Kyoto University,
Graduate School of Economics
Research Project Center Discussion Paper Series

Endogenous Phase Switch in Baumol’s Service Paradox Model

Hiroaki Sasaki

Discussion Paper No. E-10-010

Research Project Center
Graduate School of Economics
Kyoto University
Yoshida-Hommachi, Sakyo-ku
Kyoto City, 606-8501, Japan

December 2010

Endogenous Phase Switch in Baumol’s Service
Paradox Model
Hiroaki SASAKI
Graduate School of Economics, Kyoto University, Yoshida-honmachi, Sakyo-ku, Kyoto
606-8501, Japan; e-mail: sasaki@econ.kyoto-u.ac.jp

Abstract
This paper develops a two-sector model that considers Baumol’s service paradox. The
paper simultaneously incorporates two ideas about technological progress in the model:(1) the consumption of services contributes to human capital accumulation and (2) the
production of manufacturing leads to technological progress due to learning-by-doing.
Accordingly, productivity growth in both services and manufacturing is endogenously
determined. We show that initially, a shift in the employment share toward the services
sector decreases the per capita real GDP growth rate, but at some point in time, the shift
begins to increase the growth rate. Therefore, we observe an endogenous phase switchfrom a phase where the employment shift toward services depresses the economy toanother where the employment shift promotes the economy.
Keywords: service paradox; structural change; endogenous productivity growth
JEL classification: J21; J24; O11; O14; O30

1

1 Introduction
Baumol (1967) predicts that in developed countries, the services employment share tends toincrease. He explains this as follows. There are two sectors—a progressive sector (manufacturing) and a stagnant sector (services)—in the economy. Suppose that the productivity
growth in services is lower than that in manufacturing. Suppose also that the ratio of manufacturing output to services output is constant. Then, though the price of services relative
to that of manufacturing continues to rise, the demand for services keeps increasing because
the output ratio (i.e., consumption ratio) is constant. Since the productivity growth in services is lower than that in manufacturing, more employment in services is necessary to meetthe increasing demand for services, which results in a rise in the services employment share.
Given the productivity growth diﬀerential and the constant demand ratio, the tendency toward a services economy is inevitable.1) Indeed, we can easily see the employment shift
toward services in developed countries.
In the same paper, Baumol makes an important prediction: as the employment share
shifts toward services, the per capita real GDP growth rate will decline. He explains this asfollows. The per capita real GDP growth rate is given by a weighted average of the productivity growth in manufacturing and the productivity growth in services with the weight being
the corresponding employment share. Suppose that the employment share shifts toward services. Because the employment share of the services sector in which productivity growth
is lower increases, the per capita real GDP growth continues to decline and converges tothe productivity growth in services in the end. Hence, if the shift in the employment sharetoward services is inevitable, then the decline in the per capita real GDP growth is also
inevitable.
However, does the shift in employment share toward services necessarily lower the percapita real GDP growth? For this issue, there are some theoretical contributions.
Pugno (2006) considers that the consumption of services augments human capital à laLucas (1988). The consumption of health care and education services will lead to humancapital accumulation. Accordingly, the consumption of services increases the productivityof workers, thereby resulting in an increase in the productivity of both manufacturing andservices. This means that productivity growth is endogenized. He incorporated this humancapital accumulation eﬀect into Baumol’s model, and showed that if this eﬀect is relatively
strong, the employment shift toward services increases, and not decreases, the per capita
1) In Baumol’s (1967) model, labor is fully employed. In contrast, based on Pasinetti’s (1993) pure labor
economy model, Notarangelo (1999) presents a model with unemployment. She shows that given the productivity growth diﬀerential and price-non-elastic demand, employment shifts toward services as in Baumol(1967).

2real GDP growth.
De Vincenti (2007) reaches a conclusion similar to Pugno (2006). He assumes thatboth the growth rate of productivity in manufacturing and that in services are increasing
functions of the employment share of services, and then, incorporates these specifications
into Baumol’s model. He also showed that the employment shift toward services increases
the per capita real GDP growth. The basic idea of De Vincenti (2007) is similar to that of
Pugno (2006).2)
These two theoretical studies pay attention to the external eﬀects from the consumption and production of services and endogenize the productivity growth. If the productivity growth is exogenously given and the productivity growth in services is lower than thatin manufacturing, then the employment shift toward services necessarily decreases the percapita real GDP growth. However, if the productivity growth is endogenously determined,
then the employment shift toward services does not necessarily decrease the per capita real
GDP growth.
Unlike in these researches, there exist theoretical studies wherein services are used asintermediate inputs for manufacturing. Oulton (2001) shows that if services are used asintermediate inputs for manufacturing, the employment shift toward services raises the percapita real GDP growth. In Oulton’s model, services are only used as intermediate inputs,
and hence, not used as final demand. In contrast, Sasaki (2007) shows that if services areused as both intermediate inputs and final demand, the employment shift toward services
decreases the per capita real GDP growth in the long run.3) In these two studies, productivity
growth is exogenously given as in Baumol (1967).
The above studies pay attention to whether the employment shift toward services increases or decreases the per capita real GDP growth. In contrast, unlike in these studies,some theoretical studies show that the per capita real GDP growth is constant even if theemployment share shifts toward services. These studies attempt to make structural changes
compatible with Kaldor’s (1961) stylized facts. Kaldor asserts that in developed countries,
per capita real GDP growth is almost constant in the long run and there is no downward
tendency.
Kongsamut, Rebelo, and Xie (2001) build a three sector (agriculture, manufacturing, and2) However, under certain conditions, there occurs an inverted U-shaped relationship between the service
employment share and the per capita real GDP growth. That is, initially, the per capita real GDP increases withthe employment shift toward services, but at some point in time, it begins to decrease with the employment
shift. This will be explained in detail in Section 4.
3) If the elasticity of substitution between labor input and services input in manufacturing is suﬃciently larger
than unity, then it is possible that the per capita real GDP growth increases up until some point in time withthe employment shift toward services. However, even in that case, the per capita real GDP growth decreases
in the long run.

