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Structural Change in a Multisector Model of Growth
By L. RACHEL NGAI AND CHRISTOPHER A. PISSARIDES*
Economic growth takes place at uneven
rates across different sectors of the economy.
This paper has two objectives related to this
fact: (a) to derive the implications of different
sectoral total factor productivity (TFP)
growth rates for structural change, the name
given to the shifts in industrial employment
shares that take place over long periods of
time; and (b) to show that even with ongoing
structural change, the economy’s aggregate
ratios can be constant. We refer to the latter as
aggregate balanced growth. The restrictions
needed to yield structural change consistent
with the facts and aggregate balanced growth
are weak restrictions on functional forms that
are frequently imposed by macroeconomists
in related contexts.
We obtain our results in a baseline model
of many consumption goods and a single cap-
ital good, supplied by a sector that we label
manufacturing. Our baseline results are con-
sistent with the existence of intermediate
goods and many capital goods under some
reasonable restrictions. Production functions
in our model are identical in all sectors except
for their rates of TFP growth, and each sector
produces a differentiated good that enters a
constant elasticity of substitution (CES) util-
ity function. We show that a low (below one)
elasticity of substitution across final goods
leads to shifts of employment shares to sec-
tors with low TFP growth. In the limit, the
employment share used to produce consump-
tion goods vanishes from all sectors except
for the one with the smallest TFP growth rate,
but the employment shares used to produce
capital goods and intermediate goods con-
verge to nontrivial stationary values. If the
utility function in addition has unit intertemporal
elasticity of substitution, during structural change
the aggregate capital-output ratio is constant and
the aggregate economy is on a balanced growth
path.
Our results contrast with the results of Cris-
tina Echevarria (1997), John Laitner (2000),
Francesco Caselli and Wilbur Coleman II
(2001), and Douglas Gollin, Stephen Parente,
and Richard Rogerson (2002), who derived
structural change in a two- or three-sector
economy with nonhomothetic preferences.
Our results also contrast with the results of
Piyabha Kongsamut, Sergio Rebelo, and Dan-
yang Xie (2001) and Reto Foellmi and Josef
Zweimuller (2005), who derived simulta-
neous constant aggregate growth and struc-
tural change. Kongsamut, Rebelo, and Xie
(2001) obtain their results by imposing a re-
striction that maps some of the parameters of
their Stone-Geary utility function onto the
parameters of the production functions, aban-
doning one of the most useful conventions of
modern macroeconomics, the complete inde-
pendence of preferences and technologies.
Foellmi and Zweimuller (2005) obtain their
results by assuming endogenous growth
driven by the introduction of new goods into
a hierarchic utility function. Our restrictions
are quantitative restrictions on a conventional
CES utility function that maintains the inde-
pendence of the parameters of preferences
and technologies.
Our results confirm William J. Baumol’s
(1967) claims about structural change. Baumol
divided the economy into two sectors, a “pro-
gressive” one that uses new technology and a
“stagnant” one that uses labor as the only input.
He then claimed that the production costs and
prices of the stagnant sector should rise indefi-
nitely, a process known as “Baumol’s cost dis-
* Ngai: Centre for Economic Performance, London School
of Economics and CEPR (e-mail: l.ngai@lse.ac.uk); Pissar-
ides: Centre for Economic Performance, London School of
Economics, CEPR, and IZA (e-mail: c.pissarides@lse.ac.uk).
We have benefited from comments received at several presen-
tations (the CEPR ESSIM 2004 meetings, the SED 2004
annual conference, the NBER 2004 Summer Institute, the
2004 Canadian Macroeconomic Study Group, and at several
universities), and from Fernando Alvarez, Francesco Caselli,
Antonio Ciccone, Nobu Kiyotaki, Robert Lucas, Nick Oulton,
Danny Quah, Sergio Rebelo, Robert Shimer, Nancy Stokey,
Richard Rogerson, Jaume Ventura, and two anonymous refer-
ees. Funding from the CEP, a designated ESRC Research
Centre, is acknowledged.
429
ease,” and labor should move in the direction of
the stagnant sector.
1
In the more recent empirical literature, two
competing explanations (which can coexist)
have been put forward for structural change:
our explanation, which is sometimes termed
“technological” because it attributes structural
change to different rates of sectoral TFP
growth; and a utility-based explanation, which
requires different income elasticities for differ-
ent goods and can yield structural change even
with equal TFP growth in all sectors. Baumol,
Sue Anne Batey Blackman, and Edward N.
Wolff (1985) provide empirical evidence at the
two-digit industry level, consistent with our
model. Irving B. Kravis, Alan W. Heston, and
Robert Summers (1983) also present evidence
that favors the technological explanation, at
least when the comparison is between manufac-
turing and services. Two features of their data
satisfied by the technological explanation pro-
posed in this paper are: (a) relative prices reflect
differences in TFP growth rates; and (b) real
consumption shares vary a lot less over time
than nominal consumption shares.
2
Our model
is also consistent with the observed positive
correlation between employment growth and
relative price inflation across two-digit sectors
3
and with historical OECD evidence presented
by Simon Kuznets (1966) and Angus Maddison
(1980) for one-digit sectors.
4
Section I describes our model of growth with
many sectors and Sections II and III, respec-
tively, derive the conditions for structural
change and balanced aggregate growth. In Sec-
tions IV and V, we study two extensions of our
baseline model, one where consumption goods
can also be used as intermediate inputs and one
where there are many capital goods. The Ap-
pendix discusses the implications of one more
extension and differences in capital intensities
across sectors, and contains proofs of the main
results.
I. An Economy with Many Sectors
The baseline economy consists of an arbi-
trary number of m sectors. Sectors i 1, ... ,
m 1 produce only consumption goods. The
last sector, which is denoted by m and labeled
manufacturing, produces both a final con-
sumption good and the economy’s capital
stock. We derive the equilibrium as the solu-
tion to a social planning problem. The objec-
tive function is
(1) U
冕
0
e
t
vc
1
, ... , c
m
dt,
where
0, c
i
0 are per capita consump
-
tion levels, and the instantaneous utility func-
tion v is concave and satisfies the Inada
conditions. The constraints of the problem are
as follows.
