DP2015-36

Inter-Industrial Factor Alloca tionin the Short and Long Runs
in C ha mberlinian Monopolistic
Competition
Hiroshi GOTO
September 16, 2015

Inter-Industrial Factor Allocation in the Short and Long Runs in
Chamberlinian Monopolistic Competition
Hiroshi Goto∗
Research Institute for Economics & Business Administration, Kobe University, 2-1, Rokkodai-cho,Nada-ku, Kobe, Hyogo, 657-8501, Japan

Abstract
This study reviews the allocation of factors among industries distinguishing between the
short run and long run in a monopolistically competitive world. Without any conditions
on scale economy, non-homotheticity, and factor intensity, it is shown that the long-run
equilibrium characterized by factor-price equalization between industries is always locallystable. In addition, the direction of non-homothetic bias does not have qualitative eﬀects
on the allocation of factors, while it works as key elements to determine the direction of
the change in relative industry size. The direction of the change in factor allocation mainly
depends on the gross substitutability of factors which is magniﬁed by non-homotheticity andscale economies.
JEL classiﬁcation: F11, L13, L16
Keywords: Non-Homotheticity, Scale Economy, Factor Intensity, Elasticity of Substitution∗

TEL: +81-78-803-7270, FAX: +81-78-803-7059, E-mail: hgoto@rieb.kobe-u.ac.jp

11. Introduction
In the past three decades, the Dixit and Stiglitz (1977) type of monopolistic competition
model has been used widely in economic analysis. This enables us to investigate easily theissue of ﬁrm-level scale economies in a general equilibrium framework. In particular, therich implications for industrial structure between countries or regions have been studied indetail in the new trade theory and new economic geography.1 However, the standard models
for this have tended to assume that production factors are instantaneously mobile between
industries, and to ignore the fact that they may be speciﬁc to particular use, at least in the
short run.
There are a number of studies that have found substantial inter-industrial wage gaps
remain even after controlling for individual and job characteristics, human-capital variables,and geographic location (e.g., Dickens and Katz, 1987; Krueger and Summers, 1988; Edin
and Zetterberg, 1992). These ﬁndings suggest that in the short run, factors may be speciﬁc
to particular use and the adjustment is only gradual, even if factors move between industries
to cancel out diﬀerences in factor reward in the long run.2 This distinction between the shortand long runs has not been the main focus in the standard models used in the new trade
theory and new economic geography.
On the other hand, as early as 1936, Haberler (1936) discussed international trade distinguishing between the short and long runs. Thereafter, in the 1970s, traditional trade theory
studied how such short-run factor speciﬁcity changes the eﬀects of varying good prices andfactor endowments on factor rewards and how these are adjusted to the long-run equilibrium(e.g., Mayer, 1974; Mussa, 1974; Amano, 1977; Neary, 1978a, 1978b).3 However, beginning
1
2

See, for example, Helpman and Krugman (1985), Fujita et al. (1999), and Combes et al. (2008).
There are a few empirical studies that directly test the adjustment speed of inter-industrial factor

allocation. Morrison and Berndt (1981) show quasi-ﬁxity of inputs using annual US manufacturing datafrom 1952 to 1971. Helliwell and Chung (1986) estimate the speed of adjustment of the capital-energy ratio
and show that the factor ratio changes by only 18 percent per year. In addition, Sneessens and Drèze (1986)
ﬁnd the capital–labor ratio adjusts by only 27 percent per year.
3
These studies assume perfect competition and constant returns technology for characterizing industries.
However, there are some considerable doubts about the appropriateness of perfect competition and constant returns technology for industrial characteristics, because a not-negligible number of studies show that2

with Krugman (1979), during the 1980s, the short-run factor speciﬁcity and adjustment process of factor allocation among industries began to receive less attention in the new trade
theory, which widely uses the Dixit–Stiglitz type of monopolistic competition.4 In the 1990s,
the new economic geography, pioneered by Krugman (1991), became an analytical framework to investigate how factor allocation between regions is determined and adjusted, while
it pays little attention to factor allocation between industries.
This study aims to review the allocation of factors among industries distinguishing between the short and long runs in monopolistically competitive world. In particular, weaddress the adjustment process leading to a stable allocation of factors between industries
and the response of factor allocation to factor endowment changes. To this end, we develop
a general equilibrium model that consists two industries characterized by the Dixit–Stiglitz
type of monopolistic competition. Each industry uses two factors (capital and labor) thatmove between two industries gradually under a general form of production technology that
may not be homothetic.
This setting of the model is more natural compared to that of traditional trade theory.
According to Hall (1988), multiple US industries have markups that diﬀer across theseindustries. For example, the markup rate in the construction industry is 2.196, but is 3.300
in the ﬁnance, insurance, and real estate industries.5 In addition, some studies suggest thatassuming homotheticity on production technology may not be appropriate. Christensen andeconomies of scale exist in many industries. For example, Christensen and Greene (1976) conﬁrm economies
of scale estimating the translog cost function in the US electric power industry. In addition, Noulas etal. (1990) estimate a translog cost function in the US banking industry, and show that the industry has
economies of scale. If economies of scale prevail in industries, analysis using constant returns technology is
not appropriate. Moreover, the assumption of perfect competition is also not appropriate, since economies
of scale make the perfect competition unsustainable. In fact, Hall (1988), using data from 1953 to 1984,
ﬁnds that some US industries have marginal costs well below price, which suggests that these industries are
not in perfect competition.
4
See, for example, Krugman (1980), Helpman (1981) and Horn (1983). Krugman (1981) analyzes a modelin which labor is speciﬁed for a particular industry, but does not consider the adjustment of labor allocation
between industries.
5
Hall (1988) estimates these markups using data of US industries from 1953 to 1984.

3

Greene (1976) reject homotheticity in production using cross-sectional data for US ﬁrms
producing electric power. Kim (1992) shows that a non-homothetic production function
is more consistent with US manufacturing from 1947 to 1971. Thus, in this study, we
assume that all industries under monopolistic competition have diﬀerent markups and nonhomothetic production technologies.
Some similar approaches have been studied previously. Horn (1983) constructs a twoindustry, two-factor model in which one industry is in perfect competition with constantreturns-to-scale technology while in the other industry, the Dixit–Stiglitz type of monopolisticcompetition prevails and ﬁrms exhibit internal increasing returns to scale with a general formof technology. Horn (1983) investigates the relationship between relative factor endowments
and relative industry size, and shows that the eﬀect of change in relative factor endowmentson relative industry size cannot be predicted without the following special circumstance:
non-homothetic bias is towards the factor that is used intensively on average.6 Chao and
Takayama (1987, 1990) analyze the stability of long-run equilibrium characterized by thezero-proﬁt condition of ﬁrms using the same setting as Horn (1983). They show that a
key condition is non-homothetic bias toward the factor that is used intensively on average
to ensure the stability of long-run equilibrium characterized by the zero-proﬁt condition.
However, these studies pay little attention to short-run factor speciﬁcity and allocation offactors between industries.
In this study, without requiring any condition on non-homotheticity and factor intensity,
we show that local stability of long-run equilibrium characterized by factor price equalization
between industries is ensured. The direction of the non-homothetic bias does not havequalitative eﬀects on the allocation of factors, while it works as key elements to determine
the direction of change in the relative industry size measured by output, which is caused bychange in factor endowments. On the other hand, the response of factor allocation to factor
endowment changes depends on the “gross” substitutability of factors which is magniﬁed by
non-homotheticity and scale economies. It is shown that an increase in capital endowments

6

In addition, Helpman (1981) undertakes a similar investigation, but uses the Lancaster (1980) type ofpreference for the diﬀerentiated goods.

