Discussion Paper No.346
The Heckscher-Ohlin-Samuelson Model andthe Cambridge Capital Controversies

Kazuhiro Kurose & Naoki Yoshihara

March 31, 2016

TOHOKU ECONOMICS RESEARCH GROUPGRADUATE SCHOOL OF ECONOMICS AND
MANAGEMENT TOHOKU UNIVERSITY
27-1 KAWAUCHI, AOBA-KU, SENDAI,
980-8576 JAPANThe Heckscher—Ohlin—Samuelson Model and the Cambridge
Capital Controversies∗
Kazuhiro Kurose†

Naoki Yoshihara‡

March 31, 2016

Abstract
This paper examines the validity of the factor price equalisation theorem (FPET) in relation tocapital theory. Additionally, it presents a survey of the literature on Heckscher—Ohlin—Samuelson (HOS)
models that treat capital as a primary factor, beginning with Samuelson (1953). Furthermore, this paper
discusses the Cambridge capital controversy, which contends that marginal productivity theory does nothold when capital is assumed to be as a bundle of reproducible commodities instead of as a primaryfactor. Consequently, it is shown that under this assumption, the FPET does not hold, even when there
is no reversal of capital intensity. This paper also demonstrates that the recent studies on the dynamic
HOS trade theory generally ignore the diﬃculties posed by the capital controversies and are thereby able
to conclude that the FPET holds even when capital is modelled as a reproducible factor. Our analysis
suggests that there is a need for a basic theory of international trade that does not rely on factor priceequalisation and a model that formulates capital as a bundle of reproducible commodities.
JEL Classification Code: B51, D33, F11.
Keywords: factor price equalisation, capital as the bundle of reproducible commodities, reswitchingof techniques, capital reversing.

∗ An earlier version of this paper was presented at a workshop held at Hitotsubashi University and the 2016 URPE at ASSA
meetings in San Francisco. We are grateful to A. Takamasu for his comment at the workshop. We also thank E. Burmeister, T.
Sasaki, H. Sasaki, and H. Sato for their helpful comments.
† Graduate School of Economics and Management, Tohoku University, Kawauchi 27-1, Aobaku, Sendai 980-8576, Japan.
E-mail: kurose@econ.tohoku.ac.jp‡ Department of Economics, University of Massachusetts Amherst, Thompson Hall, 200 Hicks Way, Amherst, MA
01003, USA; The Institute of Economic Research, Hitotsubashi University, Naka 2-1, Kunitachi, Tokyo 186-0004, Japan;
and School of Management, Kochi University of Technology, Tosayamada, Kami-city, Kochi 782-8502, Japan. E—mail:n_yoshihara_1967@yahoo.co.jp11

Introduction‘Globalisation’ is undoubtedly one of the key words that characterises the 21st Century. It would be definedsuﬃciently as the integration of markets for goods, services, capital, and labour (which had formerly been
segmented by political barriers) into a single ‘world market’, although the various definitions of ‘globalisation’are proposed as Wolf (2004) discusses.
The driving force behind modern globalisation was, at first, the establishment of a free trade system
under the International Monetary Fund and the General Agreement on Tariﬀs and Trade (IMF-GATT),
which gradually reduced tariﬀs after the end of the WWII. Thereafter, the collapse of the IMF’s fixedexchange rate system and its replacement by the flexible exchange rate system further drove globalisation.
Additionally, the relaxation and abolishment of various regulations, which were the product of the counterKeynesian revolution, resulted in the free international movement of capital. Finally, the WTO was founded
and it established comprehensive rules for international transactions of goods, finances, information (i.e.
communication), intellectual property, and services.1
Almost all economic theories have been supportive of globalisation. Indeed, the classical economists of
the 19th century, such as Smith, Ricardo, J.S. Mill, and Marx, asserted that open economies were superior toclosed ones.2 Moreover, nowadays, neo-classical economics also assert the superiority of an open economy byestablishing the Heckscher—Ohlin—Samuelson (HOS) model. As a basis of supporting globalisation, economists
typically refer to the potential gains from trade (Anderson, 2008) enjoyed by every economic agent under open
economies. To explain the potential gains from trade, Ricardo’s theory of ‘comparative advantage’ remains
one of the cornerstones of international economics, while neo-classical economics also argue ‘comparativeadvantage’ within the HOS framework.
However, there are several diﬀerences between neo-classical and classical (including Marx) models on
gains from trade; the former supposes that every country is faced with a common set of techniques, but
diﬀer in terms of factor endowments, while the latter assumes that each country is endowed with its own
techniques which may vary from each other. Note that in the modern economy, globalisation actively promotes
the international movement of not only goods, services, capital, and labour, but also of information andknowledge, which allows everyone to access common information and knowledge of production technology
from anywhere, at least in the long run. In order to capture such a feature of the modern economy, it would
be admissible to assume that every country is faced with a common set of techniques, which is formalized
by a common production possibility set, as in the HOS model. It should be noted, however, that access to
information and knowledge of production technology does not necessarily imply that every country can use
them eﬀectively. In order for a country to use a technique, it must have the necessary capital formation andlabour force. Given the imperfection of international factor markets, the choice of a technique is dependent
on the country’s factor endowments.
In order to analyse globalisation with the HOS model, the validity of a set of theorems (i.e. the HO
theorem, factor price equalisation theorem, the Stolper-Samuelson theorem, and the Rybczynski theorem)
must be examined. Although the discovery of the Leontief Paradox (Leontief, 1953, 1956) precipitated suchan examination, we focus on the factor price equalisation theorem (FPET), which is the cornerstone of the
HOS model. According to this theorem, the equilibrium international price, as determined by free trade,ensures the equalisation of factor prices. Thus, it is important to determine whether or not factor prices tend
to converge in modern globalisation.
In their analysis of the US current account imbalance, Obstfeld and Rogoﬀ (2005) reveal that the income
return on US-owned assets exceeded that on US liabilities by an average of 1.2% a year from 1983 to 2003.
Furthermore, the return on US foreign investments, including capital gains, exceeded that on US liabilities
by a remarkable 3.1% during the same period. If the return is regarded as a measurement of factor price,
how can this persistent diﬀerence be explained? This paper seeks to answer this question by reviewing thedevelopment of the HOS model.3
1 See Wolf (2004) concerning the history of the world economy’s construction in detail.
2 However, Malthus and List exceptionally criticise the free trade doctrine. It is well known that Malthus (1815) criticises thefree trade system for its eﬀects on food security and the stability of prices. List (1904) also criticises free trade for its failure toprotect infant industries.
3 While we have a special interest in capital, several studies examine the inexplicable relationship between international trade
and wage disparities; see, for example, Kurokawa (2014).

2

We pay particular attention to the relationship between the theoretical development of the HOS model andthe outcome of the Cambridge capital controversies, which revealed that the neo-classical principle of marginal
productivity does not, in general, hold. The neo-classical production function treats capital as a primaryfactor of production, and thus, its amount is given independently of the price system. If capital is treated asa bundle of reproducible commodities, however, the neo-classical theory does not hold. Assuming the neoclassical production function, the rate of profit maintains a one-to-one correspondence with a technique. If
capital consists of a bundle of reproducible commodities, however, a technique may correspond to some rate ofprofit. This phenomenon is termed the reswitching of techniques. Additionally, if capital consists of a bundleof reproducible commodities (unlike the principle of marginal productivity argues), then the monotonicallydecreasing relationship between capital intensity and the rate of profit does not generally hold. In otherwords, capital intensity may rise as the rate of profit increases, a phenomenon termed capital reversing. As
we shall argue later, the outcome of the controversies may be used to re-examine the validity of the HOS
model as it assumes the neo-classical production function and as capital is a primary factor. Neo-Ricardians
who were influenced by Sraﬀa (1960), such as Steedman, Metcalfe, and Mainwaring, have thus far conducted
the majority of such re-evaluations. By using the Leontief model with alternative techniques, they assert that
if capital consists of heterogeneously reproducible commodities, then the FPET does not necessarily hold.
The neo-Ricardian arguments may provide a clear explanation for the persistent diﬀerences between the
US’ returns and those of the rest of the world. Since it is clear that in the globalised economy, capital is not a
primary factor, but rather composed of a bundle of reproducible commodities, the FPET does not necessarily
hold. Even if the globalised economy were perfectly competitive as the theory assumes, there could still be
diﬀerences in the returns.
The paper is organised as follows: Section 2 presents a brief survey of the literature on the traditional
HOS model, in which capital is treated as a primary factor. Section 3 deals with the neo-Ricardian critiquesof the FPET and presents a numerical example of the two-integrated-sector model, in which the FPET does
not hold despite a lack of sectoral capital intensity reversal. The model is based on the Leontief model withalternative techniques. It should be noted that our example is more rigorous than that presented by Metcalfeand Steedman (1972). Section 4 reviews neo-classical economists’ counterarguments against the critiques
popularized by the controversies. We show that Burmeister (1978), who presented the most influential formof the HOS model after the controversies, merely assumed that the inconvenient phenomena emphasized by
the controversies would not occur. Section 5 presents our concluding remarks.
Throughout the paper, we assume that international trade does not incur any costs (e.g. transportation
costs and tariﬀs) except for the direct cost of production; there is no perfect specialisation, and as such, every
country produces all commodities; and that there is no joint production unless otherwise stated.

2

The HOS Model with Capital as a Primary Factor of Production

In this section, we examine the traditional HOS model in which capital is treated as a primary factor of
production. Although Heckscher and Ohlin (1991) define the structure of comparative advantage as thediﬀerence in countries’ factor endowments and put forward a prototype of the FPET, it is Samuelson (1953)
who formalises it by using the general equilibrium theory.
Subsequently, Gale and Nikaido (1965) and Nikaido (1968) develope the Samuelsonian formulation;
Samuelson (1966a) and Nikaido (1972) define capital intensity as the relative share of factor costs; andMas-Collel (1979a,b) further developes Nikaido’s (1972) formulation. Kuga (1972) characterises the FPET
without using a Jacobian matrix for the cost function; Blackorby et al. (1993) extend Kuga (1972) by allowing
for decreasing returns to scale and intermediate goods.

2.1

Samuelson (1953)

Following Samuelson (1948, 1949), wherein he proved the FPET with a two-country, two-commodity, twofactor model, Samuelson (1953) extends the theorem by using the general equilibrium model. By simplifying
Samuelson’s (1953) model, wherein there are n commodities and n primary factors, we can define the equilibrium condition as follows:3

p 5 wA (w) ,[p − wA (w)] X = 0,
A (w) X = V,

(1)
(2)
(3)

where p ≡ [pi ] , w ≡ [wi ] , X ≡ [Xi ] , V ≡ [Vi ] ∈ Rn denote the vector of commodity prices, factor prices, output, and factor endowments. A (w) is the physical input coeﬃcient matrix under which the unit cost is
minimised given the primary factor price vector, w; and thus, wA (w) denotes the unit cost function. (1) isthe condition that allows for competitive equilibrium prices; (2) is the condition for the commodity market’s
equilibrium; and (3) is the condition which establishes the full utilisation of factors.
nP
Let us assume the neo-classical production function, Xj = fj (V1j , · · · , Vnj ), where
Vij = Vi .4 As thej=1function is homogeneous of degree one, it can be rewritten as follows:
1 = fj (a1j , · · · , anj ) , where aij ≡

Vij
.
Xj

(4)

It should be noted that the function satisfies the following assumptions:∂fj (a1j , · · · , anj )wi=
for i, j = 1, · · · , n,
pj
∂aij

(5)

∂fj
= 0.
∂aij

³ 0
´
¡ 00
¢
0
00
, · · · , Vnj
The production set satisfies the free disposal condition. For Vj0 ≡ V1j , · · · , Vnj , Vj00 ≡ V1j, and∀λ ∈ (0, 1), moreover, the following is satisfied:
¡
¢
¡ ¢
¡ ¢
fj λVj0 + (1 − λ) Vj00 = λfj Vj0 + (1 − λ) fj Vj00 .

(6)(6) indicates that fj is a concave function. If all commodities are produced and all factors are utilised inevery industry, then the equality holds in both (1) and (5).
nP
Let us denote the unit cost function as follows: c (w) ≡ wA (w) = [cj (w)], where cj (w) ≡wi aij (w).
i=1

If the neo-classical production function is assumed, then function c (w) has the following properties:5
Assumption 2.1.1: c (w) is diﬀerentiable with respect to w.
Assumption 2.1.2: c (w) is a homogeneous function of degree one.
Assumption 2.1.3: c (w) is concave with respect to w.
Assumption 2.1.4: c (w) is monotonically increasing with respect to w.

The FPET holds that the cost function, c: w 7→ p, is global univalent. Let us consider a simple casewherein n = 2. In this case, the FPET’s assumption of no factor intensity reversal causes the factor prices
to equalise in free trade equilibrium. The factor intensity of industry 1 is given by a11 (w) /a21 (w) and thatof industry 2 is a12 (w) /a22 (w). An absence of factor intensity reversal means that½
∀w ≥ 0, a11 (w) a22 (w) − a12 (w) a21 (w) > 0 or
(7)
∀w ≥ 0, a11 (w) a22 (w) − a12 (w) a21 (w) < 0.
4 See Burmeister and Dobell (1970, pp. 8—12) with respect to the neo-classical production function in detail.
5 See, for example, Mas-Colell et al. (1995, p. 141).