The labor force is exogenous and growing
at rate
, and the aggregate capital stock is
endogenous and defines the state of the econ-
omy. Sectoral allocations are controls that
satisfy
(2)
冘
i 1
m
n
i
1;
冘
i 1
m
n
i
k
i
k,
where n
i
0 is the employment share and k
i
0 is the capital-labor ratio in sector i, and k 0
is the aggregate capital-labor ratio. There is free
mobility for both factors.
All production in sectors i 1, ... , m 1is
consumed, but in sector m production may be
either consumed or invested. Therefore:
(3) c
i
F
i
n
i
k
i
, n
i
i m;
(4) k
˙
F
m
n
m
k
m
, n
m
c
m
k,
1
Baumol controversially also claimed that as more
weight is shifted to the stagnant sector, the economy’s
growth rate will be on a declining trend and eventually
converge to zero. This claim contrasts with our finding that
the economy is on a balanced-growth path. We get our
result because we include capital in our analysis, ironically
left out of the analysis by Baumol (1967, 417) “primarily for
ease of exposition ... that is [in]essential to the argument.”
2
See Rodney E. Falvey and Norman Gemmell (1996) for an
update of some of their results. Falvey and Gemmell find a unit
income elasticity and a small (negative) price elasticity for ser-
vices in a cross section of countries, consistent with our results.
3
These correlations are shown in the working paper
version of this paper, Ngai and Pissarides (2004).
4
Kuznets (1966) documented structural change for 13
OECD countries and the USSR between 1800 and 1960, and
Maddison (1980) documented the same pattern for 16 OECD
countries from 1870 to 1987. They both found a pattern with
the same general features as the predictions that we obtain
when the ranking of the average historical TFP growth rates is
agriculture followed by manufacturing followed by services.
430 THE AMERICAN ECONOMIC REVIEW MARCH 2007
where
0 is the depreciation rate. Production
function F
i
( 䡠 , 䡠 ) has constant return to scale,
positive and diminishing returns to inputs, and
satisfies the Inada conditions.
The social planner chooses the allocation of
factors n
i
and k
i
across m sectors through a set
of static efficiency conditions:
(5) v
i
/v
m
F
K
m
/F
K
i
F
N
m
/F
N
i
i.
The allocation of output to consumption and
capital is chosen through a dynamic efficiency
condition:
(6) v˙
m
/v
m
F
K
m
,
where F
N
i
and F
K
i
are the marginal products of
labor and capital in sector i.
5
By (5), the rates
of return to capital and labor are equal across
sectors.
In order to focus on the implications of dif-
ferent rates of TFP growth across sectors, we
assume production functions are identical in all
sectors except for their rates of TFP growth:
(7) F
i
A
i
Fn
i
k
i
, n
i
; A
˙
i
/A
i
i
; i.
With these production functions, we show in the
Appendix that static efficiency and the resource
constraints (2) imply
(8) k
i
k; p
i
/p
m
v
i
/v
m
A
m
/A
i
; i,
where p
i
is the price of good i in the decentral
-
ized economy.
The utility function has constant elasticities
both across goods and over time:
(9) vc
1
, ... , c
m
1
1
1
;
冉
冘
i 1
m
i
c
i
1/
冊
/ 1
,
where
, ,
i
0 and ¥
i
1. Of course, if
1, v ln
, and if 1, ln
¥
i1
m
i
ln c
i
. In the decentralized economy, de-
mand functions have constant price elasticity
and unit income elasticity. With this utility
function, (8) yields
(10)
p
i
c
i
p
m
c
m
冉
i
m
冊
冉
A
m
A
i
冊
1
⬅ x
i
i.
The new variable x
i
is the ratio of consumption
expenditure on good i to consumption expendi-
ture on the manufacturing good, and will prove
useful in the subsequent analysis. The intuition
behind this formula is in terms of price elastic-
ities, given that all goods have unit income
elasticity. The ratio of consumption expenditures
is a weighted average of the ratio of the weights of
each good in the utility function and of their relative
prices. A higher price ratio p
i
/p
m
raises the ratio of
expenditure on good i to good m by one minus their
common price elasticity.
We also define aggregate consumption ex-
penditure and output per capita in terms of
manufacturing:
(11) c ⬅
冘
i 1
m
p
i
p
m
c
i
; y ⬅
冘
i 1
m
p
i
p
m
F
i
.
Using static efficiency, we derive
(12) c c
m
X; y A
m
Fk,1,
where X ¥
i1
m
x
i
.
II. Structural Change
We define structural change as the state in
which at least some of the labor shares are
changing over time, i.e., n˙
i
0 for at least some
i. We derive in the Appendix (Lemma A2) the
employment shares
(13) n
i
x
i
X
冉
c
y
冊
i m,
(14) n
m
x
m
X
冉
c
y
冊
冉
1
c
y
冊
.
The first term on the right side of (14) parallels
the term in (13) and so represents the employ-
ment needed to satisfy the consumption demand
for the manufacturing good. The second brack-
eted term is equal to the savings rate and rep-
5
The corresponding transversality condition is lim
t3
k
exp(
0
t
(F
k
m
) d
) 0.
431VOL. 97 NO. 1 NGAI AND PISSARIDES: STRUCTURAL CHANGE IN A MULTISECTOR MODEL OF GROWTH
resents the manufacturing employment needed
to satisfy investment demand.
Conditions (13) and (14) drive our structural
change results. To see the intuition behind them,
note that by aggregation over all i, we obtain
that in our economy the employment share used
to produce consumption goods is equal to c/y,
and the employment share used to produce cap-
ital goods is 1 c/y. Conditions (13) and (14)
state that the same holds for each sector i. From
(10) and (12), the consumption expenditure
share of each sector is p
i
c
i
/p
m
c x
i
/X. So the
employment share of consumption good i is
the consumption share of good i multiplied by
the employment share of total consumption.
Equivalently, the employment share of con-
sumption good i is the average propensity to
consume good i : n
i
p
i
c
i
/p
m
y.
Condition (13) has the important implica-
tion that the growth rate of two sectors’ relative
employment depends only on the difference be-
tween the sectors’ TFP growth rates and the
elasticity of substitution between goods:
(15)
n˙
i
n
i
n˙
j
n
j
1
j
i
i, j m.