3

services) neoclassical growth model, and show that along the generalized balanced growthpath, the per capita real GDP growth is constant although the employment share of each sector continues to change.4) When deriving the result, they use a non-homothetic preference,
which yields an endogenous structural change. Iscan (2010) modifies Kongsamut et al.’s(2001) model and examines how the interaction between the productivity growth diﬀerential à la Baumol, and Engel’s law explains the long-run tendency of the services employment
share in the US. He concludes that two thirds of the movements of the services employment
share in reality can be explained by the two eﬀects. These models, in contrast to Baumol’s
model that considers only labor input, consider capital accumulation.5)
As explained above, there are three types of views on the relationship between the employment shift toward services and the per capita real GDP growth.6) A question then arises
as to which view is consistent with reality.
Hartwig (2010) empirically treats this problem. He conducted an empirical analysis
with regard to how the expenditure shift toward services such as health care and education aﬀects the per capita real GDP growth rate. He analyzes 18 OECD countries during
the period 1970–2005, and concludes that the expenditure shift lowers the per capita real
GDP growth rate, which is broadly consistent with Baumol’s (1967) statement.7) Nordhaus(2008) empirically analyzes the US to ascertain Baumol’s statement. As a result, he concludes that the employment shift toward services lowers the per capita real GDP growth. In
contrast to these studies, Maroto-Sánchez and Cuadrado-Roura (2009) empirically investigate 37 OECD countries during the period 1980–2005, and conclude that the employment
shift toward services has a positive eﬀect on the per capita real GDP growth.
As stated above, diﬀerent empirical studies have obtained contrasting results. There areseveral reasons why this is so. For example, the diﬀerence in the obtained results may be
due to the use of diﬀerent data for each empirical study. Moreover, there are many types ofservices, and accordingly, while some services have high productivity growth, others have
low productivity growth.
However, in this paper, we propose that there is a nonlinear relationship between the4) For the result where the per capita GDP growth becomes constant in spite of the structural changes, seealso Ngai and Pissarides (2007), Acemoglu and Guerrieri (2008), and Foellmi and Zweimüller (2008).
5) Bonatti and Felice (2008) investigate a model in which capital is accumulated in both sectors, manufacturing productivity increases endogenously, and consumer preferences are non-homothetic.
6) The above explanations are based on the models of closed economy. Spilimbergo (1998) builds an open
economy model, and shows that in a transition from a closed to an open economy, owing to a comparativeadvantage eﬀect, the speed of employment shift toward services is faster in an open economy than in a closed
economy, and consequently, the per capita real GDP growth declines much further in an open economy. Matsuyama (2009) presents a three-sector model, and shows that in a closed economy, the productivity growth
diﬀerential results in structural changes, and that this is not necessarily the case in an open economy.
7) Hartwig (2010) shows that if Japan is excluded from the data sample, there is a statistically significant andlong-lasting negative impact of this expenditure shift on economic growth.

4

employment shift toward services and the per capita real GDP growth. This implies that
because the relationship between the employment shift toward services and the per capita
real GDP growth diﬀers with country, a cross-country analysis treating diﬀerent countries
together will yield an ambiguous result. To show this, we present an extended Baumol’s
model that produces the conclusions of Baumol (1967), Pugno (2006), and De Vincenti(2007).
This paper incorporates both Pugno’s (2006) idea that the consumption of services leads
to the accumulation of human capital and Arrow’s (1962) idea that the production of manufacturing leads to the productivity-increasing learning-by-doing eﬀect. Accordingly, productivity growth in both sectors is endogenized. We show that initially, a shift in the employment share toward the services sector decreases the per capita real GDP growth rate but atsome point in time, it begins to increase the per capita real GDP growth rate. Consequently,
we obtain a U-shaped relationship between the employment share of services and the percapita real GDP growth.
This result means that whether the employment shift toward services increases or decreases the per capita real GDP growth depends on whether the economy is in a downward
phase or an upward phase. Therefore, in our model, there occurs an endogenous phaseswitch from a phase where the employment shift toward services depresses the economy toanother where the employment shift promotes the economy. Suppose that some countries
lie in the downward phase while others lie in the upward phase. Then, an empirical analysis that treats these countries together is likely to yield an ambiguous result with regard to
the relationship between the employment shift toward services and the per capita real GDP
growth.
The remainder of this paper is organized as follows. Section 2 presents our model andinvestigates the mechanism that shifts the services employment, and the relationship between the employment shift and the per capita real GDP growth. Section 3 fully analyzes
the dynamics of the model using numerical simulations. Section 4 compares our results withthose obtained in previous studies. Section 5 concludes the paper.