4

If (7) is satisfied, then the FPET holds. This is because A (w) is non-singular and thus has an inverse−1
matrix. Therefore,w = pA (w) . In this case,©
ª it is shown that c (w) is bijective over the range of the costfunction, P ≡ p ∈ Rn+ | ∃w ∈ Rn+ : c (w) = p . This directly implies that c is global univalent.6
If we allow factor intensity reversal to occur (i.e. (7) is not satisfied), due to the continuity of function
c (w), a11 (w0 ) a22 (w0 )−a12 (w0 ) a21 (w0 ) = 0 holds for A (w0 ), which is chosen for w0 . If the final commodity
prices in the incompletely specialised-trade equilibrium are given by p0 = w0 A (w0 ), then A (w0 ) is singularand does not have an inverse matrix. Therefore, there are an infinite number of factor price vectors thatwould satisfy the equation; in other words, the factor prices do not equalise.
2P∂ahj (w)∂p∂cj (w)
Partially diﬀerentiating cj (w) with respect to wi yields: ∂wji = ∂w
= aij (w) +
wh ∂w
.
i
i
h=1

Given that

2
P

h=1
1
pj

2
P

h=1

wh

∂fj ∂ahj∂ahj ∂wi= 0 is obtained by diﬀerentiating (4) and

∂fj
∂ahj

=

wh
pj

is obtained from (5),∂ahj (w)= 0 is obtained. As pj > 0, we find that:∂wi∂pj= aij (w) ,∂wii, j = 1, 2.

(8)This implies that partially diﬀerentiating the price equation yields the input coeﬃcient, aij . Samuelson(1953) shows that the cost function’s non-vanishing Jacobian matrix is the suﬃcient condition for the validity
of FPET in the case of n = 2:#"∙¸
∂c1 (w)∂c2 (w)
a11 (w) a12 (w)
∂w1
1
det
= det ∂c∂w
∂c2 (w) 6= 0.
1 (w)a21 (w) a22 (w)
∂w2

∂w2

Furthermore, Samuelson (1953) extends the case of n = 2 to a more general scenario of n = 3 andconjectures the suﬃcient condition of the validity of FPET as follows:⎤⎡ ∂c (w)
1
n (w)
"
#
· · · ∂c∂w
∂w11
∂c1 (w)∂c2 (w)⎢ .
∂c1 (w)
.. ⎥
..
∂w1
⎥
1
⎢ .
(9)
6= 0, det ∂c∂w.
∂c2 (w) 6= 0, · · · , det ⎣.
. ⎦ 6= 0.
1 (w)
∂w1
∂w2
∂w2
∂c1 (w)∂cn (w)
···
∂wn
∂wn

(9) indicates that the suﬃcient condition for the validity of the FPET is that the successive principal
minors of the cost function’s Jacobian matrix be non-vanishing.
Furthermore, it is clear that the condition is also valid for the Leontief production function. This is∂pbecause (8) may simply be rewritten as ∂wji = aij , which has constant coeﬃcients. Therefore, the Jacobianmatrix has one-signed principal minors even when w changes.
Samuelson’s (1953) use of the cost function’s Jacobian matrix to characterise the condition for the validityof the FPET had a decisive impact on the direction of later research.7

2.2

The Application of Jacobian Matrix

Gale and Nikaido (1965) and Nikaido (1968) point out a major flaw in Samuelson (1953).8 Before proceedingwith our analysis, let us first define the following matrices:
Definition 2.2.1: A square matrix, A, is termed a P-matrix if all the principal minors are positive.
6 See the Appendix for the rigorous proof in the case of n = 2.
7 See Chipman (1966) for a description of other HOS models based on the general equilibrium theory.
8 Gale and Nikaido (1965) provide a counter-example for condition (9). They suppose the mapping F (x) ≡ [f (x)] as definedibelow:f1 (x1 , x2 ) = e2x1 − x22 + 3,
f2 (x1 , x2 ) = 4e2x1 x2 − x32 .

The successive principal minors can then be given as:

5

Definition 2.2.2: A square matrix, A, is termed an N-matrix if all the principal minors are negative. An
N-matrix can be further divided into two categories:i) An N-matrix is said to be of the first category if A has at least one positive element.
ii) An N-matrix is said to be of the second category if all of the elements are non-positive.
Let the mapping f : Ω → Rn satisfy the following assumptions:
Assumption 2.2.1: Ω is a closed rectangular region in Rn .9
Assumption 2.2.2: Given that the mapping f (x) ≡ [fj (x)] (x ∈ Ω, j = 1, 2, · · · , n), fj (x) is monotonicallyincreasing and totally diﬀerentiable on Ω:
dfj (x) =nX∂fj (x)j=1∂xi

dxi , (j = 1, 2, · · · , n) .

Consequently, the following theorem holds:
Theorem 1 (Gale and Nikaido, 1965; Inada, 1971; Nikaido, 1968): For a given vector, p ≡ [pj ], mapping
p = f (x) is global univalent if either (a) or (b) holds:(a) The Jacobian matrix of f (x), is everywhere a P-matrix in Ω.
(b) The Jacobian matrix is continuous and is everywhere an N-matrix in Ω.
Proof : See Nikaido (1968, pp. 370—371).
As Ethier (1984, p. 151) points out, Gale and Nikaido’s (1965) assumptions regarding the mapping, f , arequite general and their conditions for global univalence are purely mathematical. Therefore, it is necessary
to clarify the kinds of assumptions that shall be imposed on the cost function in order to capture a standard
economic environment.
Samuelson (1966a) conjectures that ‘factor intensity’ could be defined by the share of the increase in thecost of factor i relative to the increase in the cost of production per unit; in other words, for price equation,p = c (w), the factor intensity, αij , is given as:αij ≡cij (w) wi, (∀i, j = 1, . . . , n),pj∂cj (w)∂wi . αij is the share of the rate of increase in the cost of factor i relative to that in the coste ≡ [αij ] (i, j = 1, 2, · · · , n).10 Moreover, let usof producing one unit of commodity j. Let us define matrix A

where cij (w) ≡e has successive principal minors whose absolute values are bounded from below by constant,assume that A
positive numbers, δ k (k = 1, 2 · · · , n), if its rows and columns are adequately renumbered:11∂f1 /∂x1
∂f2 /∂x1∂f1= 2e2x1 > 0,
∂x1
 




∂f1 /∂x2   2e2x1−2x2
 = 2e2x1 4e2x1 + 5x22 > 0,
=
2x
2x
2
∂f2 /∂x2  8e 1 x2 4e 1 − 3x2 holds for ∀x; therefore, (9) is satisfied. However, F (0, 2) = F (0, −2) = (0, 0), which precludes global univalence.
9 A closed rectangular region is defined as follows:Ω ≡ {x| pi 5 xi 5 qi , i = 1, 2, · · · , n},where −∞ < pi < qi < +∞.
n
1 0 α = 0 and is
αij = 1 are obtained using the Euler Theorem and the homogeneity of the cost functions. Therefore, Aij
i=1

a stochastic matrix.
1 1 (10) is equivalent to (7) if there are two-commodities and two-factors.

6

¯
⎡
¯
α11
¯
¯
⎢ ..
det
¯
⎣ .
¯
¯
αk1

···
..
.
···

Consequently, the following theorem holds:⎤¯
α1k ¯¯.. ⎥¯ = δ , k = 1, 2 · · · , n.
k
. ⎦¯¯
αkk ¯

(10)

Theorem 2 (Nikaido, 1972): If cj (w) satisfies Assumption 2.1.1~2.1.4, then the price equation, p = c (w),is completely invertible for the given p > 0.12
Proof : See appendix.
Theorem 2 verifies Samuelson’s conjecture.13
Stolper and Samuelson (1941) also investigated the relationship between final commodity prices and factor
prices; thus, the approach of using the cost function’s Jacobian matrix is applied to the generalisation of the
Stolper-Samuelson Theorem. Chipman (1969) proposed the following two criteria:i) Weak Stolper—Samuelson Criterion (WSS): An increase in pj leads to a more than proportional
increase in the price of the corresponding factor wj . The price of factor wi (i 6= j) may increase, but the rateof increase is smaller than that of wj :∂ ln wj∂ ln wj∂ ln wi> 1,>.
∂ ln pj∂ ln pj∂ ln pjii) Strong Stolper—Samuelson Criterion (SSS): An increase in pj decreases all factor prices except forthat of wj :∂ ln wi< 0, if i 6= j.
∂ ln pj
e must exist and its diagonal elements must be
In order to satisfy the WSS condition, the inverse of A
£ ¤e −1 ≡ αij (i, j = 1, 2, · · · , n), thegreater than 1 and its non-diagonal elements.14 In other words, letting A
e −1 , implies
WSS condition implies αjj > 1 and αjj > αij (i 6= j). Similarly, the SSS condition, in terms of A
ij
that α < 0 (i 6= j). While Chipman (1969) proves the case of n 5 3, Uekawa (1971) and Uekawa et al.
(1972) rigorously prove the condition for the validity of the Stolper—Samuelson theorem in the case of n = 4.15
1 2 ‘Completely invertibility’ means that p = c (w) 6= c (w0 ) = p0 for arbitrarily positive vectors w 6= w0 and a unique w > 0
exists such that p = c (w) for ∀p > 0.
1 3 Samuelson’s own summary can be found in Samuelson (1967). Moreover, Stiglitz (1970) constructs a dynamic HOS modelby introducing a relationship between savings and investment, and he derives the condition for the FPET. See Smith (1984)
with respect to the dynamic HOS model in detail.
1 4 Chipman (1969) discusses the relationship between Gale and Nikaido’s (1965) condition for the FPET and the WSS condition.
When n = 2, theis equivalent to the WSS condition, but is not if n = 3. Suppose that w0 is determined for a given condition
p and π 0 = ϕ ω 0 (which is defined in the proof of Theorem 2). Let us suppose that its Jacobian matrix is given as follows:0.55 0.40 0.05
 00ϕ  = 0.05 0.50 0.45 .
0.25 0.35 0.40
 
ϕ0  0 is a stochastic matrix. Furthermore, ϕ () is the diﬀerentiable and monotonically increasing function, and all principal  minors of ϕ0 0 are positive, namely ϕ0  0 is a P-matrix and satisfies Theorem 1. This means that the FPET holds. However,
we obtain:


0.77
−2.59
2.82
 0  0 −1
3.77
−4.45 .
= 1.68ϕ −1.95 −1.68
4.64
This means that the WSS condition does not hold.
1 5 The Stolper—Samuelson theorem is similarly generalised by Inada (1971), Kemp and Wegge (1969), Morishima (1976), andWegge and Kemp (1969). See Ethier (1984) for further research on the theorem.

7

2.3

Kuga (1972)

The preceding analyses can only be applied to cases wherein the number of final commodities is equal tothat of factors. In order to overcome this limitation, Kuga (1972) uses a new approach to characterise thecondition for the FPET, which he terms the ‘diﬀerentiation method’.
Let us assume that the general production possibility frontier is given as follows:
X1 = T (V; X) ,where X1 denotes the output of commodity 1, V ∈ Rr+ is the factor endowment, and X ∈ Rn−1
is the output+
vector of commodity 2 to n. Moreover, T satisfies the following assumptions:
Assumption 2.3.1: T is positively homogeneous of degree one with respect to (V; X).
Assumption 2.3.2: T is concave with respect to (V; X).
Assumption 2.3.3: T is strictly concave with respect to X for any fixed V.
Assumption 2.3.4: T is twice diﬀerentiable with respect to (V; X).
Let the price of commodity 1 be the numéraire; then, the problem is expressed as follows:nX

max T (V; X) +pj Xj ,

(11)

j = 2, 3, · · · , n,

(12)

j=2

the solution of which is given by:
pj = −∂T (V; X),∂Xj

Thanks to the Berge maximum theorem, we can see that the set of solutions to (11), Xj , is upper hemicontinuous with respect to V for a given p. Moreover, because of Assumption 2.3.3, the set is singleton.
Therefore, the solution, Xj , is the continuous single valued function of V:
X =X (V; p) .
The price of factor i is given by:wi =∂T (V; X (V; p)),∂Vii = 1, 2, · · · , r.

(13)

The equalisation of factor prices in this model implies that factor price wi is solely dependent on thecommodity price that is determined by free trade, and thus the right-hand side of (13) is kept constant withrespect to the variation of V.
By partially diﬀerentiating (13) with respect to Vτ (τ = 1, 2, · · · , r), we obtain:n
X∂2T∂ 2 T ∂Xj
∂wi
=
+
,
∂Vτ
∂Vτ ∂Vi j=2 ∂Xj ∂Vi ∂Vττ = 1, 2, · · · , r,(14)

in matrix form this is written as

⎡ ∂w1
∂V1

⎢
where wV ≡ ⎣ ...

∂w1
∂Vr

···
..
.
···

wV = M1 + M2 XV ,
⎤
2
∂2T
T
. . . ∂V∂1 ∂V
∂V12
r
∂V1
.. ⎥, M ≡ ⎢
.. ⎥
..
⎥
⎢ ..
1
.
. ⎦
. ⎦,
⎣ .
∂wr
∂2T
∂2T
···
∂Vr
∂Vr ∂V1
∂V 2
∂wr ⎤

⎡

r

8

(15)

⎤
2
⎤
⎡ ∂X2
T
2
· · · ∂V∂1 ∂X· · · ∂X
∂V1
∂Vr
n
⎥
⎢ .
.. ⎥, X ≡ ⎢ ..
.. ⎥.
..
..
M2 ≡ ⎢
⎣ .
.
.
. ⎦ V
. ⎦
⎣ ..
∂Xn
∂Xn
∂2T
∂2T
·
·
·
· · · ∂Vr ∂Xn∂V1∂Vr∂Vr ∂X2
Similarly, partially diﬀerentiating (12) with respect to Vτ yields:⎡∂2T∂V1 ∂X2n

X ∂ 2 T ∂Xl
∂2T
+
= 0,
∂Xj ∂Vτ∂Xl ∂Xj ∂Vτ
l=2

j = 2, 3, · · · , n, τ = 1, 2, · · · , r.