But (8) implies that the growth rate of relative
prices is:
(16)
p˙
i
p
i
p˙
j
p
j
j
i
i
and so,
(17)
n˙
i
n
i
n˙
j
n
j
1
冉
p˙
i
p
i
p˙
j
p
j
冊
i, j m.
PROPOSITION 1: The rate of change of the
relative price of good i to good j is equal to the
difference between the TFP growth rates of
sector j and sector i. In sectors producing only
consumption goods, relative employment shares
grow in proportion to relative prices, with the
factor of proportionality given by one minus the
elasticity of substitution between goods.
6
The dynamics of the individual employment
shares satisfy
(18)
n˙
i
n
i
c
˙
/y
c/y
1
i
; i m;
(19)
n˙
m
n
m
冋
c
˙
/y
c/y
(1 )(
m
)
册
c/yx
m
/X
n
m
冉
c
˙
/y
1 c/y
冊冉
1 c/y
n
m
冊
,
where
¥
i1
m
( x
i
/X)
i
is a weighted aver
-
age of TFP growth rates, with the weight
given by each good’s consumption share.
Equation (18) gives the growth rate in the
employment share of each consumption good
as a linear function of its own TFP growth
rate. The intercept and slope of this function
are common across sectors, but although the
slope is a constant, the intercept is in general
a function of time because both c/y and
are
in general functions of time. Manufacturing,
however, does not conform to this rule, be-
cause its employment share is a weighted
average of two components, one for the pro-
duction of the consumption good, which con-
forms to the rule, and one for the production
of capital goods, which behaves differently.
The properties of structural change follow
immediately from (18) and (19). Consider, first,
the case of equality in sectoral TFP growth
rates, i.e., let
i
m
@i. In this case, our
economy is one of balanced TFP growth, with
relative prices remaining constant but with
many differentiated goods. Because of the con-
stancy of relative prices, all consumption goods
can be aggregated into one, so we effectively
have a two-sector economy, one sector produc-
ing consumption goods and one producing cap-
ital goods. Structural change can still take place
in this economy, but only between the aggregate
of the consumption sectors and the capital sec-
tor, and only if c/y changes over time. If c/y is
increasing over time, the investment rate is fall-
ing and labor is moving out of the manufactur-
ing sector and into the consumption sectors.
Conversely, if c/y is falling over time, labor is
moving out of the consumption sectors and into
manufacturing. In both cases, however, the rel-
ative employment shares in consumption sec-
tors are constant.
If c/y is constant over time, structural
change requires 1 and different rates of
6
All derivations and proofs, unless trivial, are collected
in the Appendix.
432 THE AMERICAN ECONOMIC REVIEW MARCH 2007
sectoral TFP growth rates. It follows imme-
diately from (16), (18), and (19) that if
(c
˙
/y) 0, 1 implies constant employment
shares but changing prices. With constant em-
ployment shares, faster-growing sectors pro-
duce relatively more output over time. Price
changes in this case are such that consump-
tion demands exactly match all the output
changes due to the different TFP growth rates.
But if 1, prices still change as before but
consumption demands are either too inelastic
(in the case 1) to match all the output
change, or are too elastic ( 1) to be sat-
isfied merely by the change in output due to
TFP growth. So if 1, employment has to
move into the slow-growing sectors, and if
1, it has to move into the fast-growing
sectors.
PROPOSITION 2: If
i
m
@i m, a
necessary and sufficient condition for structural
change is c˙/c y˙/y. The structural change in
this case is between the aggregate of consump-
tion sectors and the manufacturing sector.
If c˙/c y˙/y, necessary and sufficient condi-
tions for structural change are 1 and ?i 僆
{1, ... , m 1} s.t.
i
m
. The structural
change in this case is between all sector pairs
with different TFP growth rates. If 1, em-
ployment moves from the sector with the higher
TFP growth rate to the sector with the lower
TFP growth rate; the converse is true if 1.
Proposition 2 for 1 confirms the struc-
tural change facts identified by Baumol, Black-
man, and Wolff (1985). When demand is price
inelastic, the sectors with the low productivity
growth rate attract a bigger share of labor, de-
spite the rise in their price. From the static
efficiency results in (8) and (12), we find that
the nominal output shares (defined as p
i
F
i
/p
m
y)
are equal to the employment shares in all sec-
tors, and by (10) the nominal consumption
shares are given by x
i
/X, so the results obtained
for employment shares also hold for nominal
consumption and output shares. But real con-
sumption growth satisfies
(20) c˙
i
/c
i
c˙
j
/c
j
i
j
; i, j,
an expression also satisfied by real output shares
@i, j m.
A comparison of (15) and (20) reveals that
a small can reconcile the small changes in
the relative real consumption shares with the
large changes in relative nominal consump-
tion shares found by Kravis, Heston, and
Summers (1983). The authors concluded that
their finding is evidence in favor of the tech-
nological explanation of structural change.
More recently, Daniel E. Sichel (1997) found
the same pattern for relative output shares,
and Falvey and Gemmell (1996) found that
the real consumption share of services (a sec-
tor with low TFP growth rate) falls very grad-
ually with income, both of which are
consistent with our model when 1.
III. Aggregate Growth
We now study the aggregate growth path of
this economy, with the objective of finding a
sufficient set of conditions that satisfy struc-
tural change as derived in the preceding sec-
tion, and in addition satisfy Kaldor’s stylized
facts of aggregate growth. Recall that, for the
analysis of structural change, we imposed a
Hicks-neutral technology. It is well-known
that with this type of technology, the econ-
omy can be on a steady state only if the
production function is Cobb-Douglas. We
therefore let F(n
i
k
i
, n
i
) k
i
n
i
,
僆 (0, 1).
7
With TFP in each sector growing at some rate
i
, the aggregate economy will also grow at
some rate related to the
i
s. The following
proposition derives the evolution of the ag-
gregate economy.
PROPOSITION 3: Given any initial k(0), the
equilibrium of the aggregate economy is a path
for the pair {c, k} that satisfies the following two
differential equations:
(21)
k
˙
k
A
m
k
1
c
k
,
7
Daron Acemoglu and Veronica Guerrieri (2005) exam
-
ined the implications of different capital intensities for
economic growth and structural change. They show that
capital deepening can cause both structural change and
unbalanced growth. We examine in the Appendix the im-
plications of different capital shares and a fixed factor for
our model.