4may make labor-intensive industry relatively large measured by input, while it makes capitalintensive industry relatively large measured by output.
The remainder of this paper is organized as follows. Section 2 presents the model andintroduces measures of the degree of non-homotheticity and economies of scale in production.
We describe the adjustment process of factor allocation between industries in Section 3. The
eﬀects of change of factor endowments are investigated in Section 4. Section 5 concludes.
2. Model and Introduction of Some Key Measures
In this section, we develop the model and introduce measures of economies of scale andnon-homotheticity. Consider an economy with two industries (1 and 2). Both industry 1
and 2 produce a group of diﬀerentiated goods.
2.1. Preference and Production
All consumers share the same preference represented by following utility function:
U = X1α1 X2α2 ,0 < αi < 1, α1 + α2 = 1,

(1)where Xi represents the composite index of the consumption of diﬀerentiated goods produced
in industry i. Let us denote the consumption of each variety ωi in industry i as xi (ωi ). Then,the composite index Xi takes the form of a constant elasticity of substitution function as
follows:

[∫
Xi ≡

ρi

xi (ωi ) dωi

] ρ1

i

,

0 < ρi < 1, i = 1, 2,Ωiwhere Ωi is the set of available varieties in industry i, and ρi represents the intensity of
substitution among varieties for in industry i. In this speciﬁcation, σi ≡ 1/(1−ρi ) represents
the elasticity of substitution in industry i.
Maximizing utility given by (1) subject to the budget constraint, we obtain the following
market demand functions:
αi Yxi (ωi ) = xi (pi (ωi ), Pi , Y ) ≡
Pi

(

Pi
pi (ωi ))σi,ωi ∈ Ωi , i = 1, 2,

where Y is the income of the economy, pi (ωi ) is the price for each variety ωi produced inindustry i, and Pi is the price index of industry i deﬁned by
1
[∫
] 1−σ
i
1−σi
, i = 1, 2.
Pi ≡pi (ωi )dωiΩi5Turning to production, ﬁrms are monopolistically competitive and require labor andcapital. We assume that each ﬁrm has an increasing returns-to-scale technology and can befree to choose which varieties to produce. Then, without loss of generality, we can assume
ﬁrm ωi produces variety ωi .
All ﬁrms in the same industry i have the same technology represented by a twice differentiable, monotonically increasing and strictly quasi-concave production function. We
denote factor prices for labor and capital in industry i by wi and ri . Then, solving the cost
minimization problem, we can obtain the cost function of each ﬁrm in industry i:
Ci (ωi ) = Ci (wi , ri , xi (ωi )),ωi ∈ Ωi , i = 1, 2,

which is concave and linear homogeneous in factor prices.
Using the ﬁrst-order condition for proﬁt maximization, we obtain the well-known pricing
rule as follows:
1 ∂Ci (wi , ri , xi (ωi )), ωi ∈ Ωi , i = 1, 2.
(2)
ρi∂xi (ωi )Therefore, when output is common to every active ﬁrm in industry i, the price is also commonpi (ωi ) =to every variety in the industry i. Because we only focus on the symmetric situation, we
drop the variety label in the remainder of this paper.
2.2. Measures of Scale Economy and Non-Homotheticity
The production technologies diﬀer in two respects from the traditional Heckscher–Ohlinmodel: one is the existence of scale economy and the other is that we allow non-homotheticity.
To capture the degree of the ﬁrst diﬀerence, we deﬁne the measure of economies of scale asfollows:

Ci (wi , ri , xi ) / ∂Ci (wi , ri , xi ), i = 1, 2,
ϕi = ϕi (wi , ri , xi ) ≡xi∂xi
which is the partial elasticity of cost with respect to output.7 ϕ must take on values larger
than unity if increasing returns prevail locally, while it becomes less than unity if there are7

A similar measure of economies of scale is used in Helpman (1981), Horn (1983), and Chao and Takayama(1990). However, they use 1/ϕ as the measure. We use ϕ, which is what Helpman and Krugman (1985) use,
because it is more natural that large ϕ means strong scale economy. There are some other concepts thatmeasure the economy of scale. A detailed discussion of these concepts is found in Hanoch (1975) and Ide
and Takayama (1987).

6

decreasing returns. ϕ = 1 means that constant returns to scale prevail.
We allow free entry and exit for ﬁrms. This implies that proﬁts must be driven to zerofor any active ﬁrm. Thus, we obtainCi (wi , ri , xi ) = pi xi ,i = 1, 2,(3)which is called the Chamberlinian tangency solution. Together with the pricing rule, (3)ensures that the measure of economies of scale ϕi is equal to markup 1/ρi > 1. Thismeans that the degree of scale economy is determined independently from the supply sideat the Chamberlinian tangency solution. We can observe easily that a weaker intensity of
substitution between varieties (i.e., smaller ρi ) results in larger ϕi . Note that larger ϕi meansthat ﬁrms in industry i do not exhaust scale merit and their size become smaller than theeﬃcient level. This is because smaller ρi strengthens the monopoly power of ﬁrms. Whenρi becomes large, ﬁrm must take advantage of scale merit to decrease average costs, which
results in smaller ϕi .
Using the second-order condition for proﬁt maximization, (3) implies
ηi ≡ −

1
1
xi ∂ϕi= εi + 1 −= εi +> 0,ϕi ∂xiϕi
σii = 1, 2,

where εi is the partial elasticity of marginal cost with respect to output given as
εi ≡xi∂ 2 Ci,∂Ci /∂xi ∂x2i

i = 1, 2.

Since −ηi is the partial elasticity of the measure of economies of scale with respect to output,
the degree of scale economies in production must be decreasing locally when the output
increases at the Chamberlinian tangency solution.
Next, to capture the degree of non-homotheticity in production, we deﬁne the following
variables:µLi ≡∂ 2 Cixi,∂Ci /∂wi ∂xi ∂wi

µKi ≡

xi
∂ 2 Ci,∂Ci /∂ri ∂xi ∂ri

i = 1, 2.

µLi (µKi ) is the partial elasticity of demand for labor (capital) of industry i with respect to
output. Using the linear homogeneity in the factor prices for the cost function, we obtainthe following relationship (cf., Horn, 1983, p. 91):
θLi µLi + θKi µKi = 1/ϕi ,7i = 1, 2,where θLi (θKi ) is the distributive share of the cost for labor (capital) in the production ofindustry i given byθLi ≡wi ∂Ci,Ci ∂wiθKi ≡ri ∂Ci,Ci ∂rii = 1, 2,which must satisfy θLi + θKi = 1. If the production function is homothetic, then µLi = µKi =1/ϕi . So, we measure the degree of non-homotheticity in production as follows:8
)(
)
(
1
1
= θKi θLi (µKi − µLi )2 ≧ 0,
µLi −
δi ≡
µKi −
ϕi
ϕi

i = 1, 2.

