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Scale economies, perverse comparative statics results, the Marshallian stability and the long-run equilibrium for a small open economy
257
Economics Letters 27 (1988) 257-263 North-Holland
SCALE ECONOMIES, PERVERSE THE MARSI-IALLIAN STABILITY FOR A SMALL OPEN ECONOMY
COMPARATIVE STATICS RESULTS, AND THE LONG-RUN EQUILIBRIUM *
Toyonari IDE and Akira TAKAYAMA Southern Illinois University, Carbondale, IL 62901, USA
Received 17 February 1988
Using a simple general-equilibrium model, we show that the perverse comparative statics results can occur under scale economies. These paradoxes disappear iff the equilibrium is Marshallian stable, and the latter holds iff it is a stable long-run equilibrium under capital adjustment.
The problem of (national) scale economies and trade is by no means new. However, it has been attracting a great deal of attention recently. See, for example, Jones (1968), Herberg and Kemp (1969), Mayer (1974) Panagariya (1980, 1986), Markusen and Melvin (1981) and Helpman (1984). The literature typically utilizes a simple 2 X 2 general equilibrium framework. One focal point is various non-normal comparative statics results for the economy under variable returns to scale. For example, following Jones’ important work (1968) Panagariya (1980) writes, (i)
‘The validity of the Stopler-Samuelson theorem is neither necessary nor sufficient for the PPF to be locally strictly concave to the origin’ (p. 511). (ii) ‘The validity of the Rybczynski theorem is neither a necessary nor sufficient condition for the PPF to be locally strictly concave to the origin’ (p. 514). Although one may not have any vested interest in these two theorems, the above conclusions are somewhat discomfiting. Poor students, who first learn these two theorems in connection with the Heckscher-Ohlin-Samuelson model, are then informed with the above results. These students then may wonder what they should believe in the Stolper-Samuelson and Rybczynski theorems. We shall clarify circumstances under which the these theorems are valid. The purpose of this paper is, however, not to reconcile the Jones-Panagariya controversy. We hope to go beyond this. We argue that the non-normal comparative statics results such as perverse Stolper-Samuelson and Rybczynski effects and the perverse price-output response would all vanish, if and only if the equilibrium is Marshallian stable. ’ We further show that an equilibrium is l
’
We are indebted to Peter Neary and Winston Chang for providing us with very useful comments on our related papers (1988a, b) and for their encouragement. Avinash Dixit also provided us helpful comments. For a clarification of the common confusion between the Walrasian and the Marshallian stability, see Davies (1963) and Newman (1965) for example.
0165-1765/88/$3.50
0 1988, Elsevier Science Publishers B.V. (North-Holland)
258
T. Ide, A. Takayama
/ Scale economies. peruerse comparative statics results
Marshallian stable, if and only if such an equilibrium is a stable long-run equilibrium in which enough time is allowed so that capital is allocated to the two sectors in the respective optimal levels. 2 Thus, the comparative statics paradoxes are theoretical curiosa which will ‘almost never’ be observed in real world economies. Note that the Marshallian stability is much easier to examine than the stability of the factor market adjustment process. Also this paper indicates that in the context of an economy characterized by scale economies, Marshall’s two celebrated concepts (both appear in his Principles), the Marshallian stability and long-run (or long period) equilibrium wed happily. Our work also reminds us of Neary’s seminal work (1978a, b), which relates the comparative statics paradoxes to the stability of factor market adjustment processes for the economy under factor-market distortions. 3 We consider an economy consisting of two sectors whose production functions are specified by 4 X= XTF( L,, Y= G(L,,
K,),
0 < T-C 1,
or
X= F(L,,
T* = l/(1
Kx)T*,
- T),
(1)
Ky),
(2)
where L, and K,, respectively, denote the amounts of labor and capital used in sector i. It is assumed that F and G are homogeneous of degree one with respect to inputs. It is also assumed that scale economies prevail in sector X, where the scale merit is captured by T or T *. 5 Let a,, denote the quantity of factor i required to produce one unit of good j. The requirement that both factors are fully employed is given by, a,,X+
a,,Y=
a,,X+
L,
a,,Y
= K,
(3)
where L and K, respectively, denote the endowments of labor and capital. In addition we have the following zero profit condition: aLyw + aKyr = 1,
ULXW + aKXr =p,
(4)
where p denotes the price of good X in terms of good Y, and where w and r, respectively, denote the wage rate and the rental rate in terms of good Y. Let h, be the fraction of the ith factor used in the jth production, and B,i signify the ith cost share of the jth industry. Let 1X 1 and 18 1 be notations for the determinants of coefficients, h,,s, defined by
I h I = h,b,
- ~,,b~
I 0 I = why
- 4,4,.