433VOL. 97 NO. 1 NGAI AND PISSARIDES: STRUCTURAL CHANGE IN A MULTISECTOR MODEL OF GROWTH
(22)
c˙
c
1
m
A
m
k
1
.
We define an aggregate balanced growth path
such that aggregate output, consumption, and
capital grow at the same rate. It follows from
Proposition 3 that a necessary condition for the
existence of an aggregate balanced growth path
is that the expression (
1)(
m
)bea
constant. To show this, let
(23)
1
m
⬅
constant.
Define aggregate consumption and the capital-
labor ratio in terms of efficiency units, c
e
cA
m
1/(1
)
and k
e
kA
m
1/(1
)
, and let g
m
m
/(1
), the rate of labor-augmenting tech
-
nological growth in the capital-producing sector.
The dynamic equations (21) and (22) become
(24) c˙
e
/c
e
k
e
1
/
g
m
;
(25) k
˙
e
k
e
c
e
g
m
k
e
.
Equations (24) and (25) parallel the two dif-
ferential equations in the control and state of
the one-sector Ramsey economy, making the
aggregate equilibrium of our many-sector
economy identical to the equilibrium of the
one-sector Ramsey economy when
0, and
trivially different from it otherwise. Both
models have a saddlepath equilibrium and
stationary solutions (cˆ
e
, k
ˆ
e
) that imply bal
-
anced growth in the three aggregates. The
capital-labor ratio is growing at the rate of
growth of labor-augmenting technological
progress in the sector that produces capital
goods, g
m
. Aggregate consumption and out
-
put deflated by the price of manufacturing
goods are also growing at the same rate.
Proposition 2 and the requirement that
be constant yield the following important
proposition.
PROPOSITION 4: Necessary and sufficient
conditions for the existence of an aggregate
balanced growth path with structural change are:
1, 1;
and i 僆 1, ... , m 1 s.t.
i
m
.
Recalling the definition of
following equation
(19), Proposition 3 implies that the contribution of
each consumption sector i to aggregate equilib-
rium is through its weight x
i
in
. Because each x
i
depends on the sector’s relative TFP level, the
weights here are functions of time. So
cannot be
constant during structural change and the only
way that
can be constant is through
1,
which yields
0. In this case, our aggregate
economy in c and k becomes formally identical to
the one-sector Ramsey economy with growth rate
m
. There are two other conditions that give a
constant
and so yield balanced aggregate
growth:
i
m
@i or 1. But as Proposition
2 demonstrates, neither condition permits struc-
tural change on the balanced growth path, where
c/y is constant.
Proposition 4 requires the utility function to be
logarithmic in the consumption composite
,
which implies an intertemporal elasticity of sub-
stitution equal to one, but also to be nonlogarith-
mic across goods, which is needed to yield
nonunit price elasticities. A noteworthy implica-
tion of Proposition 4 is that balanced aggregate
growth does not require constant rates of growth
of TFP in any sector other than manufacturing.
Because both capital and labor are perfectly mo-
bile across sectors, changes in the TFP growth
rates of consumption-producing sectors are re-
flected in immediate price changes and realloca-
tions of capital and labor across sectors, without
effect on the aggregate growth path.
To give intuition for the logarithmic inter-
temporal utility function, we recall that bal-
anced aggregate growth requires that aggregate
consumption be a constant fraction of aggregate
wealth. With our homothetic utility function, this
can be satisfied either when the interest rate is
constant or when consumption is independent of
the interest rate. The relevant interest rate here is
the rate of return to capital in consumption units,
which is given by the net marginal product of
capital,
y/k
, minus the change in the relative
price of the consumption composite,
m
. The
latter is not constant during structural change. In
the case 1,
is falling over time (see Lemma
A3 in the Appendix for proof), and so the real
interest rate is also falling, and converging to
y/k
. With a nonconstant interest rate, the
434 THE AMERICAN ECONOMIC REVIEW MARCH 2007
consumption-wealth ratio is constant only if con-
sumption is independent of the interest rate, which
requires a logarithmic utility function.
8
Under the conditions of Proposition 4,
there is a steady state characterized by aggre-
gate balanced growth, in the sense that in this
steady state the aggregate ratios are constant.
In order to achieve this balance, the aggre-
gates c and y are divided by manufacturing
price, to conform to the aggregate k. If some
other price index is used as deflator, the rate
of growth of the aggregates is constant only if
the rate of growth of the price index is con-
stant, but of course the aggregate ratios are
still constant. The published aggregate series
studied by macroeconomists usually use an
average price as deflator which does not have
fixed weights. If the price index used to de-
flate national statistics is some p˜ , the pub-
lished real aggregate income is y/p˜ . If the
weights used to construct p˜ are the sector
shares, p˜ changes during structural change.
But because sector shares do not change rap-
idly over time, visually there is virtually noth-
ing to distinguish the “stylized fact” of
constant growth in reported per capita GDP
with another “stylized fact” of constant
growth in our per capita output measure.
9
Next, we summarize the dynamics of em-
ployment shares along the aggregate balanced
growth path.
PROPOSITION 5: Let sector l denote the sec-
tor with the smallest TFP growth rate when
1, or the sector with the biggest TFP growth
rate when 1. On the aggregate balanced
growth path, n
l
increases monotonically.
Employment in the other sectors is either
hump-shaped or declines monotonically. As-
ymptotically, the economy converges to an
economy with
n
*
m
ˆ
冉
g
m
g
m
冊
;
n
*
l
1
ˆ ,
where
ˆ is the savings rate along the aggregate
balanced growth path.
Proposition 5 follows immediately from
(18)–(19) and Lemma A3. Consider the case
1, the one for 1 following by a
corresponding argument. For 1, sector i
expands if and only if its TFP growth rate is
smaller than
, and contracts if and only if its
growth rate exceeds it. But if 1, the
weighted average
is decreasing over time (see
Lemma A3 in the Appendix). Therefore, the set
of expanding sectors is shrinking over time, as
more sectors’ TFP growth rates exceed
. This
feature of the model implies that sectors with
TFP growth rates below the initial
exhibit a
hump-shaped employment share, an implication
that we believe is unique to our model. These
employment shares first rise, but once
drops
down to their own
i
, they fall.
