Finnish Economic Papers – Volume 22 – Number 1 – Spring 2009
Factor Supply Changes in Small Open Economies:
Rybczynski Derivatives under Increasing
Marginal Costs*
ANDREAS PFINGSTEN
University of Münster, Department of Banking, Universitätsstr. 14–16,
D-48143 Münster, Germany;
email: [email protected]
and
REINER WOLFF
University of Fribourg, Department of Quantitative Economics, Boulevard de Pérolles 90,
CH-1700 Fribourg, Switzerland; email: [email protected]
We discuss the validity of Rybczynski’s theorem under increasing marginal costs
within firms or industries. In particular, we show that an extra supply of any factor may lead to an expansion of all sector outputs if at least one sector permits
input substitution. We provide a corresponding necessary and sufficient condition.
This condition can even be satisfied when the equilibrium is Walrasian and Marshallian stable. Our findings are also robust with respect to aggregate improvements in total factor productivity which raise the economy’s outputs beyond private returns. (JEL: D50, F11)
1. Motivation
The famous theorem by Rybczynski (1955) is
one of the single most significant insights of
modern pure trade theory. Given the standard
Heckscher-Ohlin framework of a small open
* Earlier versions of this paper have been presented at
North American and European meetings of the Econometric
Society, at meetings of the German and European Economic Associations, and at various faculty seminars, including
the Universities of Essen, Göttingen, Linz, Munich, British
Columbia/Vancouver, and Zürich (ETH). We are indebted to
C. Blackorby, W. Buhr, W.E. Diewert, T. Hens, G. Gabisch,
W. Kohler, and J. Mercenier, among others, for helpful comments. Two anonymous referees provided insightful suggestions which lead to a considerable improvement of the
­paper.
economy with two commodities and factors, the
theorem states that an extra supply of some input will induce the production sector where this
input is used more intensively to expand more
than proportionally. At the same time, the other
sector will contract. Proofs of this theorem usually rely on the convenient simplification that
scale effects can be assumed away due to constant returns in production. The underlying notion of a small economy is a relative concept.
Under globalization, an economy or economic
region may quickly become small relative to a
growing world market.
Not surprising, the theorem does not in general carry over to variations of the base para
Finnish Economic Papers 1/2009 – Andreas Pfingsten and Reiner Wolff
digm. Blackorby/Schworm/Venables (1993, p.
426) have shown that under a regime of joint
production deviations from the usual Rybczynski adjustments may even occur for constant
returns to scale. Other authors, starting from a
seminal paper by Jones (1968), have introduced
technologies with scale economies or diseconomies. An important feature of these technologies is that any improvement or deterioration in
the overall productivity of inputs is considered
as an output externality which is not within the
exploitation potential of a single firm or industry. From the perspective of any one firm,
though, the constant-returns hypothesis is maintained. The concept of perfectly competitive
markets can thus be preserved. So-called comparative-statics paradoxes may then emerge and
may sometimes persist despite supplementary
conditions. The literature also indicates that
these paradoxes do not occur, however, when
the commodity markets are Marshallian stable
(cf. Mayer (1974), Panagariya (1980), Ide/
Takayama (1988, 1990), Ingene/Yu (1991),
Ishikawa (1994), Burney (1996)).
The assumed constancy of internal returns to
production can be motivated on a-priori grounds
by a standard replication argument: A firm may
always choose to reiterate its activities – such
that, e.g., a duplication of the firm’s inputs must
result in twice as much output as before. Yet this
popular view has been criticized by Buchanan/
Yoon (1999, pp. 517) as converting a production
function into an ‘engineering tautology’. To
quote Kemp (1969, p. 154):
“The assumption of constant returns to scale
[...] has been a very convenient assumption
since it has enabled us to focus attention [...]
It is conceivable, however, that Nature is
blind to the comforts of academic economists.”
In a more recent contribution, Choi (1999, p.
18), providing a number of further references,
points out that the assumption of constant returns to scale
“[...] precludes any divergence in industrial
returns to scale, which have been recognized
as an indisputable phenomenon since the
early days of economic science.”
10
Empirical support against the constant-returns
assumption has been provided by the new literature on the total factor productivity of a large
number of (mostly manufacturing) industries in
Europe (Caballero/Lyons (1990, 1991), Oulton
(1996), Midelfart/Steen (1999), Lindström
(2000), Henriksen/Midelfart/Steen (2001)) as
well as in Canada and in the United States (Caballero/Lyons (1992), Benarroch (1997)). While
the main focus of this literature is on positive
inter-industry externalities, many authors also
found strong evidence of decreasing returns to
scale within firms or industries or within regions
(e.g., Caballero/Lyons (1990, 1991), Burnside
(1996), Lindström (2000)).1 We conclude that
decreasing returns are a legitimate and even important topic to be analyzed in the context of the
Rybczynski problem.
Therefore, our model of a small open and
competitive economy with two or more goods
and factors imposes the conjugate assumption
that firms operate under decreasing returns to
scale, i.e. with increasing marginal costs. This
economy can be interpreted as a geographic region which is exposed to global competition.
We assume that the firms in this region benefit
from agglomeration advantages and knowledge
spillovers which they perceive as an exogenous
productivity gain. The model is otherwise fairly
standard. In particular, following Woodland
(1982), our analysis is carried out entirely in
dual space. We start from (minimum) cost functions and, hence, exploit the structure and regularity imposed by duality theory irrespective of
the functional form of the underlying technology.2
In contrast to the conclusions of earlier writers and Rybczynski’s theorem notwithstanding,
in our model all outputs may benefit simultaneously from larger endowments. As a notable
exception in the literature, Ylönen (1987, pp.
214–216) already pointed out that such an outcome can arise – as a plausible implication –
1 The reader may consult Basu/Fernald (1995, 1997) for
a critical literature assessment from a methodological perspective.
2 Thereby, we need no longer refer to traditional, however less convenient, concepts such as the ‘production possibility locus’ of an economy (cf. Herberg/Kemp (1969)).
Finnish Economic Papers 1/2009 – Andreas Pfingsten and Reiner Wolff
from the Hansson/Lundahl (1983) model of a
2 × 2 economy with differentiable and homo­
geneous production functions, if the degrees of
homogeneity are less than one for both goods.
(Also cf. a respective brief remark by Hansson/
Lundahl (1983, p. 538).) We provide necessary
and sufficient conditions for our general result
which requires that at least one production sector permits input substitution. Our conditions
are illustrated by a selection of examples. These
examples cover both substitution and non-substitution technologies, as well as technologies
with a specific factor. We allow for inferior inputs like low-skilled labor, which do not exist
under constant returns, and consider factor complements when more than two factors are
present (cf. Diewert (1982, p. 569)). A further
general result, which differs from both the 2 × 2
case with constant returns and the Hansson/
Lundahl (1983) approach, states that all factor
prices may fall when any single endowment is
increased. Finally, and contrary to the literature,
we show that our observations do not rule out
the Walrasian and Marshallian stability of an
equilibrium. Equally important, our findings are
robust with respect to global shifts in total factor productivity which raise the economy’s outputs beyond private returns.
The paper is organized as follows: Section 2
provides our major results on the existence of
non-standard adjustments to endowment changes of a small competitive economy under increasing marginal costs. Two computational
examples with different parameterizations are
presented in Section 3 for the purpose of illustration. In Section 4, stability is shown. Section
5 introduces external scale effects, and Section
6 concludes.
2. A Small-Country Model with
Increasing Marginal Costs
We consider n > 2 profit maximizing sectors or
industries each producing xi > 0 (i = 1, …, n)
units of a single output from at least one input.
The economy’s total number of inputs is m > 2.
Every input shall be used for the fabrication of
some good. There are fixed local factor endowments v´ = (v1, …, vm) where vj > 0 for all j = 1,
…, m.3 The small, open economy takes world
output prices pi > 0 (i = 1, …, n) as exogenous
whereas domestic factor prices wj > 0 (j = 1,
…, m) are determined endogenously due to factor immobility. We also assume that the minimum cost in each sector i of producing xi output
units at factor prices w´ = (w1, …, wm) can be
computed from a twice continuously differen­
tiable cost function Ci :  +m+ × ++ → ++ with
the following properties satisfied for all (w, xi)
  +m+ × ++: Cost is a nondecreasing first-order
homogeneous and (weakly) concave function of
factor prices w. Marginal cost is positive and (at
least locally) increasing (because of decreasing
internal returns to scale in production).4
The last-mentioned condition will be satisfied,
for instance, if the marginal cost curve of a representative firm in a given industry and, hence,
the firm’s average cost curve are U-shaped.
Then, the profit maximizing output of this firm
is bounded from below by the positive output
level which minimizes average cost at given
factor prices. This output (the so-called efficient
scale, Mas-Colell et al. (1995, p. 144)) will always be determined along the upward-bending
part of the marginal cost curve and establishes
a discrete market-entry barrier for new competitors.
Our analysis relies on the postulate that
the economy attains an equilibrium state in
which all factors are at full employment. Hence,
Σ in=1 C wi j (w, xi) = vj for all j, where C wi j is shorthand notation for ∂Ci/∂wj and thus stands for the
quantity demanded of factor j in sector i according to Shephard’s Lemma. At the same time,
marginal costs shall equal output prices in all
sectors i: C xi i (w, xi) = pi. We assume that each
industry supports a given number of firms which
can operate at least at their efficient scale without loss. This number shall not be affected by
small parameter changes. We are thus given
m + n model equations. Total differentiation
yields:
3 By default, all vectors shall be defined as column vectors. The prime ‘ /’ denotes transposes.
4 We note for a cost function which is continuously differentiable in factor prices that the isoquants of the underlying production function do not allow for linear substitution between inputs.
11
firstly,
DChanges
is positive
definite
and,
secondly, the diagonal inverse D
Pfingsten/Wolff:
Pfingsten/Wolff:
Pfingsten/Wolff:
Factor
Factor
Factor
Supply
Supply
Supply
Changes
Changes
in Small
ininSmall
Small
Open
Open
Open
Economies
Economies
Economies
5 55exists
number of firms which can operate at positive
least at their
efficient scale without
loss. This
definite.
positive definite.
D
ber shall not be affected by small parameterWe
changes.
We
are
thus
given
m
+
model
thethe
the
present
present
present
context,
context,
however,
however,
however,
all
all
entries
alln
entries
along
along
along
thethe
diagonal
diagonal
ofD
are
Dare
are
positive.
positive.
positive.
Therefore,
Therefore,
Therefore,
−1
M
We
define
:=
C
−the
Cdiagonal
D−1of
Cof
(cf.
Gantmacher
(1990,and
pp. 45-4
ww
xw
xw
define
M
:=
Ccontext,
−
C
D
Centries
(cf.
Gantmacher
(1990,
pp. 45-46))
req
ww
xw
xw
firstly,
firstly,
DD
isDpositive
isispositive
positive
definite
definite
definite
and,
and,
and,
secondly,
secondly,
secondly,
thethe
the
diagonal
diagonal
diagonal
inverse
inverse
inverse
D DDexists
exists
exists
and
and
and
is likewise
isislikewise
likewise
thatand
det(M)
= 0. Wolff
Certainly
some
technologies
are
excluded
by this assum
ions. Total differentiation
yields: Papers
Finnish Economic
1/2009
–firstly,
Andreas
Pfingsten
Reiner
that
det(M)
= 0.definite.
Certainly
some technologies
are excluded by this assumption. For
positive
positive
positive
definite.
definite.
ample, suppose that no sector permits input5substitution such that Cww i
Pfingsten/Wolff:
Factor Supply

