the assumptions of the factor price equalization …
TRANSCRIPT
/
THE ASSUMPTIONS OF THE FACTOR PRICE EQUALIZATION THEOREM
by
STEPHEN P. MAGEE, B. A.
A THESIS
IN
ECONOMICS
Submitted to the Graduate Faculty of Texas Technological College in Partial Fulfillment of
the Requirements for the Degree of
MASTER OF ARTS
Approved
Accepted
August,
:0! 6
SOS-
No. loo
ACKNOWLEDGMENTS
I am indebted to Professor Robert L. Rouse for his direction of
this thesis, and to Professors Paul A. Samuelson and Charles P. Kindle-
berger for their helpful criticism.
I also wish to acknowledge my great debt to the man who
introduced me to mathematical economics. Professor Jarvis Witt.
ii
TABLE OF CONTENTS
Chapter Page
I. INTRODUCTION 1
II. HISTORICAL ANTECEDENTS 3
III. THE PROOF 8
IV. THE ASSUMPTIONS OF THE THEOREM 15
V. CONCLUSION 48
BIBLIOGRAPHY 50
CHAPTER I
INTRODUCTION
One of the popular controversies in international trade theory
and indeed in international economic policy is the effect of free trade
on the return to productive factors. It was argued not many years ago
that reducing U. S. tariffs, for example, would precipitate an inundation
of Japanese goods produced by cheap labor, thereby driving down wages
accruing to American labor. This popular controversy has its counter
part in th« theoretical literature as well. Samuelson's (1948 and 1949)
factor price equalization theorem was the first rigorous proof that trade
in goods alone(no factor movements) is sufficient to equalize the
returns to factors in two trading countries. There are eight conditions
which must be met, however, before price equalization can occur. The
purpose of this work is to examine these eight assumptions in the two
country, two good, two factor model. Are all of them necessary conditions
for the Samuelson proof? Are there modifications and relaxations of
these assumptions which would still permit the theorem to hold? One
purpose of the work here undertaken would seem to be to illustrate the
interplay between the very building blocks of international trade theory:
four inputs, four marginal productivities, two outputs, commodity price,
input prices and factor proportions for each country.
Originality is generally preempted by reviews of the theoretical
literature. My attempt to eliminate the assumption of zero transport
costs by input-output methods did not result in the theoretical
break-through that I had originally envisioned. Professor Samuelson
pointed out that in so far as transport costs pose no more an impedirr.ent
to intra-country than inter-country unit product costs, the theoretical
substance of the paradox posed disappears. The impact of the transport
cost assumption is thus on possible verification of the theorem. To
eliminate the assumption even by incorporation via input-output methods
would limit, although not eliminate, the geographical impact of factor
price equalization.
One curiosum of international trade theory is why a complete
proof of the factor price equalization theorem was not forthcoming
earlier. The question had been discussed for over 150 years, and yet
only in 1948 did all the discussion bear real fruit. This historiccil
aspect will be discussed in Chapter II. Chapter III will be a short
description of Samuelson's proof of the theorem. Chapter IV will be
an examination of the eight assumptions and the questions posed above,
while Chapter V will be a summary.
CHAPTER II
HISTORICAL ANTECEDENTS
The proposition thettnobility of factors between international
trading entities would equalize factor prices has never been open to
real question. For example, mass migrations of Japanese laborers to
the United States and American capital to Japan would equalize absolute
wage-rent levels in the two countries.
One economic motivation of the National Origins Act of 1924
limiting immigration to the United States was a recognition of this
obvious fact. But even when factors flows are severely if not com
pletely limited, there was the feeling in nineteenth century literature^
that goods flows might tend to equalize factor prices. This thought J
is even more important in light of the assumption that factors are
immobile internationally.
Caves (1960, p. 24) points out that this analytical problen-.
is grounded in the Ricardian comparative costs framework. Ricardo
(1821, pp. 77-79) argued that trade was justified in a two good, two
country context even if one of the countries possessed an absolute
advantage in the production of both goods. The advantage to both
supposedly lay in the specialization of production. Sar.uelson (1958)
has illustrated this principle in his example of an econoir.y containing
a lawyer and a typist. The lawyer has an absolute advantage in his
practice of the law but also happens to be a better typist than the
professional. In this case, even though one is absolutely superior in
both endeavors, it makes eminently good sense to follow Ricardo's
admonition, letting the lawyer practice law and the typist type (not
to mention the employment effect).
J
Ricardo*8 labor cost theory of vaikue was an inappropriate base
for his theory of comparative costs. But if we give his theory the
charitable two factor interpretation, then we are prepared to explain
differences in comparative costs between two countries. Heckscher '"1
(1919, pp. 497-512) showed that if two countries have the same factor
endowments and use the same technique in all branches of production,
then comparative costs (i. e. comparative advantages) are the same.
In this case there could be no gain or loss from trade. Heckscher
derived two necessary conditions for differences in comparative costs
between countries:
1. Differences in factor endowments
2iffDx€fecendBS in factor intensities in the production of different goods (if they were the same, then price ratios would be the same in the two countries).
He then stated that such differences would cause trade to "expand until
an equalization of the relative scarcity of the factors of production
among countries has occurred." (Heckscher, 1919, pp. 285-286) Thus,
Heckscher (1919) was the first in the history of the factor price
equalization controversy to formulate systematically a statement that
equalization could, in fact, occur. He proposed a proof based on the
notion that factor price equalization could occur only when the input
coefficients were fixed for each good and were identical between coun
tries. But in a fixed coefficient model, relative factor prices cannot
vary continuously with relative commodity prices (even with perfectly
competitive factor markets). This logical flaw invalidates the Heckscher
proof, particularly when one notes that fixed factor supplies and
fixed input coefficients often imply that one factor is fully
employed in each country. This is the linear programming model of
contemporary economic theory with two constraints. Factor price
equalization can occur then only if both countries produce the same
good; but this violates the principle of comparative advantage. Caves
(1960, p. 81) gives a more favorable reading to the Heckscher proof
than the one above; he acknowledges its path-breaking nature and
intuitive grasp of the mechanism of international trade.