Obviously, in the homothetic case, δi becomes zero.
2.3. Factor Proportions of Individual Firm
Finally, let us examine the relationship between factor price and factor proportion ofindividual ﬁrms, which plays an essential role in this study. We denote capital–labor ratioof an individual ﬁrm in industry i as ki ≡ (∂Ci /∂ri ) / (∂Ci /∂wi ). Then, the change in factor
proportions is given byk̂i = σpi (ŵi − r̂i ) + (µKi − µLi ) x̂i ,i = 1, 2,(4)where the circumﬂex denotes the rate of change in each variable (e.g., k̂i ≡ dki /ki ) and σpi
is the Allen’s elasticities of substitution, which is derived as output is ﬁxed as
σpi ≡

Ci
∂ 2 Ci,(∂Ci /∂wi )(∂Ci /∂ri ) ∂ri ∂wi

i = 1, 2.

The ﬁrst term of the RHS in (4) represents the substitution eﬀect which excludes the eﬀectof change in output. So, we call σpi the pure elasticity of substitution between factors.
If the production technology is homothetic, the optimal capital–labor ratio is determined
independently from the output level. Thus, the change in factor proportions is captured onlyby the ﬁrst term of the RHS in (4). However, allowing non-homotheticity develops positiveor negative eﬀects from the change in output level on the factor proportion. This clearly
depends on whether µKi is bigger or smaller than µLi . µKi > µLi (µKi < µLi ) means that a

8

A similar measure of the degree of non-homotheticity is used in Horn (1983) and Chao and Takayama(1990).

8

marginal increase in output makes the input ratio of factors biased toward capital (labor).
In this sense, we say that industry i has marginal capital-biased (labor-biased) technology ifµKi > µLi (µKi < µLi ).
The change in output level can be linked to the change in factor rewards as follows:x̂i =ϕi θKi θLi (µKi − µLi )(ŵi − r̂i ) ,ηi

i = 1, 2,(5)which is derived using the fact that the degree of scale economy ϕi (wi , ri , xi ) is ﬁxed to 1/ρi
at the Chamberlinian tangency solution. This equation (5), together with (4), reveals therelationship between factor rewards and factor proportion as
k̂i
δi ϕi
= σpi +≡ si ,(ŵi − r̂i )ηii = 1, 2.

(6)With output adjusted to the change in factor prices, we call si the gross elasticity of substitution between factors in industry i.
(4) and (6) show that the relative change in factor proportion is always magniﬁed by
the output changes brought about by the change in factor prices (i.e., si > σpi ), if nonhomotheticity prevails in production.9 In the homothetic case, this magniﬁcation eﬀect does
not arise and the gross elasticity of substitution si is equal to the pure elasticity of substitution σpi . As we see in Subsection 3.2, diﬀerences of our results from a perfectly competitive
world with constant returns technology occur mainly through this magniﬁcation eﬀect. In
addition, because diﬀerence of markups between industries develops the diﬀerent magniﬁcation eﬀects for each industry, the gross elasticity substitution may diﬀer between industries,
even if all industries have similar value of the pure elasticity substitution. As Torstensson(1996) mentioned, this implies that it is diﬃcult to rule out factor-intensity reversals inmonopolistically competitive world unless all industries have homothetic production technology.10 These facts suggest that non-homotheticity is essential to address the inﬂuence ofscale economy and monopolistic competition.
9Horn (1983) expounds why the output changes brought about by the change in factor prices always

magnify the relative change in factor proportion. See Figure 1 in Horn (1983).
10
Torstensson (1996) provide a simple example in which factor-intensity reversals occur without the difference in the pure elasticity substitution.

93. Equilibrium and Stability
In this section, we ﬁrst describe the short-run and long-run equilibrium. Then, stabilityof the equilibrium and the adjustment process are analyzed.
3.1. Equilibrium
With regard to the factor market, we assume that full employment is satisﬁed, but laborand capital are speciﬁed to each industries in the short run. In this case, factor prices may
diﬀer between industry. This situation is described by∂C1 (w1 , r1 , x1 )∂C2 (w2 , r2 , x2 )= L1 , N2= L − L1 ,∂w1
∂w2
∂C1 (w1 , r1 , x1 )∂C2 (w2 , r2 , x2 )N1
= K1 , N 2= K − K2 ,
∂r1
∂r2

N1

(7)

where Ni is the number of ﬁrms in industry i, Li (Ki ) is the amounts of labor (capital)
employed in industry i, and L (K) represents endowments of labor (capital), respectively.
Short-run equilibrium is given by (2), (3), (7), and the market-clearing condition in the
goods markets given by
N2 x2
α2= p1 ,
N1 x1
α1

(8)

where we normalize as p2 = 1. Therefore, in the short-run equilibrium, we can express therelative prices of factors, W ≡ w1 /w2 and R ≡ r1 /r2 , as a function of L1 , K1 , L, and K:
W = W (L, K; L1 , K1 ) and R = R(L, K; L1 , K1 ).
On the other hand, in the long run, labor and capital move from low-priced industry tohigh-priced industry. Thus, as long as both industries employ labor and capital, the factor
prices must be equalized between the two industries in the long run, that is, W ∗ = R∗ = 1(we use (∗) to denote the long-run equilibrium value of each variable). We are interested
in whether the short-run equilibrium is adjusted to the long-run equilibrium and how thelong-run allocation of factors between industries is determined.
Let us examine the properties of W (·) and R(·). These properties are used for the
analysis of the stability and adjustment processes. Applying comparative statics techniqueto the short-run equilibrium given by (2), (3), (7) and (8) yields the following equations (see

10

1
, ;

, ;

, ;

,

,

,

∗

Figure 1

Appendix A):
[
)
)]
(
(
L1 ∂W
1
1
1
=−
1 + λL1 θK2− 1 + λL2 θK1
−1
< 0,
W ∂L1
λL2
s
s
[
)
)]
(2
(1
K1 ∂R
1
1
1
=−
1 + λK1 θL2− 1 + λK2 θL1
−1
< 0,
R ∂K1
λK2
s2
s1
[
)
)]
(
(
K1 ∂W
1
1
1
=
λK1 θK2− 1 + λK2 θK1
−1 ,
W ∂K1
λK2
s2
s1
[
)
)]
(
(
L1 ∂R
1
1
1
=
λL1 θL2− 1 + λL2 θL1
−1 ,
R ∂L1
λL2
s2
s1