Then we may assert I X 1 5 0 and I 6’ I 5 0 according to whether k, >
the Marshallian
(1) has a rather
exemplified perfectly
by Markusen
competitive
representing
and Melvin (1981)
condition,
scale economies
other hand, (1 - a) restricts is imposed
by Panagariya
stability is done by us elsewhere
special form compared
to X=
and Panagariya
scale economies
the production
K,,
function
[Ide and Takayama To obtain of each
many
the consistency
firm but internal
is not important.
to be homothetic,
(1988b)].
X), it is useful to obtain
(1986).
must be external
in the form of (l), this distinction
See Helpman
useful conclusions
of scale economies to the industry.
(1984,
pp. 332-334).
which is not the case in Jones (1968).
as with
However, On the
Homotheticity
(1980).
It can be shown that T * is equal to ‘scale elasticity’, For a clarification
F(L,,
of various concepts
of returns
and that it is also equal to the ratio of average cost over marginal
to scale, see Ide and Takayama
(1987).
cost.
T. Ide, A. Takayama / Scale economies, peroerse comparatrue siatrcs results
usual fashion. Further, denoting the elasticities may obtain A
- eKxux~
ULX'
-
Ti,
a^,, = eL,u&
- TJ?,
of factor substitution
259
of the jth industry by a,, we
a^,, = - e,,a,;,
a^,, = e,,+.
(5)
where o = W/T (the wage-rental ratio), and where (n) signifies the rate of change. Differentiating (3) and (4), and using (5), we may obtain
x=(l-&X, A
(64
{(h,.i-h,,K)+(X,,6,+X,,S,)O},
i=
-e,,@
Q=e,&
+ T~)/lel,
+
T@/‘lel>
cZ=(jj+T&‘leI,
(7)
where 6, = X,,B,,u, + ALyBKyuy and 6, = X,,B,,u, + X,,&,,u,,. The procedure used to obtain (6) and (7) has become routine in the literature since the seminal work of Jones (1965). The economic interpretation of 6, and 8, is found in Jones (1965, p. 561). When &?= i = 0, we may obtain from (6) and (7) .
,.
x-‘= (l-
T;lx/IeI
(F+
T@,
6 = (1 - T)(6,
+ 8,)
+ T(X,,&
+ h,,S,),
where A~(l-T)IX~~e~-T(~,,6,+X,,6,).
(10)
Then letting Z = X/Y, we may obtain the following remarkable result, b/Z = A/S.
(11)
To determine the shape of the production possibility frontier (PPF), we write its functional relation as, Y = Y(X), r = Y(O), 0 = Y(X). Using Y = Y(X), the MRT between two commodities under incomplete specialization is MRT = -(dY/dX). Then we may obtain [cf. Markusen-Melvin
(1981)1, T* . MRT=p
so that
p > MRT.
(12)
Namely the ‘wedge’ between the MRT and p is constant, being equal to T *. Along the PPF, Z increases iff X increases, so that p and MRT increase or decrease depending on whether A > 0 or A -C 0. Thus the PPF is strictly concave to the origin if A > 0, and it is strictly convex to the origin if A < 0. If A = 0 for some (X, Y) > 0, the PPF changes its shape. Such points of ( X, Y) are called the inflection points of the PPF. We may cell A the inflection index. In a neighborhood X = 0, we can show A < 0. In fact, this and other results on the shape of the PPF are obtained by Herberg and Kemp (1969) and others.