  Changes
 in Small
 Open Economies
–1, the
existence
of
Dxw
regularity
oftwo
M
ismore
necesample,
that
no
sector
input
such
that
Cww
is require
aare
null
−1
−1
−1
(cf.
Also
suppose
that
thesubstitution
cost
structures
of
or
sectors
evenma
iden
We
We
We
define
define
define
MM
:=
M
:=
C
:=
CC
−permits
C
−−
CCxw
D
DC
CCxw
(cf.
Gantmacher
Gantmacher
Gantmacher
(1990,
(1990,
(1990,
pp.
pp.
pp.
45-46))
45-46))
45-46))
and
and
and
require
require
Cww Cxw
dw suppose
dv
ww
ww
ww
xw
xwD
xw(cf.






=
,
−1 C
−1 C
(1) sary
and
sufficient
for
a
unique
solution
of
(1)
case
C
D
will
not
have
full
rank.
Then,
since
M
=
−C
D
xw
xw
that
that
that
det(M)
det(M)
det(M)
=

=

0.
=

0.
Certainly
0.
Certainly
Certainly
some
some
some
technologies
technologies
technologies
are
are
are
excluded
excluded
excluded
by
by
this
by
this
this
assumption.
assumption.
assumption.
For
For
For
exexexAlso
that the cost structures of xw
two or more sectors are even identical, in wx
C
D
dx suppose dp
in
terms
changes
in
factor
rents
w
and
the present xw
context, however, all entries
along
the that
diagonal
ofof
D
positive.
Therefore,
regularity
of
Mare
will
be
violated.
Basically,
however,
our
regularity
assump
ample,
ample,
suppose
suppose
suppose
that
nono
sector
nosector
sector
permits
permits
permits
input
input
input
substitution
substitution
substitution
such
that
that
that
Cww
CCww
is sector
isnull
aanull
matrix.
matrix.
−1
 matrix.
ww
case Cxw Dample,
Cxw
willthat
not
have
full
rank.
Then,
since such
Msuch
=
−C
Dais−1
Cnull
xw–1
xw , the assu
outputs
x.6 In
all,
due
to
det(M) ≠ 0,
M
exists
−1
nates
borderline
singular
cases
ofis
little
economic
significance.
Given
the ex
Also
Also
Also
suppose
suppose
suppose
that
that
that
the
the
the
cost
cost
cost
structures
structures
structures
of
of
two
of
two
two
or
or
more
or
more
more
sectors
sectors
sectors
are
are
are
even
even
even
identical,
identical,
identical,
in
in
which
in
which
which
firstly,
D
is
positive
definite
and,
secondly,
the
diagonal
inverse
D
exists
and
likewise
ducing as x and p the vectors of sector
outputs
andwill
output
prices, Basically,
respectively,
regularity
ofcase
M
be
violated.
however,
our
regularity
assumption
onlyofel
introducing
p the
vectors of
sector
out−1
−1
−1
C
 will
−1
−1
−1
C
,athe
and
the
following
comparative-statics
responses
the
regularity
of
M
is
necessary
and
sufficient
for
unique
solution
case
case
C
C
C
D
D
D
C
C
will
will
not
not
not
have
have
have
full
full
full
rank.
rank.
rank.
Then,
Then,
Then,
since
since
since
M
M
=
M
−C
=
=
−C
−C
D
D
D
C
C
,
,
the
the
assumed
assumed
assumed
nas x and

xw
xw
xw
xw
xw
xw
xw
xw
xw
xw
xw
xw
n
k
k
1 , . . . , C n ) and
definite.
with matricespositive
C
:=
= nates
), C
:= (Cxcases
xw
ww and
puts
output
prices,
respectively,
and
wi wregularity
xfactor
w
k=1 Cww
k=1 (C
to
changes
in
therents
world
market
prices
p6 and
nan
1w
j regularity
changes
in
w significance.
and
sector
outputs
x.
Inthe
all,
due
to
det(M)
borderline
singular
of little
economic
Given
existence
of =
D
regularity
ofwith
of
M
ofM
will
Mwill
will
be
be
violated.
be
violated.
violated.
Basically,
Basically,
Basically,
however,
however,
however,
ourour
our
regularity
regularity
regularity
assumption
assumption
assumption
only
only
only
elimielimielimin
n
k
k
i
5
−1

−1
−1
−1
C
: =
Σ
C = Σ
(C ),
C
: =
tional
endowments
v
result
(cf.
Hadley
(1974,
diag(Cxi xi ). matrices
and
the
following
comparative-statics
responses
to
changes
in
the
world
ww
w
w
w
w
xw
nates
nates
nates
borderline
borderline
borderline
singular
singular
singular
cases
cases
cases
of
of
little
of
little
little
economic
economic
economic
significance.
significance.
significance.
Given
Given
Given
the
the
the
existence
existence
existence
of
of
D
of
D
D
,
,
,
=1 Cixw
We define M := C
Cxw
pp. 45-46))
requiresolution of (1) in termm
j of(cf.
ww −
k =1
kD
the
regularity
MGantmacher
is necessary (1990,
and sufficient
for and
a unique
(C 1 , …, C n ) and D : = diag (C i ).5
−1 −1
−1
pp.and
107–109)):
national endowments v result (cf. Hadley (1974, pp. 107-109)):
w
x n w
x ixthe
the
regularity
regularity
regularity
ofkof
M
of M
M
is is
necessary
is necessary
necessary
and
and
sufficient
sufficient
sufficient
for
for
a unique
aa unique
unique
solution
solution
solution
of of
(1)
of (1)
(1)
in in
terms
in terms
terms
of ofof−1
i the
1det(M)
6 for
that x an
=
0. Certainly
some in
technologies
are
excluded
byand
this
assumption.
For
exWe can verify from
substitution
matrix
C
whether
inputs
changes
factor
rents
w
and
sector
x.x.
all,
due
to
det(M)
=exists
0,exists
M ex
 two
outputs


ww
66 In
−1
−1
We industry’s
can verifyHicksian
from an
industry’s
Hicksian
changes
changes
changes
in in
factor
infactor
factor
rents
rents
rents
ww
and
wand
and
sector
sector
sector
outputs
outputs
outputs
x.6x.
In
In
all,
Inall,
all,
duedue
due
to to
det(M)
todet(M)
det(M)
= =
0,
=0,
M
0,M
M−1
exists
−1
−1
−1
dw that C
Ma null matrix. −M Cxw D
k
suppose
that
no
sector
permits
input
substitution
such
is





ww
j, i = j, areample,
complements
or
substitutes.
The
associated
matrix
element
will
come
(2)
=
substitution matrix C w wand
whether
two
i comparative-statics
the following
comparative-statics
responses
changes
the
world
market
(2) and
and
and
theinputs
the
the
following
following
following
comparative-statics
comparative-statics
responses
responses
responses
toto
to
changes
tochanges
changes
in in
the
inin
the
the
world
world
world
market
market
market
prices
prices
prices
p pp pric
dx
−D−1 Cxw M−1
D−1 + D−1 Cxw M−1 Cxw D−1
and
j,case
i ≠ j,and
are
complements
orsign
substitutes.
The
Also
that
thea cost
structures
of
two
or
more
are
even
identical,
in107-109)):
which
and
and
national
national
national
endowments
endowments
endowments
v interpretation
result
vvresult
result
(cf.(cf.
(cf.
Hadley
Hadley
Hadley
(1974,
(1974,
(1974,
pp.pp.
pp.
107-109)):
107-109)):
egative in the
firstsuppose
take
positive
inand
the
second.
This
and
national
endowments
v sectors
result
(cf.
Hadley
(1974,
pp.
107-109)):
out
 nega    Consequently:

  
associated
matrix
element
will
come
−1

−1 C ,−1the





−1
−1
−1
−1
−1
−1
−1
case
C
D
C
will
not
have
full
rank.
Then,
since
M
=
−C
D
assumed
xw of the
xw note −M
es over to the elements
already
dwdw
dw Cww . We
MM
M
−M
−MCxw
CCD
D−1
dvdv
dv
xwaggregate substitution matrix
xw
xwD
∂xxw
 sign


 −1

−1 
−1
−1
(2)(2)
(2)
=
. . .dv
tive in the first case and take adw
positive
in== 
−1
.
M
−M
C
D
(3)
=
−D
C
M
.
−1
−1
−1