(9)

where λLi ≡ Li /L (λKi ≡ Ki /K) represents the share of labor (capital) of industry i. As
is obvious from (9), ∂W/∂L1 and ∂R/∂K1 are always negative, but ∂W/∂K1 and ∂R/∂L1
can be either positive or negative according to whether the “gross” elasticity of substitution
between factors, si , is smaller than one, which depends on the degree of scale economy,
non-homotheticity and substitutability between factors in production.
3.2. Only Labor Can Move between Industries
We start out by constructing a case that only labor can move between industries graduallyin response to gap of factor rewards. This case is described in Figure 1.
At point A in Figure 1, the wage of industry 1 is higher than that of industry 2. Thus,
labor moves from industry 2 to industry 1. Because ∂W/∂L1 < 0, this movement of labor11toward industry 1 lowers W , which continues until W reaches one. Therefore, labor allocationis always adjusted to point E in Figure 1. This means that, when capital is ﬁxed in eachindustry, labor allocation at point E is globally stable regardless of the degree of scaleeconomy, non-homotheticity and substitutability between factors in production.
Suppose that labor allocation is initially given at point E in Figure 1, and then, capital
allocation exogenously changes so that industry 1 becomes more capital abundant (i.e., K1′changes to K1 > K1 ). If the gross elasticities of substitution between factors, si , are smaller
than one in both industry 1 and 2, which means that the pure elasticities of substitution,
σpi , the degree of non-homotheticity, δi , and the degree of scale economy, ϕi , are suﬃciently
small, then ∂W/∂K1 becomes positive.11 Thus, labor allocation is adjusted from point E′′

to point E through point F . On the other hand, if both s1 and s2 are larger than one,which is caused by a suﬃciently large σpi , δi , and ϕi , then we have a negative value of′′∂W/∂K1 . In this case, labor allocation is adjusted from point E to point E through point′′

F . This means that labor ﬂows out from the industry that becomes more capital abundant.
In particular, if the pure elasticities of substitution in production are bigger than one in bothindustry 1 and 2, labor always ﬂows out from the industry that capital ﬂows into, even if
both industry 1 and 2 have homothetic production technology.
Note that scale economy and monopolistic competition have inﬂuence only when nonhomotheticity prevails in production. We show a result at (B.2) that is obtained by acalculation similar to (9) in a perfectly competitive world with a Cobb–Douglas utility function and constant returns technology (see Appendix B). Clearly, both results, (9) and (B.2),
have the same form. The diﬀerence is only that (9) uses the “gross” elasticity of substitution
si instead of the “pure” elasticity of substitution σpi . If the production technologies are
homothetic in both industries, there are no magniﬁcation eﬀects and (9) is equal to (B.2).
This suggests that non-homotheticity is essential when we consider the inﬂuence of scaleeconomy and monopolistic competition.

11

Christensen and Greene (1976) shows that the elasticity of substitution between labor and capital issmaller than one in the U.S. electric power industry.

12

3.3. Both Labor and Capital Can Move between Industries
We next turn to the case in which both labor and capital move between industries, andexamine the stability of the long-run equilibrium. For the stability analysis, we postulatethat adjustment towards the long-run equilibrium occurs according to[]L̇1 = aL W (L, K; L1 , K1 ) − 1 ,][K̇1 = aK R(L, K; L1 , K1 ) − 1 ,where the dot signiﬁes the time derivative and aL and aK are the positive adjustment parameter. Around the long-run equilibrium given by W (L, K; L∗1 , K1∗ ) = R(L, K; L∗1 , K1∗ ) = 1,
this dynamic system is approximated by
  


∗
∗
∗
L̇
a W
a W
L − L1 1  =  L L1 L K1   1
,
∗
∗
∗
K̇1
aK RL1 aK RK1
K1 − K1(10)

where WL1 ≡ ∂W/∂L1 , WK1 ≡ ∂W/∂K1 , RL1 ≡ ∂R/∂L1 , and RK1 ≡ ∂R/∂K1 .
The long-run equilibrium is locally stable if and only if the determinant of the coeﬃcient∗matrix of (10) is positive, that is, WL∗1 RK> WK∗ 1 RL∗ 1 . We already know that the signs
1of WL1 and RK1 are negative. On the other hand, the sings of WK1 and RL1 cannot be
determined in general. Even so, we can determine the sign of WL1 RK1 − WK1 RL1 because,from (9), we obtain( )sign (WL1 RK1 − WK1 RL1 ) = sign σA ,(11)

σA ≡ |λ||θ| + (θL1 λK1 + θK1 λL1 )s1 + (θL2 λK2 + θK2 λL2 )s2 ,where |θ| and |λ| are deﬁned by |θ| ≡ θK1 θL2 −θL1 θK2 and |λ| ≡ λK1 λL2 −λL1 λK2 , respectively.
Jones (1965) calls σA the aggregate elasticity of substitution. However, to deﬁne σA , thepure elasticity of substitution σpi is used in Jones (1965) instead of the gross elasticity of
substitution si . If production technologies in both industries are homothetic, there are no
magniﬁcation eﬀects and si is equal to σpi . Note that elasticity of substitution in consumption
between X1 and X2 is unity. In addition, it is satisﬁed |λ||θ| + (θL1 λK1 + θK1 λL1 ) + (θL2 λK2 +θK2 λL2 ) = 1. Thus, it is seen that the aggregate elasticity of substitution is a weighted
average of the elasticity of substitution in production and demand.

13∗
Since |λ||θ| is always non-negative at the long-run equilibrium, WL∗1 RK− WK∗ 1 RL∗ 1 is1always positive, which yields the following proposition.12

Proposition 1. The long-run equilibrium is always locally stable.

Hence, around the long-run equilibrium, inter-sectoral factor mobility adjusts the shortrun equilibrium determined by (2), (3), (7), and (8) to the long-run equilibrium, regardless ofthe degree of scale economy and non-homotheticity in production. This is surprising result
because scale economy and non-homotheticity usually serve as drivers that make equilibrium
unstable. Chao and Takayama (1990) actually show that when the degrees of scale economy
and non-homotheticity are strong, the free entry-exit of ﬁrms in monopolistically competitive
industries may not lead to the Chamberlinian tangency solution characterized by the zeroproﬁt condition.13 Proposition 1 allows us to mitigate botheration which arises when we
consider stability of equilibrium in a monopolistically competitive world.
The stability of the long-run equilibrium does not depend on the degrees of scale economy
and non-homotheticity, but they play key roles in the uniqueness of the long-run equilibrium
and adjustment process. The following proposition is about uniqueness.

Proposition 2. Assume that at any long-run equilibrium, either condition (a) or (b) issatisﬁed, where:(a) : s1 < 1 and s2 < 1,(b) : s1 > 1 and s2 > 1.
Then, the long-run equilibrium given by W (L∗1 , K1∗ ; L, K) = R(L∗1 , K1∗ ; L, K) = 1 is unique
if it exists.
12

Through the seminal work of Jones (1965), |λ| and |θ| have become widely used in the literature ofinternational trade theory.
13
Chao and Takayama (1990) construct a two-sector model with a perfectly competitive sector, a monopolistically competitive sector, and non-homothetic production function and they verify the stability of
the Chamberlinian tangency solution. They derive a necessary and suﬃcient condition for stability whichstrongly depends on the degrees of scale economy and non-homotheticity. For details, see proposition 2 andthe appendix of Chao and Takayama (1990).