Heckscher's student, Bertil Ohlin (1933), approached inter
national trade from the Casselian general equilibrium viewpoint.
His important contribution to the Heckscher model was its incorpor
ation of demand. He encountered some difficulty as Caves (1960, p. 28)
points out, in "verbalizing his general equilibrium system," which
caused the Heckscher-Ohlin theorem to ignore demand in a rather )
important way. The Heckscher-Ohlin theorem, for many years one of the
most important in international trade theory, states that countries f
tend to export commodities requiring more of their relatively j
plentiful factors.
The singular absence of any reference to demand in the theorem
seems to be atavistically classical, although not without reason.
Demand considerations have always been troublesome in international
trade theory as evidenced by Ohlin's (1933, p. 38) feeling that I
"complete regional price equalization was impossible" because demand | ^ "*
for factors was localized and joint. >
The looseness of the Heckscher-Ohlin theorem has been tightened
by Caves (1960, p. 28) who shows that a "country tends certainly to
produce those commodities which require more of its relatively plentirul
factors; whether it exports these depends on the preferences of domestic
consumers for these relative to other goods." The Hecks_cher-Ohlin
theoripm would easily hold if we merely adde^_the assumption that there_
is little variation in demand relative to productive capabilit^ies between
country es_.
The relation between factor returns and free trade has a long
history in international economic theory. Bastable (1903), Cairns (187-^),
Taussig (1927), and Ellsworth (1938), have dealt extensively with the
subject. For our purposes, the most relevant work was a paper by Stolper
oy
and Samuelson (1941). Their theorem was a logical extension of the /^*^
Heckscher-Ohlin theorem: it asserts that reducing tariffs will expand"! ^ ^ ^
trade, raising the marginal product (and distributive share) of the 7 /
abundant factor while reducing the marginal product (and distributive
share) of the scarce factor in each country. The assumptionsylneedad -'
for the proof are restrictive but the proof is free of the index
number problem. In practical terms, it says that elimination of
tariffs, say, between the United States and Japan will benefit American
owners of capital and Japanese laborers. Japanese capitalists and
American laborers, conversely, would lose. These results are dependent
on at least six necessary conditions:
1. Identical linear homogeneous production functions for each good in both countries.
2. Costs of production being independent of the scale of production.
3. Perfectly inelastic supply functions for the productive factors.
4. No cessation of production of imported goods at home.
5. The U. S. has a comparative advantage in the good utilizing relatively more capital than labor.
6. Perfect competition.
'Consider the United States before trade: it produces watches and
wheat with labor and capital. Watch production is relatively labor
intensive and thus wheat production is relatively capital intensive.
If the United States has a comparative advantage in wheat production,
then the movement from autarky to free trade would be accompanied by
U. S. exports of wheat. Caves (1960, pp. 68-69) verbalizes the proof
that Stolper and Samuelson (1941) give at this point:
As the wheat industry expands, it will bid factors of production away from the(import-competing) watch industry. But the latter will release relatively much labor and little capital, compared to the proportions in which the expanding industry demands these
two factors. Therefore, there will be substitution in both industries toward the use of relatively more labor and relatively less capital. The marginal productivity of labor, which is the same in both Industries because of the assumption of perfect factor markets, will decline in both industries; that of labor will rise. With the reward to each factor equal to its marginal value productivity, the real return to capital will rise (whether we measure it In wheat or watches), and the real wage of labor will fall.
The theoretical groundwork is now complete for an examination of
the factor price equalization theorem.
CHAPTER III
THE PROOF
Before proceeding with the mechanics of the system, it will be
helpful to set forth the eight hypotheses Professor Samuelson (1949,
pp. 181-182) used in his proof.
1. There are but two countries, America and Europe.
2. They produce but two commodities, food and clothing.
3. Each commodity is produced with two factors of production, land and labor. The production functions of each commodity show "constant returns to scale," in the sense that changing all inputs in the same proportion changes output in that sMme proportion, leaving all "productivities essentially unchanged. In short, all production functions are mathematically "homogeneous of the first order" and subject to Euler's theorem^
ri
4. The law of diminishing marginal productivity holds: as any one input is increased relative to other inputs, its marginal productivity diminishes.
5. The commodities differ in their "labor and land intensities." Thus, food is relatively "land using" or "land-intensive," while clothing is relatively "labor-intensive." This means that whatever the prevailing ratio- of wages to rents, the optimal proportion of labor to land is greater in clothing than in food.
6. Land and labor are assumed to be qualitatively identical inputs in the two countries and the technological production functions are assumed to be the same in the two countries.
7. All commodities move perfectly freely in international trade, without encountering tariffs or transport costs, and with competition effectively equalising the market price-ratio of food and clothing. No factors of production can move between the countries.
8. Something is being produced in both countries of both commodities with both factors of production. Each country may have moved in the direction of specialising on the
8
commodity for which it has a comparative advantage, but it has not moved so far as to be specializing completely on one commodity.