−1
−1
−1
−1
−1
−1
−1
−1
−1



−1
−1
−1
−1
−1
−1
xw
xw
dx
dx
−D
−D
−DC
C
Cxw
M
Msectoral
D DD+ D
++DDCxw
CC
M
MCelimiCCxw
D
D
dpdp
dp
 Basically,
semi-definiteness


.
xw
xwD
regularity
of MThis
will
beto
violated.
however,
regularity
assumption
only
xw
xwM
xw
xwM
∂v
=dx
(2) negative
Cww is negative
due
the
each
thesemi-definite,
second.
interpretation
carries
over
to our of
−1 + D−1 C M−1 C
dx Consequently:
−D−1 Cxw
M−1
D−1 1/C i ).dpTherefore
−1D
xw
−1
xw
Recall
that
D
has
a
positive
diagonal
(with
elements
k
Consequently:
Consequently:
xi xi
nates
borderline
cases
ofsubstitution
little
economic
significance.
Given
the, existence of D ,
the
elements
ofsingular
the
aggregate
maitution matrix
Cww
(cf. Diewert
(1982,
p. 567)).
Positive
or
negative
entries
to Cxw
Rybczynski effects ∂x/∂v will always possess the same sign as the corresp
∂x∂x
∂x
trix
Cwware
. We
note
that inputs,
Cand
is
−1
−1
−1
(1)
 −1
−1
ww
Consequently:
the
regularity
ofalready
M isornecessary
sufficient
for Consequently:
a to
unique
solution
ofC
in
terms
of
(3)(3)
(3)negative
==
−D
−D
−D
CCxw
M
M.−1
..
ise, reveal that
there
superior
inferior
respectively,
the ∂v
making
of
xwM
−1=
to
−C
M
. Formally, xw
introducing
as O the null matrix of appropriate
∂v
∂v
xw
semi-definite, due to the negative semi-6
−1
in factorofrents
w and
outputs
all,
due
to
det(M)
=
−1
−1In
i iexists
i ).
∂x
ticular good.changes
The intensity
sectoral
scalesector
effects
isthat
measured
the
diagonal
−1
0, M
−1
Recall
Recall
Recall
that
that
Dx.
DD−1
has
has
has
aalong
positive
aProposition
apositive
positive
diagonal
diagonal
diagonal
(with
(with
elements
elements
elements
1/C
1/C
1/C
).
Therefore,
). Therefore,
Therefore,
the
the
respective
respective
respective
xi x.xi x
ix
ix
ii
definiteness
of each
sectoral
matrix
All
Rybczynski
∂x/∂v the
will
come
out positive
(3) substitution
= 1.(with
−D
C
(3)
xw Mderivatives
∂v
Rybczynski
Rybczynski
effects
∂x/∂v
∂x/∂v
∂x/∂v
will
always
always
possess
possess
the
the
same
same
same
sign
sign
sign
as as
the
asthe
the
corresponding
corresponding
entries
entries
entries
and
following
comparative-statics
responses
to
changes
in−1always
the
world
prices
pcorresponding
 will
7possess
will
If we had assumed
constant
returns
to scale,
DRybczynski
would
beeffects
aeffects
null
matrix.
Then
(1) themarket
C k the(cf.
Diewert
(1982,
p. 567)).
Positive
or
−C
M
> O.
w w
xw
toD
to
−C
to
−C
−C
M
M
. Formally,
. . Formally,
Formally,
introducing
introducing
as
as
O
asO
the
Othe
the
null
null
null
matrix
matrix
matrix
of of
appropriate
ofi appropriate
appropriate
dimension:
dimension:
dimension:
−1
xwhas
xw
xwM
negative
to CxwRecall
,v likewise,
reveal
that
that
a(1974,
positive
diagonal
(with
elements
1/C
). Therefore,
the respec
national
endowments
result
(cf.
Hadley
pp.introducing
107-109)):
d correspond and
to
(5.1)
inentries
Diewert/Woodland
(1977)
and
(A9)
in Recall
Jones/Scheinkman
xi xiand
This
proposition
provides
a necessary
sufficient
that
D–1 has
a positive
diagonal
(withcondition for an
 are 


 or inferior

there
superior
inputs,
respectiveProposition
Proposition
Proposition
1.
1.
All
1.
All
All
Rybczynski
Rybczynski
Rybczynski
derivatives
derivatives
derivatives
∂x/∂v
∂x/∂v
∂x/∂v
will
will
will
come
come
come
out
out
out
positive
positive
positive
if,
if,
and
if,
and
and
only
only
only
if,supply.
if,
if, ent
i a).
∂x/∂v
always
possess
the same
sign
the
sector
outputs
as
response
to an increase
any corresponding
single factor
T
) as well as to (6) in Chang (1979) and Rybczynski
(48)-(49) in effects
Woodland
(1982,
Section
4.7).
­ewill
lements
1/C theasinrespective
x ixi In Therefore,
 M
The
−1
−1
−1
7O.
7−1
7 Cxw D−1
M−1  good.
−M
dv
ly,todw
themaking
of
a
particular
in−C
−C
−C
M
M
>
O.
>
>
O.
xwxw
xw
−1
intuition
is
that
scale
disadvantages
on
the
sectoral
level
generate
increasi



= 
(2)
.
R
­
ybczynski
effects
∂x/∂v
will
always
possess
to −Cxw
M . Formally,
introducing as O−1the null matrix of appropriate dimension:
−1 C effects
−1 + D−1 C M−1 C
of sectoral
scale
measured
dx which
−D
M−1 is This
D
Dasand
dpspace.
e note for a cost tensity
function
is continuously
differentiable
inThis
factor
pricesthe
that
the
isoquants
ofthe
costs
ina the
economy’s
output
Fromforaentries
normative
xw
xw
xw
This
proposition
proposition
proposition
provides
provides
provides
necessary
aasign
necessary
necessary
and
and
sufficient
sufficient
sufficient
condition
condition
condition
for
for
an
an
an
increase
increase
increase
in in
all
in all
all then
same
corresponding
toviewpoint,
along
the do
diagonal
If
we
had
assumed
derlying production
function
not allowofforD.
linear
substitution
between
inputs.
–1
specialization
strategies
in
favor
offactor
selected
industries
are
less
rewarding
Proposition
1.outputs
All
Rybczynski
derivatives
∂x/∂v
will
come
out
positive
if,
and
onl
sector
sector
sector
outputs
outputs
as as
aasresponse
a–C
aresponse
response
to
to
an
to
an
increase
an
increase
increase
in
in
any
in
any
any
single
single
single
factor
factor
supply.
supply.
supply.
The
The
The
underlying
underlying
underlying
´
M
.
Formally,
introducing
as
O
the
null
x w
Consequently:
returns
to scale,
would
beintuition
a null
maubscripts attachedconstant
to C again
represent
partial D
derivatives.
inputs
canappropriate
be
across
sectors,
i.e.,
“if [. . opportunity
.] opportunity
the
isoquants are
 M−1
7is that
intuition
intuition
isisthat
that
scale
scale
scale
disadvantages
disadvantages
disadvantages
onre-allocated
on
the
onthe
the
sectoral
sectoral
sectoral
level
level
level
generate
generate
generate
increasing
increasing
increasing
opportunity
matrix
of
dimension:
−C
>
O.
xw
trix. Then (1) would correspond
to
(5.1)
in
(Ylönen
(1987,
p. From
216)).
Examples
highlighting
the(incomplete)
range
of this result
∂x costs
costs
costs
in in
the
in the
the
economy’s
economy’s
economy’s
output
output
output
space.
space.
space.
From
From
a normative
aa normative
normative
viewpoint,
viewpoint,
viewpoint,
then,
then,
then,
(incomplete)
(incomplete)
−1
(3)
= in−D
Cxw M−1 .

Diewert/Woodland (1977) This
and∂v(A9)
Jones/
specialization
specialization
specialization
strategies
strategies
strategies
in
in
favor
in
favor
favor
of
of
selected
of
selected
selected
industries
industries
industries
are
are
are
less
less
less
rewarding,
rewarding,
rewarding,
provided
provided
provided
that
that in
proposition
provides
a
necessary
and
sufficient
condition
for
an
increase
Proposition
1 All Rybczynski derivatives ∂x/∂v that
6
 C

−1
  −1 −1
Scheinkman (1977) as well as to (6) in Chang
In the terminology of the implicit function theorem, the Jacobian determinant 
C
–1 flat”
inputs
inputs
inputs
cancan
be be
be
re-allocated
re-allocated
re-allocated
across
across
sectors,
sectors,
sectors,
i.e.,i.e.,
“if“if
“if
[. . [..][.and
. the
..].] the
the
isoquants
isoquants
isoquants
are
are
relatively
relatively
flat”
flat”
−1 has a positive
iacross
will
come
out
positive
if,
only
if, are
–C
´x wrelatively
M
outputs
as
acan
response
to
an
increase
ini.e.,
any
supply.
Recall
that
D(48)–(49)
diagonal
(with
elements
1/C
Therefore,
thesingle
be
nonzero.
Both
w and x
can
then
berespective
viewed factor
as local functions
of (v,The
p), i.e.underl
functions
xi xi ).
(1979)
and
in sector
Woodland
(1982,
Sec O.7Examples
(Ylönen
(Ylönen
(Ylönen
(1987,
(1987,
(1987,
p. p.
216)).
p.>216)).
216)).
Examples
Examples
highlighting
highlighting
highlighting
thethe
the
range
range
range
of of
this
of this
this
result
result
result
areare
are
offered
offered
offered
in inin
in some neighborhood of the economy’s equilibrium state.
ishowever,
that
scalethe
the
sectoral levelentries
generate increasing

 opportu

tion 4.7). In
the ∂x/∂v
presentintuition
context,
alldisadvantages
Rybczynski
effects
will
always
possess
same sign
ason
the
corresponding
7
 C ww
 