(16)

⎤
2
T
· · · ∂X∂2 ∂X
n
⎥
⎢
..
..
..
⎥ and theIn matrix from, this is written as MT2 + M3 XV = 0, where M3 ≡ ⎢
.
.
.
⎦
⎣
∂2T
∂2T
·
·
·
2
∂Xn ∂X2∂Xn
superscript T denotes the transpose. Thanks to Assumption 2.3.3, the Hessian matrix M3 has an inverseT
16such that XV = −M−1
When this is combined with (15), we obtain:
3 M2 holds.
⎡

∂2T
∂X22

T
wV = M1 − M2 M−1
3 M2 .

(17)

In order for the FPET to hold, wV = 0 must hold:T
M1 = M2 M−1
3 M2

(18)

(17) and (18) have economic implications. Partially diﬀerentiating (12) with respect to pj yields 1 =
n
P
∂Xk
∂2T
−
∂Xk ∂Xj ∂pj , which can be rewritten in matrix form as:
k=2

I = −M3 Xp ,
⎡ ∂X2
∂p2

⎢
where I is an identity matrix of order n − 1 and Xp ≡ ⎣ ...

∂Xn
∂p2

(13) with respect to pj yields

(19)

···
..
.
···

∂X2 ⎤
∂pn

.. ⎥. Similarly, partially diﬀerentiating
. ⎦

∂Xn
∂pn

n

∂wi X ∂ 2 T ∂Xl=(i = 1, 2, · · · , r, j = 1, 2, · · · , n) ,∂pj
∂Xl ∂Vi ∂pj
l=2

which can be written in matrix form as:
wp = M2 Xp ,
⎡ ∂w1

(20)

∂w1 ⎤

· · · ∂pn.. ⎥. Consequently, M−1 = −X holds from (19), as does M = w X−1 from
..
p
2
p p
.
3
. ⎦
∂wr
∂wr
· · · ∂pn
∂p2(20). Because of (17), we therefore obtain:∂p2⎢
where wp ≡ ⎣ ...

wV = M1 + wp MT2 .

(21)

The variations in factor endowments, Vτ , tend to give rise to the variations in factor prices and output.
2
T
The elements of M1 , ∂V∂i ∂V(i = 1, · · · , r), indicate the variation of factor prices vary in response to variation
τ
∂X

∂X

in Vτ when there is no adjustment in Xj by an amount of ∂Vτj (j = 1, · · · , n) (i.e. ∂Vτj = 0). The elements

1 6 Assumption 2.3.3 says that T is strictly concave with respect to X. This implies that the Hessian matrix of T is negative
definite. A square matrix is negative definite if and only if its inverse is negative definite (Mas-Colell et al., 1995, p. 936);
therefore, M3 is invertible.

9

2

T
of M2 , ∂V∂τ ∂X, indicate the discrepancies between international prices, pj , and domestic commodity producj
∂X

∂X

∂T
, in response to variations in Vτ when there is no adjustment in ∂Vτj (i.e. ∂Vτj = 0). On thetion prices, ∂Xjcontrary, the elements of wp convey the adjustment in the wi ’s through the adjustments in the Xj ’s corresponding to the marginal discrepancies in international prices. Therefore, wp MT2 in (21) can be interpreted
as the potential amount of adjustment in the wi ’s through the Xj ’s corresponding to the discrepancies in MT2 .
Consequently, M1 and wp MT2 shall be termed the ‘direct eﬀect’ and the ‘adjustment eﬀect’, respectively. In
order for the FPET to hold in this model (wV = 0), the direct eﬀect must be just oﬀset by the adjustmenteﬀect. By summarising the above analysis, we obtain:

Theorem 3: Under Assumption 2.3.1~2.3.4, the FPET holds if and only if the direct eﬀect is oﬀset by the
adjustment eﬀect.
Kuga (1972) assures the validity of the FPET by keeping the factor price independent of the factor
endowment, which was an entirely diﬀerent approach than previous models had used.

2.4

Mas-Colell (1979a, b)

e rather than the Jacobian matrix of the cost function
Mas-Colell (1979a, b) uses the relative share matrix, A,
to characterise the condition for the FPET. In doing so, he makes the following assumptions regarding thecost function:
Assumption 2.4.1: c (w) is a continuously diﬀerentiable function and homogeneous of degree one.
Assumption 2.4.2: c : Rn++ → Rn++
Although the cost function is usually assumed to be concave with respect to w, only homogeneity is
assumed here. Because of Assumption 2.4.2, the iso-cost curve is unbounded.
As the definition of the WSS condition shows, the relative cost share is related to the cost function in thefollowing manner:
αij ≡wi ∂cj (w).
cj (w) ∂wiConsequently, the following theorem holds.
¯¯¯e ¯¯ > ε holds for
Theorem 4 (Mas-Colell, 1979a): Under Assumptions 2.4.1~2.4.2, if, for some ε > 0, ¯det A
all w ∈ Rn++ , then c (w) is a homeomorphism.
Proof : See the Appendix.
In other words, Theorem 4 implies that for all p ∈ Rn++ the equation p = c (w) has a unique solutionthat continuously depends on p.
Moreover, Mas-Colell (1979a) presented the condition that allows the cost function to be a homeomorphism when it is bounded (i.e. cj : Rn+ → Rn+ ) by utilizing the relative share matrix.
Mas-Colell’s assumptions regarding the cost function are generalisations of Nikaido (1972); the diﬀerence
lies in the fact that the space of the commodity prices and that of factors are homeomorphisms. The
linear homogeneity and concavity of the cost function allow Nikaido (1972) to claim complete invertibility,
but complete invertibility does not require that the invertible mapping be continuous. Incidentally, global
univalence does not require this continuity either.

10

2.5

Blackorby et al. (1993)

Blackorby et al. (1993) also characterise the necessary and suﬃcient conditions for the factor price equlisation,
which is a generalisation of Kuga (1982) in that they allow the possibility of joint production and decreasing
returns to scale. Note that, as Kuga (1972) does, Blackorby et al. (1993) also argue that the factor priceequlisation must be solely dependent on commodity prices and, therefore, independent of factor endowments
in all countries.
Suppose that the international economy consists of C countries, indexed as c = 1, · · · , C. Moreover, there
are M final commodities which are traded freely in international markets and N primary factors that eachcountry is endowed with. Let Xc ∈ RM denote the production vector of country c, the positive elementsof which represent the outputs and the negative elements of which represent the inputs. Vc ∈ RN denotes
factor endowments of country c. Given the transformation function T c : RM +N → R, a net output and factor
endowment vector, (Xc , Vc ), is feasible if and only if T c (Xc , Vc ) 5 0. Furthermore, it should be noted thatT c satisfies the following assumptions:¯©
Assumption 2.5.1: (i) Dc ≡ (Xc , Vc ) ∈ RM+N ¯ T c (Xc , Vc ) 5 0} is a non-empty and closed convex
set, and (0, 0) ∈ Dc ; (ii) T c is increasing in Xc ∈ RM and decreasing in Vc ∈ RN ; (iii) T c is convex in(Xc , Vc ) ∈ RM+N .
Assumption 2.5.2: T c is continuous and twice diﬀerentiable.

The transformation function is related to the production function, Gc , as follows:
T c (Xc , Vc ) = 0 ⇐⇒ Yc = Gc (Zc , Vc ) ,where Xc ≡ (Yc , Zc ). Yc and Zc denote the net output and net input, respectively. Due to Assumption
2.5.1, Gc is concave and decreasing in Zc as well as increasing in Vc . Therefore, the two expressions shown
below represent the same profit maximisation problem:
Rc (p, Vc ) = max{pXc | T c (Xc , Vc ) 5 0},c(22){py Yc + pz Zc | Yc 5 Gc (Zc , Vc )},Rc (p, Vc ) = max
c
c

(23)

X

Y ,Z

where p ≡ (py , pz ) is the price vector. Rc (p, Vc ) satisfies the same properties that the profit function
generally does; that is, Rc (p, Vc ) is a homogeneous function of degree one, non-decreasing convex, andincreasing concave in Vc .
An equilibrium factor price vector Wc ∈ RN and a corresponding equilibrium production vector Xc∗ ∈ RM
of country c = 1, · · · , C, are respectively defined as follows:
Wc
Xc∗

½

½

= ∇V Rc (p, Vc ) , if Rc is diﬀerentiable,∈ ∂V Rc (p, Vc ) , if Rc is not diﬀerentiable,(24)= ∇p Rc (p, Vc ) , if Rc is diﬀerentiable,∈ ∂p Rc (p, Vc ) , if Rc is not diﬀerentiable,(25)

where ∂i Rc (p, Vc ) denotes the sub-gradient set at (p, Vc ), where i = p, Vc .
Here, the equalisation of factor prices is defined as follows:
Definition 2.5.1 (factor price equalisation (FPE)): Equilibrium factor prices are equalised for countriesc = 1, · · · , C if and only if there exists a non-empty, open, convex subset of commodity prices Π ⊆ RM
+ , andfor each p ∈Π, there exists a profile of non-empty, open, convex subsets of factor endowments, (Γc (p))c=1,··· ,Cccsuch that for each p ∈Π there exists a vector W ∈ RN
+ such that W = ∇V R (p, V ) for each countryc = 1, · · · , C and for an arbitrary profile of factor endowments, (Vc )c=1,··· ,C ∈× Γc (p).
c=1,··· ,C11

As a preliminary step, let us introduce the following two concepts which play important roles in thismodel:¡¢
Definition 2.5.2 (Linear Segment): A vector (ψ c , δ c ) = ψ cy , ψ cz , δ c is a linear segment of Gc at (Zc , Vc ) if
and only if there exists an ε > 0 such that
Gc (Zc , Vc ) + λψ cy = Gc (Zc + λψ cz , Vc + λδ c ) ,for all λ ∈ (−ε, ε).
Definition 2.5.3 (Direction of linearity): A vector (ψ c , δ c ) ∈ RM+N is a direction of linearity of T c at
(Xc , Vc ) if and only if there exists an ε > 0 such thatT c (Xc + λψ c , Vc + λδ c ) = 0,
for all λ ∈ (−ε, ε).
While changes in factor endowments generally produce changes in the production vector, a direction of
linearity means that a change in the feasible and eﬃcient production vector, Xc , precipitated by a change
in factor endowments, δ c , is linear; in other words, ∂T c (Xc + λψ c , Vc + λδ c ) = ∂T c (Xc , Vc ). Therefore, achange in the production vector along a direction of linearity does not change the gradient vector of T c .
According to Definitions 2.5.2 and 2.5.3, it is clear that (ψ c , δ c ) is a direction of linearity of T c if and only
if (ψ c , δ c ) is a linear segment of Gc . In what follows, for c = 1, · · · , C, Π and Γc (p) are of full dimension,and as such, the notation ΓcN (p) is used to emphasise this. Here, the necessary and suﬃcient condition for
the FPE to hold is given by the following theorem.
Theorem 5: Under Assumption 2.5.1, the FPE holds for each p ∈ Π and each (Vc )c=1,··· ,C ∈×

c=1,··· ,CΓcN (p)if and only if the following conditions hold:1) there exist N vectors, (ψ i (p) , δ i (p)), for i = 1, · · · , N that are directions of linearity of T c at (Xc∗ , Vc );2) for i = 1, · · · , N , the vectors δ i (p) are linearly independent and the same for all countries;3) the mappings ψ i : Π → RM for i = 1, · · · , N are the same for all countries.

Proof : See the Appendix.
Suppose that the economy has a price vector, p, and a factor endowment of Vc ∈ ΓcN (p). Now imagine
a change in the economy’s endowment of its ith factor, δ i (p). The directions of linearity, (ψ i (p) , δ i (p)),
at each (Xc∗ , Vc ) allow the gradient vector of T c to remain constant if the net output, Xc∗ , changes by theamount of ψ i (p). Since the N independent vectors (δ i (p) for i = 1, · · · , N ) span an N dimensional space,
any change in the economy’s factor endowment can be allocated to N directions of linearity. Therefore, it ispossible for the economy to adjust production to any local change in its factor endowment so that the gradientvector, ∇T c (Xc∗ , Vc ), remains constant. Based on (24) we can see that ∂Rc /∂Vi = p (∂X c∗ /∂Vi ) = Wi .
Since Theorem 5 ensures that ∂X c∗ /∂Vi = ψ i (p) is the same for all countries, the FPE holds.
Theorem 5 implies that even though the equilibrium production vectors diﬀer from country to country(i.e. free trade is achieved), the factor prices can still be equalised. The following theorem emphasises this.
Theorem 6: Under Assumptions 2.5.1 and 2.5.2, the FPE holds for p ∈ Π and (Vc )c=1,··· ,C ∈×

c=1,··· ,CΓcN (p)if and only if there exist ψ i (p) for i = 1, · · · , N that are the same for all countries such that:∇XV T c (Xc∗ , Vc ) = −∇XX T c (Xc∗ , Vc ) Ψ,
c

c∗

c

T

c

c∗

c

∇VV T (X , V ) = Ψ ∇XX T (X , V ) Ψ,
∇X T c (Xc∗ , Vc ) Ψ + ∇V T c (Xc∗ , Vc ) Ω = 0,
12

(26)
(27)
(28)

⎡

ψ 11 (p)⎢where Ψ ≡ ⎣ ...