10
In contrast to each sector’s employment
share, once the economy is on the aggregate
balanced growth path, output and consump-
tion in each consumption sector grow accord-
ing to
(26)
F
˙
i
F
i
A
˙
i
A
i
k
˙
i
k
n˙
i
n
i
i
g
m
1
.
If 1, the rate of growth of consumption and
output in each sector is positive (provided
i
0), and so sectors never vanish, even though
their employment shares in the limit may van-
ish. If 1, the rate of growth of output may
8
After reexamining the evidence, Robert Barro and
Xavier Sala-i-Martin (2004, 13) concluded, consistent with
our model, “it seems likely that Kaldor’s hypothesis of a
roughly stable real rate of return should be replaced by a
tendency for returns to fall over some range as an economy
develops.” In our model, it is converging from above to a
positive value.
9
Nicholas Kaldor (1961, 178) spoke of a “steady trend
rate” of growth in the “aggregate volume of production.” In
Ngai and Pissarides (2004, fig. 4) we plot our series of per
capita real incomes and the published chain-weighted series
for the United States since 1929, and show that they are
virtually indistinguishable from each other.
10
Maddison (1980, 48), in his study of historical OECD
data, found a “shallow bell shape” for manufacturing em-
ployment for each of the 16 OECD countries, which can be
reproduced by our model if the manufacturing TFP growth
rate takes values between the TFP growth rates of agricul-
ture and services.
435VOL. 97 NO. 1 NGAI AND PISSARIDES: STRUCTURAL CHANGE IN A MULTISECTOR MODEL OF GROWTH
be negative in some low-growth sectors, and
since by Lemma A3
is rising over time in this
case, their rate of growth remains indefinitely
negative until they vanish.
Finally, we examine briefly the implications of
1. When
1, balanced aggregate growth
cannot coexist with structural change, because the
term
(
1)(
m
) in the Euler condition
(24) is a function of time. But as shown in the
Appendix Lemma A3,
is monotonic. As t 3 ,
converges to the constant (
1)(
m
l
),
where
l
is the TFP growth rate in the limiting
sector (the slowest or fastest growing consump-
tion sector depending on whether or 1).
Therefore, the economy with
1 converges to
an asymptotic steady state with the same growth
rate as the economy with
1.
What characterizes the dynamic path of the
aggregate economy when
1? By differen-
tiation and using Lemma A3, we obtain
(27)
˙
11
冘
i 1
m
x
i
/X
i
2
,
which is of second order compared with the
growth in employment shares in (15), given that
the
s are usually small numbers centered
around 0.02. Therefore, the rate of growth of the
economy during the adjustment to the asymp-
totic steady state with
1 is very close to the
constant growth rate of the economy with
1, despite ongoing structural change in both
economies.
IV. Intermediate Goods
Our baseline model has no intermediate
inputs and has only one sector producing cap-
ital goods. We now generalize it by introduc-
ing intermediate inputs and (in the next
section) by allowing an arbitrary number of
sectors to produce capital goods. The key
difference between intermediate goods and
capital goods is that capital goods are reus-
able, while intermediate goods depreciate
fully after one usage. The motivation for the
introduction of intermediate inputs is that
many of the sectors that may be classified as
consumption sectors in fact produce for busi-
nesses. Business services is one obvious ex-
ample. Input-output tables show that a large
fraction of output in virtually all sectors of the
economy is sold to businesses.
11
As in the baseline model, sectors are of two
types. The first type produces perishable goods
that are either consumed by households or used
as intermediate inputs by firms. We continue
referring to these sectors as consumption sec-
tors. The second type of sector produces goods
that can be used as capital. For generality’s
sake, we assume that the output of the capital-
producing sector can also be processed into both
consumption goods and intermediate inputs.
The output of consumption sector i is now
c
i
h
i
, where h
i
is the output that is used as
an intermediate good. Manufacturing output
can be consumed, c
m
, used as an intermediate
input, h
m
, or used as new capital, k
˙
.We
assume that all intermediate goods, h
i
, are
used as an input into an aggregate CES pro-
duction function (h
1
, ... , h
m
) [¥
i1
m
i
h
i
(
1)/
]
/(
1)
which produces a single
intermediate good , with
0,
i
0, and
¥
i
1. The production functions are mod
-
ified to F
i
A
i
n
i
k
i
q
i
, @i, where q
i
is the
ratio of the intermediate good to employment
in sector i and
is its input share, with
,
0 and
1. When
0, we return to
our baseline model. We show in the Appendix
that a necessary and sufficient condition for
an aggregate balanced growth path with struc-
tural change is
1, i.e., should be
Cobb-Douglas.
12
When is Cobb-Doug-
las, our central results from the baseline
model carry through, with some modifica-
tions.
The aggregate equilibrium is similar to the
one in the baseline model:
(28)
c˙
c
Ak
1/1
,
11
According to input-output tables for the United States,
in 1990 the percentage distribution of the output of two-
digit sectors across three types of usage, final consumption
demand, intermediate goods, and capital goods, was 43, 48,
and 9, respectively. In virtually all sectors, however, a large
fraction of the intermediate goods produced are consumed
by the same sector.
12
Nicholas Oulton (2001) claims that if there are inter
-
mediate goods, and if the elasticity of substitution between
the intermediate goods and labor is bigger than one, Bau-
mol’s “stagnationist” results could be overturned (in the
absence of capital). No such possibility arises with Cobb-
Douglas production functions.
436 THE AMERICAN ECONOMIC REVIEW MARCH 2007
(29)
k
˙
k
1
Ak
1/1
c
k
,
where A [A
m
(
m
)
]
1/(1
)
and
m
is the
marginal product of the manufacturing good in
. The growth rate of A is constant and equal to
m
(
¥
i1
m
i
i
)/(1
), where
i
is the
input share of sector i in . Therefore, we can
define aggregate consumption and the aggregate
capital-labor ratio in terms of efficiency units
and obtain an aggregate balanced growth path
with growth rate (
m
¥
i1
m
i
i
)/(1
), which is the sum of labor-augmenting tech-
nological growth in the capital-producing sector
and a
fraction of labor-augmenting techno-
logical growth in all sectors that produce inter-
mediate goods. Recall the aggregate growth rate
in the baseline model depended only on the TFP
growth rate in manufacturing. In the extended
model with intermediate goods, the TFP growth
rates in all sectors contribute to aggregate growth.