To be complete, we should actually add a ninth which Professor
Samuelson (1949, p. 183) added in the body of his paper: "in each
country there is assumed to be given totals of labor and land." J / ,
When these conditions exist, then real factor prices will be the
same in both countries. (Cave^ (1960, pp. 77-78) has described the proof
of this statement as a three step process. First, the production
functions must yield a transformation curve concave to the origin.
Second, there must exist a unique relationship between price commodity
ratios, production levels and factor marginal productivities. Third,
international trade must equalize product prices in the two countries^
Thus, a single price ratio in the two countries implies equal
factor ratios in the production of each good. Linear homogeneity of
the identical production functions means equality of both the marginal
productivities and hence of factor prices.
Assumptions 3 and 4 bear directly on the shape of the transformation
curve between food and clothing. With constant returns to scale, halving
the two inputs used to produce food would halve the output of food. This
is shown in Figure 1 below. /If all land and labor were devoted to food
production, the economy would be at point B. But halving the inputs
would cut food production in half (0A»1/20B). Thus we know that at worst,
the transformation of food into clothing would follow the linear BCD
curve.'' The preceding argimient rules out a curve convex to the origin
(BGD). But it would be economically inefficient to produce goods with
identical factor proportions. This follows immediately from the hypo
thesis that food is land intensive while clothing is labor intensive. Thus,
by transferring a bigger portion of labor than land to clothing production
we could do better than point C in Figure 1, say point F. We are insured
that the transformation curve is concave below at point F because
10
of diminishing marginal productivity. As more and more labor is
transferred to clothing, its marginal product declines (assumption U).
CL0THIM6
FOC7P
Figure 1 - Three possible transformation functions between food
and clothing
The concave transformation function is important for the proof ^
because it helps preclude specialization in production. If both Europe/
and America had transformation curves of BCD variety, then America
would specialize in food production and Europe in clothing production
because of relative factor endowments. But the slopes of the trans
formation curves in equilibrium would be unequal, implying inequality
of commodity prices. Thus, assumptions 3 and U are necessary ("but not
sufficient) to assure that assumption 8 holds. Assumption 7 (perfect
competition) insures us that ve are, in fact, on the transformation
curve rather than below it.
|(;<t:|!
11
/ Thus, the concave transformation curve implies that as clothing
expands, the law of increasing marginal costs of clothing in terms of
food is invoked. But increased clothing production places pressure on
the price of the factor it uses most intensively, on wages relative to
rent, A higher wages-rent ratio is accompanied by a decrease in the
proportion of labor to land in both industries. An increase in the wages-
rent ratio pushes up the price of the labor-intensive commodity
(clothing) relative to the land-intensive commodity (food). We have now
come the full circle, illustrating the monotonic relationship between
commodity price ratios, production levels and factor price ratios.
Figure 2 shows the relation between commodity and factor price ratios.
Figure 2 - The left quadrant shows an inverse relationship between P-iP and w/r. The right quadrant relates optimal labor to land. ratios (L/T) for any given wage to rent ratio (w/r).
The higher the wage/rent ratio, the lower the price of food is relative to
clothing. Once demand specifies the prices and quantities of each rood.
12
the v/r ratio is uniqueiy determined. At this value, there will be
optimal valuesodf L/T in both industries. Since clothing is relatively
labor intensive, (L/T) ia greater than (L/T)-,, Although Figure 2
illustrates only one country, the diagram for the other will be
identical since identical linear homogeneous production functions are
assumed (hypothesis 6), Trade equalizes P^/P in Europe and America.
These propositions can be illustrated mathematically as
follows (Samuelson, 19^9$ pp. 190-193).^ Food production, F, is a Joint
function of labor devoted to it, L-, and of land, T-, denoted
F=F(L^, T^)=T^f(L^/T^). (l)
The right portion of the equation is a direct result of using
production functions which are homogeneous of degree one (ass\anption 3).
The general case of a function homogeneous of degree p obeys the
following property: if both inputs are multiplied by k, then output
is multiplied by kP,
F=F(L^ T^j k^F=F(kL^, kT^).
In the first degree case, p = 1 so that doubling inputs, for example,
wou^d Just double outputs. This is essentially what we did in (l)
above. The argument was multiplied by 1/T and the function itself by
T producing no change in F. But it is a helpful formulation to
express food production as T (a scalar multiple equal to land devoted
to food production) times f(L /T^) the return to food on one unit of
land. Similarly, we can write clothing production, C, as a function
C = C(L^, T^) = T^c(L^/T^).
The marginal physical product of labor ia food is written
^^fL^ = f'(L^/T^) ,
which is simple partial differentiation of (l). Here f represents fr
^^^LF
^•P
13
marginal product of labor on one unit of land and by our assumption
of diminishing returns (hypothesis 4)
f"(L^/T^) - ^ 0 .
The marginal product of land can be obtained by differentiating (1)
or by the use of Euler*s theorem (tceatlng the marginal product of
land as a rent residual),
^ T F - Q F / J T ^ - f(L^/Tp - L^/T^ i*(L^/T^) - g(L^/Tp .
We interpret g as the "rent residual" and its marginal product i^
g'(L^/T^) - -L^/T^f"(L^/Tp^ i/
Similar relations hold for clothing:
MPP^^- ^C/^L^-c»(L^/T^)
MPP g -4SIC/<9T^ - ^^^c ' c^ " W ' ^ ^ c ^ ^ c ^ " ^<^/^c^
h'(L /T ) - - L /T c"(L /T ) . c c ^ c c c c
By considering one country at a time, we know that factor returns
will be the same in food and clothing, regardless of the measure used.
