
75
. 024
. 016
. 016
. 020
. 012
. 012
. 016
. 008
. 008
. 012
80
85
90
95
00
. 000
60
05
65
70
75
SER78
SER77
. 020
. 004
. 004
. 000
60
80
85
90
95
00
05
. 012
65
70
75
80
85
90
95
00
. 008
. 000
. 000
. 004
05
60
65
70
75
80
85
90
95
00
05
60
65
70
75
80
85
90
95
00
05
. 016
. 014
. 012
. 012
. 010
. 010
85
. 020
. 016
. 012
. 008
. 008
. 007
. 004
. 006
60
65
70
75
80
85
90
95
00
05
. 004
60
65
70
75
80
85
90
95
00
05
. 000
60
65
70
75
SER88
. 028
. 024
. 014
80
. 012
SER87
. 016
75
. 024
. 011
SER86
SER85
. 016
70
. 028
. 008
. 004
65
SER80
. 020
. 009
. 004
60
SER79
. 010
. 020
80
85
90
95
00
60
05
65
70
75
SER89
. 028
. 028
. 020
. 024
. 024
. 020
. 020
. 016
. 016
. 012
. 012
. 016
80
85
SER90
. 024
. 016
. 012
. 012
. 008
. 008
. 008
. 008
. 006
60
65
70
75
80
85
90
95
00
05
. 006
60
65
70
75
80
85
90
95
00
60
05
65
70
75
SER95
80
85
90
95
00
70
75
80
85
90
95
00
05
. 004
. 000
. 000
. 000
60
65
70
75
80
85
90
95
00
05
60
65
70
75
SER97
80
85
90
95
00
05
. 008
. 004
. 000
60
65
70
75
SER98
. 018
. 016
. 012
. 030
. 028
. 025
. 024
85
90
95
00
05
90
95
00
05
60
65
70
75
80
85
. 020
. 020
. 012
80
SER99
. 014
. 016
. 010
. 015
. 012
. 008
. 010
. 008
. 006
. 004
. 000
65
. 004
05
. 014
. 004
60
. 008
. 004
SER96
. 020
. 000
05
05
. 008
60
. 011
60
. 016
05
. 004
00
85
. 012
. 002
SER57
. 012
60
. 008
95
80
SER30
. 004
. 014
. 006
. 012
90
75
. 016
SER48
. 004
. 016
85
70
. 020
. 008
60
05
. 006
. 008
80
65
. 024
. 010
. 000
. 008
. 012
75
60
05
. 006
. 010
. 016
70
00
. 000
65
. 012
. 008
. 012
. 008
65
95
. 004
60
SER29
. 012
. 008
. 012
60
90
. 008
SER38
. 010
. 016
. 000
60
05
. 012
. 016
60
. 012
. 000
90
85
. 014
. 005
85
80
. 016
SER65
. 020
. 004
00
. 016
. 018
SER47
. 006
. 004
80
95
. 020
60
05
. 008
. 010
75
90
. 004
. 010
. 015
70
75
. 008
. 020
65
70
. 010
. 025
60
65
. 012
SER94
. 020
. 016
. 000
85
. 000
. 014
. 020
SER93
. 030
. 008
80
. 005
. 008
SER84
. 008
95
. 006
. 012
05
. 024
. 012
90
. 014
SER46
. 004
60
. 016
85
. 009
60
05
80
. 008
SER83
SER92
. 012
75
. 016
. 016
SER74
. 024
SER91
. 020
80
75
. 010
60
05
. 020
05
75
. 002
60
. 000
65
SER82
. 016
70
. 004
. 008
60
70
. 025
. 016
. 006
SER73
. 014
. 004
65
70
. 020
SER28
. 020
. 008
. 016
. 008
65
. 008
60
05
. 018
. 012
60
. 012
. 002
60
. 020
. 016
60
. 010
. 020
SER72
. 020
05
. 004
. 000
. 012
. 004
SER71
. 024
00
. 006
. 004
. 004
. 004
. 002
85
95
. 008
. 002
. 008
. 006
. 004
80
90
. 010
. 002
. 012
. 016
. 006
75
85
65
SER20
. 006
SER37
. 008
SER64
. 008
70
80
60
. 024
. 008
. 030
60
. 006
. 016
05
. 010
65
05
. 004
. 014
. 008
60
00
. 012
SER55
. 012
. 010
. 008
. 000
95
. 012
. 020
SER63
. 012
. 004
90
. 014
05
-.004
60
05
SER62