14

=0
=0

=0
=0

(a) s1 < 1 and s2 < 1(b) s1 > 1 and s2 > 1

Figure 2

Proof See Appendix C.
We can determine the sign of WK1 and RL1 unambiguously by either condition (a) or (b)of Proposition 2. If the pure elasticities of substitution, σp1 and σp2 , are bigger than one, thenthe gross elasticities of substitution, s1 and s2 , are also bigger than one, and hence, condition(b) must be satisﬁed, regardless of the degree of scale economy, ϕi , and non-homotheticity,
δi , in production. Even if the pure elasticities of substitution are smaller than one, the large
magniﬁcation eﬀects that are caused by large ϕi and δi may make si bigger than one. To
satisfy condition (a), the pure elasticities of substitution and magniﬁcation eﬀects must be
suﬃciently small. Clearly, in the non-homothetic case, the large-scale economy works tohold condition (b). At the Chamberlinian tangency solution, the degree of scale economy,
ϕi , is equal to the markup 1/ρi . Thus, satisfying condition (b) becomes easy for the smaller
elasticity of substitution between varieties in consumption, which is represented by smaller
ρi , and leads to larger the markups.
Figure 2 describes a phase diagram of the adjustment process around the long-run equilibrium corresponding to conditions (a) and (b) of Proposition 2. Consider the case s1 < 1
and s2 < 1, which implies WK1 > 0 and RL1 > 0. Point A in Figure 2(a) corresponds to a
deviation from the long-run equilibrium, in which capital is excessively employed in industry
15

1. This case is equivalent to exogenous movement of capital from industry 2 to industry
1. At ﬁrst, because movement of capital from industry 2 to industry 1 makes the wage ofindustry 1 relatively high compared to that of industry 2, labor ﬂows into industry 1 (seeFigure 1), while capital ﬂows out due to RK1 < 0. After crossing line L̇1 = 0, both laborand capital ﬂow out from industry 1 until allocations of labor and capital between industries
reach the long-run equilibrium. This case arises when the pure elasticities of substitution in
production are smaller than 1 and the degrees of non-homotheticity and scale economy inproduction are suﬃciently small (i.e., σpi , δi , and ϕi are suﬃciently small).
On the other hand, if s1 > 1 and s2 > 1 hold, we have WK1 < 0 and RL1 < 0, whicharises when σpi is bigger than 1 or δi and ϕi are suﬃciently large. Point B in Figure 2(b)corresponds to a deviation from the long-run equilibrium as point A in Figure 2(a). In
contrast to the case s1 < 1 and s2 < 1, labor ﬂows out, at ﬁrst, from industry 1, since
movement of capital from industry 2 to industry 1 makes the wage of industry 1 relatively
low compared to that of industry 2. After crossing line L̇1 = 0, labor ﬂows into industry 1
while capital continues to ﬂow out.
Therefore, according to which of conditions (a) or (b) of Proposition 2 is satisﬁed, theadjustment process changes sharply, even if the long-run equilibrium is always locally stablewithout any conditions. As shown earlier in this subsection, condition (b) is easy to besatisﬁed more in circumstances in which large degrees of non-homotheticity and scale economy prevail. In the homothetic case, only the pure elasticity of substitution is important to
identify the adjustment process.
4. Capital Inflow from Outside the Economy
We consider the eﬀects of capital inﬂows from outside of economy–eﬀects of exogenous
increase of capital endowment. In this case, the initial long-run equilibrium is broken, andfactors are reallocated between industries toward a new long-run equilibrium. In this section,
we assume that marginal cost does not decrease as output increases (i.e., εi ≧ 0) and bothfactor is superior (i.e., µLi > 0 and µKi > 0).

16

4.1. Deﬁnition of Marginal (Average) Factor Intensity
The long-run equilibrium is given by (2), (3), (8), and∂C1 (w, r, x1 )∂C2 (w, r, x2 )
+ N2
= L,
∂w
∂w
∂C1 (w, r, x1 )∂C2 (w, r, x2 )
N1
+ N2
= K.
∂r
∂r

N1

Taking the total diﬀerential of the long-run equilibrium condition yields   0ŵ
−ςL ςL0
λL1 µL1λL2 µL2 λL1 λL2       ςK −ςK 0 λK1 µK1 λK2 µK2 λK1 λK2   r̂  K̂        τL1 τK1 −1
ε1000   p̂1   0        τL2 τK2 00
ε200   x̂1  =  0  ,       θL1 θK1 −1 1/ϕ1 − 1000   x̂2   0        θL2 θK2 0
0
1/ϕ2 − 1 0
0  N̂1   0    
0
N̂2
0
0
1
1
−1
1
−1

(12)

where ςL , ςK , τLi , and τKi are deﬁned byςL ≡ λL1 θK1 σp1 + λL2 θK2 σp2 ,
τLi ≡

wi
∂ 2 Ci,∂Ci /∂xi ∂wi ∂xiςK ≡ λK1 θL1 σp1 + λK2 θL2 σp2 ,

τKi ≡

ri
∂ 2 Ci,∂Ci /∂xi ∂ri ∂xi

i = 1, 2.

τLi (τKi ) represents the partial elasticities of marginal cost for the wage (rental) rate in eachindustry i, which is in the following relationship with µ, θ, and ϕ:τji = ϕi θji µji ,i = 1, 2, j = L, K.

Thus, it is satisﬁed that τLi +τKi = 1. In particular, τji must be equal to θji in the homotheticcase.
Let us deﬁne |µ| by |µ| ≡ µK1 µL2 − µL1 µK2 . Using |µ|, we introduce the concept ofmarginal factor intensity. This concept was introduced ﬁrst by Jones (1968). Consider
the isoquant map measuring capital on the vertical axis. Then, industry i is said to be
marginally capital intensive (labor intensive) if its expansion path is steeper (gentler) at
the production point than that of industry j. This means, intuitively, that an expansionof output in industry i would require extra capital (labor) per unit of extra labor (capital)
17

compared to industry j. Following Chao and Takayama (1990), we capture this concept by
the following deﬁnition: Industry 1 is marginally capital-intensive compared to industry 2 if
and only if |µ| > 0, while industry 1 is marginally labor-intensive compared to industry 2, if
and only if |µ| < 0.
If industry j has homothetic technology, this deﬁnition implies that industry i is marginallycapital intensive if µKi > µLi . This is equivalent to the condition that industry i becomes
marginally capital intensive in Chao and Takayama (1990), in which industry j is characterized by perfect competition and constant returns technology. Thus, our deﬁnition of
marginal factor intensity includes that of Chao and Takayama (1990).
Consider the isoquant map again. We measure capital on the vertical axis. In contrast
to the concept of marginal factor intensity, following Jones (1968) and Chao and Takayama(1990), we call industry i capital intensive (labor intensive) compared to industry j in an
average sense if industry i has a steeper (gentler) ray from the origin passing the production
point than that of industry j.14 According to this concept, we use the following deﬁnition:
Industry 1 is capital intensive compared to industry 2 in an average sense, if and only if
K1 /L1 > K2 /L2 while industry 1 is labor intensive compared to industry 2 in average sense,
if and only if K1 /L1 < K2 /L2 .
4.2. Eﬀects on Prices
First, we explore the eﬀects on the price side of the economy. Solving (12), we obtain()
θK2 (ϕ2 − 1)θL2 (µK2 − µL2 )θK2 (ϕ2 − 1)µK2 + ε2
ŵ
=
+1 =
> 0,
σA
η2
σA
η2
K̂
(
)
(13)
r̂
θL2 (ϕ2 − 1)θK2 (µK2 − µL2 )θL2 (ϕ2 − 1)µL2 + ε2
=
−1 =−
< 0,
σA
η2
σA
η2
K̂
Similarly, we can derive the following relationship:θK1 (ϕ1 − 1)µK1 + ε1
> 0,
σA
η1
K̂
r̂ − pˆ1
θL1 (ϕ1 − 1)µL1 + ε1
=−
< 0,
σA
η1
K̂