This is expressed
Pf • ^LF ' c • ^LC ^-^^-
P^ . MPP,j,j, = P^ . MPP^^ ; i^^i
or in terms of our previous notation
This we can write implicitly
(P^/P^)f'(L^/Tp - c'(L^/T^) - 0
(P^/P^)[f(L^/Tp - L^/T^f'(L^/T^)] - [c(L^/T^) - L^/T^c'(L^/T^^^ „ Q.
14
Our system is now reduced to two equations and the three
unknowns L./T-, L /T and P./P . But combining demand conditions I z c c f c
in the two coimtries gives two additional equations specifying
P^ and P . This leaves four equations and four unknowns, closing
the system. The proof is complete. /
p;r
CHAPTER IV
THE ASSUMPTIONS OF THE THEOREM
This chapter is a piecemeal dissection of the hypotheses on
which the factor price equalization theorem rests. ' It is an exercise
to determine their several necessities and the extent to which they
can be modified, ceteris paribus, without violating the conclusion of
the theorem. As each is examined, the assumption is made that the
others contlnua to hold. Assumption 2 and the first part of
Assumption 3 are considered jointly for convenience. The hypotheses
follow the same order as the listing in Chapter III.
1. There are but two countries, America and Europe.
i
This assumption is not at all necessary since in the general
equilibriiim context of N countries we can merely add an equal number
of equations and unknowns to the two country case and return to equi
librium as Caves (1960, p. 82) has shown. Tinbergen (1949), on the
other hand, points out that adding more countries simply increases
the likelihood of specialization, which, as we shall see below,
usually leads to an inequality of factor prices. This follows from
the observation that for any given set of production relationships
between two goods, there exists an upper bound on the deviation in
factor endowment ratios between the countries (given that no country
specializes). But the greater the number of countries involved, the
greater the likelihood of specialization. In Figure 3 factor prices
would be equalized in countries 1, 2, and 3, but the addition of
country 4 disrupts the relationship since it specializes in good X.
In general, however, the N country case does not logically invali
date the theorem.
15
16
COWiXr h •y+
Figure 3 - The four country model
Laursen (1952, p. 551) would possibly attribute to wide ranges of
population density the situation depicted in Figure 3.
2. They produce but two commodities, food and clothing.
3. Each commodity is produced with two factors of
production, land and labor.
These two assumptions will be considered together for convenience—
the second part of assumption 3, (3"), will be dealt with separately.
Caves (1960, p. 77) adds a further qualification to 3' which is implicit
in Samuelson*s presentation: the factors are given and fixed in quantity
17
for each countrx*. Samuelson (1948, p. 174) also implies full
employment of the factors. Given N goods and M factors, we will
consider five cases.
Case a - The number of goods produced equals the number of factors utilized (N - M). Here N refers to goods and M to factors.
Laursen (1952, p. 548) points out that the essense of factor-
price equalization under free trade is a coupling of two systems: the
points of contact between these systems are uniform commodity prices
and identical production functions. So long as these "points of
contact" are maintained (i. e. commodity movements prevent differential
factor endowments from being reflected in differences in marginal
productivities), the generalized N - M case does not alter the
conclusions of the theorem.
McKenzie (1955) uses vector and activity analysis in the case
where N «» M = 3. If X and Y. are inputs and outputs respectively th ^ ® " ®
in the s activity vector (i * j = 1,2,3,goods), with w and p being
the respective factor and goods prices, then we can specify the zero
profit- condition, (inputs X. are negative).
S_v-y- + £w,x. - 0. (3)
We will deal with the unit activity level ( S Ix 1 = 1). Thus we can
define a set S of factor input vectors (X.) whose components sum to -1
This set is shown in the equilateral triangle in Figure 4a. Since the
altitude of the triangle is 1, the i component of X = (x^, x^, x^) is
the negative of the perpendicular distance from X to the side opposite
the i^^ vertex: (if X—1/3, -1/3, -1/3), we are at the center of
triangle 4b).
'yrr'' *^
18
^ek 5 ^ unrf
"*Hij >€rj7\av\c
The vector space K (at left) is the pw
set of all price-wage vectors at which factor prices are equalized.
'^i.0r\>O)
7}-- (^.a-i)
(-l.0>0)
Figure 4b - The vector space denoting the components of a three tuple vector (perpendicular .distance to the respective side).
19
Let R be a normalixed vector of factor supplies for the k^" country.
McKenzie then shows that for any given price-wage vector (p, w) we
can define a subset K of all elements of S which are linear com-pw
binations of the s's such that the zero profit condition (3) holds. 1 2 If the endowment ratios for the two countries (R and R ) lie inside
the subset K then factor prices are equalized at pw. At the same
price but a different wage, say pw*, we have a new subset K ,. These pw
two subsets do not overlap and are separated by the hyperplane H.
Thus, any given (p,w) is the factor price equalization vector
if and only if the factor endowments of the countries involved (2 or more) lie in the subset K .
pw
The general N good, M facaocrcase where N - M, has been con
sidered by Samuelson (1953). His notation has been utilized heavily
in what follows. The N goods are denoted X., X., . . . , X^ and are produced by M factors of production V^, V^ V^. V^. stands
for the amount of the j*^^ input used by the i^^ industry. The production
functions for the N goods can be written
X. - xi(V.,, V,., . . . , V. ) (i - 1, 2, . . . , N). (4) 1 il» '12' • • • » 'im
The coefficients of production are denoted a^ » V^./X^ and are
inputs v.. per unit output of X^.
Perfect competition and intra-country factor mobility assures
that the value marginal productivity of each factor will equal its wage.