All matrix and vector inequalities shall hold element-wise.
Cxw
CCxw
ww
xw
   CCww
 
In positive.
In
the
Inthe
the
terminology
terminology
terminology
of the
ofofthe
the
implicit
implicit
implicit
function
function
function
theorem,
theorem,
theorem,
thethe
the
Jacobian
Jacobian
Jacobian
determinant
determinant
determinant
will

−1 . Formally,
 C
 xw
 will
 will
along
the diagonal
are
CCxw
D DD
costsofinDthe
space.
From
aprovides
normative
viewpoint,
then,
(incompl
xw
toentries
−Cxw M
introducing
aseconomy’s
O the nulloutput
matrix
ofproposition
appropriate
dimension:
This
a p),
necessary
and
be
be
nonzero.
be
nonzero.
nonzero.
Both
Both
Both
w
and
w
w
and
and
x
can
x
x
can
can
then
then
then
be
be
viewed
be
viewed
viewed
as
local
as
as
local
local
functions
functions
functions
of
(v,
of
of
(v,
p),
(v,
i.e.
p),
i.e.
functions
i.e.
functions
functions
which
which
which
are
are
defined
are
defined
defined
Therefore, firstly, D is positive definite and, secspecialization
strategies
in
favor
of condition
selected
industries
are lessin rewarding,
for
all sector provided
in some
ininsome
some
neighborhood
neighborhood
neighborhood
ofsufficient
the
ofof
the
the
economy’s
economy’s
economy’s
equilibrium
equilibrium
equilibrium
state.
state.
state.an increase
–1 exists
ondly, the diagonal
inverse Dderivatives
and
isand
Proposition
1. All Rybczynski
∂x/∂v
will
comeshall
out
positive
if, and only if,
7 77
AllAll
matrix
Allmatrix
matrix
andand
vector
vector
vector
inequalities
inequalities
inequalities
shall
shall
hold
hold
hold
element-wise.
element-wise.
element-wise.
outputs
as
a
response
to
an
increase
in
any sininputs can be re-allocated across sectors, i.e., “if [. . .] the isoquants
are relatively fl
likewise
 M−1positive
−C
> O.7 definite.
gle
factor
supply.
The
underlying
intuition
is
xw
−1C´ (cf. GantWe define M : = Cww – C
D
(Ylönen
(1987,
p.
216)).
Examples
highlighting
the
range
of
this
result
are offere
xw
x w
that scale disadvantages on the sectoral level


macher
(1990, pp.provides
45–46))a and
requireand
that
This proposition
necessary
sufficient condition for an increase in all


6
66
generate increasing opportunity costs in theC
C
ww
xw 
6
det(M) ≠ 0.
some
technologies
In
ofany
the implicit
functionsupply.
theorem,The
the Jacobian
determinant  
sector
outputs Certainly
as a response
to the
an terminology
increase inare
single
factor
underlying
economy’s
output space.
From a normative
Cxw
D 
excluded by this assumption.
For example,
supbe nonzero. Boththe
w and
x can viewpoint,
then be
viewedthen,
as
local
functions
of (v,specialization
p), i.e. functions which are de
(incomplete)
intuition
is that
scale disadvantages
sectoral
level
generate
increasing
opportunity
pose that
no sector
permits input on
substitution
strategies
in favor then,
of selected
industries are less
costs
theCeconomy’s
space.suppose
From that
a normative
viewpoint,
(incomplete)
suchinthat
matrix.
Also
ww is a null output
7
in some neighborhood of the economy’s equilibrium state.
All matrix and vector inequalities
shall hold
element-wise.
rewarding,
provided
that inputs
can be re-allo-
the cost structures
of two
or more
sectorsindustries
are
specialization
strategies
in favor
of selected
areacross
less rewarding,
provided
cated
sectors, i.e.,
“if [...]that
the isoquants
even identical, in which case C xwD −1C ´x w
inputs can be re-allocated across sectors, i.e., “if [.are
. .] relatively
the isoquants
relatively
flat”
flat” are
(Ylönen
(1987,
p. 216)). Exwill not have full rank. Then, since M =
(Ylönen
(1987,
Examples
highlighting
– C D–1
C´ , p.
the216)).
assumed
regularity
of M will the range of this result are offered in
xw
x w
6 In the terminology of the implicit function theorem, the


be violated. Basically, however, our regularity
 Cww Cxw 
6

 will
In
the
terminology
of
the
implicit
function
theorem,
the
Jacobian
determinant
will be nonzero.
Jacobian
determinant
assumption only eliminates borderline singular
 Cxw
D 
Bothofw(v,
and
can functions
then be viewed
local
functions of (v, p),
of little
economic
becases
nonzero.
Both w
and x can significance.
then be viewed Given
as local the
functions
p),x i.e.
whichasare
defined
i.e. functions which are defined in some neighborhood of the
economy’s equilibrium state.
5 Subscripts
7 All matrix and vector inequalities shall hold elementAll
matrix and
vector to
inequalities
shall holdpartial
element-wise.
attached
C again represent
derivatives.
wise.
in some neighborhood of the economy’s equilibrium state.
7
12
Finnish Economic Papers 1/2009 – Andreas Pfingsten and Reiner Wolff
amples highlighting the range of this result are
offered in the next section. Any such example
must allow for input substitution in at least one
production sector, as we shall now prove. We
will make use of the following observation:
Lemma 1 If no technology supports input substitution, then Cxw > O.
Proof Consider the cost function Ci (w, xi) of an
arbitrary sector i. By definition of this function
and because of Shephard’s Lemma, Ci (w, xi) =
ΣjwjC wi j (w, xi) for all positive factor prices and
outputs. Hence, C xi i (w, xi) = ΣjwjC wi j xi (w, xi). As
input substitution is not permitted, the factor
demands C wi j (w, xi) and, hence, the second-order
derivatives C wi j xi (w, xi) are all independent of w.
Now suppose, per absurdum, that Cwk xi (w, xi)
< 0 for some input k. Then, for every wk that
is sufficiently large, we would obtain C xi i (w, xi)
< 0, which contradicts the fact that marginal
cost is always positive. Finally, notice that
Cxi wk (w, xi) = Cwk xi (w, xi) according to Young’s
Theorem. 
Remark 1 Suppose that no technology supports
input substitution. Then the matrix inequality
–C´x wM–1 > O cannot be satisfied.
Proof As no technology permits input substitution, C iw w = O for all sectors i. Hence, by
definition, C ww = O. This implies M =
–Cxw D−1 C´x w. Now let E : = (eij) denote the
m × m identity matrix such that E = MM−1 =
CxwD−1 (–C´x wM–1), where Cxw > O by Lemma
1. Therefore, if all entries to the matrix product
in brackets were positive, eij = 0 (i ≠ j) would
require that the entire row i of Cxw vanishes,
contradicting eii = 1. Consequently, not all elements of –C´x wM–1 can be positive at the same
time. 
Our remark confirms that scale effects do not
matter, i.e., the standard Rybczynski outcome
persists, if input substitution is ruled out in all
industries. In this case, expanding the economy’s outputs altogether would require more of
every, and not just one, input.
We now briefly address the adjustments in
factor prices. In the standard Heckscher-Ohlin
framework of two commodities and factors with
constant returns to scale, the price of the factor
of which an extra supply has become available
will decrease whereas the other factor price will
increase (cf. Woodland (1982, p. 79)). In our
model economy, factor price responses to endowment changes are given by M–1 (cf. equation
(2)). By definition, M (and hence M–1) is negative definite and will thus always possess a
negative diagonal.8 Together with the examples
in the following section this proves:
Proposition 2 The own-price effects of arbitrary supply shocks are normal, whereas for i ≠ j
the adjustments ∂wi /∂vj may be of either sign. In particular, and different from both the
standard set-up and the Hansson/Lundahl (1983)
approach with homogeneous production functions, all factor prices may decrease as any single endowment increases. A definite statement
can be made in the special case of an economy
that produces its outputs from just two inputs.
This statement is based on the occurrence of
inferior factors like, e.g., low-skilled labor:
Remark 2 Suppose there are two inputs to production in all of the economy. Also suppose that
at least one industry uses an inferior factor.
Then, it follows from the inequality –C´x wM–1
> O that the factor price responses ∂wi /∂vj (i ≠ j)
cannot be normal.
Proof To begin with, introduce as e the first ndimensional unit vector, i.e. e´ = (1 0 ... 0). Assume that –C´x wM–1 > O, and hence –M–1 Cx w
> O for the− product transpose. Thereby, M−1b
> o, where b : = –Cxwe holds the negative of the
first column of Cxw (and o stands for the null
vector of length 2). Without loss of generality,
suppose that this column’s first entry is negative, indicating an inferior input in sector one.
Hence, b1 > 0, while b2 < 0. At this point, recall
that M–1 is negative definite and thus has a neg8 Note that r´ Mr = r´ C r – r´ C D–1 C´ r < 0 for all
ww
xw
xw
r ≠ o of length m, since Cww is negative semi-definite and
–1
D is positive definite (with C´ xwr possibly equal to the null
vector). Hence, r´ Mr < 0 by the assumed regularity of M.
(Cf. Diewert/Woodland (1977, p. 392) for a similar argument in the context of constant returns to scale.)
13
Finnish Economic Papers 1/2009 – Andreas Pfingsten and Reiner Wolff
ative diagonal. Consequently, we conclude from
M–1 b > o that the second element in the first
row of M–1 takes a negative value (and so does
the first entry to the second row by the symmetry of M–1). 
Proposition 2 and, in particular, Remark 2 are
less surprising if residual profits are accounted
for. As a consequence of decreasing returns and
marginal cost pricing, the economy can generate positive profits. These profits also can be
maintained since we assume a given number of
firms per industry, e.g., when low-level market
entry may not cover average cost. Then, profits
possibly rise in each single industry if a factor
supply shifts upwards:
Corollary 1 Consider again the n × 2 economy
presented in Remark 2. Hence, suppose that
both factor prices decrease in response to an
expansion of a single endowment. Then all sectors will enjoy higher profits.
Proof Let Πi (pi, w) : = maxxi > 0 {pi xi – Ci (w, xi)}
denote the profit function of an arbitrary sector i.
By Hotelling’s Lemma, Π wi j (pi, w) = –C wi j (w, xi)
for both factor prices (j = 1, 2). Total differentiation of Πi (pi, w) with respect to w thus gives
dΠi (pi, w) = –C wi 1 (w, xi) dw1 –C wi 2 (w, xi) dw2. As
at least one of the two factor demands C wi j (w, xi)
in sector i must be strictly positive, we conclude
that the profit of sector i increases when both
factor prices fall. 
It is important to note that the division of profits
depends on firm ownership which we have not
modeled, as we did not include a domestic or
international capital market. Therefore, the consequences of profits on the distribution of incomes within the economy and, potentially,
outside must remain indeterminate. However,
given that world-market output prices are considered as exogenous, the economy’s Rybczynski derivatives will not be affected. In view of
the standard reciprocity relation (cf. Samuelson
(1953/1954)), one might presume that non-Rybczynski responses of outputs to endowment
changes always go along with non-StolperSamuelson effects of output price variations on
factor prices. As to these effects, however, the
14
distribution of residual incomes, maybe to omitted factors, becomes an essential issue. Yet, a
complete exploration of the comparative statics
associated with output price variations and, in
particular, of the Stolper-Samuelson derivatives
under decreasing returns to scale is beyond the
scope of the present paper.
3. Computational Examples
We now present two computational examples of
technologies and related cost functions, respectively, which shall further illustrate our results
under different scenarios. Our first example is
that of a two-sector economy with two inputs.
The sectoral cost functions correspond to a production process with a fixed factor intensity in
one sector and to a substitutional technology in
the other. We consider different equilibrium
states in which one input can also be inferior. In
all three states, all Rybczynski derivatives come
out positive, along with both standard and nonstandard factor price responses.
We then assume, in our second example, that
sector 2 employs a third input that is specific to
this sector. We consider two different parameterizations of the sectoral cost function which
illustrate the occurrence of complementary and
inferior inputs. Once again, all Rybczynski derivatives are positive. In comparison, a standard
Ricardo-Viner two-sector model with sectorspecific inputs and constant returns (cf. Jones
(1971)) predicts that a rise in the supply of a
specific factor in one sector will increase this
sector’s equilibrium output, while the other sector will shrink. Also in more general models,
where strictly sector-specific factors are replaced by mobile so-called ‘extreme’ inputs, not
all Rybczynski derivatives can be positive at the
same time (cf. Ruffin (1981) and Jones/Easton
(1983)). In our second example, again, not ev­
ery factor price response is normal.
Example 1 Let n = m = 2 and consider both the
cost function C1 (w1, w2, x1) = (0.1w1 + 0.05w2)
x111.26, which comes from a production process
with a fixed factor intensity, as well as the cost
function C 2 (w 1, w 2, x 2) = w 1 x 112. 2 + 2√w 1w 2 x 112. 1
– w2x112. 3, which belongs to a substitutional
9
Pfingsten/Wolff: Factor Supply Changes in Small Open Economies
Finnish Economic Papers 1/2009 – Andreas Pfingsten and Reiner Wolff
results, depending on the level of η:
Pfingsten/Wolff:
Changes
in Small Openlengthy
Economies
calculations (to be obtained 9from the
technology,
overFactor
someSupply
domain
in the neighbor1. Letof
η=
. We find: assumed equilibrium authors upon request) lead essentially to the subhood
the1.0economy’s