···
..
.
M
ψ 1 (p) · · ·⎤ψ 1N (p).. ⎥, Ω is an identity matrix of order N defined by Ω ≡ (δ , · · · , δ ), the ele1
N
. ⎦

ψM
N (p)

h c
i
∂T
∂T c
∂T c
· · · ∂X
ments of which, δ i , are the basis vectors for i = 1, · · · , N , ∇X T c (Xc∗ , Vc ) ≡ ∂X
, ∇V T c (Xc , Vc ) ≡
c
c
c
∂X
1
2
M
h c
i
∂T
∂T c
∂T c
c
c∗
· · · ∂V
, Vc ) , ∇XV T c (Xc∗ , Vc ), and others denote the Hessian matrices
c , and ∇XX T (X
∂V1c
∂V2c
N
c
of T .
Proof : See the Appendix.
(26) and (27) demonstrate that substitutability of the factor endowments, ∇VV T c (Xc∗ , Vc ), and theinteraction between endowments and commodities (∇XV T c (Xc∗ , Vc )) are determined by ∇XX T c (Xc∗ , Vc )and ψ i (p), i = 1, · · · , N . This implies that there is no restriction to the substitutability between commodities.
Therefore, if the price of commodities, p, changes, then the equilibrium production vector can vary fromcountry to country. Since ψ i (p) are the same for all countries, however, the production vector’s response tochanges in factor endowments must be the same for every country.
Furthermore, this model can be also applied to the case wherein Π and Γc (p) are not of full dimension,that is, when N dimensional space cannot be spanned while there are N primary factors. Let ΓcK (p) forp ∈ Π denote a K dimensional subset of endowment space where K 5 N . Furthermore, we assume thatΓcK (p) is convex.
Definition 2.5.4: The vectors δ i (p) for i = 1, · · · , K are said to locally span ΓcK (p) at Vc ∈ ΓcK (p) if theyspan the K dimensional aﬃne subset that contains ΓcK (p).
As is shown by Theorems 5 and 6, and given the above concept, factor prices equalise in the K dimensional
subspace (Blackorby et al., 1993).
Blackorby et al. (1993) is more general than Kuga (1972) and Mas-Collel (1979a, b) in that the assumptionof production technique allows for decreasing returns to scale, joint production, and the existence of inputs
other than primary factors (i.e. intermediate goods). In order to derive the condition for factor priceequalisation, it is crucial that the transformation functions, T c , are directions of linearity. Kuga (1972) issimilar to Blackorby et al.’s (1993) in that they both allow the number of final commodities to diﬀer fromthe number of primary factors and both of their conditions for factor price equalisation hold that factor
prices are solely dependent on commodity prices and independent of factor endowments. It should be noted,
however, that Kuga’s definition of factor price equalisation diﬀers slightly from that used by Blackorby et al.;specifically, the former is stronger than the latter. Additionally, both models rely on diﬀerent mechanisms toequalise factor prices. Kuga (1972) uses the method presented in Theorem 3 because the direct eﬀect would
be oﬀset by the adjustment eﬀect, not because the gradient vector of the transformation function would
remain constant in the face of changing factor endowments. The necessary and suﬃcient condition derived
from Blackorby et al. (1993) is weaker than Kuga (1972) in that the class of production economies supposedin the former is broader than that in the latter and the definition of factor price equalisation is weaker.
It is unclear how broad the class of production economies that satisfy Blackorby et al. (1993) necessary
and suﬃcient conditions are. However, we can check whether or not the factor prices equalise in a production
economy by using Theorem 5.

3

The HOS Model with Reproducible CapitalIn the previous section, capital is a primary factor of production. As the classical economists and Marxemphasised, however, the capitalist economic system has been based on the establishment of industrialised
societies since the 19th century. An essential feature of this system is that capital follows a circuit between
a monetary form of value and heterogeneously reproducible commodities.
The Cambridge capital controversies revealed that it would be problematic to introduce the idea ofcapital as a bundle of reproducible commodities into neo-classical economic theory. These controversies
13arose in the 1960’s and 1970’s between the neo-classical economists who resided mainly in Cambridge, Mass.
(e.g. Samuelson, Solow, Modigliani, Burmeister, Meade, and Hahn) and those who resided in Cambridge,
UK (J. Robinson, Pasinetti, Garegnani, Kaldor, and Sraﬀa). The primary sources of the controversies werethe concept of capital, the logical validity of the neo-classical production function and principle of marginal
productivity.
The controversies brought to light several problematic issues with the HOS model, namely the questionsof whether ‘reswitching of techniques’ or ‘capital reversing’ could occur when capital is taken as a bundle ofreproducible commodities. The former is a phenomenon in which one technique could correspond to somerate of profit and the latter is that the decreasing monotonicity between the rate of profit and the capitalintensity would not necessarily hold. These phenomena imply that the properties of the neo-classical costfunction may not hold.
As Sraﬀa (1960) shows, when capital consists of a bundle of reproducible commodities, the price of capitalmust be determined simultaneously with the price structure and the rate of profit, and as such, the capitalendowment cannot be formulated independently of the income distribution, like V in the previous section.
When countries are allowed to choose their techniques, one technique might correspond to some rate of profit.
If the rate of profit is regarded as the factor price, then it would imply that there is a possibility that thecommodity price and the factor price are not global univalent. Therefore, it is unclear whether or not factor
price equalisation can still be characterised by the theorems described in the previous section when capitalconsists of a bundle of reproducible commodities.

3.1

The Cambridge Capital Controversies

First, let us briefly review the most relevant issues highlighted by the Cambridge controversies.17
This paper will focus on ‘capital reversing’. According to the neo-classical production function, the wageprofit curve (or factor price frontier) is convex toward the origin, there is a one-to-one correspondence between
the rate of profit and a technique, and capital intensity is monotonically decreasing with respect to the rate ofprofit (i.e. the marginal productivity of capital principal is at work). In the neo-classical production function,
capital is, of course, a primary factor.
Samuelson (1962) attempts to apply the aforementioned properties of the neo-classical production function
to the case where capital consists of a bundle of reproducible commodities. He constructs a simple modelin which one kind of consumption commodity is produced by using labour and capital, which is, itself,
reproducible. This technique is characterised by fixed coeﬃcients. Moreover, this model assumes capital
to be heterogeneous; it is indexed by capital α, β, γ · · · . Therefore, we cannot produce capital, α, by using
labour and other capital, like β, γ · · · . Furthermore, it is assumed that the capital-labour ratio used to produce
capital, α, is technologically given and is the same as the capital-labour ratio for the consumption commodity
when it is produced by using capital α. The capital-labour ratio to reproduce capital β is diﬀerent from theratio to reproduce capital α, but it is the same as the capital-labour ratio for the consumption commodity
when it is produced by using capital β. The same assumption is imposed on the reproduction of all capital.
In this case, the wage-profit curves that correspond to the diﬀerent capitals become straight. This isbecause, thanks to the assumptions that the techniques are represented by fixed coeﬃcients and both consumption commodity and capital require the same capital-labour ratio, the price structure remains unaﬀectedby changes in the income distribution. The technological relationship concerning income distribution is described by the envelope of all the straight wage-profit lines. The envelope is convex to the origin.
Samuelson concludes that the envelope, which he obtains from heterogeneous capital, could suﬃciently
approximate the wage-profit curve that is obtained if one assumes that capital is a primary factor. The
approximation of the production function is termed the ‘surrogate production function’, and the approximatedcapital is termed ‘surrogate capital’.
If Samuelson were correct, then the principle of marginal productivity of capital could be utilised even if
capital were heterogeneous. This is because there is a one-to-one correspondence between the rate of profit
and the technique. The ‘non-switching’ theorem proven by Levhari (1965) holds that one technique would
1 7 The controversies covered a wide range issues; see Blaug (1975), Cohen and Harcourt (2003), Harcourt (1972), and Pasinetti(2000) for further details on their scope.

14

not correspond to the same rates of profit throughout the entire economic system, which seems to further
support Samuelson’s conclusion.
However, Samuelson’s conclusion relies heavily on the assumption that the capital-labour ratio of capital production is the same as that of consumption production, which is a singularly peculiar assumption.
Pasinetti (1966) is the first to produce a counter-example to Levhari (1965)’s non-switching theorem; asa result, it is made clear that the surrogate production function had no general foundation for economic
analysis.18 Without the assumption, capital reversing can occur; in other words, capital intensity may not
be a monotonically decreasing function of the rate of profit, which contradicts the principle of marginal
productivity.
Moreover, some rates of profit may correspond to one technique, which is termed the ‘reswitching oftechniques’. Suppose that α and β are alternative techniques available in an economic system and r denotes
the rate of profit; the reswitching of techniques occurs if α is the cost minimising technique at r ∈ [0, r1 ],β is the cost minimising technique at r ∈ [r1 , r2 ] where r1 < r2 , and α is the cost minimising technique atr ∈ [r2 , Rα ] where Rα is the maximum rate of profit under technique α.19 This is also a phenomenon that isinconsistent with the principle of marginal productivity. Moreover, it is made clear that the reswitching oftechniques can take place regardless of whether the techniques are decomposable or not.20
The Cambridge capital controversies provide suﬃcient reason to doubt the validity of theorems derived
from the HOS model, which relies on a cost function derived from the neo-classical production function.

3.2

The Model with Reproducible Capital: the case of n = 2

In the 1970s and 1980s, the HOS model was criticised by Mainwaring, Metcalfe, and Steedman exclusively on
the basis of the critiques raised by the controversies (Metcalfe and Steedman, 1972, 1973; Mainwaring, 1984;
Steedman, 1979). Although they present a number of numerical examples and criticised the FPET, they do
not rigorously argue the (in)validity of the FPET under the Leontief production model. Consequently, we
shall derive a theorem that states that the FPET is valid under the Leontief production model with n = 2.
In this case, however, the necessary and suﬃcient conditions for factor price equalisation will be extremely
restrictive. Therefore, we shall show that the theorem’s implications may be interpreted as the impossibility
theorem.
The basic premise of the model is that labour is only a primary factor and that all physical input iscomposed of reproducible commodities. One technique is represented by the Leontief production model;
consequently, the equilibrium price of commodity j (j = 1, · · · , n) is given as follows:
pj = lj w + (1 + r)nXaij pi ,

i=1

where lj > 0, aij = 0, w = 0, r = 0 denote the labour coeﬃcient, physical coeﬃcient, the wage rate, and therate of profit, respectively. For simplicity, we assume that capital is circulating. In general, there are some
Leontief techniques available for the production of commodity j. The criterion for the choice of techniques
is that it minimise the production cost given a certain price system. Suppose that the following is the cost
minimising technique under price system (p, w, r):
1 8 See the set of papers published in the ‘Paradoxes in Capital Theory’ symposium in the Quarterly Journal of Economics :Bruno et al. (1966), Garegnani (1966), Levhari and Samuelson (1966), Morishima (1966), Pasinetti (1966), Samuelson (1966b).
1 9 See Pasinetti (1977, chap. 6). There were three types of reactions from neo-classical economics against the critiques levelledby the neo-Ricardians; the first was to describe the phenomenon as a ‘paradox’, ‘perverse’, ‘exceptional’, ‘inconvenient’, or‘anomalous’, that is, contending that the phenomenon was rarely observed in reality and therefore irrelevant (Blaug, 1975;
Samuelson, 1966b); the second was to attempt to investigate the conditions under which capital reversing or reswitching oftechniques would not take place (Burmeister and Dobell, 1970; Burmeister, 1980); the third was to assert that the neo-Ricardianmodel was merely a special case of the intertemporal general equilibrium model and that it could consequently be freed fromthe neo-Ricardian critiques (Hahn, 1982). See Pasinetti (2000) concerning this topic in detail.
2 0 Burmeister (1980, pp. 114—115) asserts that the reswitching of techniques is an irrelevant phenomenon within the field of
neo-classical economics. This is because it does not necessarily accompany the paradoxical behaviour of consumption. Neoclassical economic thought maintains that the steady state consumption level per capita is a monotonically decreasing functionof the rate of profit. This simple relation is a parable derived from a one-commodity model. According to Burmeister, any
phenomenon which is not inconsistent with the neo-classical parable does not vitiate the essence of neo-classical economics.

15

´
³
(aij (p, w, r))i=1,...,n , lj (p, w, r) for ∀j = 1, · · · , n.

Then, the following theorem is valid.

Theorem 7: In the case of n = 2 under the Leontief production model, the commodity price and the rateof profit are global univalent if and only if there is no capital intensity reversal.
Proof : See the Appendix.
The relationship between the wage-profit curve and relative price is such that:dp
l1 (p, w, r) {1 − (1 + r) a22 (p, w, r) + (1 + r) l2 (p, w, r) a21 (p, w, r)} d2 w1,=−
dr
2a21 (p, w, r)
dr2
which implies:
sign

µ ¶
µ 2 1¶
dp
d w
=
−sign
.
dr2
dr

As Mainwaring (1984) points out, the relative capital intensity determines the sign of dp
dr , which in turn,
determines the form of the wage-profit curve as far as the two-commodity Leontief production model isconcerned. In the two-commodity Leontief production model, if the numéraire industry is more capitalintensive (labour intensive) than the other industry, then the relative price will be a decreasing (increasing)
function of the rate of profit and the wage-profit curve will be concave (convex) to the origin.
2 (p,w,r)a22 (p,w,r)
Under a convex production set, the technical change that reverses the size of l1 (p,w,r)a12 (p,w,r)+ll2 (p,w,r)
2 (p,w,r)a21 (p,w,r)
and l1 (p,w,r)a11 (p,w,r)+l
is not peculiar at all. Therefore, in some limited cases, factor intensity
l1 (p,w,r)reversal may not occur. Therefore, we may interpret Theorem 7 as a de facto impossibility theorem of factor
price equalisation.