If commodity prices are p^, . . . , p , and factor prices w^, i 1 . , w^,
then equilibrium requires ^
„ ip Jxi(a,, \J^\i (i • 1. 2. • • • . « (5) (j - 1. 2 l)-
20
or equivalently its dual
p. - a -w + . . . + a. w ' i il 1 im m
The latter formulation has the following linear programming inter
pretation. If the cost of producing good X. exceeds its market value
^i il 1 ^12^2 + . . . + a. w ) in the optimal solution, then the
im m * '
good is not produced (X « 0). Equality of price and unit cost in the
dual implies that X. » 0.
The most general system would consider factor supplies a
function of all good and factor prices. For simplicity Professor
Samuelson has chosen to represent factor supplies as perfectly inelastic so hhat V . , . . . , V are constants, i ' m
^^ij " ^Ij^l •*• *2j^2 + • • • + a^jX^ ^Vj (j - 1, 2, . . . ,m).(6)
The equality in (6) denotes the fact that all non-free factors must be
fully employed. If w. becomes zero, the ^ replaces the • sign in the
j equation and the j factor is free.
Finally, we can aggregate total demands and supplies (in their
most general representation) and write them in the following functional
form:
X. - D"''(p,, . . . , p , WT , . . . ,w ) (i » 1, 2, . . . N). (7a) 1 1 n 1 m
X. = S^(p^, . . . , Pj ,w , . . .,w^) (j - 1, 2, . . . M).(7b)
But Walras's Law assures us that there are at most only N + M -1
independent equations since^p.Di ^£yj.S^ is the familiar earning-
expenditure identity of classical economics.
We are prepared now to count equations and unknowns to insure
the logical completeness of the system.
21
UNKNOWNS
a*s
w's
P's
X»s
NM
M
N
N
EQUATIONS
(4) N
(5) NM
(6) M
(7a and6)N4M-l
NM+2M+2N NM+2M+2N-1
At first blush the system seems to be underdetermined since there is
one more unknown than there are independent equations. But it must
be remembered that (7a) and (7b) are homogeneous of degree zero: our
equations depend only on ratios of p*s and w*s so that equations and
unknowns are, in fact, equal. Samuelson (1953, p. 11) suggests that
we could also handle the problem by assigning a numeraire (p, • 1 or w. =
or making some "non-homogeneous monetary assumption" such as
5 P .X. " a constant MV. The effect of the latter operations is to
determine the absolute as opposed to the relative price level.
The method of counting equations and unknowns is inadequate in
considering the "existence" problem of general equilibrium. Dorfman,
Solow, and Samuelson (1958, pp. 346-389) have invoked the Kakutani
fixed point theorem as a rigorous final step in the proof that linear
programming solutions to general systems such as the one above do exist.
Case b - The number of goods produced exceeds the number of factors utilized (N > M).
Laursen (1952; ,p. 552) shows that factor prices are still
equalized and the probability of specialization is reduced as N grows
relative to M. Samuelson (1953, p. 7) analogously argues that we have
N-M degrees of freedom in the geographical production pattern for any
given world totals to be produced. The conclusions of Samuelson's
theorem are generally inapplicable, however, in the following case.
1)
22
Case c - The number of factors exceeds the number of goods produced (N < M).
Vanek (1960, p. 634) and many others show that in general
factor prices will not be equalized when N<M, although it is possible,
He describes the geometrical situation required for factor price
equalization in the two good, three factor case (X and Y goods;
K, L'and T factors). Consider two boxes (two countries) in three
dimensions (Figure 5). 0 . and 0 ^ coincide while 0 . and 0 2* ^^^
respective origins in countries 1 and 2, do not.
^^unfrij 2.
Figure 5 - The three factor, two good model
The factor price equalization theorem holds only if both contract
curves (connecting 0^^ and 0^^ ^^^ °yl ^^ °y2 respectively) are in
one surface, passing through 0^^, 0^^, and 0^^ ^^ °y2 ''^^''^ ^^
generated by a family of lines, all passing through 0^^. It goes
without saying that this condition is generally not fulfilled
empirically.
23
If Europe and America have different endowments of capital,
then the relative value of L and T in the area with the larger K is
enhanced relative to the other.
Case d - Input-output relationships exist between the goods produced
Vanek (1963) considers the case where there are two goods
(X and Y) and two factors (T and L) and where X is input in producing
Y and Y is an input in producing X. The input coefficients are fixed
(a^—X required per unit Y; a^—Y required per unit X). If a.-a^'O,
we have the usual case depicted at point E in country 1 in Figure 6a.
V
Figure 6a - Transformation curves for the two-good input-output case
&• A' ^4
O^
^ d<7uri4v-u 1
Figure 6b - The Edgeworth-Bowley representation of two-good input-output case
24
In Figure 6b, one unit of X is denoted by the U isoquant and
the price of X is 0 A in terms of labor, while Y' price is 0 A* in
X s * y
terms of labor. Then we introduce input-output relationships; the
isoquants shift outward from the respective origins. Since a.^ is
larger, the relative decrease (largest isoquant shift) is the largest
in the Y good. Since relative prices are equalized in the new situation
(Ob/Ob* in country 1 equals 0 b/0 b* in country 2), then so are
factor prices (slope b'n* in country 1 equals slope of b'n' in country 2)
Thus, Samuelson*s conclusions are vitiated in the input-output case.
One way to eliminate the zero transport cost assxomption would be
to consider transportation a final good and treat it as an input in the
"production" of other internationally traded goods. We must assume,
however, that unit domestic shipping costs are no less than unit inter
national shipping costs (not unrealistic since per mile costs are lower
for non-bulk ocean shipping). But this assumption would effectively
limit factor price equalization to "enclave" type production and
consumption items.