T. Ide, A. Takayama
260
Define
the supply
acceptable Denoting
price
price
/ Scale economies, perverse comparative statics results
as the price
for producers).
Then
at which
producers
p discussed
the supply price by ps, we may then replace
&s/Z = A/6,
where
are just
breaking
in the previous
section
even (the is the
minimal
supply
p by ps, and (11) may be written
price.
as
6 > 0.
(11’)
When we plot ps over Z (measuring Z on the horizontal axis), the p,-curve may not be upward-sloping in the usual way. It depends on the sign of A. It is also possible that the ps-curve is downward-sloping PPF corresponds
in one region of Z and upward-sloping to the shape of the ps curve.
To close the model, commodity price ratio
in another
region of Z. The shape of the
we assume that the country is a small open which prevails in the world market, where
economy, and let p* p* is an exogenously
constant for the country. We assume that p*-line intersects the p,-curve country will completely specialize in the production of one commodity).
at least once (otherwise We focus our attention
such interior equilibrium points. We say that the price-output response is normal depending on whether or not a small increase in the relative price of one commodity increase
in its output
following Proposition A -C 0.
(for given factor
endowments).
Then
from (11)
the on
or perverse, leads to an
we may at once
obtain
the
result, where p is taken parametrically. I.
The price-output
To investigate the validity following assumptions.
response
is normal
or perverse
of the Stolper-Samuelson
depending
and Rybczynski
Assumption I. In an economy with a fixed set of factor endowments, must increase the (average) cost or producing each commodity. Assumption 2. At constant commodity demand for each factor of production. These
assumptions
obtain
the following
Proposition
be the given
2. 6
correspond
prices, the expansion
to ‘Assumptions
upon
theorems,
an increase
of any industry
2 and 3’ imposed
whether
in Jones
A > 0 or
we impose
the
in any factor price
results in an increased
(1968).
Then
we may
result.
Under Assumption
I, the Stolper-Samuelson
on whether A > 0 or A -C 0. Under Assumption depending on whether A > 0 or A x 0.
relation
2, the Rybczynski
is normal relation
or perverse is normal
depending or perverse
Panagariya (1980) is concerned with limited cases in which Assumptions 1 and 2 do not hold. ’ If these do not hold, the perversity is not complete, i.e., complete reversals of normal sign patterns need not hold. 6 The S-S relation is said to be normal if (at constant factor endowments) a small increase in the relative commodity price raises the real reward to the factor which is used in that sector more intensively and decreases the real reward of the other factor. The Rybczynski relation is said to be normal if (at constant commodity prices), a small increase in one factor increases the output of the commodity which uses the increased factor more intensively and decreases the other output. If these relations are completely reversed, we say the S-S and R relations are perverse. ’ For details, see Ide and Takayama (1988a).
T. Ide, A. Takayama
261
/ Scale economies, perverse comparative statics results
We now show that these perverse comparative statics are necessarily associated with the Marshallian instability of a particular equilibrium. To this end, we postulate the following Marshallian adjustment process for Z > 0 (with constant factor endowments):
.2= a[p*/p(z) - 11= +(z>,
(13)
a > 0.
Namely, the output of X increases relative to that of Y, if and only if the world demand price p* exceeds the domestic supply price ps. The demand price is the maximum price that consumers are willing to pay. Recall that we assumed that there exists a finite value Z* > 0 which satisfies +(Z*)=O, where Z* signifies an incomplete specialization equilibrium. For simplicity, assume away the knife-edge case of +‘( Z *) = 0. Then Z * is asymptotically (locally) stable if and only if $‘(Z*) < 0. Then using (11’) we may obtain the following result. Proposition
3.
The equilibrium point Z* is Marshallian
stable, if and only if A > 0.
Combining this with Propositions 1 and 2, we may at once obtain the following results, where we impose Assumptions 1 and 2. response, the Stolper-Samuelson Proposition 4. The price-output if and only if the equilibrium point is Marshallian stable.
and Rybczynski
relations are normal
In the Marshallian short-run equilibrium, capital is sector specific, and in the Marshallian long-run equilibrium, the capital stock in each sector is adjusted to the optimal scale. Assume that one factor, say labor, is freely and quickly mobile between the sectors, and that it achieves the inter-sectoral equality of the wage rate instantaneously. Such a state is called the short-run equilibrium. Namely, the fast moving factor is called ‘labor’. In the short-run, capital is sector specific, and rental rate is not equalized. Let rx and rY, respectively, denote the rental rate in sectors X and Y in the short-run. We then postulate the following adjustment process: ix=
b[r,/r,-
l],
b> 0.