3.3

The Neo-Ricardian Critique of the HOS Model

The previous subsection demonstrates that factor intensity could be easily reversed when capital consisted of
reproducible commodities. This leads us to wonder whether we can show that factor prices will not necessarilyequalise in the absence of factor intensity reversals if capital consists of a bundle of reproducible commodities.
In order to answer this question, we utilise a two-integrated-sector model.21 Both Sectors 1 and 2 arecomposed of consumption good and capital good producing industries. Sector 1’s consumption good producing industry is termed Industry 1 and its capital good producing industry is Industry 2; similarly, theconsumption good producing industry of Sector 2 is Industry 3, and the capital good producing industry of
Sector 2 is Industry 4.
Let us assume that Industry 1 has three available techniques:αα(aα11 , a21 , l1 ) = (0.38, 0.63, 0.06) ,³´
aβ11 , aβ21 , l1β = (0.4188, 0.424, 0.265) ,(aγ11 , aγ21 , l1γ ) = (0.52, 0.01, 0.65) .

aιij , ljι denote the amount of commodity i and labour that are required to produce a unit of commodity junder technique ι (ι = α, β, γ). On the other hand, Industry 2 has only one available technique:(a12 , a22 , l2 ) = (0.08, 0, 1) .
2 1 Takamasu (1991) attempts to criticise the FPET by using a numerical example. Unfortunately, however, the capital intensity

reversal occurs in his example. Therefore, he fails to criticise the FPET. Our forthcoming numerical example is a modificationof his example.

16

Figure 1 depicts the envelope of the wage-profit curves that were obtained under each technique. The
vertical axis of the figure represents the wage rate in terms of the commodity produced by Industry 1, w1 .
Insert Figure 1 here.
There are four switching points; technique α is chosen if 0 5 r 5 r1 ∼= 0.18; technique β is chosen
if r1 5 r 5 r2 ∼= 0.317; technique γ is chosen if r2 5 r 5 r3 ∼= 0.503; technique β is chosen again if
r3 5 r 5 r4 ∼= 0.9003;and technique α is chosen again if r4 5 r 5 Rα ∼= 1.066, where Rα is the maximum
rate of profit in Sector 1. The reswitching of techniques occurs.
Let w1ι (r) and k1 (r) denote the wage rate measured by the consumption commodity produced in Sector
1 under technique ι and the capital intensity in terms of the consumption good, respectively. k1 (r) is defined
as follows:22¯ α¯⎧¯ dw1 (r)¯⎪|¯
¯ , if r = 0,
r=0
⎪
dr
⎪
⎪
α
α
⎨ w1 (0)−w1 (r), if 0 < r 5 r1 and r4 5 r 5 Rα ,r
k1 (r) =(29)
w1β (0)−w1β (r)⎪⎪, if r1 5 r 5 r2 and r3 5 r 5 r4 ,⎪r⎪⎩w1γ (0)−w1γ (r), if r2 5 r 5 r3 ,rThe switch from γ to β at r = r3 and that from β to α at r = r4 does not adhere to the monotonicallydecreasing relationship between the rate of profit and capital intensity; in other words, there is capital
reversing.
With respect to Sector 2, let us assume that Industry 3 has two technique alternatives:¡ δ δ¢a33 , a43 , Lδ3 = (0.2, 0.485, 0.03) ,(a²33 , a²43 , L²3 ) = (0.3, 0.41, 0.02) .
On the other hand, Industry 4 has only one available technique:(a34 , a44 , L4 ) = (0.29, 0, 1.61) .
Letting w2 be the wage rate in terms of Sector 2’s consumption commodity (the product of Industry 3),
Figure 2 depicts the wage-profit curves.
Insert Figure 2 here.
In Sector 2, ε switches to δ at r = r5 ∼= 0.205 but the techniques do not reswitch and there is no capital
reversing. Using the same procedure as described in (29), we can obtain capital intensity in terms of Sector
2’s consumption commodity, which is denoted as k2 (r).
Let p1 and p2 be the price of Sector 1’s and Sector 2’s consumption commodities, respectively. In order
to compare the capital intensity of both sectors, it must first be measured in terms of the same commodity
price. Let k2 (r) be the capital intensity of Sector 2 in terms of Sector 1’s consumption commodity. Then,1k2 (r) = k2 (r) × pp21 is, by definition, given. Since the wage rate is uniform in both sectors, pp21 = w
w2 , which1
implies that k2 (r) = k2 (r) × w
w2 .
Table 1 presents a summary of the above model.
Insert Table 1 here.
2 2 See Garegnani (1970).

17

Table 1 shows that Sector 2 is always more capital intensive than Sector 1, which means that no capitalintensity reversal occurs in this model (see Figure 3 as well). Moreover, it shows that the relative price,
p2 /p1 , is not a monotonic function of the rate of profit (see Figure 4). This means that the relative price andthe rate of profit are not global univalent.
Insert Figures 3 and 4 here.
In other words, the FPET does not necessarily hold if capital consists of a bundle of reproducible commodities. This is a problem that neo-classical economists, who treat capital merely as a primary factor,
cannot neglect.
Our numerical example featured four commodities, two of which are, so to speak, the intermediate goods,
while the other two are consumption goods that are usually traded in the international market. Here, we
argue that, with respect to the two consumption goods, there is global univalence between the relative priceand the rate of profit. In this sense, the numerical example is essentially equivalent to a two-good, two-factor
model (the factor is not primary but reproducible here). The setting of the numerical example is parallel tothe model used in the proof of Theorem 7. Notwithstanding, if capital consists of a bundle of reproduciblecommodities, then the global univalence between the relative price and the rate of profit may not necessarily
be ensured.
Note that Dixit (1981, pp. 291-292) argues that the existence of non-tradable goods as circulating
capital inputs other than the two tradable, final consumption goods causes the impossibility to derive a‘simple condition’23 ensuring the univalent relation between prices of commodities and factor prices, since
the expression for the elasticity of a wage and profit rate frontier involves indirect eﬀects working throughinduced changes in prices of non-tradable goods. In comparison to Dixit’s (1981) claim, what we havediscussed above provides a strengthening of the impossibility of the univalence relation, since the standard
condition of no capital intensity reversal is satisfied by the economy constructed in this subsection. It may
suggest a more fundamental source of the impossibility of the univalence relation, which remains to be a
future agenda.

4

The HOS Model after the Controversies

In regard to the introduction of capital as a bundle of reproducible commodities into the HOS model,
Samuelson first argues:
Now suppose there are uniform diﬀerences in factor intensity, so that for some two goods thatare simultaneously produced in both countries, say goods 1 and 2, p1 (r) /p2 (r) = p12 (r) is amonotone, strictly increasing (or decreasing) function of r [the interest rate]. Then, the interest
rate will be equalized by positive trade in those goods alone’ (Samuelson, 1965, p. 49).
Bliss (1967) criticises Samuelson (1965), arguing that the problem is the condition for the monotonic
relationship between the relative price and the rate of profit. However, Samuelson said nothing of this.24
In light of Bliss’ (1967) and Metcalfe and Steedman’s (1972, 1973) critiques, Samuelson (1975) acknowledges the possibility that the FPET might not hold globally but rather just locally when capital consists of abundle of reproducible commodities (i.e. the local factor equalisation theorem). However, Samuelson (1975,
p. 351) believes that Metcalfe and Steedman’s (1972, 1973) warning is non-academic and thus gives little
credence to it. Just as the neo-classicals argue about the Cambridge capital controversies, he contends thatthe phenomena described by Metcalfe and Steedman are unlikely to occur in reality.
Following the neo-Ricardian critiques, Burmeister (1978) constructs the most rigorous model with reproducible capital.
2 3 Here, the simple condition seems to be the standard condition of no capital intensity reversal.
2 4 See Samuelson (1978) for more on this point.

18

4.1

Burmeister (1978)

As Samuelson frankly admits, the FPET does not necessarily hold if capital consists of a bundle of reproducible commodities. Therefore, stronger conditions must be imposed on the model in order for the FPET
to hold.
Burmeister (1978) specifies the conditions by using a P-matrix, which generalised the Stolper—Samuelson
Theorem put forward by Chipman (1969), Inada (1971), and others. Inada (1971) modifies the SSS condition
defined in Section 2.2 as follows:
SSS-I Condition: All the diagonal elements of A−1 are positive and all the non-diagonal elements arenegative.
SSS-II Condition: All the diagonal elements of A−1 are negative and all the non-diagonal elements are
positive.
Here, A has no restriction £except¤ a square and non-negative matrix, while the SSS condition in Section 2.2
e −1 = αij .
is characterised by A
Burmeister (1978) assumes that there are m consumption goods, n reproducible capital goods, and hprimary factors, where h 5 m. First, let us consider an economy in which there is no opportunity to choose
techniques. The rate of profit is denoted by r, the capital good price vector by p ≡ [pi ] (i = 1, 2, · · · n),
the consumption good price vector by s ≡ [si ] (i = n + 1, n + 2, · · · , n + m), and the primary factor pricevector by w ≡ [wi ] (i = 1, 2, · · · , h). Furthermore,
the capital coeﬃcient matrix is represented by A ≡⎤⎡a11 · · · a1n a1,n+1 · · · a1,n+m
⎥
⎢ ..
..
..
..
..
..
⎦, and the primary factor coeﬃcient matrix is represented by e ≡
⎣ .
.
.
.
.
.
an1 · · ·e11 · · ·
⎢ ..
..
⎣ .
.
⎡

ann an,n+1 · · · an,n+m⎤
e1n e1,n+1 · · · e1,n+m
⎥
..
..
..
..
⎦.
.
.
.
.
eh1 · · · ehn eh,n+1 · · · eh,n+m
Consequently, the price equation is given as follows:[p, s] = we+ (1 + r) pA,(30)

where [p, s] = [p1 , · · · , pn , sn+1 , · · · , sn+m ]. It is assumed that m consumption goods and at least one capital
good are freely traded internationally and that consumption good 1 is adopted as the numéraire (sn+1 = 1).
Following Sraﬀa’s (1960) terminology, consumption goods are the ‘non-basic’ goods under the above
assumptions, and as such, the production condition of those ‘non-basic’ goods has no eﬀect on the rate ofprofit or the prices of the ‘basic’ goods.25 This implies that, without losing generality, we can assume thatm = h. As such, matrices A and e can be compiled as follows:⎤⎡a11 · · · a1n a1,n+1 · · · a1,n+h
⎥
⎢
..
..
..
..
..
..
⎥
⎢
.
.
.
.
.
.
⎥ ∙
⎢
¸
⎢ an1 · · · ann an,n+1 · · · an,n+h ⎥⎥ = A1 A2 .
A≡⎢
⎢ e11 · · · e1n e1,n+1 · · · e1,n+h ⎥
A3 A4
⎥
⎢
⎥
⎢
..
..
..
..
.
.
.
.
⎦
⎣
.
.
.
.
.
.
eh1

···

ehn

eh,n+1···eh,n+hLet us assume that A is non-singular and defined as follows:
∙
¸
−1
B1 B2
B≡A =
.
B3 B4

2 5 In order to properly analyse techniques and income distributions, it is important to clearly distinguish between basic andnon-basic goods. According to Sraﬀa (1960), a ‘basic’ good is a commodity that is directly or indirectly required for the
production all commodities, while ’non-basic’ goods encompass all other commodities.

19

B1 , B2 , B3 , B4 denote a square matrix of order n, a (n × h) matrix, a (h × n) matrix, and a square matrixof order h, respectively.
hisn+hLetting s ≡ 1, ssn+2
,
·
·
·
,
sn+1 be the consumption good price vector when sn+1 = 1 allows us to rewriten+1(30) as follows:[p, s] = [(1 + r) p, w] A, or−1[p, s] A= [p, s] B = [(1 + r) p, w] .

(31)
(32)

Here, s is regarded as a given vector, as it is assumed to be determined by free trade. Because of theassumption that A is non-singular, sB3 = w holds, which implies that primary factor prices will equalise.
By transforming the first n equations of (32), we obtain:p [B1 − (1 + r) I] = −sB3 ,(33)

where I is an identity matrix of order n. Because of (33), we can confirm that, given the consumption goodprice vector, the capital good price vector uniquely determines the rate of profit if the price of capital goods
is a monotonic function of the rate of profit, which validates the FPET. As such, we obtain the following
theorem:
Theorem 8: If the production of m capital goods and h (5 m) consumption goods is satisfied by the SSS-II(the SSS-I) condition, then dp
dr < (>) 0.
Proof : See the Appendix.
Theorem 8 implies that, given A and the consumption good price vector, the capital good price vector isa monotonic function of the rate of profit if the SSS-I or SSS-II condition is satisfied. Under this condition,
the prices of not only primary factors but also of capital goods are equalised.
Next, let us consider an economy in which there is a choice of techniques, as in the neo-classical production
function. The relationship between factor rent, qi , and capital good prices, pi , can be obtained in equilibrium:qi
pi = 1+r. Diﬀerentiating (31) with respect to r yields:
∙
¸ ∙
¸
dp ds
dA
dq dw
A + [q, w]
,
=
,
,
dr dr
dr dr
dr
where q ≡ [qi ] (i = 1, 2, · · · , n). Since we assume the neo-classical production function, [q, w] dA
dr = 0 holds;
therefore, we obtain:
∙
¸
∙
¸
dp ds
dq dw
,
B=
,
.
(34)
dr dr
dr dr
−1

We assume that B = A is ensured for all the possible technique choices. In other words, we assume thatwhile the price structure changes as the optimally chosen techniques change (and thus the elements of Achange as well), the change in techniques is limited only in cases where A is non-singular.
As before, we can confirm that, given the consumption good price vector, the capital good price vector
uniquely determines the rate of profit if the price of capital goods is a monotonic function of the rate ofprofit. If we can confirm this, then we can say that the prices of capital goods equalise even when there is achoice of techniques. Consequently, we obtain the following theorem:
Theorem 9: If all countries produce n capital goods and h (5 m) consumption goods subject to the neoclassical production function (4), and if the SSS-II (or SSS-I) condition is satisfied at every feasible factor
price (q1 , · · · , qn , w1 , · · · , wh ) where qi = (1 + r) pi , then dp
dr < (>) 0.
Proof : See the Appendix.
20

Theorem 9 implies that, given the consumption good price vector, the capital good price vector is amonotonic function of the rate of profit if capital goods are produced on the basis of the neo-classicalproduction function and the SSS-I or SSS-II condition is satisfied. This means that there is a one-one
correspondence between the capital good prices and capital rental rates, and thus, capital good rental rates
are internationally equalised under the equilibrium price system. On the other hand, due to the implicit
assumption that B3 always exists, there is a one-to-one correspondence between the primary factor pricevector and the consumption good price vector, and as such, factor prices equalise.
On first inspection, the economic meaning of the SSS-I and SSS-II conditions seems unclear. The SSS-I
condition implies that A−1 is a Minkowski matrix, and the SSS-II condition implies that A−1 is a Metzler
matrix. As Uekawa et al. (1972) point out, the SSS-I and the SSS-II conditions are equivalent to the SSS-I000
and the SSS-II conditions, respectively.
SSS-I0 Condition: The inverse of the non-negative matrix A ≡ [aij ] is a Minkowski matrix if and only if, for
any non-empty proper subset J of N = {1, 2, · · · , n} and any given xi > 0, i ∈ J C , there exists xi > 0, i ∈ J,
such thatX

aij xi >

i∈J

X

X

i∈J C

aij xi <

i∈J

X

i∈J C

aij xi for j ∈ J,
aij xi for j ∈ J C .