The employment shares (13) and (14) are
now modified to
(30) n
i
x
i
X
冉
c
y
冊
i
; i m;
(31) n
m
冋
x
m
X
冉
c
y
冊
m
册
冉
1
c
y
冊
.
For the consumption sectors, the extra term in
(30) captures the employment required for pro-
ducing intermediate goods.
i
is the share of
sector i’s output used for intermediate purposes
and
is the share of the aggregate intermediate
input in aggregate output. For the manufactur-
ing sector, the terms in the first bracket parallel
those of the consumption sectors. The second
term captures the employment share for invest-
ment purposes.
Our results on structural change now hold for
the component of employment used to produce
consumption goods, ( x
i
/X)(c/y). The definition
of x
i
and X is the same as in the absence of
intermediate goods. The contribution of inter-
mediate goods to sectoral employment dynam-
ics is the addition of the constant employment
share
i
, with no impact on the other two
components of employment. Following on from
this, the asymptotic results in Proposition 5 are
also modified. Asymptotically, the employment
share used for the production of consumption
goods still vanishes in all sectors except for the
slowest growing one (when 1), but the
employment share used to produce intermediate
goods,
i
, survives in all sectors.
V. Many Capital Goods
In our second extension we allow an arbitrary
number of sectors to produce capital goods. We
study this extension with the baseline model
without intermediate inputs.
We suppose that there are
different capital-
producing sectors, each supplying the inputs
into a production function G, which produces a
capital aggregate that can be either consumed or
used as an input in all production functions F
i
.
Thus, the model is the same as before, except
that now the capital input k
i
is not the output of
a single sector but that of the production func-
tion G. The Appendix derives the equilibrium
for the case of a CES function with elasticity
,
i.e., when G [¥
j1
m
j
(F
m
j
)
(
1)/
]
/(
1)
,
where
0,
m
j
0 and F
m
j
is the output of
each capital goods sector m
j
. G now replaces the
output of the “manufacturing” sector in our
baseline model, F
m
.
It follows immediately that the structural
change results derived for the m 1 consump-
tion sectors remain intact, as we have made no
changes to that part of the model. But there are
new results to derive concerning structural
change within the capital-producing sectors.
The relative employment shares across the
capital-producing sectors satisfy
(32) n
m
j
/n
m
i
m
j
/
m
i
A
m
i
/A
m
j
1
;
i, j 1, ... ,
.
(33)
n˙
m
j
n
m
j
n˙
m
i
n
m
i
1
m
i
m
j
;
i, j 1, ... ,
.
These equations parallel (13) and (15) of the
baseline model and the intuition behind them is
the same.
When there are many capital goods, the A
m
of
437VOL. 97 NO. 1 NGAI AND PISSARIDES: STRUCTURAL CHANGE IN A MULTISECTOR MODEL OF GROWTH
the baseline model is replaced by G
m
j
A
m
j
for
each sector m
j
, where G
m
j
denotes the sector’s
marginal product in the production of aggregate
capital, and A
m
j
is the sector’s TFP level. This
term measures the rate of return to capital in the
jth capital-producing sector, which is equal
across all
sectors because of the free mobility
of capital. In the Appendix we derive the ag-
gregate growth rate
(34)
m
冘
j 1
j
m
j
;
j
⬅
m
j
A
m
j
1
冒
冉
冘
i 1
m
i
A
m
i
1
冊
,
which is a weighted average of TFP growth
rates in all capital-producing sectors. The dy-
namic equations for c and k are the same as in
the baseline model, given the new definition
of
m
.
If TFP growth rates are equal across all
capital-producing sectors, c and k grow at a
common rate in the steady state. But then all
capital producing sectors can be aggregated into
one, and the model reduces to one with a single
capital-producing sector. If TFP growth rates
are different across the capital-producing sec-
tors and
1, there is structural change within
the capital-producing sectors along the transi-
tion to the asymptotic state. Asymptotically,
only one capital-producing sector remains. In
the asymptotic state, c and k again grow at a
common rate, so there exists an asymptotic ag-
gregate balanced growth path with only one
capital-producing sector.
A necessary and sufficient condition for the
coexistence of an aggregate balanced growth
path and multiple capital-producing sectors
with different TFP growth rates is
1. The
reason for this result is that a balanced aggre-
gate path requires a constant
m
, which is un
-
attainable if the relative TFP levels in the
capital-producing sectors are allowed to influ-
ence it. From (34), the influence of the produc-
tivity levels disappears only when
1. The
aggregate growth rate in this case is
m
/(1
),
where
m
¥
j1
m
j
m
j
. Using (32), the relative
employment shares across capital-producing
sectors are equal to their relative input shares in
G. There is no structural change within the
capital-producing sectors, their relative employ-
ment shares remaining constant independently
of their TFP growth rates.
The model with 1 and
1 has clear
contrasting predictions about the relation be-
tween the dynamics of sectoral employment
shares and TFP growth (or relative prices). Sec-
tors that produce primarily consumption goods
should exhibit a well-defined linear relation be-
tween their employment share growth and their
TFP growth rate; sectors that produce many
intermediate goods should still have a positive
linear relation, but less well-defined because of
the constant term due to the production of in-
termediate goods. But sectors that produce pri-
marily capital goods should exhibit no linear
relation at all between their employment share
growth and their relative TFP growth rate.
VI. Conclusion
We have shown that different TFP growth
rates across industrial sectors predict sectoral
employment changes that are consistent with
the facts if the substitutability between the final
goods produced by each sector is low. Balanced
aggregate growth requires, in addition, a loga-
rithmic intertemporal utility function. Underly-
ing the balanced aggregate growth there is a
shift of employment away from sectors with a
high rate of technological progress toward sec-
tors with low growth, and eventually, in the
limit, all employment converges to only two
sectors, the sector producing capital goods and
the sector with the lowest rate of productivity
growth. If the economy also produces interme-
diate goods, the sectors that produce these
goods also retain some employment in the limit,
which is used to produce the intermediate
goods.