2 Model
2.1 Basic components
First, we specify the firms’ behavior. Consider an economy with two sectors: manufacturing(m) and services (s). Both sectors produce outputs using only labor inputs. We specify the5

production function of each sector as follows:
Qm = Am Lm ,(1)

Qs = As Ls,(2)

where Qi denotes the output (i = m, s); Ai , the labor productivity; and Li , the employment.
We assume that labor is fully employed. From this, we have Lm + L s = L, where L denotes
the population of the labor force.
Suppose that labor is perfectly free to move between the two sectors. Then, the nominal
wages in both sectors are equalized. We denote the wage as w. From the profit-maximizing
and zero-profit conditions, we obtain the following equations:w,
Amw
ps = .
As

pm =

(3)
(4)

Consequently, the prices are equal to unit labor costs.
Next, we specify the consumers’ behavior. We assume that a representative consumer
solves the following optimization problem:[ 1 σ−1]σ
1σ−1 σ−1σσσσ
max u = α cm + (1 − α) (c s + γ),

(5)s.t. pm cm + p s c s = w,

(6)cm , c s

where ci denotes the per capita consumption (ci = Ci /L); σ, the elasticity of substitution
between the two types of consumption; α, a parameter governing the weight of expenditure
for manufacturing; and γ, a positive parameter governing home production. Such a nonhomothetic specification is also adopted in Iscan (2010).8) When γ = 0, the preference ishomothetic, and hence, the income elasticities of the manufacturing consumption and theservices consumption are unity. When γ > 0, the preference is non-homothetic, and hence,
the income elasticity of the manufacturing demand is less than unity and that of services
demand is greater than unity. Introducing a positive γ does not aﬀect the dynamics of theemployment share very much. However, it does aﬀect the dynamics of the output ratio.
Baumol (1967), when drawing the conclusion that the employment shift toward services
decreases the per capita GDP growth, assumes that the output ratio remains constant over8) The CES utility functions with γ = 0 are adopted in Quibria and Harrigan (1996), Spilimbergo (1998),
and De Vincenti (2007).

6

time.9) The assumption of γ > 0 corresponds to Baumol’s assumption.
By solving the maximization problem comprising equations (5) and (6), we obtain thefollowing demand functions for manufacturing and services:
( )σ


α
ps


( )σ  w −
pm
γ

1−α
pm
p s 
α

Cm =

( )σ + γ L,
1 − α pm αps ps +

pm
1−α
pm
( )σ
α
ps
w−
pm
γ
1−α
pm
Cs =
( )σ · L.
α
ps
ps +
pm
1−α
pm

(7)

(8)

The market clearing conditions for manufacturing and services are given by
Qm = C m ,(9)

Qs = C s.

(10)

Substituting equations (1) and (7) in equation (9), and substituting equations (2) and (8)in equation (10), we obtain the each sector’s employment share:γ
LmAs= αAσ−1
,
m ·
L
αAσ−1
+
(1
− α)Aσ−1
m
s
γ
1+
γ
Ls
As
= (1 − α)Aσ−1
·
.
−
s
σ−1
L
αAσ−1
As
m + (1 − α)A s
1+

(11)

(12)

Here, let us assume that the growth rate of the labor productivity of each sector is exogenously given by ri and that rm > r s as in Baumol (1967):
gAm ≡ rm > r s ≡ gAs .

(13)

Hereafter, we denote the growth rate of a variable x as g x , where g x = ẋ/x = (dx/dt)/x.
Then, using equations (3) and (4), we find that the relative price of services, p s /pm , increases
indefinitely (Baumol’s cost disease). Moreover, by examining equations (11) and (12), we
obtain the following proposition:
Proposition 1. Suppose that the growth rate of labor productivity in services is lower than
9) For the empirical plausibility of Baumol’s assumption, see Sasaki (2007) and the literature cited therein.

7

that in manufacturing. If the elasticity of substitution between the consumption of manufacturing and the consumption of services is less than unity, then the employment share ofservices L s /L will increase in the long run.
Proof. See appendix A.

Proposition 1 is the same as that in Baumol (1967). As mentioned above, Baumol assumes that the output ratio is constant. In contrast, we use a CES and non-homothetic
preference.
The output of each sector is given by


γ


1
+

 σ
A
s
 L,
Qm = αAm ·
σ−1 

αAσ−1

m + (1 − α)A s


γ


1
+


A
s
 L.
Q s = (1 − α)Aσs ·
−
γ
σ−1

αAσ−1
+
(1
−
α)A

m
s

(14)

(15)

From these equations, we obtain the output ratio (i.e., the consumption ratio):γAs(1 − α)Aσs ·−γ
σ−1
αAm + (1 − α)Aσ−1
Qs
s
=
.
γ
Qm
1+
As
αAσm ·σ−1
αAm + (1 − α)Aσ−1
s
1+

(16)

Examining equation (16), we obtain the following proposition:
Proposition 2. Suppose that the growth rate of labor productivity in services is lower thanthat in manufacturing. Suppose also that the elasticity of substitution between the consumption of manufacturing and the consumption of services is less than unity. If γ = 0, the output
ratio Q s /Qm will decline in the long run. The output ratio declines in the long run even if
γ > 0, but at a moderate rate.
Proof. See appendix B.