SSS-II0 Condition: The inverse of the non-negative matrix A ≡ [aij ] is a Metzler matrix if and only if, forJ ⊂ N and any given wj > 0, j ∈ J C , there exists wj > 0, j ∈ J, such thatX

wj aij <

j∈J

X
j∈J

X

j∈J C

wj aij >X

j∈J C

wj aij for i ∈ J,
wj aij for i ∈ J C .

According to Uekawa et al. (1972), the SSS-I0 and SSS-II0 conditions bear the following economic implications:
SSS-I0 Condition: Suppose that commodities are randomly grouped into two composite commodities, J
and J C , and let xi be the output of the ith commodity. Then, for any non-trivial J and any set of outputs
xi > 0, i ∈ J C , there exists xi > 0, i ∈ J, such that more of the jth factor (j ∈ J) and less of the jth factor(j ∈ J C ) is used to produce the composite commodity J than to produce J C .

SSS-II0 Condition: Suppose that the primary factors are randomly grouped into two composite factors,
J and J C , and let wj be the price of factor j. Then, for any non-trivial J and any set of factor prices,
wj > 0, j ∈ J C , there exists wj > 0, j ∈ J, such that the composite factor J contributes less to the cost of
production of the jth commodity (j ∈ J) and more to the cost of production of the jth commodity, j ∈ J C .
Therefore, the SSS-I and SSS-II conditions characterise the factor intensity. Clearly, these conditions are
extremely strong.
Unlike the traditional HOS model, Theorem 9 treats capital as a bundle of reproducible commodities.
This begs the question as to how Burmeister’s model is related to the outcome of the Cambridge capital
controversies, which we can evaluate by simplifying the model.
The simplest case of the model features one consumption good, one capital good, and one primary factor(i.e. labour).26 In this case, (31) is rewritten as follows:
2 6 In fact, the modern dynamic HOS model features the same structure. See, for example, Chen (1992), Nishimura andShimomura (2002, 2006), and Bond et al. (2011, 2012).

21

[p, s] = [(1 + r) p, w] A,
∙
¸
a
a12
A ≡ 11
.
l1
l2

(35)

Despite the fact that capital goods are reproducible, (35) is a de facto one-good model with respect tothe determination of factor prices because the consumption goods are non-basic goods. Since dp
dr > 0 holds
because of (35), the model maintains a one-to-one correspondence between the rate of profit (or factor rental)
and the price of capital goods, given the price of consumption goods.27 This implies that the factor prices
equalise. However, it is obvious that the simplified model simply avoids the diﬃculties pointed out by the
Cambridge capital controversies because it is a de facto one-good model.
In other words, Burmeister’s (1978) model is structured so as to circumvent the issues pointed out in
the Cambridge capital controversies as it assumed away several economic environments. For instance, unlikein the Leontief production model, there are never reproducible goods, like corn, which can be used as both
capital and consumption goods.
The simplest example in which there exist commodities that can be used as both capital and consumption
goods is a two-good economy in which both commodities are basic goods. In this case, we obtain:
∙
¸
a
a12
A ≡ 11> 0; L ≡ (l1 , l2 ) > 0.
a21 a22
By applying Burmeister (1978) to this case, we find that A is given as follows:⎡⎤
a11 a12 0
A ≡ ⎣ a21 a22 0 ⎦ .
l1l2 0Here, det A = 0, which means that A is singular. Therefore, we cannot obtain the property that is ensuredby Theorems 8 and 9.

5

Concluding Remarks

In this paper, we surveyed the HOS model in order to determine how it was modified in reaction to the
neo-classical economists’ critiques in the Cambridge capital controversies.
Burmeister (1978) makes one of the most significant contributions to the HOS model in light of thecontroversies. He derives the conditions for factor price equalisation under the assumption that there exist
reproducible capital goods. The modern dynamic HOS models that feature reproducible capital goods, suchas Chen (1992), Nishimura and Shimomura (2002, 2006), and Bond et al. (2011, 2012), have essentially thesame structure as Burmeister (1978). As previously mentioned, these models use assumptions to avoid thephenomena which the controversies highlighted, and thus, they exclude many economic environments that
could arise in reality.
If there is no room to consider capital as a bundle of reproducible commodities rather than a primaryfactor, then the validity of the HOS model should be carefully reconsidered. First, if capital is treated as abundle of reproducible commodities, then capital intensity reversal may easily arise. Second, even if there is
no reversal of capital intensity, the global univalence between the rate of profit and the relative price may nothold. In other words, even in a two-good model, the FPET may not necessarily hold if capital is treated as abundle of reproducible commodities. All of these arguments suggest that it is necessary to construct a basic
theory of international trade that does not rely on factor price equalisation and treats capital as a bundle ofreproducible inputs.
2 7 In this case, not only the price of capital goods but also that of consumption goods has a one-to-one correspondence withwl1
the rate of profit. Due to p = 1−(1+r)a, we obtain:
11ds
wl1 a12 [1 − (1 + r) a11 ] + (1 + r) wl1 a12 a11
> 0.
=
dr
[1 − (1 + r) a11 ]222

6References

Anderson, J. (2008) International trade theory, in Nurlauf, N., Blume, L. (eds.) The New Palgrave Dictionaryof Economics, Second Edition, Vol. 4, Basingstoke, Macmilan, pp. 516—522.
Berger, M. (1977) Nonlinearity and Functional Analysis, New York, Academic Press.
Blackorby, C., Schworm, W., Venables, A. (1993) Necessary and suﬃcient conditions for factor price equalization, Review of Economic Studies, 60, pp. 413—434.
Blaug, M. (1975) The Cambridge Revolution: Success or Failure?: A Critical Analysis of Cambridge Theories
of Value and Distribution, London, Institute of Economic Aﬀairs.
Bliss, C. (1967) Reviews of The Collected Scientific Papers of Paul A. Samuelson Vol. 2, Economic Journal,
77, pp. 338—345.
Bond, E., Iwasa, K., Nishimura, K. (2011) A dynamic two country Heckscher—Ohlin model with nonhomothetic preferences, Economic Theory, 48, pp. 171—204.
Bond, E., Iwasa, K., Nishimura, K. (2012) The dynamic Heckscher—Ohlin model: A disgrammatic analysis,
International Journal of Economic Theory, 8, pp. 197—211.
Bruno, M., Burmeister, E., Sheshinski, E. (1966) The nature and implications of the reswitching of techniques,
Quarterly Journal of Economics, 80, pp. 526—553.
Burmeister, E. (1978) An interest rate and factor price equalization theorem with nontraded commodities,
Journal of International Economics, 8, pp. 1—9.
Burmeister, E. (1980) Capital Theory and Dynamics, Cambridge, Cambridge University Press.
Burmeister, E., Dobell, A., (1970) Mathematical Theories of Economic Growth, New York, Macmillan.
Chen, Z. (1992) Long-run equilibria in a dynamic Heckscher—Ohlin model, Canadian Journal of Economics,
25, pp. 923—943.
Chipman, J. (1965) A survey of the theory of international trade: Part 1, The classical theory, Econometrica,
33, pp. 477—519.
Chipman, J. (1966) A survey of the theory of international trade: Part 3, The modern theory, Econometrica,
34, pp. 18—76.
Chipman, J. (1969) Factor price equalization and the Stolper-Samuelson theorem, International Economic
Review, 10, pp. 399—406.
Cohen, A., Harcourt, G. (2003) Retrospective: What happened to the Cambridge capital theory controversies?, Journal of Economic Perspectives, 17, pp. 199—214.
Dixit, A. (1981) The export of capital theory, Journal of International Economics, 11, pp. 279—294.
Ethier, W. (1984) Higher dimensional issues in trade theory, in Jones, R., Kenen, P. (eds.) Handbook ofInternational Economics, Vol. I, Amsterdam, North Holland, pp. 131—184.
Gale, D., Nikaido, H. (1965) The Jacobian matrix and global univalence of mappings, Mathematische Annalen,
159, pp. 81—93.
Garegnani, P. (1966) Switching of techniques, Quarterly Journal of Economics, 80, pp. 554—567.
Garegnani, P. (1970) Heterogeneous capital, the production function and the theory of distribution, Review
of Economic Studies, 37, pp. 407—436.
Hahn, F. (1982) The neo Ricardians, Cambridge Journal of Economics, 6, pp. 353—374.
Harcourt, G. (1972) Some Cambridge Controversies in the Theory of Capital, Cambridge, Cambridge University Press.

23

Heckscher, E., Ohlin, B. (1991) Heckscher—Ohlin Trade Theory, Cambridge (Mass.), MIT Press.
Inada, K. (1971) The production coeﬃcient matrix and the Stolper-Samuelson condition, Econometrica, 39,
pp. 219—239.
Kemp, M., Wegge, L. (1969) On the relation between commodity prices and factor rewards, International
Economic Review, 10, pp. 407—413.
Kuga, K. (1972) The factor-price equalization theorem, Econometrica, 40, pp. 723—736.
Kurokawa, Y. (2014) A survey of trade and wage inequality: Anomalies, resolutions and new trends, Journalof Economic Surveys, 28, pp. 169—193.
Leontief, W. (1953) Domestic production and foreign trade; The American capital position re-examined,
Proceedings of the American Philosophical Society, 97, pp. 332—349.
Leontief, W. (1956) Factor proportions and the structure of American trade, Review of Economics andStatistics, 38, pp. 386—407.
Levhari, D. (1965) A nonsubstitution theorem and switching of techniques, Quarterly Journal of Economics,
79, pp. 98—105.
Levhari, D., Samuelson, P. (1966) The nonswitching theorem is false, Quarterly Journal of Economics, 80,
pp. 518—519.
List, F. (1904 [1841]) The National System of Political Economy, Translated by Lloyd, S., London, Longman,Green.
Mainwaring, L. (1984) Value and Distribution in Capitalist Economies: An Introduction to Sraﬃan Economics, Cambridge, Cambridge University Press.
Malthus, T. (1815) The Grounds of an Opinion on the Policy of Restricting the Importation of Foreign Corn,London, John Murray.
Mas-Colell, A. (1979a) Two propositions on the global univalence of systems of cost function, in Green J.,
Scheinkman, J. (eds.) General Equilibrium, Growth and Trade: Essays in Honour of Lionel McKenzie, Vol.
I, New York, Academic Press, pp. 323—331.
Mas-Colell, A. (1979b) Homeomorphism of compact, convex sets and the Jacobian matrix, SIAM Journal on
Mathematical Analysis, 10, pp. 1105—1109.
Mas-Colell, Whinston, M., Green, J. (1995) Microeconomic Theory, Oxford, Oxford University Press.
Metcalfe, J., Steedman, I. (1972) Reswitching and primary input use, Economic Journal, 82, pp. 140—157(reprinted in Steedman (1979), pp. 15—37).
Metcalfe, J., Steedman, I. (1973) Heterogeneous capital and the Heckscher—Ohlin—Samuelson theory of trade,in Parkin, M. (ed.) Essays in Modern Economics, London, Longman, pp. 50-60 (reprinted in Steedman(1979), pp. 64—76).
Morishima, M. (1966) Refutation of the nonswitching theorem, Quarterly Journal of Economics, 80, pp.
520—525.
Morishima, M. (1976) The Economic Theory of Modern Society, Cambridge, Cambridge University Press.
Nikaido, H. (1968) Convex Structures and Economic Theory, New York, Academic Press.
Nikaido, H. (1972) Relative shares and factor price equalization, Journal of International Economics, 2, pp.
257—263.
Nishimura, K., Shimomura, K. (2002) Trade and indeterminancy in a dynamic general equilibrium model,
Journal of Economic Theory, 105, pp. 244—259.
Nishimura, K., Shimomura, K. (2006) Indeterminancy in a dynamic two-country model, Economic Theory,29, pp. 307—324.
24