Case e - Supplies of productive factors are responsive to factor prices
In the case where factors are not in perfectly inelastic supply
with respect to factor prices, we can still have factor-price
equalization. The most general case can be seen in McKenzie's (1955)
diagram (4a): for different factor price vectors (p,w) we would simply
get factor endowment ratios which are more divergent for each country
than in the fixed factor case. So long as there existed some (pw) for
which the subset K contained the vectors R^ and R^, our conclusions pw
are unchanged.
nwrnm-'^
25
I L
Figure 7 - Variable factor supplies: A and B are points in autarky; C and D are free trade points with fixed factors; E and F are possible equilibrium points when factors are variable
Notice, however, that the likelihood of complete specialization
is increased in the variable factor case because original factor endow
ments tend to become even more divergent as the price of the abundant
factor increases with trade (Caves, 1960, p. 104). As country 1
produces more X in trade, P. rises and P^ falls, causing the T/L ratio
to fall (Figure 7). Thus, country 1 produces even more X for export
and country 2 even more Y. The factor prices remain equal although a
higher elasticity of the factor prices could easily have led to
specialization.
1 3. The production functions of each commodity show "constant returns to scale," in the sense that changing all inputs in the same proportion changes output in the same proportion, leaving all "productivities" essentially unchanged. In short, all production functions are mathematically "homogeneous of the first order" and subject to Euler's theorem.
26
4. The law of diminishing marginal productivity holds: as any one input is increased relative to other inputs, its marginal productivity diminishes.
Samuelson (1949, pp. 184-185) uses assumption 4 to assure the
concavity from below of the production possibility curve. Constant or
increasing marginal productivity would allow specialization and non-
factor price equalization. Thus our main concern will be with
assumption 3". Laing (1961) has shown on the Lerner diagram that
increasing and decreasing returns to scale will prevent factor price
equalization. For example, he says that "if commodities have opposite
returns in each country and if each commodity has opposite sorts of
returns in the two countries," then only commodity prices can be
equalized (Laing, p. 343). He uses a portion of the Samuelson-Johnson
factor-price, commodity-price diagram to illustrate this result. In the
three cases in Figure 8, we have the CR line depicting constant returns
to scale while good Y is subject to decreasing returns in country 1
(to left of CR) and increasing returns in dountry 2 (to right of CR).
Although trade can never equalize factor prices, we can get an idea of
the direction of movement by looking at the situation before trade
occurs (X is L intensive and Y is T intensive).
In case 3"a, country 2 has the highest commodity price ratios and
factor price ratios prior to trade (B>A for both ratios). In this case,
the factor prices converge during the dynamic process but overshoot in
final equilibrium.
27
3" OL
3U
3"c
Figure 8 - Three factor price-commodity price patterns
28
In case 3"b, country 1 has the highest commodity price ratio
and factor price ratio prior to trade (A>B for both ratios). Here
factor prices converge but fall short of equality with trade.
In case 3"c, country 2 has a higher commodity price ratio and
lower factor price ratio. In this case, factor prices diverge with
trade.
In case 3"d, the export industries of both countries have
increasing returns. This is a subset of case 3"a above, where factor
prices first converge, but then diverge.
In In case 3"e, the export industries of both countries have
decreasing returns. This is a subset of case 3"b above, where factor
prices converge but fall short of equalization.
It is obvious that assumption 7 (perfect competition) is also
violated with increasing returns. The theorem clearly fails to hold in
these cases.
Minhas (1962) has attacked the production function problem in
ammore general way. He defines what is called a "homohypallagic
production function" of the form
V = [A T +a L ]-l/Bx (for good X; similarly for Y). « X X
V is the deflated or real value added in the X industry, T and L are
the two factors of production and A , B and a are parameters. The
elasticity of substitution J^= p~^ is assumed to be constant throughout,
He shows that the relative factor intensities of two industries X and Y
are independent of the factor prices only ifj*x -/v (this is a most
important conclusion in dealing with factor reversals). For the ful
fillment of this condition, Minhas shows that "it is sufficient that
all industries be subject either to a Cobb-Douglas production function
where/j,» J"-i / or to a fixed coefficient production function where
J'^'zXf ^ 0 (pp. 142-143). ItS^t-Stf . then there will inevitably be
a reversal at some value of w/r. The diagrammatic presentation of the
29
factor reversal phenomenon will be considered under assumption of 5.
In general the homohypallagic production function (or any function which
does not reflect constant return to scale) will not suffice in
equalizingffacfcorpprices.
5. The commodities differ in their labor and land intensities. Thus, food is relatively land intensive while clothing is relatively labor intensive. This means that whatever the prevailing ratio of wages to rents, the optimal proportion of labor to land is greater in clothing than in food.
This assumption has been the one most discussed in the literature
because of its implication of non-reversible factor intensities. The
first sentence implies that the production functions of the two good,
two factor case must not be identical. If they are identical for X and Y,
trade will not equalize factor prices (See Figure 9). This is Professor
Samuelson's(1948, p. 175) case where production functions are
identical and trade does not occur because of equal commodity price
ratios.
r L
country 2
courftVy \
Figure 9 - Identical production functions
30
Laursen (1952) p. 547) points out that in the special case where
relative factor endowments are identical and production function are the
same, we may still have trade because of differing demands (Figure 10).
In any case, factor prices are equalized (regardless of trade).
cooi h
T
OK^2, <-f>or\\r\^ 2.
COOT^Ti^l
v^JI.