(14)
Namely, the sectoral gap of the rental rate induces capital to move between the sectors. In the long-run, we have rx = rY = r. Using the model described earlier, we may express the RHS of (14) as a function of K, alone. Define the function 1c,by,
+(Kx) = b[rx(Kx)/r,(Kx) - 11.
(15)
The long-run equilibrium value of K, is defined by 4 (K; ) = 0. Assume away the knife-edge case of #‘(K,*) = 0. Then we may assert that the long-run equilibrium is asymptotically (locally) stable if and only if +‘( K:) < 0. We further impose the following assumption. Assumption 3. At constant commodity prices, the expansion of any sector results in an increased demand for each factor of production. Under this assumption, we may obtain the following result.
262
T. I&
A. Takayama
/ Scale economies, perverse comparative statics results
Proposition 5. The Marshallian stability the adjustment process due to intersectoral
condition (A > 0) holds, if and only if the equilibrium capital mobility is stable.
under
Namely, the equilibrium is Marshallian stable, if and only if it is a stable ong-run equilibrium in which a long enough time is allowed so that the allocation of capital between the two sectors is adjusted to the respective optimal levels. Combining this with Proposition 4 we may obtain the following result at once, where we impose Assumptions l-3. Proposition 6. The normal price-output response relation, the (normal) Stolper-Samuelson theorem and the (normal) Rybczynski theorem holds, if and only if the equilibrium under the adj.ustment process due to intersectoral capital mobility is stable. Suppose that both labor and capital are sector specific so that we have wx # w r (as well as rx # rY), where wx and w r, respectively, denote the wage rate in sectors X and Y. In this case, we may postulate the following adjustment process:
L=
4 WL,,
G=P[
PXCL,,
K,)/w,(L,,
K,) - 11= @(Lx, K,),
K,)/P,(L,,
K,)
- 11 = *(Lx,
K,),
where LY> 0 and p > 0. The long-run equilibrium is then defined by @( Lz, K,* ) = 9( L$ K,* ) = 0. Performing the Taylor approximation and considering the linear approximation system (LAS), we may then obtain the following result, where we impose Assumption 3. Proposition 7. The Marshallian stability stability of the LAS described above.
condition
(A > 0) is necessary for, but not sufficient
for the
The factor adjustment process such as (16) is investigated by Neary (1978a) in the context of an economy characterized by factor-market distortions. Proposition 7 indicates that unlike his case, the correspondence between the Marshallian stability and the normal comparative statics results is not complete in the present model with scale economies when both factors are allowed to adjust simultaneously.
References Davies, David G., 1963, A note on Marshallian versus Walrasian stability conditions, Canadian Journal of Economics and Political Science 29, Nov., 535-540. Helpman, Elhanan, 1984, Increasing returns, imperfect markets, and trade theory, in: R.W. Jones and P.B. Kenen, eds., Handbook of international economics (North-Holland, Amsterdam) 325-365. Herberg, Horst and Murray C. Kemp, 1969, Some implications of variable returns to scale, Canadian Journal of Economics 2, Aug., 403-415. Ide, T. and A. Takayama, 1987, On the concept of returns to scale, Economics Letters 23, 329-334. Ide, T. and A. Takayama, 1988a, Scale economies and the Marshallian stability in the pure theory of international trade, Unpublished manuscript, Jan. (Southern Illinois University, Carbondale, IL). Ide, T. and A. Takayama, 1988b, Marshallian stability and factor-market distortions in a small open economy, Unpublished manuscript, Jan. (Southern Illinois University, Carbondale, IL). Jones, Ronald W., 1965, The structure of simple general equilibrium models, Journal of Political Economy 73, Sept., 557-572.