results,
depending
on the level of η:
state w1 = 1.5,
w2 = 1 0.408
and 
x1 = 1, x2 = η.
Some sequent
−0.272
0.126 2.098
0.066
0
results,
on the level
Cwwdepending
=
, ofCη:
xw =
0.408
−0.612
1. Let η =1.0 . We find:
M =
Cww =

 −18.900 −0.127 ,
−0.272
0.408
−0.127 −0.682
,
0.408 −0.612
2. Let η =1.15 . Then:
M

−18.900 −0.127


=
,
−0.127
−0.317
Cww =
0.476
−0.682
0.476
−0.714
,
2. Let η =1.15 . Then:

0.245
,
 −22.906
=
M
Cww =
−0.317
0.476
0.245 −0.775
,
0.476 −0.714
3. Let η =1.5 . This gives: 
M
=
Cww =
−22.906
0.245
−0.425
0.638
0.245
 ,
−0.775
0.638
−0.957
,
3. Let η =1.5 . This gives: 
 −33.215
=
M
Cww =
1.465
,
−0.425
0.638
1.465 −1.045
,
0.638 −0.957
M−1 =
Cxw =
−1
M
0.063

 −0.053
0.010

0.126
0.010 2.098
−1.468,
0.063 0.047

−0.053
=
Cxw =
,
0.047
0.010

0.010 2.145
−1.468
0.126
,
0.063 0.011
D=




Cxw =
−1
−0.044
−0.014
0.126
0.063
−0.014
,
D=
−0.065
D=


.

.
0


.


.


.

0.006
 0.091
0.111
0
0.206
,
0.049
0.006
 0.083
=
0.066
0
0.094
, 0.043
0
0.206
M−1 = 0.006 0.083
−Cxw
0.094
0.043
0.066
0
D=
,
0
0.154
,


−0.032 −0.045


=
,
M
Cxw =

=
0.066
−Cxw M−1
 ,
−1.294
2.237

−Cxw M−1
0.126 2.145
−0.014
−1.294,
0.063 0.011
M−1 = 
,
0.239
0.006

 0.091
−Cxw M−1 =
0.066
0
0.111
D=
, 0.049
0
0.239
,


−0.044 −0.014


−1
M =
,
Cxw =
0

0.007
 0.070
=
0.066
0
0.069
, 0.034
0
0.154
−Cxw M−1
0.126
2.237
−0.045
−1.020 ,
0.063 −0.065
D=
−1 is positive.Consequently, ∂x
 to −C 
 i /∂vj is positive

In all threecases, every entry
xw M
−33.215
−0.032
1.465
−0.045
0.007
0.070
−1

−1
, Man
=increase in any single
, −C
=
.
M i,j= = 1,2. In other words,
xw Mendowment
for all
factor
1.465 −1.045
−0.045 −1.020
0.069 implies
0.034 an
increase in all sector outputs, i.e., no sector will shrink. At the same time, in the first
−1 is positive.
In all both
three factors
cases, every
entry to and
−Cxw
Mmatrix
Consequently,
∂xi /∂vfactor
j is positive
case,
are superior
the
M−1 exhibits
the standard
price
cases,
every
entry
–C
´x wM–1 is
forInallalli,jthree
=In1,2.
In other
words,
increase
in inputs
any
single
factor
implies
an first exExample
2 This endowment
is an
outgrowth
of our
responses.
the
second
case,
thetoan
superiority
of
is maintained.
However,
all factor
positive.
Consequently,
∂x
/∂v
is
positive
for all
ample.
We
takesame
n = 2,time,
m = 3.
Thefirst
first sector
i j i.e., no sector
−1
increase
in
all
sector
outputs,
will
shrink.
At
the
in
the
prices will now decrease since all elements of M are negative (cf. Proposition 2). This
i, j = 1, 2. In other words, an increase in any sin- shall
only use factors 1 and 2 and shall thereby
case, both
factors
are superior
the matrix
M−1 in
exhibits
the standard
factor
price
holds
again
in the third
case, where
factor
2 is
the second
industry
(cf.
Remark
gle
factor
endowment
implies
anand
increase
in
allinferior
once more
possess
the same
technology and
responses.
In the
case,will
the shrink.
superiority
of inputs
is maintained.
However,
factorHowever,
sector
outputs,
i.e.,second
no sector
At the
2).
cost function,
respectively,
as all
before.
same
in the
first case,
both
areofsuthe second sector
employs a
pricestime,
will now
decrease
since
all factors
elements
M−1now
are suppose
negative that
(cf. Proposition
2). This
Example
2.
ThisMis–1 exhibits
an outgrowth
of our first example. We take n = 2, m = 3.
perior
and
the
matrix
the
standard
specific
third
input
such
that
sectoral
holds again in the third case, where factor 2 is inferior in the second industry (cf. Remarkminimum
The first
sector
shall only
use second
factors case,
1 andthe
2 and
shall
once local
more representation
possess the of the
factor
price
responses.
In the
cost
hasthereby
an extended
2).
11. 2
11. 1 2 superiority
of
inputs
is
maintained.
However,
all
form
C (wHowever,
same technology and cost function, respectively, as
before.
suppose
that1 w2 x 2 –
1, w2, w3, xnow
2) = w
1 x 2 + 2√w
11. 3
11. 1
factor
prices
will
now
decrease
since
all
elew
x
+
(σ√w
w
+ τ√w
w
+ ψw
) x
.
The
2 example.
1 minimum
2take
2 cost
3 m
2 sectoral
2. employs
This is an
outgrowth
our first
We
2,
=3an
3. 2
n =
the Example
second sector
a specific
thirdofinput
such
that
has
ments of M–1 are negative (cf. Proposition 2).
­economy’s equilibrium
position 1.1
shall be at√
2
1.2
The first sector
shall only use offactors
1 and
more
− x = 1,
extended
form
C 2(wand
w2shall
, w3 ,atxthereby
) =
w1once
x +
2 wpossess
1 ,tained
2w
1 w2 x2 the
This
holdslocal
againrepresentation
in the third case,the
where
factor
and
1 = 1.5,2w2 = 1, w3 = 1.5
1
√
√
1.3
1.1
same
technology
and
as
before.The
However,
now
suppose
x2 inferior
+ (σ w
τcost
w2function,
w3industry
+ ψw3respectively,
) x(cf.
The economy’s
equilibrium
position
shallthat
be parame2w
in1 wthe
x2 = 1.5.
following
are
interesting
2is
2 + second
2 . Remark
2). at sector
terizations
of σ,following
τminimum
and ψ are
(computational
the
second
employs
third
such
that
cost
has an details
attained
w1 = 1.5,
w2 = a1,specific
w3 = 1.5
and input
x1 = 1,
x2 =
1.5.sectoral
The
interesting
√
can
be
acquired
from
the
authors):
2
1.2
extended local representation of the form C (w , w , w , x ) = w x + 2 w w x1.1 −
1
2 can
3 be
2 acquired
1 2 from the1 authors):
2 2
parameterizations of σ, τ and ψ (computational details
√
√
1.3
1.1
w2 x2 + (σ w1Factor
w2 + τSupply
w2 wChanges
) xSmall
economy’s
be
Pfingsten/Wolff:
Open
Economiesequilibrium position shall 10
3 + ψw3in
2 . The
1. Let (σ, τ, ψ) = (−0.1, 1.5, −0.54). We get:
attained at w1 = 1.5, w2 = 1, w3 = 1.5 and x1 = 1, x2 = 1.5. The following are interesting