Ox lanJz
Figure 10 - Identical production and factor endowments
In general, however, the difference in the production function causes
one of the goods to be labor intensive and the other to be land inten
sive. The T/L ratio in the labor intensive industry will be less than
the country's factor endowment ratio of T/L while the ratio in the lad
intensive will exceed that of the country involved. Thus, regardless
of the factor price ratio, one of the goods is always labor intensive
while the other is land intensive. As mentioned earlier, Mi nhas has
shown the factor intensiveness of the two goods to be independent of
the factor price ratios only in the case where the elasticities of sub
stitution between the two factors are equal (jx -J^y)- Thus, he has
generalized considerably the restrictions that must be placed on produc
tion functions to prevent factor reversals (although assuT ption 3
above still limits us to the Cobb-Douglas function in the constant
returns cases) since the linear homogeneous production function is
31
J". d u . simply one of many which satisfies the condition of x *=
There are a number of geometric approaches to the reversal
problem. One of the most fruitful has been Lerner*s diagram (1952),
In the Figure 11, we can simultaneously measure the T/L ratios, com
modity price ratios (in terms of the factors) and the factor price
ratios. In diagram 11a, we can depict factor price ratios in
country 1 before trade as
?J?^ - Oa^/Oa^ Ob^/Ob^.
«.»* b,»
t'\^"tJ«-'
cCrh:
(Li^<
Figure 11 - The Lerner diagrams
32
Country 2's factor price ratios are P /P = 0a2,/0ag - Ob^/Ob^.
The price of Y in terms of factor L is Oa^ in country 1 and Oa^ in
country 2, while the price of X is Ob^ in country 1 and Oa^ in country 2.
Since P /Py - Ob^/Oa^ Oa^/Ob^ country 2 will export X and country 1
will export Y, causing X's price to rise in country 2 and Y*s price to
rise in country 1. When commodity prices become equal
(Oa,-Ob. Oa 2 » Ob^), factor prices are also equal (Figure lib).
If factor intensities for the two goods are reversed for different
w/r ratios, then factor price equalizations will not occur if the
countries lie on opposite sides of OS in factor intensity. In this case,
the elasticities of substitution differ ( J*x :jt/y ). In country 1, X
is T intensive while in country 2 it is L intensive (See Figure lie).
The ratio of goods prices is equalized at unity but factor prices are
not.
Johnson (1958, chapter 1) borrows Professor Samuelson*s diagram
(1949, p. 188n.) to illustrate the reversal process (Figure 12).
I
€rwlocjmervt$ rahos •for ^
r,
Figure 12 - The Johnson diagram
33
Just as in the Lerner case, country 1 exports Y and country 2 exports
1 1 2 2 good X (P^ /Py P^ /p ) equalizing factor prices at (w/r)Q when we
are in the range to the left of A (without specialization). If aggregate
factor endowment ratios are separated by an odd number of reversal
(r^ and say r Oiji then export commodities will have the same factor
intensity moving factor prices in the same directions. If the number
of reversals is even, then a Kommodity does not reverse its factor
intensiveness in the two countries (r and say r«„). Factor prices are
convergent or divergent depending on whether or not the labor intensive
commodity is exported by the labor intensive country. Since it is not
in our case, prices diverge.
A reversal of sorts is described by Lancaster (1957, pp. 37-38)
who uses a theorem of "inverse points" to show the case where the factor
described as a labor in country 1 is equivalent in its economic aspects
to land in country 2 (Figure 13) . The endowment of one country is just
the inverse of the other country: for any point P on countyy I's
contract curve there exists a Q on country 2*s, such that commodity prices
are equal but productivities are reciprocal (mppj /mpp_2 l/moo )
In this case, trade does not occur and factor prices remain unequal.
• • counircj
mppT'
coon fry*
Figure 13 - The Lancaster diagram
'•'i't^
34
Pearce (1959, pp. 725-732) takes a rather topological approach
to the problem: we will consider, however, only a few key equations
in his system. In investigating the general equilibrium approach, he
first establishes the zero profit conditionCDiplxij * t"'4j fj">-i-.-TV) where
the left side of the equation deals with the factor prices and i fac
tors used in producing good j whose prices ( i ) and outputs (4*3)
are described on the right. In the two good case we will have multiple
solutions to these equations if for non-zero values for the pj and qj
there exist non-zero Xij(i^^ factor used in producing j^^ good) such that
That is, given the X's, we will have two equations and two variables
(dp and dp^). We can express this relationship as
' ^ X t-x. 1 I.
and ll'' 21 are In the non-reversal case, the slopes X2 2 22
unequal so that the onlyysolution to equations ® is dp^ » dp^ = 0
(See Figure 14). If for-sdmetfiactorcprtceiratiolwe do have a reversal,
then the slopes are X^i/X2i - -^12/^22 ^ ®*^°^ ^^ ^® Lerner diagram
(Figure 14b).
*^p»-")<r. ^p'
^^"^J^^ Figure 14a
35
Figure 14b
dpz
<)pi
Figure 15 - Geometric conditions for multiple solution
For the three dimensional case (3 factors and 3 goods) we will
have three planes in the dp^, dp«, dp^ space related to the factor
"intensities" of the three goods. If these three planes intersect in
a line or a plane then wacwill have a multiple solution and reversals
occur. In Figure 15, they intersect in a line implying a reversal. In
general, however, three planes intersect in only a point.
Thus the general condition precluding factor reversals is that
points 0, R and S should never lie on the same straight line. If they
do, a reversal has occurred. In the three factor, three goods case we
can extend this result to say that if points 0, R, S, and say, T lie in
the same plane, a reversal will occur (Pearce and James, 1951-1952, p. 114)
Thus, if K, L and T are the factors of production and 1, 2, and 3 are
goods, then there will not be a reversal unless points (K , L^, T ),
(K^.L^, T^), and (K^, L^, T^) all lie in the same plane for some factor
price ratio.