ŵ − pˆ1

14

=

(14)

In traditional trade theory, this case is just called as industry i is capital intensive (labor intensive)

compared to industry j.

18where we use
p̂1

) θ (
))
θK2 (
K1
(ϕ2 − 1)µK2 + ε2 −(ϕ1 − 1)µK1 + ε1
η2
η1
(
)
1
θK1 θL1 (µK1 − µL1 )θK2 θL2 (µL2 − µK2 )=−|θ| + (ϕ1 − 1).
+ (ϕ2 − 1)
σA
η1
η2

1
=
σA
K̂

(

(15)

(13) and (14) indicate the eﬀect of a change of capital endowment on real factor rewards.
By the inﬂow of capital from outside the economy, real factor rewards improve in labor but
not capital regardless of the degree of scale economy, non-homotheticity, and substitutability
between production factors.15 In addition, this depends on neither marginal nor averagefactor intensity. Thus, capital inﬂow always works in labor’s favor but operates against
capital owners. Analogously, we can see that labor inﬂow from outside the economy works
in favor of capital owners but operates against labor.
From (15), a suﬃcient condition to ensure that p1 /p2 falls is given by|θ| > 0 and

µK2
µK1
≧1≧
.
µL1
µL2

The ﬁrst term in this condition states that industry 1 is capital intensive in an averagesense.16 On the other hand, the second term means that industry 1 has marginally capitalbiased technology while industry 2 has marginally labor-biased technology, which implies thatindustry 1 is marginally capital intensive compared to industry 2. Similarly, we see that a
suﬃcient condition that p1 /p2 rises is that industry 1 is labor intensive in an average sense
and industry 1 has marginally labor-biased technology but industry 2 has marginally capitabiased technology, which implies that industry 1 is marginally labor intensive compared
to industry 2. Thus, to determine whether p1 /p2 falls or rises unambiguously, the nonhomothetic bias of each industry must be towards the factor which is used intensively in an
average sense. This requires that average factor intensity and marginal factor intensity movein the same direction, that is, |θ||µ| ≧ 0.171516Note that we normalize p2 to unity.
Note that the condition that industry 1 is capital intensive (labor intensive) compared to industry 2 isequivalent to the condition that |θ| is positive (negative).
17
A similar result is obtained by Horn (1983) in which there is only one diﬀerentiated goods industry.

19

The eﬀect of a change of capital endowment on P1 /P2 can be revealed by (15) and()θK1 θL1 (µL1 − µK1 )λL1 θK1 s1 + λL2 θK2 s2 + |θ|λL2 +
,
η1
(
)
N̂2
1
θK2 θL2 (µK2 − µL2 )=λL1 θK1 s1 + λL2 θK2 s2 − |θ|λL1 −
.
σA
η2
K̂
N̂1

1
=
σA
K̂

(16)

(15) and (16) yield the following equation:P̂1 − P̂2
K̂

1 N̂1
1 N̂2
−
1 − σ2 K̂
K̂ 1 − σ1 K̂)
1 (=−|θ| (λL1 ϕ2 + λL2 ϕ1 ) + (ϕ1 − ϕ2 ) (λL1 θK1 s1 + λL2 θK2 s2 ) ,
σA
=

p̂1

+

(17)

which states that if industry i is capital intensive in an average sense and has a greater degreeof scale economy than that of industry j, an inﬂow of capital from outside the economy lowers
the price index of industry i relative to that of industry j. We have seen that the degreeof scale economy ϕi is equal to the markup 1/ρi at the Chamberlinian tangency solution.
Thus, a large markup that is caused by lower substitutability between varieties works as adevice that lowers the price index relatively due to an increase in production resources.
4.3. Eﬀects on Output
Next, let us turn to the eﬀect of the capital inﬂow on output. It is easy to show that(13) and (14) give()
ϕ1
ŵ − p̂1r̂ − p̂1
ϕ1 θL1 θK1 (µK1 − µL1 )
=
θL1
+ θK1
=
,
ϕ1 − 1
σA
η1
K̂
K̂
K̂
)
(
x̂2
ϕ2
ŵ
r̂
ϕ2 θL2 θK2 (µK2 − µL2 )=θL2 + θK2=.
ϕ2 − 1
σA
η2
K̂
K̂
K̂
x̂1

(18)

This simply shows that a single ﬁrm’s size in an industry that has marginally capital-biased(labor-biased) technology increases (decreases) with an increase in capital endowment. Factor intensity and scale economy do not inﬂuence the sign of x̂i . The sign of x̂i mainly depends
on the direction of non-homothetic bias. In particular, in the homothetic case, ﬁrm size neverchanges even if factor endowments change.
On the other hand, what can we say about the relative industry size? It is well known
that an increase in capital endowment makes the capital-intensive industry relatively large inthe usual Heckscher–Ohlin framework, which is known as the Rybczynski theorem. However,20

in our model, we characterize all industries by monopolistic competition instead of perfectcompetition. In addition, there are scale economies and non-homotheticity in all industries.
Thus, an increase in capital endowment need not to always increase capital intensive industry
compared to labor intensive industry. We now check this point.
Let us measure industry i’s size, Yi , by the sum of output, that is, Yi ≡ Ni xi . Then,(8) reveals that the change in the ratio of industry size, Y1 /Y2 , is negatively related to thechange of p1 , as follows:Ŷ1 − Ŷ2

=−

p̂1

.
(19)
K̂
K̂
We cannot see whether an inﬂow of capital from outside the economy expands the capitalintensive industry rather than the labor-intensive industry, in general. To determine thesign of Ŷ1 − Ŷ2 unambiguously, we need to suppose that each industry’s non-homothetic
bias is toward the factor that is used intensively in an average sense (see (15)). As alreadyshown, this condition requires that average and marginal factor intensity move in the same
direction, that is, |θ||µ| ≧ 0. Thus, non-homotheticity plays an essential role in the directionof the change in the ratio of industry size. If both industries have homothetic technology, thecapital-intensive industry always becomes bigger relatively to the labor-intensive industry
due to the inﬂow of capital. This case makes (19) equivalent to the equation (11) of Jones(1965).
This result is obtained measuring industry size by sum of output. However, it may beconsidered that we should measure industry size by the quantity index instead of the sum of
output, because we regard Pi as a price index of industry i. Thus, we also check the eﬀectof the capital inﬂow on the ratio of industry size measuring industry i’s size by the quantity
index Xi .
The preference on the composite indexes of diﬀerentiated goods, X1 and X2 , is deﬁned
by the Cobb–Douglas form. Thus, the change in relative industry size is given as follows:X̂1 − X̂2K̂=−P̂1 − P̂2
K̂.