T. Ide, A. Takayama
/ Scale economies, perverse comparaiive statics results
263
Jones, Ronald W., 1968, Variable returns to scale in general equilibrium theory, International Economic Review, Oct., 261-272. Markusen, James M. and James M. Melvin, 1981, Trade, factor prices, and the gains from trade with increasing returns to scale, Canadian Journal of Economics 14, Aug., 450-469. Marshall, Alfred, 1920, Principles of economics, eighth ed. (Macmillan, London). Mayer, Wolfgang, 1974, Variable returns to scale in general equilibrium theory: A comment, International Economic Review 15, Feb., 225-235. Neary, J. Peter, 1978a, Dynamic stability and the theory of factor market distortion, American Economic Review 68, Sept., 671-682. Neary, J. Peter, 1978b, Short-run capital specificity and the pure theory of international trade, Economic Journal 88, Sept., 488-510. Newman, Peter, 1965, The theory of exchange (Prentice-Hall, Englewood Cliffs, NJ). Panagariya, Arvind, 1980, Variable returns to scale in general equilibrium theory once again, Journal of International Economics 10, Nov., 499-526. Panagariya, Arvind, 1986, Increasing returns, dynamic stability and international trade, Journal of International Economics 20, Feb., 43-63.
Economics Letters 27 (1988) 257-263 North-Holland
SCALE ECONOMIES, PERVERSE THE MARSI-IALLIAN STABILITY FOR A SMALL OPEN ECONOMY
COMPARATIVE STATICS RESULTS, AND THE LONG-RUN EQUILIBRIUM *
Toyonari IDE and Akira TAKAYAMA Southern Illinois University, Carbondale, IL 62901, USA
Received 17 February 1988
Using a simple general-equilibrium model, we show that the perverse comparative statics results can occur under scale economies. These paradoxes disappear iff the equilibrium is Marshallian stable, and the latter holds iff it is a stable long-run equilibrium under capital adjustment.
The problem of (national) scale economies and trade is by no means new. However, it has been attracting a great deal of attention recently. See, for example, Jones (1968), Herberg and Kemp (1969), Mayer (1974) Panagariya (1980, 1986), Markusen and Melvin (1981) and Helpman (1984). The literature typically utilizes a simple 2 X 2 general equilibrium framework. One focal point is various non-normal comparative statics results for the economy under variable returns to scale. For example, following Jones’ important work (1968) Panagariya (1980) writes, (i)
‘The validity of the Stopler-Samuelson theorem is neither necessary nor sufficient for the PPF to be locally strictly concave to the origin’ (p. 511). (ii) ‘The validity of the Rybczynski theorem is neither a necessary nor sufficient condition for the PPF to be locally strictly concave to the origin’ (p. 514). Although one may not have any vested interest in these two theorems, the above conclusions are somewhat discomfiting. Poor students, who first learn these two theorems in connection with the Heckscher-Ohlin-Samuelson model, are then informed with the above results. These students then may wonder what they should believe in the Stolper-Samuelson and Rybczynski theorems. We shall clarify circumstances under which the these theorems are valid. The purpose of this paper is, however, not to reconcile the Jones-Panagariya controversy. We hope to go beyond this. We argue that the non-normal comparative statics results such as perverse Stolper-Samuelson and Rybczynski effects and the perverse price-output response would all vanish, if and only if the equilibrium is Marshallian stable. ’ We further show that an equilibrium is l
’
We are indebted to Peter Neary and Winston Chang for providing us with very useful comments on our related papers (1988a, b) and for their encouragement. Avinash Dixit also provided us helpful comments. For a clarification of the common confusion between the Walrasian and the Marshallian stability, see Davies (1963) and Newman (1965) for example.