Proposition 2 states the long-run tendency of the output ratio. As mentioned earlier,Baumol (1967) assumes that the output ratio is constant, that is, Qm /Q s = constant. On
the other hand, we specify the utility function and assume that σ < 1 and γ > 0. Theseassumptions approximately correspond to Baumol’s assumption.
8

2.2 Human capital accumulation and learning-by-doing
In the preceding section, the growth rate of labor productivity for each sector is exogenously
given. Here, in contrast, we endogenize the productivity growth of both sectors.
We assume that each sector’s labor productivity is given by
Am = T m · h,(17)

A s = h,(18)

where T m denotes the measure of productivity specific to manufacturing and h denotes thelevel of human capital. Human capital h aﬀects both sectors’ productivities because humancapital is accumulated into workers who are free to move between the two sectors. Fromequations (17) and (18), we obtain the production functions Qm = T m (hLm ) and Q s = hL s .
Next, based on the idea of Pugno (2006), we assume that human capital is accumulated
through the consumption of services:ḣ = δc s = δ

hL s
,
L

δ > 0,(19)

where δ denotes a parameter governing the eﬃciency of human capital accumulation. In
this specification, human capital accumulation is a linear function of accumulated humancapital, which produces sustainable per capita real GDP growth.10) Pugno (2006) considers
two diﬀerent cases: in one case, the consumption of services unintentionally leads to humancapital accumulation whereas, in the other, agents intentionally consume services to accumulate human capital. In this paper, we adopt the unintentional human capital accumulationcase for ease of analysis.
Finally, we assume that the manufacturing-specific productivity T m is an increasing function of knowledge stock Km :T m = Kmϕ ,ϕ > 0,(20)

where ϕ denotes the elasticity of T m with respect to Km . Following the idea of Arrow (1962),
we assume that the knowledge stock depends on the production experience that is accumulated until now; we specify the knowledge stock as follows:[∫ t
Km = exp]
Lm (τ)dτ .
−∞ L(τ)(21)10) However, the eﬀect of human capital accumulation might weaken as the consumption of services increases. For this case, see appendix C.

9Note that the production experience Km is measured by Lm /L. We use the manufacturing
employment share, and not the level of manufacturing employment, to ascertain that the
dynamics of the model holds even when the labor force grows. In addition, we use manufacturing employment, and not manufacturing output, for the following two reasons.11)
First, in our model, labor is the sole factor of production, and then, an increase in output
has a one-to-one relationship with an increase in employment. Therefore, for simplicity, wemeasure production experience by the employment of manufacturing, not by the output of
manufacturing. Second, with our specification, the productivity growth in each sector leads
to a function that depends only on employment share. For this reason, we can easily compare our model with Pugno’s (2006) and De Vincenti’s (2007) models, in which productivity
growth is a function that depends only on employment share.
From equations (20) and (21), the flow of T m leads to( L )m
Ṫ m = ϕ
Tm.
L

(22)

Diﬀerentiating equations (17) and (18) with respect to time and substituting equations(19) and (22) in the resultant expressions, we obtain the growth rate of labor productivity foreach sector as follows:
(
Lm )
Ls
gA s = δ 1 −
=δ ,
(23)
L
L
Lm
Ls
gAm = δ + (ϕ − δ)
= ϕ − (ϕ − δ) .
(24)
L
L
From these equations, we always have gAm ≥ gAs .12) Therefore, we obtain the result that
corresponds to Baumol’s (1967) assumption, rm > r s .
The productivity growth of services is an increasing function of the employment shareof services. We now turn to the productivity growth of manufacturing. Let us pay attention
to the values of δ and ϕ. If ϕ > δ, gAm is a decreasing function of L s /L. If, in contrast, ϕ < δ,
gAm is an increasing function of L s /L. Recall that δ denotes the eﬃciency of human capital accumulation with respect to services consumption and ϕ denotes the elasticity of the
manufacturing-specific productivity with respect to manufacturing production. Therefore,
which eﬀect dominates determines whether the growth rate of manufacturing labor productivity becomes a decreasing or an increasing function of the employment share of services.
Figures 1 and 2 show the relationships between gAi and L s /L.
In Pugno (2006) and De Vincenti (2007), the growth rates of productivity for both sec11) For the case where production experience is measured by the output of manufacturing, see appendix D.
12) If ϕ = δ, the growth rate of the manufacturing labor productivity is δ, which is constant.