0.987  ,
D=
0.288
0.516

parameterizations
τ and
ψ (computational details
can be acquiredfrom the authors):
0.126 2.179
−0.399 of σ,
0.638
−0.026

Cww = 
0.638
−1.674

0.478  ,
M


−22.164
−9.231
−0.279
−0.279
0.364
−0.296
=  −9.231
−Cxw M−1 =

−6.149
0.003
0.005


0.364  ,
0.003

.

Cxw =  0.063
1. Let (σ, τ,−0.026
ψ) = (−0.1,
1.5,−0.293
−0.54). We get:
0.478


M−1 = 
0
−0.172
0.288
0.516
0.026

−0.659
−1.083
0.066
0
0
0.221


−1.083  ,
−5.199
,
15

−0.399
0.638

Cww =  0.638 −1.674
−0.026


0.478  ,

0.126
2.179



Cxw =  0.063 0.987  ,
D=

0.066
0
0
0.221
0
0.026Reiner Wolff
−0.026 Papers
0.4781/2009
−0.293– Andreas Pfingsten
Finnish Economic
and
M


−22.164
−9.231
−0.279
−0.279
0.364
−0.296
=  −9.231
−Cxw M−1 =



0.364  ,
−6.149
0.003
0.005
0.003
0.076
0.050
0.076



M−1 = 
−0.172
0.288
0.516
0.288
−0.659
−1.083
−0.423
0.638
−0.003
−0.003
0.022
−0.012

Cww =  0.638 −0.990
M


=
−32.404
0.745
−0.061
−Cxw M−1 =



0.022  ,
−0.061
0.745


0.023  ,
−1.052
−0.012
0.023
0.005
0.066
0.095
0.070
0.035
0.053

,


−1.083  ,
−5.199
.
2. Let (σ, τ, ψ) = (−0.01, 0.07, −0.02). Accordingly:

0.516


0.126
2.231
0
0.004

D=

0.119



Cxw =  0.063 −0.016  ,


−0.032
−0.020
0.119
−1.743
M−1 =  −0.020
−1.002
0.066
0
0
0.157

,

−1.743  ,
−84.536
.
We find once more again that −Cxw M−1 is element-wise positive. In both cases, factors
in our2 model
equilibrium
prices and
We
find3 once
more again that
–C´x wto
M–1
ele- that
1 and
are complementary
inputs
theis second
industry
as indicated
by thefactor
respecoutputs
are
always
(locally)
stable
once the
ment-wise
positive.
In
both
cases,
factors
1
and
tive negative off-diagonal entries to Cww . (Recall from our notation in Section 2 that
market-clearing
process
operates
according
to
3 are complementary
inputs
to
the
second
in1
Cww =2 C
+ C2ww by
and
thatnegative
C1ww vanishes.)
The second
case illustrates
that princisome plain,
yet intuitively
appealing,
dustry
asww
indicated
theobserve
respective
the simultaneous
occurence
an inferior
input
the same sector is possible if there are
off-diagonal
entries
to Cwwof
. (Recall
from
our2 inples.
1 + C2 and
Pfingsten/Wolff:
Factor
Supply
Changes
inare
Small
Open
Economies
We suggest
two
standard
types
of adjustment
notation
in
Section
2
that
C
= C
ww
ww notwwallFactor
more than two factors. In both
cases,
factorSupply
price
responses
normal.
Pfingsten/Wolff:
Changes
in
Small
Open
Economies
observe that C1ww vanishes.) The second case processes in the factor and commodity markets,
­illustrates that the simultaneous occurence of an respectively (cf. Mayer (1974)). To begin with,
yet intuitively
intuitively
appealing,
principles.
4 Stability
yet
appealing,
principles.
that rent variations are proportional
inferior
input 2 in the same sector
is possible
if we propose
by positive
scalars
kj to the processes
amount ofin excess
there are more than two factors.We
In both
cases,
suggest
two
standard
types
of adjustment
adjustment
the factor
factor and
and com
co
suggest two standard types of
processes in
the
This Walrasian
price-tâtonnenot
factor
pricesimple
responses
areWe
normal.
Theall
above
cases,
as they
are,
demonstrate input
clearlydemand.
that non-Rybczynski
outcomes
markets, respectively
respectively (cf.
(cf. Mayer (1974)).
(1974)). To
To begin
begin with,
with, we
we propose
propose that
that rent
rent vv
markets,
mentMayer
shall be complemented
in theAcgoods
marmay emerge in a standard 2 × 2-setting if increasing
marginal
costs are assumed.
are proportional
proportional by
by positive
positive
scalars
to the
the
amount mechanism:
of excess
excess input
input deman
deman
kets
by ascalars
Marshallian
adjustment
are
kkjj to
amount
of
cording to Samuelson’s correspondence principle outputs
(cf. Samuelson
(1947)),
however,
such
evolve
in
the
direction
of
the
difference
Walrasian
price-tâtonnement
shall
be
complemented
in
the
goods
markets
by
Ma
4. Stability
Walrasian price-tâtonnement shall be complemented in the goods markets by aaMa
comparative-statics results are meaningful only if between
stability world
of the equilibrium
is guarandemand
prices
and
domestic
adjustment mechanism:
mechanism: outputs
outputs evolve
evolve in
in the direction
direction of
of the
the difference
difference betwee
betwe
prices
(inusually
termsthe
of
marginalwith
cost
of proThe
cases, simple
asadjustment
they
are, demon- supply
teed.above
The argument,
so-called
comparative-statics
paradoxes
were
associated
demand prices
prices and
and domestic
supply
prices
(in terms
terms
of marginal
marginal
cost
of production
productio
demand
supply
(in
of
duction),
againprices
by some
positive
factors ofcost
pro-of
strate
clearly that non-Rybczynski
outcomes domestic
unstable equilibria and would otherwise vanish, has been put forward repeatedly in the
by some
some positive
positive
factors
of proportionality
proportionality
Assuming
for simplicity
simplicity
that kkj =
=l
portionality
li. Assuming
for simplicity
that that
may emerge in a standard 2 × 2-setting
if inby
factors
of
lli.. Assuming
for
j
literaturemarginal
on international
(e.g. According
Mayer (1974),k9j = l
Ide/Takayama
(1988,i 1990)). For 9this
creasing
costs aretrade
assumed.
i = 1 for all j and i, we obtain:
all
j
and
i,
we
obtain:
9
all j and i, we obtain:
reason,
it is important
to note thatprinciple
in our model
prices and outputs are
to
Samuelson’s
correspondence
(cf. equilibrium factor
n

n

i
Samuelson
(1947)),
however,
such
comparative- process
(4)
(4)
ẇ
=
C
(w,xxi))to
−some
vj for
all j ,
i
always (locally)
stable
once the
market-clearing
operates
according
plain,
ẇjj =
Cwwjj(w,
i − vj for all j ,
statics results are meaningful(4)
only if stability of
i=1
i=1
the equilibrium is guaranteed.
(5) The argument, (5) ẋẋi =
= ppi −
−C
Cixi i(w,
(w,xxi)) for
for all
all ii,, (5)
i
i
i
xi
so-called comparative-statics paradoxes were
usually associated with unstable
equilibria
andLemma
where
Shephard’s
hasShephard’s
been used
used in
in
(4). has been used in
where
Lemma
where
Shephard’s
Lemma
has
been
(4).
would otherwise vanish, has been put forward (4).
Associatedtrade
with the
the adjustment
adjustment model
model (4)-(5)
(4)-(5) isis the
the following
following linear
linear approx
appro
repeatedly in the literature on international
Associated
with
system (cf.
(cf. Takayama
Takayama (1985,
(1985,
p.‘·311))
which
has been
been evaluated
evaluated
around
an equ
eq
(e.g. Mayer (1974), Ide/Takayama
(1988,
9 The dot
’311))
indicates
total differentiation
with respectaround
to
system
p.
which
has
an
∗ ):note
1990)). For this reason, it is state
important
time.
(w∗∗ , xto
∗
state (w , x ):
16
(6)
(6)






ẇ
w − w∗
 ẇ 
 = A
 w − w∗

ẋ = A
x − x∗
ẋ
with
the
Jacobian
matrix
A
defined
as:
with the Jacobian matrix A defined as:


Cww
 Cww
x − x∗






Cxw
C
xw 
n
ẇj = 
C ii j (w,evolve
xi ) − in
vj the
for direction
all j ,
stment mechanism:
outputs
of the difference between world
ẇj = i=1 Cw
wj (w, xi ) − vj for all j ,
and prices and domestic
supply
prices
(in
terms
of
marginal
cost of production), again
i=1
ẋi = pi − Cxii i (w, xi ) for all i ,
ome positive factors
. Assuming
for simplicity
kj =–liAndreas
= 1 for Pfingsten and Reiner Wolff
ẋi = piof−proportionality
Cxi (w, xi ) forlFinnish
i , Economic
iall
Papersthat
1/2009
9
and
i,
we
obtain:
d’s Lemma has been used in (4).
the adjustment model (4)–(5)
We thus find, unlike Mayer (1974) and Ide/
d’s Lemma hasAssociated
been usedwith
in (4).
n

is the following
linear
approximation
system
Takayama
(1988, 1990), that (seemingly) parai
with the adjustment
model
(4)-(5)
is
the
following
linear
approximation
ẇmodel
Cwj (w,isxithe
) − following
vj for all jlinear
,
j =
with the adjustment
(4)-(5)
approximation
(cf.
Takayama
(1985,
p.
311))
which
has
been
doxical
comparative-statics
effects may be obi=1
kayama (1985, p. 311)) which has been evaluated around
an equilibrium
∗, x∗): anserved
kayama (1985,
p. 311))
around
equilibrium
i been evaluated
evaluated
around
state
(w
even
in
well-behaved
settings with loẋi which
= pi an
−has
Cequilibrium
(w,
x
)
for
all
i
,
i
xi




cally stable equilibria. The following differ
∗ 
ẇ  beenused
w−
ences in the approach taken by these authors on
re Shephard’s
Lemma
in w
(4).
∗ 
 ẇhas
 = A w −
w
(6)  ẋ  = A  x − x∗ 
the one hand and our model on the other hand
∗
Associated with the adjustment
model
is the following linear
approximation
ẋ
x − x(4)-(5)
appear
to be crucial. Firstly, we assume that
ian (cf.
matrix
A
defined
as:
em
Takayama
(1985,
p.
311))
which
has
been
evaluated
around
an
equilibrium
with
the Jacobian
matrix A defined as:
prices
equal
marginal rather than average costs
ian matrix A
defined
as:


∗
∗


and
that
firms
operate at least at their efficient
e (w , x ):
Cww  Cxw


Cww
Cxw w. − w∗
A := 
scale
without
loss.
Secondly, Mayer and Ide/
ẇ



(7) A := −C
−D

=

A .
Takayama impose several restrictions upon the
−Cẋxw
−D
∗
xw
x−x
equilibrium outcomes of their models, followepared to introduce our next proposition:
epared
to introduce
next
proposition:
the Jacobian
A defined
as: to introduce our next ing Jones (1968). They thereby neglect inferior
Wematrix
are our
now
prepared


proposition:
inputs and
3. All eigenvalues
of A are negative
real C
numbers. The equilibrium
statealso require that an increase in any
Cww
xw 
3. All eigenvalues of A are negative
real numbers.
The equilibrium state

A
:=
.
factor
price
must raise the average equilibrium
ble.
−Cxw of
−D
ble.
Proposition 3 All eigenvalues
A are negative production cost of each commodity. Therefore,
∗, x∗) is
real
numbers.
Therelated
equilibrium
state (w
certain
non-obvious cases of technologies and
d znow
denote
an
eigenvalue
and
respectively,
of
A. Hence,
are
prepared
to introduce
our
nexteigenvector,
proposition:
d z denote
an
eigenvalue
and related
eigenvector,
respectively, of A. Hence,
stable.
endowment
hermore, partition z into two vectors z1 and z2 of suitable length
such thatchanges are excluded from their aphermore, partition
z intotwoofvectors
z2real
of suitable
such
that
position
3. All eigenvalues
A are znegative
numbers.length
The
equilibrium
state to our results, the cases al1 and
proach.
According


 z denote an eigenvalue
Proof Let λ and
x∗ ) is stable.
Cww Cxw   z1  and re- lowing for non-standard Rybczynski responses
Cww
Cxw  ofz1A. Hence, of industry outputs in otherwise stable equilibria
z Azlated
= eigenvector,
(z1 z2 ) 
respectively,

 z 
z Az = (z1 z2 ) −Cxw
−D
2
f. Let λ andAz =
z denote
an
eigenvalue
and
relatedzeigenvector,
respectively,
A. Hence,
λz. Furthermore,
into
−Cxw partition
−D
z2 two vecmust beofamong
them.


=
z
C
z
−
z
Dz
tors z1 and
z2 of1 zsuitable
such
thatz2 of suitable length such that
ww
2 length
1
2 two
= λ z. Furthermore,
partition
into
vectors
z
and
1
= z C z − z Dz
1
ww 1
2

2


= λ z z .
Cww Cxw
z
= λz z.
  1  5. External Scale Effects
(8) z Az = (z1 z2 ) 

−C
−D
z2 567)). Also recall
xw
(1982, p.
w is a negative semi-definite matrix (cf. Diewert
(cf. Diewert
(1982, p. 567)).The
Also
recall
w is a negative semi-definite matrix
recent
literature on total factor productivity