Obstfeld, M., Rogoﬀ, K. (2005) Global current account imbalances and exchange rate adjustments, Brookings
Papers on Economic Activity, 2005, pp. 67—123.
Pasinetti, L. (1966) Changes in the rate of profit and switches of techniques, Quarterly Journal of Economics,
80, pp. 503—517.
Pasinetti, L. (1977) Lectures on the Theory of Production, New York, Columbia University Press.
Pasinetti, L. (2000) Critique of the neoclassical theory of growth and distribution, BNL Quarterly Review,
215, pp. 383—431.
Samuelson, P. (1948) International trade and the equalisation of factor prices, Economic Journal, 58, pp.
163—184.
Samuelson, P. (1949) International factor price equalisation once again, Economic Journal, 59, pp. 181—197.
Samuelson, P. (1953) Prices of factors and good in general equilibrium, Review of Economic Studies, 21, pp.
1—21
Samuelson, P. (1962) Parable and realism in capital theory: The surrogate production function, Review of
Economic Studies, 29, pp. 193—206.
Samuelson, P. (1965) Equalization by trade of the interest rate along with the real wage, in Baldwin, R., etal. (eds.) Trade, Growth, and the Balance of Payments: Essays in Honor of Gottfried Haberler, Amsterdam,
North-Holland Publishing Company, pp. 35—52.
Samuelson, P. (1966a) 1965 postscript, in Stiglitz, J. (ed.) The Collected Scientific Papers of Paul A Samuelson, Vol. II, Cambridge (Mass.), MIT Press, p. 908.
Samuelson, P. (1966b) A summing up, Quarterly Journal of Economics, 80, pp. 568—583.
Samuelson, P. (1967) Summary on factor-price equalization, International Economic Review, 8, pp. 286—295.
Samuelson, P. (1975) Trade pattern reversals in time-phased Ricardian systems and international eﬃciency,
Journal of International Economics, 5, pp. 309—363.
Samuelson, P. (1978) Interest rate equalization and nonequalization by trade in Leontief-Sraﬀa models,
Journal of International Economics, 8, pp. 21—27.
Smith, A. (1984) Capital theory and trade theory, in Jones, R., Kenen, P. (eds.) Handbook of International
Economics, Vol. I, Amsterdam, North Holland, pp. 289—324.
Sraﬀa, P. (1960) Production of Commodities by means of Commodities: Prelude to a Critique of Economic
Theory, Cambridge, Cambridge University Press.
Steedman, I. (1979) Fundamental Issues in Trade Theory, London, Macmillan.
Stiglitz, J. (1970) Factor price equalization in a dynamic economy, Journal of Political Economy, 78, pp.
456—488.
Stolper, W., Samuelson, P. (1941) Protection and real wages, Review of Economic Studies, 9, pp. 58—73.
Takamasu, A. (1991) Neo-Ricardian Trade Theory, Sobunsya, in Japanese.
Takayama, A. (1985) Mathematical Economics, 2nd edition, Cambridge, Cambridge University Press.
Uekawa, Y. (1971) Generalization of the Stolper—Samuelson theorem, Econometrica, 39, pp. 197—217.
Uekawa, Y., Kemp, M., Wegge, L. (1972) P-and PN-Matrices, Minkowski-, and Metzler-Matrices, and generalizations of the Stolper—Samuelson and Samuelson—Rybczynski theorems, Journal of International Economics,
3, pp. 53—76.
Wegge, L., Kemp, M. (1969) Generalizations of the Stolper—Samuelson and Samuelson—Rybczynski theorems
in terms of conditional input-output coeﬃcients, International Economic Review, 10, pp. 414—425.
Wolf, M. (2004) Why Globalization Works, New Haven, Yale University Press.
25

7

The Appendix

In the Appendix, we show the rigorous proofs of the theorems and a numerical example.

7.1

The Proof of the FPET in the two-good and two-country model

Proof : In order to prove the global univalence of the cost function, c, it is suﬃcient to show that, for priceequation p = wA (w), the relative price, p01 (w) ≡ pp12 , and w are global univalent. In other words, when
the factor prices change in such a way that w → w+ M w ≡ (w1 + M w, w2 − M w) for M w > 0, it must be
confirmed that p01 (w) and w are global univalent if p01 (w) → p01 (w+ M w) is monotonic. It is necessary to∂p01 (w)∂p01 (w)∂p01 (w)∂p01 (w)show that ∂w
M w − ∂w
M w > 0 or ∂w
M w − ∂w
M w < 0 always holds for ∀p01 (w). Thanks to1
2
1
2∂p01 (w)a11 (w) p2 − p1 a12 (w)w2 (a11 (w) a22 (w) − a12 (w) a21 (w))
=
=
and
2
2
∂w1
(p2 )
(p2 )
∂p01 (w)a21 (w) p2 − p1 a22 (w)w1 (a12 (w) a21 (w) − a11 (w) a22 (w))
=
=
,
2
∂w2
(p2 )
(p2 )2
we obtain:∂p0 (w)∂p01 (w)
Mw− 1
M w > 0 ⇔ ∀w ≥ 0, a11 (w) a22 (w) − a12 (w) a21 (w) > 0 and∂w1∂w2∂p01 (w)∂p0 (w)∀w ≥ 0,
Mw− 1
M w < 0 ⇔ ∀w ≥ 0, a11 (w) a22 (w) − a12 (w) a21 (w) < 0.
∂w1
∂w2
∀w ≥ 0,From (7), we obtain:∀w ≥ 0,∂p0 (w)∂p01 (w)∂p0 (w)∂p01 (w)
Mw− 1
M w > 0; or ∀w ≥ 0,
Mw− 1
M w < 0.
∂w1
∂w2
∂w1
∂w2

This implies that p01 (w) is monotonic in relation to the change in factor prices w → (w1 + M w, w2 − M w).
In other words, p01 (w) and w are global univalent. This means that the FPET holds under condition (7).

7.2

The Proof of Theorem 2First, we shall prove the following lemma.
∂fLemma 1: Let f : Rn → Rn be a continuous and partially diﬀerentiable mapping, where fij ≡ ∂xji . Supposethat there are 2n positive numbers mk , Mk , for k = 1, 2 · · · , n, such that the absolute values of the successive
principal minors satisfy:¯⎤¯⎡¯
f11 · · · f1k ¯¯¯¯⎢.. ⎥¯ 5 M , for k = 1, 2, · · · , n,..
mk 5 ¯det ⎣ ...
k
.
. ⎦¯¯
¯
¯
fk1 · · · fkk ¯in the whole of Rn . Then, the system of equations, f (x) = a, has a unique solution in Rn for any given
vector, a > 0.
Proof : See Nikaido (1972).
In order to prove Theorem 2, it is useful to transform the variables in the cost function as follows:ω ≡ ln w = [ln wi ] and π ≡ ln p = [ln pi ]. Therefore, π = ln c (eω ) ≡ ϕ (ω). Function ϕ is continuous and∂ ln pcij (w)wi
wii ∂cj= αij holds. In other words,
diﬀerentiable in Rn . Therefore, ∂ ln wji = w
cj ∂wi = cj × cij (w) =pj
26∂ ln pj
∂ ln wiis the share of factor i’s growth rate relative to the growth rate of the production cost of commodity
e is the Jacobian matrix of π = ϕ (ω). As such, we can prove Theorem 2.
j. Therefore, matrix A

Proof : The system of equations f (x) = a in Lemma 1 corresponds to ϕ (ω) = π. Because of fij = αij = 0,e is a stochastic matrix, which implies
δ k in (10) can serve as mk in the Lemma. As previously mentioned, AnP
that αij ∈ [0, 1] and thatαij = 1. Since the determinant is a polynomial of its elements, it is clear thati=1

the principal minors are bounded from above, which justifies the existence of Mk . Therefore, the system of
equations ϕ (ω) = π satisfies Lemma 1. This immediately shows that ϕ (ω) = π has a unique solution for any
given vector, π > 0. Therefore, c (w) = p has a unique solution for any given vector, p > 0.

7.3

The Proof of Theorem 4

Proof : Here, Function π (ω) = ln c (eω ) from Theorem 2 is used. From Assumption 2.4.2, we know thate = [αij ], where Jπ (ω) is the Jacobian matrix of π (ω). Since, as previously
π : Rn → Rn and Jπ (ω) = A
e is a stochastic matrix (which implies αij ∈ [0, 1]), the absolute values of all elements of Jπ (ω)
mentioned, A¯¯¯
e ¯¯ > ε > 0. Let F : Rn → Rn be a continuous andare uniformly bounded. Furthermore, we assume ¯det A
°
°
°
−1 °
diﬀerentiable function. The suﬃcient condition for °(JF (x)) ° 5 k is that i) |JF (x)| = ε > 0, and ii) thenabsolute values of all elements of JF (x) must be uniformly° bounded,°where kk is a norm defined in R as°
−1 °kT k ≡ maxkxk=1 kT xk. Moreover, if det JF (x) 6= 0 and °(JF (x)) ° 5 k for some k > 0, then F will bea homeomorphism (Berger, 1977, p. 222). Therefore, π : Rn → Rn is a homeomorphism, which means thatc : Rn++ → Rn++ is also a homeomorphism.

7.4

The Proof of Theorem 5First, we shall present some lemmas necessary for the proof.
Lemma 2: Let F : D ⊂ Rk → R be a concave function and β ∈ ∂F (X0 ) and β ∈ ∂F (X1 ). Then,for ∀μ ∈ [0, 1], β ∈ ∂F (Xμ ) where Xμ = μX0 + (1 − μ) X1 . Furthermore, for ∀μ ∈ [0, 1], F (Xμ ) =F (X0 ) + β (Xμ − X0 ) so that {β (Xμ − X0 ) , (Xμ − X0 )} is a linear segment at X0 .
Proof : Since β is the sub-gradient of F at X0 and X1 , the following inequalities hold for all X ∈D:F (X) 5 F (X0 ) + β (X − X0 ) ,F (X) 5 F (X1 ) + β (Xμ − X1 ) .

Multiplying the inequalities by μ and 1 − μ, respectively, yields:
F (X) 5 μF (X0 ) + (1 − μ) F (X1 ) + β (X − Xμ ) ,(36)

for ∀μ ∈ [0, 1]. On the other hand, the concavity of F ensures that
F (Xμ ) = μF (X0 ) + (1 − μ) F (X1 ) .

(37)

From (36) and (37), we know that F (X) 5 F (Xμ ) + β (X − Xμ ), which implies that β ∈ ∂F (Xμ ). Consequently, we obtain:
F (X0 ) 5 F (Xμ ) + β (X0 −Xμ ) and F (Xμ ) 5 F (X0 ) + β (Xμ −X0 ) ,from which F (Xμ ) = F (X0 ) + β (Xμ −X0 ). Therefore, {β (Xμ − X0 ) , (Xμ − X0 )} is a linear segment at
X0 .
Lemma 3: Factor prices equalise for p ∈Π, Vc ∈ Γc (p) if and only if
27

Rc (p, Vc ) = R0c (p) +NX

Ri (p) Vic ,

(38)

i=1

where Vc ≡ [Vic ] for c = 1, · · · , C. In this case, Rc is diﬀerentiable so thatW = ∇V Rc (p, Vc ) = [R1 (p) , · · · , RN (p)] .

(39)

for all countries.
Proof : Necessity. If the profit function takes the form presented in (38), then it is diﬀerentiable and (39)holds, which is consistent with (24).
Suﬃciency. From Definition 2.5.1, (ΓcN (p))c=1,··· ,C is defined for p ∈ Π where Π ⊆ RM
+ is a non-emptycNcand open convex set. Moreover, W ∈ R+ exists and W = ∇R (p, V ) , for ∀c = 1, · · · , C, for an arbitrary
profile (Vc (p))c=1,··· ,C ∈× ΓcN (p). For c = 1, · · · , C, i = 1, · · · , N, and Vc (p, Vc ) ∈ ΓcN (p),c=1,··· ,CRc (p, Vc + (δ ci , 0−i )) − Rc (p, Vc )
δ i →0(Vic + δ ci ) − Vic

Wi = limcholds, where 0−i stands for the vector of order N − 1 excluding the ith element. Since, from Definition
2.5.1, the assumption for Lemma 2 is satisfied by W as shown above, the following is satisfied for arbitraryc
Vc , V ∈ ΓcN (p):
³
´
³
´
c
c
Rc (p, Vc ) = Rc p,V + W Vc − V .
Since W is independent of Vc , Ri : Π → R+ exists for i = 1, · · · , N . Since Wi = Ri (p), we obtain:
N
³
³
´ X
´
c
c
Ri (p) Vic − V i , for ∀c = 1, · · · C.
Rc (p, Vc ) = Rc p,V +
i=1

³
´ P
N
c
c
c
By treating V as the vector fixed in ΓcN (p), we obtain: R0c (p) ≡ Rc p,V −Ri (p) V i .
i=1

¡
¢
¡
¢
Lemma 4: Suppose that F : D ⊆ Rk → R is concave and βδ 1 , δ 1 and βδ 2 , δ 2 at X are linear segments
where β ∈ ∂F (X) in X ∈ D. Then, (βδ, δ) is also a linear segment at X for δ = μδ 1 + (1 − μ) δ 2 for allμ ∈ [0, 1].
Proof : Let us define X1 = X+λ1 δ 1 and X2 = X+λ2 δ 2 where λ1 ∈ (−ε1 , ε1 ) and λ2 ∈ (−ε2 , ε2 ). Then,β ∈ ∂F (X1 ) and β ∈ ∂F (X2 ) hold based on Definition 2.5.2. Since Lemma 2 can be applied, we find thatβ ∈ ∂F (Xμ ) holds, where Xμ is the convex combination of X1 and X2 . Thanks to Lemma 2 F (Xμ ) =F (X) + β (Xμ −X0 ) holds, which means that (βδ, δ) is a linear segment of F in X.
Lemma 5: Let Rc be the profit function defined by (23). A linear segment of Rc in Vc at (p, Vc0 ) is
equivalent to a linear segment of Gc in (Zc , Vc ) at (Zc0 , V0c ), where Zc0 is the profit maximiser in (23) given(p, Vc0 ) for some p ∈ Π.
Proof : If Rc has a linear segment (Wδ, δ) at (p, Vc0 ), then W = ∇V Rc (p, Vc0 + λδ) for all λ ∈ (−ε, ε) forsome ε > 0. Letting p = (py , pz ),∇V Rc (p, Vc0 ) =N
X∂Gc (Zc0 , V0c )pyi= py ∇V Gc (Zc0 , V0c ) .
∂V
i
i=1