0165-1765/88/$3.50
0 1988, Elsevier Science Publishers B.V. (North-Holland)
258
T. Ide, A. Takayama
/ Scale economies. peruerse comparative statics results
Marshallian stable, if and only if such an equilibrium is a stable long-run equilibrium in which enough time is allowed so that capital is allocated to the two sectors in the respective optimal levels. 2 Thus, the comparative statics paradoxes are theoretical curiosa which will ‘almost never’ be observed in real world economies. Note that the Marshallian stability is much easier to examine than the stability of the factor market adjustment process. Also this paper indicates that in the context of an economy characterized by scale economies, Marshall’s two celebrated concepts (both appear in his Principles), the Marshallian stability and long-run (or long period) equilibrium wed happily. Our work also reminds us of Neary’s seminal work (1978a, b), which relates the comparative statics paradoxes to the stability of factor market adjustment processes for the economy under factor-market distortions. 3 We consider an economy consisting of two sectors whose production functions are specified by 4 X= XTF( L,, Y= G(L,,
K,),
0 < T-C 1,
or
X= F(L,,
T* = l/(1
Kx)T*,
- T),
(1)
Ky),
(2)
where L, and K,, respectively, denote the amounts of labor and capital used in sector i. It is assumed that F and G are homogeneous of degree one with respect to inputs. It is also assumed that scale economies prevail in sector X, where the scale merit is captured by T or T *. 5 Let a,, denote the quantity of factor i required to produce one unit of good j. The requirement that both factors are fully employed is given by, a,,X+
a,,Y=
a,,X+
L,
a,,Y
= K,
(3)
where L and K, respectively, denote the endowments of labor and capital. In addition we have the following zero profit condition: aLyw + aKyr = 1,
ULXW + aKXr =p,
(4)
where p denotes the price of good X in terms of good Y, and where w and r, respectively, denote the wage rate and the rental rate in terms of good Y. Let h, be the fraction of the ith factor used in the jth production, and B,i signify the ith cost share of the jth industry. Let 1X 1 and 18 1 be notations for the determinants of coefficients, h,,s, defined by
I h I = h,b,
- ~,,b~
I 0 I = why
- 4,4,.
Then we may assert I X 1 5 0 and I 6’ I 5 0 according to whether k, >
the Marshallian
(1) has a rather
exemplified perfectly
by Markusen
competitive
representing
and Melvin (1981)
condition,
scale economies
other hand, (1 - a) restricts is imposed
by Panagariya
stability is done by us elsewhere
special form compared
to X=
and Panagariya
scale economies
the production
K,,
function
[Ide and Takayama To obtain of each
many
the consistency
firm but internal
is not important.
to be homothetic,
(1988b)].
X), it is useful to obtain
(1986).
must be external
in the form of (l), this distinction
See Helpman
useful conclusions
of scale economies to the industry.
(1984,
pp. 332-334).
which is not the case in Jones (1968).
as with
However, On the
Homotheticity
(1980).
It can be shown that T * is equal to ‘scale elasticity’, For a clarification
F(L,,
of various concepts
of returns
and that it is also equal to the ratio of average cost over marginal
to scale, see Ide and Takayama
(1987).
cost.
T. Ide, A. Takayama / Scale economies, peroerse comparatrue siatrcs results
usual fashion. Further, denoting the elasticities may obtain A
- eKxux~
ULX'
-
Ti,
a^,, = eL,u&
- TJ?,
of factor substitution
259
of the jth industry by a,, we
a^,, = - e,,a,;,
a^,, = e,,+.
(5)
where o = W/T (the wage-rental ratio), and where (n) signifies the rate of change. Differentiating (3) and (4), and using (5), we may obtain
x=(l-&X, A
(64
{(h,.i-h,,K)+(X,,6,+X,,S,)O},
i=
-e,,@
Q=e,&
+ T~)/lel,
+
T@/‘lel>
cZ=(jj+T&‘leI,
(7)
where 6, = X,,B,,u, + ALyBKyuy and 6, = X,,B,,u, + X,,&,,u,,. The procedure used to obtain (6) and (7) has become routine in the literature since the seminal work of Jones (1965). The economic interpretation of 6, and 8, is found in Jones (1965, p. 561). When &?= i = 0, we may obtain from (6) and (7) .
,.
x-‘= (l-
T;lx/IeI
(F+
T@,
6 = (1 - T)(6,
+ 8,)
+ T(X,,&
+ h,,S,),
where A~(l-T)IX~~e~-T(~,,6,+X,,6,).