Our results are consistent with the observa-
tion of simultaneous growth in the relative
prices and employment shares of stagnant sec-
tors such as community services, with the near-
constancy of real consumption shares when
compared with nominal shares. It is also con-
sistent with the long-run evidence of Kuznets
(1966) and Maddison (1980) concerning the
decline of agriculture’s employment share, the
rise and then fall of the manufacturing share,
and the rise in the service share. The key re-
quirement for these results is again a low sub-
stitutability between final goods. Of course, at a
438 THE AMERICAN ECONOMIC REVIEW MARCH 2007
finer sector decomposition, the elasticity of sub-
stitution between two goods may reasonably
exceed unity, as for example between the output
of the sector producing typewriters and the out-
put of the sector producing electronic word pro-
cessors. Our model in this case predicts that
labor would move from the sector with low TFP
growth to the one with high TFP growth. The
approach that we suggested for intermediate and
many capital goods, namely the existence of
subsectors that produce an aggregate that enters
the utility or production function, is an obvious
approach to the analysis of these cases. Within
the subsectors, there is structural change toward
the high TFP goods, but between the aggregates
the flow is from high to low TFP sectors.
A
PPENDIX:PROOFS
LEMMA A1: Equations (2), (5), and (7) imply equation (8).
PROOF:
Defining f(k) F(k, 1), and omitting subscript i, (7) implies F
K
Af(k) and F
N
A[ f(k)
kf(k)]. So F
N
/F
K
f(k)/f(k) k, which is strictly increasing in k. Hence, (5) implies k
i
k
m
@i
m, and together with (2), results follow.
LEMMA A2: @i m, n
i
satisfy (13) and (18), and n
m
satisfies (14) and (19).
PROOF:
n
i
follows from substituting F
i
into (10), and n
m
is derived from (2). Given x˙
i
/x
i
(1 )(
m
i
) and X
˙
/X (1 )(
m
), the result follows for n˙
i
, i m. Using (2),
n˙
m
冘
im
n˙
i
冉
.
c/y
冊
c/y
1 n
m
1
冉
c/y
X
冊
冘
im
x
i
i
,
so result follows for n˙
m
by substituting n
m
.
PROPOSITION 3
PROOF:
Use (2) and (8) to rewrite (4) as
k
˙
/k A
m
k
1
冉
1
冘
im
n
i
冊
c
m
/k
.
But p
i
/p
m
A
m
/A
i
and by the definition of c,
k
˙
/k A
m
k
1
c/k
.
Next,
is homogenous of degree one:
冘
i 1
m
i
c
i
冘
i 1
m
p
i
c
i
m
/p
m
m
c/p
m
.
But
m
m
(
/c
m
)
1/
and c c
m
X; thus
m
m
/(1)
X
1/( 1)
and v
m
m
(
m
/(1)
X
1/( 1)
)
1
c
, so (6) becomes (22).
439VOL. 97 NO. 1 NGAI AND PISSARIDES: STRUCTURAL CHANGE IN A MULTISECTOR MODEL OF GROWTH
LEMMA A3: d
/dt 0 N 1.
PROOF:
Totally differentiating
as defined in equations (18) and (19),
d
/dt
冘
i 1
m
x
i
/X
i
冉
x˙
i
/x
i
冘
j 1
m
x˙
j
/X
冊
;
1
冘
i 1
m
x
i
/X
i
冋
m
i
冘
j 1
m
( x
j
/X)(
m
j
)
册
;
1
冉
2
冘
i 1
m
( x
i
/X)
i
2
冊
1
冘
i 1
m
x
i
/X
i
2
.
Since the summation term is always positive, the result follows.
Capital Shares and Fixed Factors.—We now discuss the structural change results (equations (8)
and (13)–(17)) when capital shares are different across sectors and there is a fixed factor of
production in at least one sector. The production function is F
i
A
i
k
i
i
z
i
1
i
i
n
i
, where
i
is labor
share and z
i
Z
i
/N
i
is a fixed factor per unit of labor. Suppose
i
i
1 for i 1, and
1
1
1, i.e., the fixed factor is used in sector 1 only. Static efficiency implies
n
i
/n
j
i
/
j
i
/
m
p
i
/p
j
1
, i, j m,
so the result that relative employment shares grow in proportion to relative prices, equation (17), is
independent of different factor shares and the existence of a fixed factor. Our other results are
modified as follows. The static efficiency condition (8) is replaced by
k
i
i
k
m
, p
i
/p
m
A
m
/A
i
m
/
i
k
m
m
i
z
i
i
i
1
;
i
⬅
m
i
i
m
,
i
⬅
i
i
j
1
i
.
Different capital shares add the term (
m
i
)k
˙
m
/k
m
in (16). In a growth equilibrium with k
m
growing, lower
i
is another reason for higher relative price in sector i. Combining the relative price
and relative employment equations, different capital shares add the term (1 )(
j
i
)k
˙
m
/k
m
in
(15). The existence of a fixed factor modifies (15) to
1 1
1
1
1
n˙
1
n
1
n˙
j
n
j
1
j
1
1
j
1
k
˙
m
k
m
, j m.
If n
1
is falling, then the presence of a fixed factor implies that n
1
falls at a faster rate. Finally, (13)
and (14) are modified to
n
i
x
i
X
c
y
冘
j
i
j
n
j
, i m; n
m
x
m
X
c
y
冘
j
i
j
n
j
1
c
y
冘
j
i
j
n
j
,
where c Xc
m
, y A
m
k
m
m
¥
i
(
m
n
i
/
i
), and x
i
(
i
/
m
)
[
i
k
m
m
i
z
i
i
i
1
( A
m
/A
i
)]
1
. The new
system implies n
1
, ... , n
m
can be solved simultaneously.
440 THE AMERICAN ECONOMIC REVIEW MARCH 2007
Intermediate Goods.—@i, F
i
A
i
n
i
k
i
q
i
,
,
僆 (0, 1),
1. We have
(A1) F
m
c
m
h
m
k k
˙
, F
i
c
i
h
i
, i m.