10

tors are increasing functions of the employment share of services. In our model, in contrast,
the manufacturing productivity growth is either increasing or decreasing in the services employment share. Nevertheless, as we will explain later, this does not aﬀect our main results.
What is important is that the productivity growth diﬀerential gAm − gAs decreases with theshift in the employment share toward services: gAm − gAs = ϕ[1 − (L s /L)]. This finding
is consistent with the empirical data shown in Bosworth and Triplett (2007), who calculate
the labor productivity growth and the total factor productivity growth for manufacturing andservices in the US. They show that the productivity growth diﬀerentials between the two
sectors decrease with time.13)[Figures 1 and 2 to be inserted here]

2.3 Per capita real GDP growthNow, we derive the per capita real GDP growth rate. To consider economic growth in realterms, we must eliminate the eﬀect of price variations. The per capita real GDP growth rate
g can be defined as follows:g(t) =pm (t)Q̇m (t) + p s (t)Q̇ s (t)Lm (t)
L s (t)= gAm (t)+ gAs (t).
pm (t)Qm (t) + p s (t)Q s (t)
L(t)
L(t)

(25)

This implies that g is the weighted average of the growth rate of labor productivity for eachsector with the weight being the corresponding employment share.
In our model, g is calculated as follows:LmLsg = (gTm + gh )
+ gh
L
L
(
)2
(
)
L s 2ϕ − δ
4ϕ − δ
=ϕ
−
+δ
.
L
2ϕ
4ϕ

(26)
(27)

Hence, g is a quadratic function with respect to the employment share of services. Examining equation (27), we obtain the following proposition:
Proposition 3. If 2ϕ < δ, then the per capita real GDP growth increases with the employment shift toward services. If 2ϕ > δ, then the per capita real GDP growth decreases until
the services employment share reaches (L s /L)∗ = 2ϕ−δ, and from then onward, it increases
2ϕwith the employment shift toward services, finally converging to g = δ.
13) They estimated that the productivity growth diﬀerentials in terms of the labor productivity are 1.3 (1987–
1995), 0.7 (1995–2000), and 0.5 (2000–2005), and those in terms of the total factor productivity are 1.3(1987–1995), 1 (1995–2000), and 0.4 (2000–2005).

11

Proposition 3 states that there exists a U-shaped relationship between the employment
share of services and the per capita real GDP growth. Suppose that the initial value of theservices employment share is suﬃciently small. Then, as the services employment share increases, the per capita real GDP growth decreases. However, when the services employment
share exceeds the threshold value given by (L s /L)∗ , the per capita real GDP growth increases
and finally converges to g = δ. Whether the relationship between L s /L and g leads to a Ushaped curve depends on the relative size of the eﬀect of human capital accumulation due
to the consumption of services. If this eﬀect is relatively strong, the shift in the employment
share of services continues to increase the per capita real GDP growth monotonically. On
the other hand, if the eﬀect is relatively weak, we have a U-shaped relationship. Figures 3–5
show the relationship between the services employment share and the per capita real GDP
growth for both cases.
[Figures 3, 4, and 5 to be inserted here]
In our model, the growth rate of labor productivity in manufacturing is always larger
than that in services, which is the same as in Baumol (1967). Nevertheless, there exists aphase in which the shift in the employment share toward services increases the per capita
real GDP growth, which is interesting.
As explained above, whether ϕ is larger than δ or vice versa determines whether gAm is anincreasing or a decreasing function of L s /L. If ϕ > δ, the manufacturing productivity growth
is a decreasing function of the services employment share. In this case, we necessarily have
2ϕ > δ, and hence we obtain a U-shaped relationship between L s /L and g in Proposition 3(see Figure 5). If ϕ < δ, the manufacturing productivity growth is an increasing function of
the services employment share. Even then, we can have 2ϕ > δ, which produces a U-shaped
relationship between L s /L and g (see Figure 4). If δ → 0, g becomes a decreasing functionof L s /L, which produces the same result as in Baumol (1967) that a shift in the employment
share toward services decreases the per capita real GDP growth.
Why do we obtain such a U-shaped relationship? Rewriting equation (25), we obtain thefollowing relations:Lsg = gAm − (gAm − gAs )| {z } L+(
Ls ) Ls= g Am − ϕ 1 −.
| {z L} L

(28)
(29)

gAm −gA s

Ceteris paribus, as long as gAm > gAs , a rise in L s /L has a negative eﬀect on the per capita
real GDP growth (see equation (28)). Note that in our model, the productivity growth dif12

ferential gAm − gAs is a decreasing function of the services employment share. Hence, as theemployment share shifts toward services, the negative eﬀect of the increase in L s /L on the
per capita real GDP growth diminishes. As stated above, the first term in the right-hand side(RHS), gAm , can be an increasing or decreasing function of the services employment share.
However, in either case, we always have gAm > 0. Therefore, when the services employment
share is small, the negative eﬀect of the second term in the RHS is relatively large, and hencea shift in the employment share toward services decreases the per capita real GDP growth.
On the other hand, when the services employment share is large, the positive eﬀect of thefirst term in the RHS is relatively large, and hence a shift in the employment share toward
services increases the per capita real GDP growth. Using the same reasoning, we can explain why the speed of switch from a downward phase to an upward phase is faster whenϕ < δ than when ϕ > δ (see Figures 4 and 5).