From (17), we can see that if the gap of the degree of scale economies (markup) between
industries is small, the economy has a relatively large capital intensive industry as capitalendowment increases. On the other hand, if the gap of factor intensity between industries21

is small, then an increase in capital makes the industry that has a larger degree of scaleeconomies (markup) relatively large. Thus, the relative size of the two industries is determined by a tug of war of two gaps (factor intensity and degree of scale economies). Unlike
the case in which we measure industry size by sum of output, the gap of degree of scaleeconomies (markup) plays an essential role rather than non-homotheticity. In fact, even if
the technologies of both industries are homothetic, the sign of the change in the relative sizeof the industries still depends on the gap of degrees of scale economies (markup) between
industries, but if there is no gap of degrees of scale economies (markup), the inﬂow of capitalfrom outside the economy always increases the size of the capital-intensive industry relative
to that of the labor-intensive industry.
4.4. Eﬀects on Factor Allocation
Finally, we turn to the eﬀect of the capital inﬂow from outside the economy on factor
allocation. From here, without loss of generality, we assume that industry 1 is labor intensive
in an average sense compared to industry 2, that is, K1 /L1 < K2 /L2 . This implies that both
|θ| and |λ| are negative.
We ﬁrst check the change of labor allocation. Because L1 = N1 (∂C1 /∂w) and laborendowments do not change, using (13), (16) and (18) yields the following relationship:N̂1
λL2 L̂2x̂1
r̂ − ŵ
=+ µL1 + θK1 σp1
λL1 K̂
K̂
K̂
K̂
K̂
)
λL2 (
=
|θ| (1 − s1 ) + θK2 (s2 − s1 )
σA
(
))
(
θK2 s1λL2=|θ| + θK1 s2−.
σAθK1 s2

L̂1=−

Therefore, we can determine the signs of L̂1 /K̂ and L̂2 /K̂ unambiguously if either condition′′(a) or (b) given by′(a) : s2 ≦ s1 < 1,′(b) : s2 ≧ s1 > 1,′

is satisﬁed. When condition (a) is satisﬁed, then labor moves from industry 1, which is labor
intensive, to industry 2, which is capital intensive, due to the increase of capital endowment,′

while labor moves from industry 2 to industry 1 when condition (b) is satisﬁed. The latter22

result may be surprising because an increase in capital endowments usually expands capitalintensive industry (i.e., industry 2).
Similarly, we can determine the signs L̂1 /K̂ and L̂2 /K̂ if()
θK2 s1(c) : |θ|−>0
θK1 s2is satisﬁed. If this condition is satisﬁed, an increase in capital endowments causes labor
movement from the labor-intensive industry to the capital-intensive industry. Because we
assume that industry 2 is capital intensive in an average sense, |θ| is negative. This meansthat condition (c) requires θK2 /θK1 < s1 /s2 . Thus, s1 must be larger than s2 , because
θK2 /θK1 is bigger than one. Therefore, for an increase in capital endowments to cause
labor movement to the capital-intensive industry, the labor-intensive industry must have asuﬃciently large gross elasticity of substitution between factors compared to the capitalintensive industry. This is because the increase in capital endowments makes the capital
price relatively lower than the labor price, which causes a substitute from labor to capitalin each industry. If the gross elasticity of substitution is too large in the capital intensive
industry, then labor moves from the capital-intensive industry to the labor-intensive industry.
We turn to capital allocation. When industry 2 is capital intensive in an average sense,
we can state that an increase in capital endowments always increases the capital employment
of industry 2, because we have
K̂2
K̂

=

)
L̂2
s2
λL2
λL1 (
+
=
s2 +
θK1 s1 + θL2 s2 − |θ| ,
σA
σA
σA
K̂

which means that the capital–labor ratio in industry 2, k2 , always rises. On the other hand,
we cannot determine whether the capital employment of industry 1 increases or decreases,
while the capital–labor ratio in industry 1 must rise, which is shown byK̂1 − L̂1
K̂

=

s1
.
σA

Figure 3 depicts the shift and adjustment process from the initial long-run equilibrium
to the new one in the Edgeworth box, which is accompanied by the change in capital en′′′

dowments under conditions (a) and (b) . The origin of industry 2 shifts from O2 to O2 in
response to the increase in capital endowments. If capital–labor ratios in both industries do
23

∗

∗
∗

∗

∗

∗

=0
=0
=0

∗

∗

∗

∗

=0
∗

∗
∗

∗

(a) s2 ≦ s1 < 1(b) s2 ≧ s1 > 1

Figure 3

not change, then the long-run equilibrium shifts from the initial one, which given by pointe, to point c. This shift of the production point corresponds to the Rybczynski theorem
which is derived by holding prices constant. However, in our model, prices must adjust to
the change in factor endowments. In this case, an increase in capital endowments raises
the capital–labor ratio ki in both industries regardless of the degree of scale economies and′non-homotheticity. Thus, the new equilibrium, which is given by point e , must be in the′′area between the lines O1 a and O2 b. Figure 3(a) corresponds to condition (a) , in whichlabor moves to the capital-intensive industry 2 from the labor-intensive industry 1.18 Note′that condition (a) implies condition (a). Thus, the adjustment path to the new long-run
equilibrium becomes that which corresponds to Figure 2(a). On the other hand, under con′dition (b) , labor moves from the capital-intensive industry 2 to the labor-intensive industry′

1, which is described in Figure 3(b). Because, condition (b) implies the condition (b), theadjustment path to the new long-run equilibrium becomes the same as Figure 2(b).
Obviously, the eﬀects of the change in factor endowments on factor allocation do notdepend on technology bias and factor intensity in a marginal sense. The eﬀects on factor
allocation depend mainly on the “gross” elasticity of substitution between factors and factor

18′

Note that the employment of capital in industry 1 may decrease under condition (a) .