(10)
Then letting Z = X/Y, we may obtain the following remarkable result, b/Z = A/S.
(11)
To determine the shape of the production possibility frontier (PPF), we write its functional relation as, Y = Y(X), r = Y(O), 0 = Y(X). Using Y = Y(X), the MRT between two commodities under incomplete specialization is MRT = -(dY/dX). Then we may obtain [cf. Markusen-Melvin
(1981)1, T* . MRT=p
so that
p > MRT.
(12)
Namely the ‘wedge’ between the MRT and p is constant, being equal to T *. Along the PPF, Z increases iff X increases, so that p and MRT increase or decrease depending on whether A > 0 or A -C 0. Thus the PPF is strictly concave to the origin if A > 0, and it is strictly convex to the origin if A < 0. If A = 0 for some (X, Y) > 0, the PPF changes its shape. Such points of ( X, Y) are called the inflection points of the PPF. We may cell A the inflection index. In a neighborhood X = 0, we can show A < 0. In fact, this and other results on the shape of the PPF are obtained by Herberg and Kemp (1969) and others.
T. Ide, A. Takayama
260
Define
the supply
acceptable Denoting
price
price
/ Scale economies, perverse comparative statics results
as the price
for producers).
Then
at which
producers
p discussed
the supply price by ps, we may then replace
&s/Z = A/6,
where
are just
breaking
in the previous
section
even (the is the
minimal
supply
p by ps, and (11) may be written
price.
as
6 > 0.
(11’)
When we plot ps over Z (measuring Z on the horizontal axis), the p,-curve may not be upward-sloping in the usual way. It depends on the sign of A. It is also possible that the ps-curve is downward-sloping PPF corresponds
in one region of Z and upward-sloping to the shape of the ps curve.
To close the model, commodity price ratio
in another
region of Z. The shape of the
we assume that the country is a small open which prevails in the world market, where
economy, and let p* p* is an exogenously
constant for the country. We assume that p*-line intersects the p,-curve country will completely specialize in the production of one commodity).
at least once (otherwise We focus our attention
such interior equilibrium points. We say that the price-output response is normal depending on whether or not a small increase in the relative price of one commodity increase
in its output
following Proposition A -C 0.
(for given factor
endowments).
Then
from (11)
the on
or perverse, leads to an
we may at once
obtain
the
result, where p is taken parametrically. I.
The price-output
To investigate the validity following assumptions.
response
is normal
or perverse
of the Stolper-Samuelson
depending
and Rybczynski
Assumption I. In an economy with a fixed set of factor endowments, must increase the (average) cost or producing each commodity. Assumption 2. At constant commodity demand for each factor of production. These
assumptions
obtain
the following
Proposition
be the given
2. 6
correspond
prices, the expansion
to ‘Assumptions
upon
theorems,
an increase
of any industry
2 and 3’ imposed
whether
in Jones
A > 0 or
we impose
the
in any factor price
results in an increased
(1968).
Then
we may
result.
Under Assumption
I, the Stolper-Samuelson
on whether A > 0 or A -C 0. Under Assumption depending on whether A > 0 or A x 0.
relation
2, the Rybczynski
is normal relation
or perverse is normal
depending or perverse
Panagariya (1980) is concerned with limited cases in which Assumptions 1 and 2 do not hold. ’ If these do not hold, the perversity is not complete, i.e., complete reversals of normal sign patterns need not hold. 6 The S-S relation is said to be normal if (at constant factor endowments) a small increase in the relative commodity price raises the real reward to the factor which is used in that sector more intensively and decreases the real reward of the other factor. The Rybczynski relation is said to be normal if (at constant commodity prices), a small increase in one factor increases the output of the commodity which uses the increased factor more intensively and decreases the other output. If these relations are completely reversed, we say the S-S and R relations are perverse. ’ For details, see Ide and Takayama (1988a).
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261
/ Scale economies, perverse comparative statics results
We now show that these perverse comparative statics are necessarily associated with the Marshallian instability of a particular equilibrium. To this end, we postulate the following Marshallian adjustment process for Z > 0 (with constant factor endowments):
.2= a[p*/p(z) - 11= +(z>,
(13)
a > 0.