3 Full dynamics and numerical examples
From the above analysis, we obtained three propositions. However, the analysis is not complete. When deriving Propositions 1 and 2, we assume that the growth rates of labor productivity for both sectors are exogenously given. When deriving Proposition 3, we assume
that the employment share of services increases with time. In our model, the productivity
growth and the employment share are dependent on each other. Therefore, we must analyze
the dynamics of the employment share and productivity growth simultaneously.
The dynamics are summarized as follows:
Ls
Ȧ s = δA s ,[L
Ls ]Ȧm = Am ϕ − (ϕ − δ)
,
L

(30)
(31)

γ
Ls
γ
As
= (1 − α)Aσ−1
·
− .
s
σ−1
σ−1
L
αAm + (1 − α)A s
As
1+

(32)

Substituting equation (32) in equations (30) and (31), we obtain a system of two diﬀerential
equations with respect to A s (t) and Am (t).
Because the above system is nonlinear and somewhat complicated, we use numericalsimulations to analyze the dynamics. By examining the dynamics of Am and A s , we canknow the behaviors of the other endogenous variables. Here, we consider only the case ofσ < 1 in which the employment share of services increases with time. We set the parameters
as in Table 1. In Case 1, we have ϕ > δ, and in Case 2, we have both ϕ < δ and 2ϕ > δ.
13

Moreover, we set the initial values of both Am and A s to unity.
[Table 1 to be inserted here]
Figures 6–11 show the results of the numerical simulations.14) These figures give thetime path of each variable when the elasticity of substitution σ varies from 0.1 to 0.9 inintervals of 0.1.
In Cases 1 and 2, the services employment share increases with time, and the per capita
real GDP growth decreases initially and then increases.15) However, there is a marked difference between Figures 7 and 9. The switching time from a downward phase to an upward
phase is shorter in Figure 9 than in Figure 7. This is because the threshold values (L s /L)∗are diﬀerent between Cases 1 and 2—(L s /L)∗ = 0.7 in Case 1 and (L s /L)∗ = 0.375 in Case2—which are shown in Figures 6 and 8 as horizontal real lines. As can be seen from Figures
6 and 8, the time path of the services employment share does not diﬀer much. However, thespeed with which the per capita real GDP growth switches from a downward phase to an
upward phase is faster in Case 2 than in Case 1 because (L s /L)∗ is smaller in Case 2 than inCase 1.
The size of the elasticity of substitution substantially aﬀects the time path of the output
ratio Q s /Qm . As mentioned in Proposition 2, the output ratio declines in the long run. However, the speed of the decline depends crucially on the size of the elasticity of substitution.
As Figures 10 and 11 show, in both Cases 1 and 2, the smaller the elasticity of substitution,
the more moderate will be the speed of the decline in the output ratio.
[Figures 6, 7, 8, and 9 to be inserted here]

4 Comparison and discussion
In this section, we compare our results with those obtained in previous studies. In the comparison, we focus our attention on the mechanism through which various results are generated. Note that the notations of parameters in other models are adjusted as per our model asmuch as possible.

14) For the numerical simulations, we use Mathematica 7.
15) In this paper, we focus our attention on the per capita real GDP growth. On the other hand, we can also
consider the growth rate of the level of utility u. From numerical simulations, we find that in Case 1, the growthrate of u shows a time path similar to the growth rate of g. In contrast, in Case 2, it is possible that the growthrate of u continues to increase with time: the growth rate of u is not U-shaped. In either case, the time paths ofthe growth rates of g and u coincide if we compute for a suﬃciently long time.

14

In Baumol (1967), the per capita real GDP growth rate is given by
gBaumol = rm

Lm
Ls
+ rs
L
L

= rm − (rm − r s )

(33)
Ls
.
L

(34)

If rm > r s and L s /L increases, then gBaumol decreases with time and finally converges to
gBaumol = r s .
In Pugno (2006), the production function of manufacturing is given by Qm = (hLm )erm t .16)
That is, we have T m = erm t , where rm is exogenously given. The production function ofservices is the same as that in our model. The growth rates of labor productivity are given
by gAm = rm + δL s /L and gAs = δL s /L. The per capita real GDP growth is given by(L s ) Lm ( L s ) L s
gPugno = rm + δ
+ δ
L L
L L
Ls
= rm + (δ − rm ) .
L

(35)
(36)

Note that in Pugno’s (2006) model, we always have gAm > gAs . If rm < δ, that is, the exogenous growth rate of labor productivity for manufacturing exceeds the eﬃciency of humancapital accumulation, then gPugno continues to decline with the shift in the employment sharetoward services. This is the same result as that in Baumol (1967). However, if rm < δ, thatis, the eﬃciency of human capital accumulation exceeds the exogenous growth rate of laborproductivity for manufacturing, then gPugno continues to rise with the shift in the employment
share toward services. In either case, we obtain a linear relationship between L s /L and g.
Moreover, the per capita real GDP growth converges to gPugno = δ.
In De Vincenti (2007), the growth rates of labor productivity for both sectors are given
by gAm = α(L s /L)β and gAs = δ(L s /L).17) Here, α, β, and δ are all positive constants. The percapita real GDP growth is given by( L )β L( L )Lss
gDV = α+ δLL
L L
( L )2Lss+αwhen β = 1.
= (δ − α)
L
L
s

m

(37)
(38)

De Vincenti (2007) makes an assumption that corresponds to 0 < β < 1. However, thisassumption is not essential, and as such, we assume that β = 1 for ease of exposition. In16) In fact, the production function is given by Qm = b(hLm )erm t , that is, a constant term b is added. However,this term is not essential, and hence, we use b = 1 for simplicity.
17) The productivity growth in De Vincenti (2007) is written in a general form, but for ease of exposition, it
is written in a specific form in our paper.