= zwill
1 − z2 Dz2definite, and we may conclude
1 Cww
ive definite. Therefore, −D
beznegative
provides
empirical
evidence for the existence of
ive definite. Therefore, −D will
be
negative
definite,
and
we
may
conclude
= λ z z .
scale effects in terms of knowledge spillovers
dicates total differentiation with respect to time.
dicates total differentiation with respect to time.
between production sectors and in terms of
production externalities which are not
(cf. Diewert (1982, p. 567)). Also recall that D other
D is positive definite. Therefore, −D will be negative definite, and
we may
within
the conclude
direct exploitation potential of a sinis positive definite. Therefore, −D will be negagle
firm
or
industry (e.g., Caballero/Lyons
The dot ‘·’ indicates
total
differentiation
with
respect
to
time.
tive definite, and we may conclude from (8) that
(1990),
Benarroch
(1997), Midelfart/Steen
all eigenvalues of A must be real and non-posin/Wolff: Factor Supply Changes in Small Open Economies
12context, a firm which is subject
(1999)).
In
this
tive. Regularity of D implies det(−D) ≠ 0, while
det(M) ≠ 0 by assumption (cf. Section 2). There­ to increasing marginal cost appears to operate
under constant or even increasing returns to
fore
) that all eigenvalues
of A must be real and non-positive. Regularity of D implies
scale if there are inter-industry shifts in produc) = 0, while (9)
det(M) = 0 by assumption (cf. Section 2). Therefore
tivity. We now show that our results are robust
with respect to a simple class of production ex det(A) = det(−D) det(Cww − Cxw D−1 Cxw ) ternalities. Our analysis once more sticks as
closely as possible to the static model of perfect
= det(−D) det(M)
competition.
We can
startnot
from
(cf. pp.
Gantmacher
(1990,det(A)
pp. 45–46))
requires
ntmacher (1990,
45-46)) requires
= 0. Consequently,
zero
be the notion that the output xi of
det(A) ≠ 0. Consequently, zero can not be an each arbitrary industry i is a function fi (·, E, V)
nvalue of A. Thus all eigenvalues of A must be negative reals which
is sufficient
eigenvalue of A. Thus all eigenvalues of A must of several input factors and of proxies E and V
∗
∗
w∗ ) to be stable
(see
Takayama
(1985,
p.
310)).
be negative reals which is sufficient 
for (x , w ) of activity-based externalities and technical
progress, respectively (Caballero/Lyons (1990)).
to be stable (see Takayama (1985, p. 310)). 
thus find, unlike Mayer (1974) and Ide/Takayama (1988, 1990), that (seemingly)
As E and V are by assumption out of control of
Note
that Cww
is a negative
semi-definite
matrix
e that Cww is
a negative
semi-definite
matrix
(cf. Diewert
(1982, p. 567)). Also recall
ical comparative-statics effects may be observed even in well-behaved settings
cally stable equilibria. The following differences in the approach taken by these
on the one hand and our model on the other hand appear to be crucial. Firstly, we
that prices equal marginal rather than average costs and that firms operate at least
efficient scale without loss. Secondly, Mayer and Ide/Takayama impose several
17
Finnish Economic Papers 1/2009 – Andreas Pfingsten and Reiner Wolff
industry i, their respective output elasticities
can be locally normalized to equal unity (cf.
Caballero/Lyons (1990, pp. 808–809), Basu/
Fernald (1995, p. 169) for a formal exposition),
in which case the industry production function
has a local representation of the form xi = αhi (·),
where α : = EV. From the viewpoint of a single
firm or industry, increases in α are perceived as
exogenous downward shifts of the (upwardsloping) marginal cost curve. We assume that α
is a non-decreasing function of the economy’s
aggregate economic activity as measured by the
levels of total endowments. Recall that these
endowments are at full employment by assumption. They can thus serve as a means of transmission of global knowledge spillovers and
­localized agglomeration forces across industries. In the literature on new economic geography and on endogenous economic growth, this
role is often attributed to the stock of physical
capital (Romer (1986)) and to the ‘human capital’ or skill of a typical worker (Lucas (1988)).
Then, λxi = h (·) with λ : = 1/α. Hence, λ decreases if external scale advantages have a positive effect on the total factor productivity in
sector i. For simplicity, we pretend that this effect is the same in every industry.
Now let yi : = λxi for all industries i. Also, assume without loss of generality that λ = 1 in the
economy’s initial equilibrium. With this change
in notation, our results in Sections 2–4 can all
be retained. Furthermore:
Proposition 4 The signs of all positive Rybczynski derivatives are preserved under a decrease of λ.
Proof Suppose there is an extra supply of resources which generates an aggregate scale effect dλ < 0. By dyi = dxi + xidλ and xi > 0, we
conclude that dxi > 0 whenever dyi > 0. Hence,
the signs of all positive derivatives ∂yi /∂vj carry
over to ∂xi /∂vj. 
It even turns out that there is more room for
a positive output response in all industries to an
expansion of endowments, since output xi will
also increase if dyi < 0 as long as | dyi | < xi | dλ |.
Again, we think that there is nothing ‘paradoxical’ about this result. In contrast, our findings
18
are still intuitively plausible and consistent with
a profit-maximizing behavior of firms or industries in a competitive environment.
6. Conclusion
The purpose of this paper was to examine the
comparative statics of endowment changes in a
small open and competitive economy with a
finite number of goods and factors. We assumed
that this economy is exposed to diseconomies
of scale, and thus to increasing marginal costs,
within each industry. This assumption was motivated by a critical assessment of the standard
replication argument for constant returns, both
from a theoretical and an empirical perspective.
Scale disadvantages create increasing opportunity costs in the economy’s output space.
From a normative viewpoint, then, (incomplete)
specialization strategies in favor of selected industries are less rewarding, provided that inputs
can be re-allocated across sectors. Common intuition thus suggests that all production sectors
should benefit from an extra supply of any
scarce resource. We proved as our major result
that this conjecture, although in conflict with
Rybczynski’s theorem, can hold true, indeed, if
the production technologies permit input substitution. In contrast to the literature, we showed
that our result is fully compatible with the Walrasian and Marshallian stability of the under­
lying equilibrium. Our findings also turned out
to be robust with respect to global shifts in total
factor productivity which raise the economy’s
outputs beyond private returns.
As the Rybczynski theorem was developed in
the context of the Heckscher-Ohlin or factor
proportions theory of international trade, empirical tests of this theory in the literature may
shed some light on the relevance of the nonRybczynski responses we have derived. A seminal book is Leamer (1984). Alas, he argues that
it is “... wildly optimistic to recover the Ryb­
czynski coefficients from data on trade and endowments by a regression of trade on a subset
of excess factor supplies” (Leamer (1984, p.
159)). Therefore, he opts for an estimation of a
reduced-form model. Unfortunately, estimates
Finnish Economic Papers 1/2009 – Andreas Pfingsten and Reiner Wolff
of the Rybczynski coefficients cannot be easily
constructed from this model. However, Harrigan
(1995), who builds on Leamer’s work (cf. also
the more recent contribution by Feenstra
(2004)), concludes from a Heckscher-OhlinVanek version of the factor proportions model
that differences in output levels across countries
can be explained linearly by differences in national factor endowments. His estimation results
indicate that capital has a positive equilibrium
effect on all ten outputs considered (cf. Harrigan (1995, Table 4, pp. 135–136)). Thus, we
find a non-Rybczynski output response with respect to an important input. This demonstrates
at least that the outcome of our model is more
than just a theoretical fiction.
References
Basu, S., and J.G. Fernald (1995). “Are Apparent Productive Spillovers a Figment of Specification Error?” Journal of Monetary Economics 36, 165–188.
Basu, S., and J.G. Fernald (1997). “Returns to Scale in
U.S. Production: Estimates and Implications.” Journal
of Political Economy 105, 249–283.
Benarroch, M. (1997). “Returns to Scale in Canadian
Manufacturing: An Interprovincial Comparison.” Canadian Journal of Economics 30, 1083–1103.
Blackorby, C., W. Schworm, and A. Venables (1993).
“Necessary and Sufficient Conditions for Factor Price
Equalization.” Review of Economic Studies 60, 413–
434.
Buchanan, J.M., and Y.J. Yoon (1999). “Generalized Increasing Returns, Euler’s Theorem, and Competitive
Equilibrium.” History of Political Economy 31, 511–
523.
Burney, N.A. (1996). “Stability Conditions in General
Equilibrium Theory under Variable Returns to Scale.”
Academia Economic Press 24, 337–350.
Burnside, C. (1996). “Production Function Regressions,
Returns to Scale and Externalities.” Journal of Monetary
Economics 37, 177–201.
Caballero, R.J., and R.K. Lyons (1990). “Internal versus
External Economies in European Industry.” European
Economic Review 34, 805–826.
Caballero, R.J., and R.K. Lyons (1991). “External Effects
and Europe’s Integration.” In European Integration:
Trade and Industry. 34–51. Eds. L.A. Winters and A.J.
Venables. Cambridge: Cambridge University Press.
Caballero, R.J., and R.K. Lyons (1992). “External Effects
in U.S. Procyclical Productivity.” Journal of Monetary
Economics 29, 209–225.
Chang, W.W. (1979). “Some Theorems of Trade and General Equilibrium with Many Goods and Factors.” Econometrica 47, 709–726.
Choi, J.-Y. (1999). “Factor Growth, Urban Unemployment
and Welfare under Variable Returns to Scale.” International Economic Journal 13, 17–34.
Diewert, W.E. (1982). “Duality Approaches to Microeconomic Theory.” In Handbook of Mathematical Economics. Vol. II, 535–799. Eds. K.J. Arrow and M.D. Intriligator. Amsterdam: North-Holland.
Diewert, W.E., and A.D. Woodland (1977). “Frank
Knight’s Theorem in Linear Programming Revisited.”
Econometrica 45, 375–398.
Feenstra, R.C. (2004). Advanced International Trade:
Theory and Evidence. Princeton: Princeton University
Press.
Gantmacher, F.R. (1990). The Theory of Matrices. Vol. I,
2nd ed., 6th ptg. New York: Chelsea.
Hadley, G. (1974). Linear Algebra. 6th ptg. Reading/Mass.:
Addison-Wesley.
Harrigan, J. (1995). “Factor Endowments and the International Location of Production: Econometric Evidence for
the OECD, 1970–1985.” Journal of International Economics 39, 123–141.
Hansson, G., and M. Lundahl (1983). “The Rybczynski
Theorem under Decreasing Returns to Scale.” Scandinavian Journal of Economics 85, 531–540.
Henriksen, E., K.H. Midelfart Knarvik, and F. Steen
(2001). Economies of Scale in European Manufacturing
Revisited. CEPR Discussion Paper No. 2896, London.
Herberg, H., and M.C. Kemp (1969). “Some Implications
of Variable Returns to Scale.” Canadian Journal of Economics 2, 403–415.
Ide, T., and A. Takayama (1988). “Scale Economies, Perverse Comparative Statics Results, the Marshallian Stability and the Long-Run Equilibrium for a Small Open
Economy.” Economics Letters 27, 257–263.
Ide, T., and A. Takayama (1990). “Marshallian Stability
and Long-Run Equilibrium in the Theory of International Trade with Factor Market Distortions and Variable
Returns to Scale.” Economics Letters 33, 101–108.
Ingene, C.A., and E.S.H. Yu (1991). “Variable Returns to
Scale and Regional Resource Allocation Under Uncertainty.” Journal of Regional Science 31, 455–468.
Ishikawa, J. (1994). “Revisiting the Stolper-Samuelson and
Rybczynski Theorems with Production Externalities.”
Canadian Journal of Economics 27, 101–111.
Jones, R.W. (1968). “Variable Returns to Scale in General
Equilibrium Theory.” International Economic Review 9,
261–272.
Jones, R.W. (1971). “A Three-Factor Model in Theory,
Trade, and History.” In Trade, Balance of Payments and
Growth. 3–21. Eds. J.N. Bhagwati, R.W. Jones, R.A.
Mundell, and J. Vanek. Amsterdam: North-Holland.
Jones, R.W., and S.T. Easton (1983). “Factor Intensities
and Factor Substitution in General Equilibrium.” Journal
of International Economics 15, 65–99.
Jones, R.W., and J.A. Scheinkman (1977). “The Relevance of the Two-Sector Production Model in Trade
Theory.” Journal of Political Economy 85, 909–935.
Kemp, M.C. (1969). The Pure Theory of International
Trade and Investment. Englewood Cliffs/N.J.: PrenticeHall.
Leamer, E.E. (1984). Sources of International Comparative
Advantage: Theory and Evidence. 2nd ptg. Cambridge/
Mass.: MIT Press.
Lindström, T. (2000). “External Economies in Procyclical
Productivity: How Important are They?” Journal of Economic Growth 5, 163–184.
Lucas, R.E., Jr. (1988). “On the Mechanics of Economic
Development.” Journal of Monetary Economics 22, 3–
42.
Mas-Colell, A., M.D. Whinston, and J.R. Green (1995).
19
Finnish Economic Papers 1/2009 – Andreas Pfingsten and Reiner Wolff
Microeconomic Theory. New York: Oxford University
Press.
Mayer, W. (1974). “Variable Returns to Scale in General
Equilibrium Theory: A Comment.” International Economic Review 15, 225–235.
Midelfart Knarvik, K.H., and F. Steen (1999). “Self-reinforcing Agglomerations? An Empirical Industry
Study.” Scandinavian Journal of Economics 101, 515–
532.
Oulton, N. (1996). “Increasing Returns and Externalities in
UK Manufacturing: Myth or Reality?” Journal of Industrial Economics 44, 99–113.
Panagariya, A. (1980). “Variable Returns to Scale in General Equilibrium Theory Once Again.” Journal of International Economics 10, 499–526.
Romer, P.M. (1986). “Increasing Returns and Long-Run
Growth.” Journal of Political Economy 94, 1002–1037.
Ruffin, R.J. (1981). “Trade and Factor Movements with
Three Factors and Two Goods.” Economics Letters 7,
177–182.
20
Rybczynski, T.M. (1955). “Factor Endowments and Relative Commodity Prices.” Economica 22, 336–341.
Samuelson, P.A. (1947). Foundations of Economic Analysis. Cambridge/Mass.: Harvard University Press.
Samuelson, P.A. (1953/54). “Prices of Factors and Goods
in General Equilibrium.” Review of Economic Studies
21, 1–20.
Stolper, W., and P.A. Samuelson (1941). “Protection and
Real Wages.” Review of Economic Studies 9, 58–73.
Takayama, A. (1985). Mathematical Economics. 2nd ed.
Hinsdale: Dryden.
Woodland, A.D. (1982). International Trade and Resource
Allocation. Amsterdam: North-Holland.
Ylönen, S. (1987). “Factor Demand and Substitution under
Decreasing Returns to Scale. An Application to the Rybczynski Theorem.” Scandinavian Journal of Economics
89, 209–216.