The planner’s problem is similar to the baseline with (A1) replacing (3) and (4), {h
i
, c
i
, q
i
}
i1, ... ,
m
as additional controls, and ¥
i1
m
n
i
q
i
(h
1
, ... , h
m
) as an additional constraint, where is
homogenous of degree one,
i
0, and
ii
0. The static efficiency conditions are
(A2) v
i
/v
m
F
K
m
/F
N
i
F
N
m
/F
N
i
F
Q
m
/F
Q
i
i
/
m
; i,
which implies k
i
k, q
i
, and p
i
A
m
/A
i
@i,so
y A
m
k
,
冘
i 1
m
i
h
i
冘
i 1
m
m
p
i
h
i
m
h; h ⬅
冘
i 1
m
p
i
h
i
.
Optimal conditions for h
m
and q
m
imply
m
A
m
k
1
1, so h
y and (A1) is
k
˙
A
m
k
冉
1
冘
im
n
i
冊
h
m
c
m
k h1
/
c
k.
The dynamic efficiency condition is v˙
m
/v
m
A
m
k
1
(
), so
(A3) c˙ /c
h/
k
, k
˙
/k 1
h/
k c/k
.
Constant c˙/c requires constant h/k, and constant k
˙
/k requires constant c/k. Thus, h
˙
/h must be constant.
To derive constant h
˙
/h, consider a CES (¥
i1
m
i
h
i
(
1)/
)
/(
1)
; then (A2) implies z
i
p
i
h
i
/h
m
(
i
/
m
)
( A
m
/A
i
)
1
, @i.So
h Zh
m
,
m
m
/
1
Z
1/
1
,
A
m
k
m
/
1
Z
1/
1
1/1
,
where Z ¥
i1
m
z
i
. Hence,
h /
m
A
m
k
1/1
m
/
1
Z
1/
1
/1
,
and so
1
h
˙
/h
m
k
˙
/k
冉
冘
i 1
m
(z
i
/Z)
i
m
冊
,
constant if ¥
i1
m
z
i
i
is constant. Given
i
differs across all i, constancy requires
1, so
写
i 1
m
h
i
i
, Z 1/
m
, z
i
i
/
m
i.
(A2) implies h
m
i1
m
(z
i
A
i
/A
m
)
i
and so
m
m
/h
m
i1
m
(
i
A
i
/A
m
)
i
. But
[
A
m
k
m
]
1/(1
)
,soh /
m
(
A
m
k
)
1/(1
)
m
/(1
)
. (A3) becomes
c˙ /c
Ak
/1
1
; k
˙
c
k 1
Ak
/1
,
441VOL. 97 NO. 1 NGAI AND PISSARIDES: STRUCTURAL CHANGE IN A MULTISECTOR MODEL OF GROWTH
where A [A
m
(
m
)
]
1/(1
)
. Define
c
e
⬅ cA
1
/1
; k
e
⬅ kA
1
/1
;
⬅ A
˙
/A.
We have
[
m
¥
i1
m
i
(
i
m
)]/(1
)
m
(
¥
i1
m
i
i
)/(1
), and
c˙
e
/c
e
k
e
1/1
g; k
˙
e
1
k
e
/1
c
e
gk
e
,
which imply the existence and uniqueness of an ABGP with growth rate
g ⬅ 1
/1
冉
m
冘
i 1
m
i
i
冊
冒
1
.
We obtain n
i
using F
i
c
i
h
i
, @i m, i.e.,
A
i
n
i
k
p
i
p
i
c
i
h
i
x
i
c
m
z
i
h
m
cx
i
/X
i
h.
Substitute p
i
and h to obtain n
i
y cx
i
/X
i
y, so (30) and (31) follow.
Many Capital-Producing Sectors.—@j, F
m
j
A
m
j
n
m
j
k
m
j
, which together produce good m through
G
冋
冘
j 1
m
j
(F
m
j
)
1/
册
/
1
,
m
j
0,
0,
冘
j 1
m
j
1.
The planner’s problem is similar to the baseline model with (4) replaced by
k
˙
G c
m
k
and (k
m
j
, n
m
j
)
j1, ... ,
as additional controls. The static efficiency conditions are
F
K
i
/F
N
i
F
K
m
j
/F
N
m
j
, i m, j,
so k
i
k
m
j
k. Also
G
m
j
/G
m
i
F
K
m
i
/F
K
m
j
A
m
i
/A
m
j
, i, j,
which implies n
m
j
/n
m
i
(
m
j
/
m
i
)
( A
m
i
/A
m
j
)
1
and grows at rate (1
)(
m
i
m
j
). Let n
m
¥
j1
n
m
j
; we have n
m
n
m
1
¥
j1
(
m
j
/
m
1
)
( A
m
1
/A
m
j
)
1
. Next,
p
i
v
i
/v
m
A
m
/A
i
, i m,
where A
m
G
m
1
A
m
1
. Thus, n
i
/n
j
and p
i
/p
j
are the same as in the baseline.
To derive the aggregate equilibrium, note that G ¥
j1
F
m
j
G
m
j
A
m
k
n
m
,soc˙/c and k
˙
/k are the
same as the baseline, so the equilibrium is the same as the baseline if
m
A
˙
m
/A
m
is constant, which
we now derive. Given
G
m
1
m
1
G/F
m
1
1/
; G/F
m
1
冋
冘
j 1
m
j
(A
m
j
n
m
j
/(A
m
1
n
m
1
))
1/
册
/
1
,
442 THE AMERICAN ECONOMIC REVIEW MARCH 2007
using the result on n
m
j
/n
m
1
we have G/F
m
1
[¥
j1
m
j
(
m
1
A
m
1
)
1
A
m
j
(
1)
]
/(
1)
, thus A
m
G
m
1
A
m
1
[¥
j1
m
j
A
m
j
(
1)
]
1/(
1)
,so
m
冘
j 1
j
m
j
,
j
⬅
m
j
A
m
j
1
冒
冉
冘
j 1
m
j
A
m
j
1
冊
,
constant if (
1) ¥
j1
j
(
m
j
m
)
2
0, i.e., if (1)
m
i
m
j
, @i, j,or(2)
1. If (1) is true,
the model reduces to only one capital-producing sector. Thus, coexistence of multiple capital-
producing sectors and an ABGP requires (2), i.e., G
j1
(F
m
j
)
j
and
m
¥
j1
m
j
m
j
.
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