24

intensity in an average sense. In particular, it should be noted that if the “gross” elasticities
between factors are large in both industries and these diﬀerences between industries is small,which is a condition to rule out factor-intensity reversals, the shares of capital and labor, λKi
and λLi , may increase in the labor-intensive industry due to increasing in capital endowments.
In fact, if we set s1 = s2 > 1, then we obtain)λL2 (|θ| (1 − s1 ) + θK2 (s2 − s1 ) > 0,
σA
K̂1
)
λK2 (
λ̂K1
=
|θ| (1 − s1 ) + θL2 (s1 − s2 ) > 0,
σA
K̂
L̂1

=

where industry 1 is a labor-intensive industry in an average sense. In this case, an increasein capital endowments may make the labor-intensive industry relatively large measured by
the input of factors, while it makes the capital-intensive industry relatively large measured
by output.
5. Conclusion
In this study, we constructed a two-industry, two-factor model to analyze factor allocation
among industries. In particular, by characterizing both industries as the Dixit–Stiglitz typeof monopolistic competition and allowing a general form of the production function, we
examined the eﬀects of scale economies and non-homotheticity of production technology on
factor allocation.
The results showed that the long-run equilibrium characterized by factor-price equalization between industries is always locally stable without any condition on scale economy,
non-homotheticity, and factor intensity. This result stands in sharp contrast to the factthat the non-homothetic bias toward the factor that is used intensively on average is a key
condition to ensure the stability of the long-run equilibrium characterized by the zero-proﬁt
condition, which is in line with Chao and Takayama (1987, 1990). In addition, the directionof non-homothetic bias is not important for determining the direction of the change in factor allocation. The direction of the change in factor allocation caused by change in factor
endowments mainly depends on the gross substitutability of factors, which is magniﬁed by
non-homotheticity and scale economies. For example, if the gross elasticity of substitution
25

between factors is less than one in all industries and becomes large in the labor-intensive
industry compared to the capital-intensive industry, then an increase in capital endowments
causes labor to move to the capital-intensive industry from the labor-intensive industry. Itshould be noted that if the gross elasticity between factors is bigger than one in both industries and has only small a diﬀerence between industries, which is a condition for absenceof factor-intensity reversals, then the increase in capital endowments may make the laborintensive industry relatively large measured by input of factors. These results suggest that if
factors can move between countries or regions as in the framework of the new economic geography, industrial structure and agglomeration pattern may strongly depend on the conditionof the gross substitutability of factors.
The analysis of this study is limited as there is only one region. To address the allocationof factors between regions, we must establish multiple regions. Expanding the model into
multiple regions will give us more interesting views like the agglomeration pattern. However,if there are multiple regions, we need to consider trade, which makes the analysis diﬃcult
because the economy is not closed in one region. We leave it for future research to address
the issue of multiple regions.
Appendix A.
Taking the total diﬀerential of the short-run equilibrium condition given by (2), (3), (7),and (8), we obtain









L̂1
ŵ

 1    1
 ŵ2   λL2 (L̂ − λL1 L̂1 ) 

  

  

 r̂1  
K̂1

  

   1
 r̂2   λ (K̂ − λK1 K̂1 )
   K2
  J  p̂1  = 
,
0

  

  

 x̂1  
0

  

  

 x̂2  
0

  

  

N̂1  
0

  
N̂2
0

26

(A.1)

where the circumﬂex denotes the rate of change in each variables (e.g., ŵi ≡ dwi /wi ), and J
is the Jacobian matrix given by−θ σ
0θK1 σp1
00µL10 K1 p1
0−θK2 σp20
θK2 σp200µL2 θL1 σp1
0
−θL1 σp1
0
0
µK1
0



0
−θL2 σp2 00µK20θL2 σp2
J =  τL1
0
τK1
0
−1
ε1
0



0
τL2
0
τK2
0
0
ε2


 θL1
0
θK1
0
−1 1/ϕ1 − 1
0



0
θL2
0
θK2
0
0
1/ϕ2 − 1

0
0
0
0
1
1
−1


1
0
1
0
0
0
0
0
1

0



1


0


1


0 ,


0


0


0

−1

where τLi and τKi are given by
τLi ≡

wi
∂ 2 Ci,∂Ci /∂xi ∂wi ∂xi

τKi ≡

ri
∂ 2 Ci,∂Ci /∂xi ∂ri ∂xi

i = 1, 2.

θ, µ, τ , and ϕ have the following relationship:
τji = ϕi θji µji ,i = 1, 2, j = L, K.

Thus, τLi + τKi = 1 (i = 1, 2). Solving (A.1), then, we obtain (9).
Appendix B.
If the economy is in a perfectly competitive world, the utility function is Cobb–Douglas
and constant returns to scale prevail in both industries, then (A.1) is rewritten as follows:  


L̂1
ŵ
−θ σ
0
θK1 σp100 1 0
  1  K1 p1
   1
0−θK2 σp20
θK2 σp2
0 0 1   ŵ2   λL2 (L̂ − λL1 L̂1 )     
 θL1 σp1
0
−θL1 σp10
0 1 0   r̂1  K̂1     10θL2 σp2
0−θL2 σp2 0 0 1   r̂2  =  λK2 (K̂ − λK1 K̂1 ) .
    







 θL1
p̂
0
θK1
0
−1 0 00 1   




0
θL2
0
θK2
0 0 0  X̂1  
0  


X̂2
0
0
0
0
1 1 −10(B.1)
27

Solving (B.1), then, we obtain
(
(
[
)
)]
1
1
L1 ∂W
1
=−
1 + λL1 θK2− 1 + λL2 θK1
−1
< 0,
W ∂L1
λL2
σp2
σp1
(
(
[
)
)]
K1 ∂R
1
1
1
=−
1 + λK1 θL2− 1 + λK2 θL1
−1
< 0,
R ∂K1
λK2
σp2
σp1
[
(
)
(
)]
1
K1 ∂W
1
1
=
λK1 θK2− 1 + λK2 θK1
−1 ,
W ∂K1
λK2
σp2
σp1
(
[
(
)
)]
1
1
L1 ∂R
1
=
− 1 + λL2 θL1−1 .
λL1 θL2
R ∂L1
λL2
σp2
σp1

(B.2)

Appendix C.
L̇1 = 0 line, K1 = f (L, K; L1 ), and K̇1 = 0 line, K1 = g(L, K; L1 ), are deﬁned by
W (L, K; L1 , f (L, K; L1 )) ≡ 1 and R(L, K; L1 , g(L, K; L1 )) ≡ 1, respectively. Then, we
obtain∂f (L, K; L1 )
WL1
=−
,
∂L1
WK1

∂g(L, K; L1 )
RL
=− 1.
∂L1
RK1

Suppose that condition (a) in Proposition 2 is satisﬁed. Then, both WK1 and RL1 arepositive. Since we already know both WL1 and RK1 are always negative (see (9)), this makes
both ∂f /∂L1 and ∂g/∂L1 positive. Moreover, in this case, we obtain)(( )∂f (L, K; L∗1 ) ∂g(L, K; L∗1 )−= sign σA > 0.
sign
∂L1
∂L1
Thus, we can conclude that both the L̇1 = 0 and K̇1 = 0 lines become upward-sloping curves
that intersect once at most.
On the other hand, if condition (b) in Proposition 2 is satisﬁed, then both WK1 and RL1
become negative, and thus, both the L̇1 = 0 and K̇1 = 0 lines are downward-sloping curves.
Furthermore, since we obtain)(( )∂f (L, K; L∗1 ) ∂g(L, K; L∗1 )−= −sign σA < 0,
sign
∂L1
∂L1
In this case too, the L̇1 = 0 and K̇1 = 0 lines intersect once at most.
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