15

addition, he makes an assumption that corresponds to α > δ, which we also adopt.18)
If α > δ and α < 2δ, gDV continues to increase with L s /L. If α > 2δ, gDV increases
with L s /L initially, but once L s /L reaches α/[2(α − δ)], gDV begins to decline with L s /L.
Therefore, we obtain an inverted U-shaped relationship between L s /L and g, which is shownin Figure 12. In either case, the per capita real GDP growth converges to gDV = δ.
[Figure 12 to be inserted here]
The diﬀerence between the results in our model and De Vincenti’s (2007) model lies inthe diﬀerences in the specification of the productivity growth for manufacturing because thespecification of the productivity growth for services in both models is identical.19) In his
model, when α > δ, the productivity growth diﬀerential gAm − gAs becomes an increasing
function of the employment share of services (recall equation (28)). This means that thelarger the employment share of services, the stronger will be the negative eﬀect of the risein L s /L on the per capita real GDP growth.20) Therefore, it is possible that the shift in theemployment share toward services decreases the per capita real GDP growth in the long run.

5 Concluding Remarks
In this paper, we built a model in which productivity grows endogenously for both manufacturing and services, and investigated the relationship between the shift in the employment
share toward services and the per capita real GDP growth. We adopted two ideas simultaneously: (1) the consumption of services contributes to human capital accumulation and (2)the learning-by-doing eﬀect is present in manufacturing.
Our analysis shows that initially, the employment shift toward services decreases the percapita real GDP growth rate, but from some point in time onward, it gradually increases the18) If, in contrast, we assume that α < δ, it is possible that the growth rate of labor productivity for services
exceeds that for manufacturing, which seems to contradict the observation in the real world. Hence, we assume
that α > δ.
19) In fact, by modifying De Vincenti’s model a little, we obtain a U-shaped relationship between L s /L andg. We add a constant term to the productivity growth in manufacturing: gAm = α(L s /L)β + ε, where ε is apositive constant. If α < δ, α < ε, and α + ε > δ, then gDV is a convex downward parabola with a positiveaxis. Accordingly, we obtain a U-shaped relationship. However, this specification seems rather arbitrary. In
our model, we use a more reasonable specification to obtain a U-shaped relationship between L s /L and g.
20) By modifying our model a little, we obtain an inverted U-shaped relationship between L s /L and g similar
to what De Vincenti has obtained. Suppose that the manufacturing productivity is given by Am = T m h1+β ,where β is a positive constant. Then, the production function of manufacturing yields Qm = T m hβ (hLm ). This
specification implies that human capital accumulation exerts a positive externality in manufacturing and that β
captures the extent of this externality. If β < 1, we obtain the same results as those in the text. In contrast, if
β > 1, we obtain an inverted U-shaped relationship between the services employment share and the per capita
real GDP growth depending on conditions. Nevertheless, the case in which the externality is larger than unity
seems rather extreme.

16

growth rate. This result is consistent with both Baumol’s (1967) view that the employment
shift toward services decreases the per capita real GDP growth and Pugno’s (2006) viewthat the employment shift toward services increases the per capita real GDP growth. Accordingly, our model can explain the two views simultaneously. As the employment shareof services increases, the learning-by-doing eﬀect in manufacturing decreases. On the other
hand, as the employment share of services increases, the human capital accumulation eﬀect
due to the consumption of services increases. The interaction of these two opposing eﬀects
produces a U-shaped relationship between the employment share of services and the percapita real GDP growth rate.
This result suggests that when conducting a cross-country analysis to ascertain the relationship between the employment shift toward services and the per capita real GDP growth,
we must pay considerable attention as the relationship diﬀers depending on whether a country is in a downward phase or an upward phase. Consequently, if we treat all countries
together, we will obtain an ambiguous relationship between the employment shift toward
services and the per capita real GDP growth.
Finally, if the services employment share continues to increase, then the per capita real
GDP growth converges to the services productivity growth in the long run. Therefore, topromote economic growth, we must raise the services productivity, which is a characteristic
common to all models that appear after Baumol’s (1967) insightful contribution.

17

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18

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19

Appendix
A Proof of proposition 1
Let us rewrite equation (11) as follows:
)
γ
α 1+
As
Lm
=
( )1−σ .
LAm
α + (1 − α)
As
(

(A-1)

First, the numerator decreases as A s increases. Second, Am becomes larger than A s with time
when rm > r s , and then, the denominator increases with time because 1 − σ > 0. For thesereasons, Lm /L decreases with time, which implies that L s /L increases with time.

B Proof of proposition 2
Let us denote the terms that simultaneously appear in both the denominator and the numerator in equation (16) as Θ. Then, we can rewrite the output ratio as follows:(1 − α)ΘAσs − γ
Qs
=
,
Qm
αΘAσm

γ
As
where Θ ≡.
σ−1αAm + (1 − α)Aσ−1
s
1+

(B-2)

Given that rm > r s , Am becomes larger than A s with time. Θ, though not constant, appears
simultaneously in both the denominator and the numerator, and thus has no eﬀect on thelong-run trend of the output ratio. Accordingly, the denominator becomes larger than the
numerator in the long run, which implies that Q s /Qm declines in the long run.
Next, we divide both the denominator and the numerator with Aσs :γ(1 − α)Θ − σ
AQs=( )σ s
Qm
Am
αΘ
As

(B-3)

When γ > 0, that is, the preference is non-homothetic, γ/Aσs in the numerator decreases with
A s . This increases the numerator. When γ = 0, this eﬀect vanishes, and the denominatorincreases with time. As such, when γ > 0, the decreasing eﬀect on Q s /Qm is weaker ascompared to when γ = 0, which implies that the long-run decline in Q s /Qm is more moderate

20when γ = 0.