From the two equations shown above, we obtain:28

W = ∇V R

c(p, Vc0 + λδ) =NX∂Gc (Zc (λ) , V0c + λδ)
pyi
,
∂Vi
i=1

= py ∇V Gc (Zc (λ) , V0c + λδ) ,where Zc (λ) is the profit maximiser at (p, V0 + λδ). Let (Zc1 , V1c ) ≡ (Zc (λ) , V0 + λδ) for all λ ∈ (−ε, ε).
Since
Ãµ
!
µ
¶
¶
c
c
c
c
c
c
y ∂G (Z , V )y ∂G (Z , V )ycccp ∇G (Z , V ) =
pj
, pi
∂Zj
∂Vj
j=1,··· ,M
i=1,··· ,Nand∂Rc∂Gc (Zc , Vc )= pyj
+ pzj = 0 (∀j = 1, · · · M ) ,∂Zj∂Zj
as such, we obtain:
µ
¶
c
c
c
y ∂G (Z , V )p ∇Z G (Z , V ) = pj
= −pz .
∂Zj
j=1,··· ,M
y

c

c

c

Therefore, py ∇Gc (Zc , V0c ) = (−pz , W) holds. Similarly, py ∇Gc (Zc (λ) , V0c + λδ) = (−pz , W) also holds.
Consequently, we obtain:(−pz , W) = py ∇Gc (Zc0 , V0c ) = py ∇Gc (Zc1 , V1c ) .
¡¢From Lemma 2, therefore, we obtain: (−pz , W) = py ∇Gc Zcμ , Vμc , where Zcμ = μZc0 + (1 − μ) Zc1 , Vμc =μV0c + (1 − μ) V1c for ∀μ ∈ [0, 1]. Here, letting β = (−pz /py , W/py ) and δ = (Zc1 , V1c ) − (Zc0 , V0c ),we see that(βδ, δ) is a linear segment of Gc at (Zc0 , V0c ).
In order to prove the converse, suppose that (ψ y , ψ z , δ) is a linear segment of Gc at (Zc0 , V0c ). As such,the following holds:
Gc (Zc0 , V0c ) + λψ y = Gc (Zc0 + λψ y , V0c + λδ) .
Therefore, the following relationships must hold for all λ ∈ (−ε, ε) for some ε > 0:py ∇Z Gc (Zc0 + λψ x , V0c + λδ) = py ∇Z Gc (Zc0 , V0c ) = −pz ,andpy ∇V Gc (Zc0 + λψ x , V0c + λδ) = py ∇V Gc (Zc0 , V0c ) = W.

Because of ∇V Rc (p, Vc0 ) = py ∇V Gc (Zc0 , V0c ), W=∇V Rc (p, Vc0 ). Similarly, ∇V Rc (p, Vc0 + λδ) =p ∇V Gc (Zc0 + λψ c , V0c + λδ) holds for ψ z , thus satisfying py ψ y = Wδ−pz ψ z . Therefore, W = ∇V Rc (p, Vc0 + λδ)holds for all ∀λ ∈ (−ε, ε). This means that (Wδ, δ) is a linear segment of Rc in V at (p, Vc0 ).
yNow, let us proceed to the proof of Theorem 5.
Proof : Necessity. Assume that the factor prices are equalised. As described in Lemma 3, the profit
function, Rc , has N linear segments at any point (p, Vc ) ∈ Π × ΓcN (p). Let us denote these linear segmentsas (Ri (p) , δ i ), where δ i for i = 1, · · · , N can be chosen as the standard basis vector for RN . Furthermore, let
Zc∗ be the equilibrium production vector given (p, Vc0 ). From Lemma 5, we see that if (Ri (p) , δ i ) is a linear
segment for Rc at (p, Vc ) there exists a vector, (ψ ci , δ i ), that is a linear segment of Gc at (Zc∗ , Vc ). Therefore,
at any point (p, Vc ) ∈ Π × ΓcN (p), there exist N linear segments (ψ ci (p, Vc ) , δ i ) for Gc at (Zc∗ , V0c ). Since(ψ i , δ i ) are linear segments of Gc , they are directions of linearity of T c .

29

3:

Since δ i for i = 1, · · · , N are the standard basis vectors, the following holds for λ ∈ (−ε, ε) from Lemma
Rc (p, Vc + λ (δ i , 0−i )) = Rc (p) + λRi (p) .

Moreover, from the definitions of the profit function and linear segments, we can see that:p {Xc∗ + λψ ci (p, Vc )} = Rc (p) + λRi (p) .

Since pXc∗ = Rc (p) if λ = 0, we obtain:pψ ci (p, Vc ) = Ri (p) .

(40)

This implies that ψ ci (p, Vc ) is independent of Vc ∈ ΓcN (p) and is the(40) must be satisfied for ∀p ∈ Π.
same for all countries.
Suﬃciency. We assume that production function Gc is N linear segments (ψ i (p) , δ i ) for i = 1, · · · , N ,where δ i are standard basis vectors. From Lemma 5, Rc has N linear segments in V given p. For i = 1, · · · , N ,cctherefore, Rc (p, Vfor ∀λ ∈ (−ε, ε) for some ε > 0. Based on Lemma−i¡ 1 + λ (δ i , 0¢ )) = R (p) + λRi (p) holds
δ4, it is clear that N R (p) , N is a linear segment of Rc in V at (p, Vc ). In other words, the following holdsfor ∀λ ∈ (−ε/N, ε/N ) for some ε/N > 0:
Rc (p, Vc + λδ) = Rc (p, Vc ) + λNX

Ri (p) δ i .

i=1

By applying Lemma 2 and using the same logic as that used in the proof of Lemma 3, we obtain what follows:
Rc (p, Vc ) = R0c (p) +N
X

Ric (p) Vic .

i=1

Here, ψ i :Π → R, for i = 1, · · · , N , are common to all countries and, as shown in (40), pψ i (p) = Ric (p) , c =1, · · · , C, holds for i = 1, · · · , N . In other words, we obtain:Rcc(p, V ) = R0c (p) +

N
X
i=1

Ri (p) Vic , for ∀c = 1, · · · , C.

From Lemma (3), we can see that the factor prices are equalised for arbitraryµ¶c
× ΓN .

³´
p, (Vc )c=1,··· ,C ∈ Π ×

c=1,··· ,C

7.5

The Proof of Theorem 6

Proof : Necessity. We assume that factor prices are equalised. Theorem 5 shows that there exist N vectors,(ψ i (p) , δ i (p)), for i = 1, · · · , N , that are directions of linearity for T c . In other words, there exists ε > 0such that for λ ∈ (−ε, ε):T c (Xc∗ + λψ i (p) , Vc + λδ i ) = 0,
diﬀerentiating which with respect to λ yields:∇X T c (Xc∗ , Vc ) Ψ+∇V T c (Xc∗ , Vc ) Ω = 0.
This is simply a repetition of (28). From the above, we obtain:∇XX T c (Xc∗ , Vc ) Ψ + ∇XV T c (Xc∗ , Vc ) Ω= 0,∇XV T c (Xc∗ , Vc ) Ψ + ∇VV T c (Xc∗ , Vc ) Ω= 0.
30

The former is (26) and pre-multiplying it by ΨT , yields:
ΨT ∇XX T c (Xc∗ , Vc ) Ψ + ΨT ∇XV T c (Xc∗ , Vc ) Ω = 0.

(41)Pre-multiplying the latter by ΩT yields:ΩT ∇XV T c (Xc∗ , Vc ) Ψ + ΩT ∇VV T c (Xc∗ , Vc ) Ω = 0,transposing this yields:
ΨT ∇XV T c (Xc∗ , Vc ) Ω + ΩT ∇VV T c (Xc∗ , Vc ) Ω = 0.

(42)

Since Ω is an identity matrix, ∇VV T c (Xc∗ , Vc ) = ΨT ∇XX T c (Xc∗ , Vc ) Ψ is obtained from (41) and (42).
Suﬃciency. (26) and (27) imply the following relationships:∙¸£¤ ψ i (p)
c
c∗
c
c
c∗
c
∇XX T (X , V ) ∇XV T (X , V )= 0,δ i (p)∙¸£¤ ψ i (p)∇XV T c (Xc∗ , Vc ) ∇VV T c (Xc∗ , Vc )= 0.
δ i (p)

These equations imply that the Hessian matrix of T c has N eigenvectors that satisfy (28) and that are associated with zero eigenvalues. The eigenvectors are given by (ψ i (p) , δ i (p)), i = 1, · · · , N . These eigenvectors
are directions of linearity of T c . Therefore, Theorem 5 implies that the FPET holds.

7.6

The Proof of Theorem 7

Proof : The wage-profit curve in the case of n = 2 is given as:

w1 (r) ={1 − (1 + r) a11 (p, w, r)} {1 − (1 + r) a22 (p, w, r)} − (1 + r)2 a12 (p, w, r) a21 (p, w, r),
l1 {1 − (1 + r) a22 (p, w, r)} + (1 + r) l2 (p, w, r) a21 (p, w, r)

where w1 (r) ≡ w(r)p1 . The relative price of commodity 2 is given by:
p (r) =
Therefore, we obtain:l2 (p, w, r) {1 − (1 + r) a11 (p, w, r)} + (1 + r) l1 (p, w, r) a12 (p, w, r).
l1 (p, w, r) {1 − (1 + r) a22 (p, w, r)} + (1 + r) l2 (p, w, r) a21 (p, w, r)
dp (r)
l1 (l1 a12 + l2 a22 ) − l2 (l1 a11 + l2 a21 ).
=dr
l1 {1 − (1 + r) a22 } + (1 + r) l2 a21

If the techniques are productive, then the denominator is positive for all feasible rates of profit. Therefore, thesign of dp
dr is solely dependent on the numerator. The relative price and the rate of profit have the following
relationship if the techniques are productive:dp
a12 (p, w, r) + pa22 (p, w, r)
a11 (p, w, r) + pa21 (p, w, r)
S0⇔
S
.
dr
l2 (p, w, r)
l1 (p, w, r)
21 (p,w,r)
22 (p,w,r)Here, a11 (p,w,r)+pa
is the capital intensity of industry 1 and a12 (p,w,r)+pa
is that of industry
l1 (p,w,r)l2 (p,w,r)
2. In other words, whether the relative price is monotonically increasing or decreasing with respect to therate of profit is dependent on the relative size of the capital intensities. The relative price is a monotonicallydecreasing function with respect to the rate of profit if and only if industry 1 is more capital intensive thanindustry 2. Conversely, it is monotonically increasing if and only if industry 2 is more capital intensive than31

industry 1.28 In other words, the relative price and the rate of profit have no monotonic relationship (i.e.
global univalent) if and only if capital intensity reversal takes place at the threshold of the relative price.

7.7

The Proof of Theorem 8

Proof : Proving the theorem for the SSS-II condition will be suﬃcient. Since we assume that −sB3 < 0holds and matrix [B1 − (1 + r) I] has all positive diagonal elements and all negative oﬀ-diagonal elements,
the solution for (33) is p > 0 for r ∈ [0, r∗ ) where r∗ is the maximum rate of profit as determined by the−1
Frobenius root of B1 (Takayama, 1985, p. 393). This implies that [B1 − (1 + r) I] < 0. Diﬀerentiatingdp(33) with respect to r, yields dr [B1 − (1 + r) I] − p = 0. In other words, we obtain:
dp−1= p [B1 − (1 + r) I] < 0.
dr
We can prove the theorem for the SSS-I condition in a similar manner.

7.8

The Proof of Theorem 9

Proof : Proving the theorem for the SSS-II condition will be suﬃcient. The first n equations of (34) are
dp
given as dq
dr = p + (1 + r) dr , which implies that:dsdp
dp
B1 + B3 = p + (1 + r)
.
dr
dr
dr
ds
= 0. Consequently, we obtain:
Since consumption prices are exogenously given, dr

dp[B1 − (1 + r) I] = p.
dr−1

If the SSS-II condition is satisfied, then [B1 − (1 + r) I]< 0. Therefore, we obtain:dp= p [B1 − (1 + r) I]−1 < 0.
dr
We can prove the theorem for the SSS-I condition in a similar manner.

2 8 Whether the relative price is monotonically increasing or decreasing can be determined by the technical coeﬃcients, independently of the price:dp
l1 (p, w, r) a12 (p, w, r) + l2 (p, w, r) a22 (p, w, r)
l1 (p, w, r) a11 (p, w, r) + l2 (p, w, r) a21 (p, w, r)
S 0 ⇐⇒S.
dr
l2 (p, w, r)
l1 (p, w, r)

32

Figure 1: The wage-profit curve of sector 1

Figure 2: The wage-profit curve of sector 2
p2
r
w1
w2
k1
k2
p1
0
0.826 0.854 0.966 1.451 1.565
0.1 0.692 0.705 0.981 1.336 1.36
0.2
0.58 0.577 1.004 1.073 1.392
0.3 0.488 0.481 1.015 1.021 1.125
0.4 0.407 0.396 1.0295 0.797 1.075
0.5 0.328 0.319 1.0277 0.796 1.015
0.6 0.258 0.250 1.030 0.894 0.967
0.7 0.193 0.187 1.0320 0.859 0.924
0.8 0.133 0.128 1.0323 0.827 0.884
0.9 0.076 0.074 1.031 0.798 0.846
1.0 0.029 0.023 1.238 0.796 0.978
1.04 0.011 0.004 2.723 0.783 2.118
Table 1: The real wage rate, relative price, and capital intensity

33

Figure 3: The capital intensity (real line shows for sector 2 and dashed line for sector 1)

Figure 4: The relative price34