Namely, the output of X increases relative to that of Y, if and only if the world demand price p* exceeds the domestic supply price ps. The demand price is the maximum price that consumers are willing to pay. Recall that we assumed that there exists a finite value Z* > 0 which satisfies +(Z*)=O, where Z* signifies an incomplete specialization equilibrium. For simplicity, assume away the knife-edge case of +‘( Z *) = 0. Then Z * is asymptotically (locally) stable if and only if $‘(Z*) < 0. Then using (11’) we may obtain the following result. Proposition
3.
The equilibrium point Z* is Marshallian
stable, if and only if A > 0.
Combining this with Propositions 1 and 2, we may at once obtain the following results, where we impose Assumptions 1 and 2. response, the Stolper-Samuelson Proposition 4. The price-output if and only if the equilibrium point is Marshallian stable.
and Rybczynski
relations are normal
In the Marshallian short-run equilibrium, capital is sector specific, and in the Marshallian long-run equilibrium, the capital stock in each sector is adjusted to the optimal scale. Assume that one factor, say labor, is freely and quickly mobile between the sectors, and that it achieves the inter-sectoral equality of the wage rate instantaneously. Such a state is called the short-run equilibrium. Namely, the fast moving factor is called ‘labor’. In the short-run, capital is sector specific, and rental rate is not equalized. Let rx and rY, respectively, denote the rental rate in sectors X and Y in the short-run. We then postulate the following adjustment process: ix=
b[r,/r,-
l],
b> 0.
(14)
Namely, the sectoral gap of the rental rate induces capital to move between the sectors. In the long-run, we have rx = rY = r. Using the model described earlier, we may express the RHS of (14) as a function of K, alone. Define the function 1c,by,
+(Kx) = b[rx(Kx)/r,(Kx) - 11.
(15)
The long-run equilibrium value of K, is defined by 4 (K; ) = 0. Assume away the knife-edge case of #‘(K,*) = 0. Then we may assert that the long-run equilibrium is asymptotically (locally) stable if and only if +‘( K:) < 0. We further impose the following assumption. Assumption 3. At constant commodity prices, the expansion of any sector results in an increased demand for each factor of production. Under this assumption, we may obtain the following result.
262
T. I&
A. Takayama
/ Scale economies, perverse comparative statics results
Proposition 5. The Marshallian stability the adjustment process due to intersectoral
condition (A > 0) holds, if and only if the equilibrium capital mobility is stable.
under
Namely, the equilibrium is Marshallian stable, if and only if it is a stable ong-run equilibrium in which a long enough time is allowed so that the allocation of capital between the two sectors is adjusted to the respective optimal levels. Combining this with Proposition 4 we may obtain the following result at once, where we impose Assumptions l-3. Proposition 6. The normal price-output response relation, the (normal) Stolper-Samuelson theorem and the (normal) Rybczynski theorem holds, if and only if the equilibrium under the adj.ustment process due to intersectoral capital mobility is stable. Suppose that both labor and capital are sector specific so that we have wx # w r (as well as rx # rY), where wx and w r, respectively, denote the wage rate in sectors X and Y. In this case, we may postulate the following adjustment process:
L=
4 WL,,
G=P[
PXCL,,
K,)/w,(L,,
K,) - 11= @(Lx, K,),
K,)/P,(L,,
K,)
- 11 = *(Lx,
K,),
where LY> 0 and p > 0. The long-run equilibrium is then defined by @( Lz, K,* ) = 9( L$ K,* ) = 0. Performing the Taylor approximation and considering the linear approximation system (LAS), we may then obtain the following result, where we impose Assumption 3. Proposition 7. The Marshallian stability stability of the LAS described above.
condition
(A > 0) is necessary for, but not sufficient
for the
The factor adjustment process such as (16) is investigated by Neary (1978a) in the context of an economy characterized by factor-market distortions. Proposition 7 indicates that unlike his case, the correspondence between the Marshallian stability and the normal comparative statics results is not complete in the present model with scale economies when both factors are allowed to adjust simultaneously.
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