C Alternative specification for human capital accumulation
In the text, based on Pugno (2006), we assume that human capital accumulation is a linear
function of the consumption of services. However, it is possible that the eﬀect of humancapital accumulation weakens as the consumption of services increases. In this case, we canuse the following specification:ḣ = δcψs ,0 < ψ ≤ 1.

(C-4)

Pugno (2006) corresponds to the case of ψ = 1 in this specification.
The productivity of each sector and the per capita GDP growth are given by( A L )ψ
s s

,
[L(
]
( L )ψ
Ls )s
ψ−1Ȧm = Am ϕ 1 −
+δ
As ,
L
L
[
]
( L )ψ
Lm (Ls )L s ( L s )ψ ψ−1
s
ψ−1
g=
ϕ 1−
+δ
As
+
δ
As .
L
L
L
L
L
Ȧ s = δ

(C-5)
(C-6)
(C-7)

Using numerical simulations, we find that the per capita GDP growth decreases with theemployment share of services.

D Alternative specification for learning by doing
In equation (21) of the text, production experience is measured by the employment of manufacturing, not by the output of manufacturing. Let us see what happens if we measureproduction experience by the output of manufacturing, not the employment of manufacturing. If we use Qm /L instead of Lm /L, we have[∫ t
Km = exp]
Qm (τ)dτ .
−∞ L(τ)21(D-8)

Using this equation, we obtain the growth rate of productivity of manufacturing as follows:
gAm = ϕAm − (ϕ − δ)

Ls
.
L

(D-9)

In this case, analytical treatment becomes much complicated. By conducting numericalsimulations, we find that in both Cases 1 and 2, the employment share of services increases
with time. Moreover, we find that in both Cases 1 and 2, the growth rate of per capita real
GDP increases at first, decreases from some point in time on, and then converges to g = δ.
That is, we obtain an inverted U-shaped relationship, not a U-shaped relationship, between
the employment share of services and the growth rate of per capita real GDP.
The reason why we obtain the above result is that the productivity growth diﬀerential
gAm − gAs that appears in equation (28) becomes an increasing function, not a decreasing
function, of L s /L. This implies that the productivity growth diﬀerential increases with time.
However, this is not consistent with the empirical finding of Bosworth and Triplett (2007).

22

Figures and tableg Am , g A s

g Am , g A s
φ

δ

g Am

g Am

δ
φ

gAs

gAs

O

1

O

Ls
L

1

Ls
L

Figure 1: Relationship between gAi and L s /L Figure 2: Relationship between gAi and L s /Lwhen ϕ < δwhen ϕ > δ

23

g

δ

φ
O

1

L s /L

Figure 3: Relationship between g and L s /L when 2ϕ < δ
g

g

δ
φ

δ

φ

O

2φ−δ
2φ

1

O

L s /L2φ−δ
2φ

1

L s /L

Figure 4: Relationship between g and L s /L Figure 5: Relationship between g and L s /Lwhen 2ϕ > δ and ϕ < δwhen ϕ > δ

24

Table 1: List of parametersαγϕδCase 1 0.5 0.5 0.05 0.03
Case 2 0.5 0.5 0.04 0.05
LsL

g

1.0

Σ = 0.1

0.9

0.034

0.8
0.032

0.7
Σ = 0.9

0.6

0.030

Σ = 0.1

0.5
0.028

0.4
200

400

600

800

1000

Σ = 0.9

t

200

400

600

800

1000

t

Figure 6: Dynamics of the services employ- Figure 7: Dynamics of the real GDP per
capita growth (σ < 1 and ϕ > δ)
ment share (σ < 1 and ϕ > δ)g

LsL
Σ = 0.1

0.9

0.048

Σ = 0.1

0.046

0.8

0.044

0.7

Σ = 0.9

0.042

0.6

0.040

0.5

Σ = 0.9

0.038

0.4
200

400

600

800

1000

200

t

400

600

800

1000

t

0.034

Figure 8: Dynamics of the services employ- Figure 9: Dynamics of the real GDP per
capita growth (σ < 1, 2ϕ > δ, and ϕ < δ)
ment share (σ < 1, 2ϕ > δ, and ϕ < δ)
QsQm

QsQm

0.8

0.8

Σ = 0.1

Σ = 0.1

0.6

0.6

200

400

600

800

1000

t
200

0.2

0.0

400

600

800

1000

t

0.2

Σ = 0.9

0.0

Σ = 0.9

Figure 10: Dynamics of the output ratio Figure 11: Dynamics of the output ratio(σ < 1 and ϕ > δ)(σ < 1, 2ϕ > δ, and ϕ < δ))
25

gDV

δ

O

α
2(α−δ)

1

L s /L

Figure 12: Relationship between g and L s /L in De Vincenti (2007)26