. 016
85
05
. 008
. 000
. 000
60
SER61
. 020
80
. 006
. 002
60
75
. 014
. 008
. 006
. 002
70
. 008
. 008
. 008
. 004
75
SER36
. 012
. 010
. 006
65
. 014
SER54
. 020
. 012
. 012
60
00
. 008
SER53
SER52
. 016
05
95
. 010
. 011
. 006
00
00
90
. 012
. 007
95
95
85
. 016
. 014
. 010
90
90
80
. 013
. 006
85
85
. 024
. 008
. 007
80
70
. 000
SER45
. 010
. 008
75
65
. 004
. 016
05
. 012
. 009
. 010
70
80
75
. 009
. 010
. 016
SER44
. 018
65
75
70
. 011
. 015
. 008
SER43
. 012
60
70
. 000
65
. 012
. 020
. 010
60
. 013
. 008
65
. 005
60
SER19
. 000
. 016
SER35
. 004
. 013
. 009
60
05
. 006
. 014
. 011
05
. 010
SER42
. 012
00
. 007
. 020
. 006
. 006
60
05
95
. 014
SER27
. 008
. 000
. 012
-.004
95
90
. 008
. 012
. 000
90
85
. 012
. 004
. 008
85
80
. 004
. 010
. 010
80
75
. 010
. 012
. 004
75
70
. 013
60
05
. 016
. 008
70
65
. 016
. 000
60
. 010
. 000
60
SER18
. 014
. 012
65
05
. 020
SER26
. 016
. 014
. 016
00
. 004
. 018
SER34
. 020
60
05
80
95
. 008
SER25
. 020
SER33
SER32
SER31
. 018
60
80
. 008
90
. 012
. 008
. 000
85
. 020
. 000
60
05
. 004
. 002
80
. 024
. 004
. 000
. 008
. 004
75
SER17
. 012
. 008
70
. 008
. 010
. 006
65
. 015
. 004
. 000
60
05
SER16
. 020
SER24
. 016
80
SER15
. 012
. 012
. 000
60
. 008
. 012
60
. 018
. 012
. 016
SER23
. 016
. 016
70
SER14
. 024
70
65
SER13
. 024
65
. 000
60
. 024
60
SER10
. 035
. 030
. 008
. 02
SER09
. 035
. 020
. 012
. 04
SER08
. 024
. 016
. 05
. 004
. 002
60
65
70
75
80
85
90
95
00
05
. 005
. 002
60
65
70
75
80
85
90
95
00
05
. 004
. 000
. 000
60
65
70
75
80
85
90
95
00
05
60
65
70
75
80
85
90
95
00
05
60
65
70
75
80
85
Figure A1. Industrial Dynamics of the 99 Newly-Defined Industries in US: 1958-2005
Note: The time-series plots of the employment shares for the 99 newly-defined industries within
the manufacturing sector are presented following an increasing order of capital intensity. That is,
the leftmost plot in the top row is Industry 1, which is most labor-intensive by construction. The
second plot in the top row is Industry 2, which is the second most labor-intensive industry, and
then followed by Industry 3, and so forth. The rightmost plot in the bottom row is industry 99, the
most capital-intensive industry.
Data Source: NBER-CES Manufacturing Data Set
1
industry 0
industry 1
industry 2
industry 3
industry 4
industry 5
0.9
L=1
gamma=0.8
lumda=1.1
a=1.5
0.8
0.7
Industrial Ouput
0.6
0.5
0.4
0.3
0.2
0.1
0
0
2
4
6
8
10
Capital Endowment E
12
14
16
Figure A2. How Industrial Output Changes with Capital Endowment under
General CES function when λ > 1
18
20

Endowment Structures, Industrial Dynamics, and Economic Growth

Transcription

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