36
Consider three identical production spheres in the KLT space
whose centers lie on a straight line parallel to the KT plane. Their
centers are on a line whose projection in the KT plane forms the base
of an equilateral triangle with the apex at the origin. Any line parallel
to the line through the origin of the spheres and tangent to their ex
terior on the side nearest the origin satisfies the ORST condition above
(any line and a point suffice to specify a plane) and implies that a
factor reversal occurs. A two dimensional projection of this result
can be seen in Figure 16. In country 1 the slopes of the tangent lines
are the same in the KT direction (with K and T's factor prices equalized
there); the ratios of factor prices K and L, and L and T are different,
however, and the reversal is evident. Pearce shows in the mathematical
appendix (1951, p. 120) that the necessary condition for the non-
reversal case is that of non-vanishing Jacobian.
Figure 16 - The Pearce and James diagram
37
An interesting possibility where the production functions
became identical in a short range but did not reverse might be
depicted in Figure 17. This is an interesting geometric case al
though to the mathematician, it is very likely of the persona non
grata variety.
\
T L
O
r
no -irade bo4 -.-fflLC^cr prices eq\)oJiic<^ if counhry i oind 2.
»1 l! t
11 It I
Figure 17 - A near reversal
In general, what can be said about many factors and the
importance of factor reversals? Caves (1960, pp. 91-92) imputes to
38
Harrod the conclusion that if there are N factors of production, the
nximber of factor intensity comparisons for two commodities is
NI/2I (N-2)I which rises more than proportionately to N. Thus, with
multiple factors the probability of intensity reversals increases.
Harrod (1958, pp. 245-250) emphasizes that factor reversals are
of great significance empirically. A much more thorough study is the
one by Minhas (1962). He shows that the mere possibility of factor
reversals may be of little consequence unless they occur in the relevant
factor-price range. His studies indicate that this is, in fact, what
has occurred in U. S. - Jfipanese trade.
6. Land and labor are assumed to be qualitatively identical inputs in the two countries and the technological production functions are assumed to be the same in the two countries.
This section will deal only with the case in which the production
functions of the goods differ between countries. Minhas (1962, p. 155)
Introduces a natural efficiency parameter and into his production
function by setting A + a = ^ "^^ and A^ ^^ 'djc: thus j
Consider the case where ^^/'^ , >yx2^Xy2' ^^^^ implies that the
price ratio (P /P ) to the right of R will lie south of RS (the equal
efficiency case) in Figure 18. If the efficiency differences are
significant, the directions of trade may even change.
39
l-S^
Figure 18 - Efficiency differences
In a sense one can think of the factor reversal as an alteration
of the production function: the same good is produced at different
factor-price ratios with vastly different T/L ratios.
Findlay and Grubert (1959) have analyzed the case where neutral
technological progress occurs in one of two industries. Bardhan has shown
that this inter-temporal difference in production functions due to tech
nical progress ±&'i perfectly analogous to the interspatial efficiency
differentials growing out of variable production functions between
countries. If instead of time periods 1 and 2 in the technical progress
sense, we have countries 1 and 2 with differing production functions,
then we can see the factor price differences which will emerge. The
casesiiliustrated are listed below (See the increasing and decreasing
returns cases above).
In Figure 19a we have countries 1 and 2 being equally efficient
in good Y but country 2 has an advantage in good X and will specialize
in it. Goods prices are equalized at Ob although factor prices are not
(Pj /P - OA1/OA^^^ - 0C1/0C^2)> .
The price of labor is higher in the country which specializes in the
labor intensive commodity.
Wtv^
40
J^*u>4rt
Figure 19a - The Lerner diagram showing efficiency differences
^ I w l r f
Figure 19b - The Lerner diagram showing efficiency differences
In Figure 19b both are equally efficient in X but country 2 has
an advantage in good Y, causing it to specialize. Goods prices are
equalized at OA, but land in country 2 has a higher return than in
country 1.
41
Clemhout <1963) shows that differences in production functions
are identical to the artificial efficiency arising out of tariff
Imposition. Hence, the previous analysis is of general applicability,
7. All commodities move perfectly freely in international trade, without encountering tariffs or transport cost, and with competition effectively equalizing the market price-ratio of food and clothing. No factors of production can move between the two countries.
Kindleberger (1963, p. 141) has shown that the tariffs and trans
port costs will cause a divergence in commodity price ratios in
international trade.
T
o,
Figure 20a - The Kindleberger diagrams
^N V J
pJ'icc oi V in line c^ L b afc»ovc tariff - Oy B
L Figure 20b - The Kindleberger diagrams
42
This can readily be seen in the case of good Y in Figure 20a. Either
a tariff or transport cost causes it to have a price of 0 A in country 2
and price 0 C in country 1. Under frictionless trade the price would
have been 0 B in both countries which would have equalized factor prices.
Thus these impediments cause country 2 to export less Y to country 1.
In Figure 20b the price divergences are measured w/r to L.
The second part of the assumption deals with the competitive
aspects of international trade. Baldwin (1948) deals with the case where
country 1 is a monopolist or price maker while country 2 is a price
taker. First he derives the offer curve for country 2 which has a com
parative advantage in good X (which is exported in normal demand conditions)
(Figure 21).
X H
n \i
Figure 21 - Baldwin's monopoly case
43
In Figure 22 country 1 has a transformational curve R V upon which
we can superimpose country 2* offer curves (since country 2*8 offer
curves are independent of country l*s production). Thus, if we choose
point o, then to the left of o, country 2 will offer X for Y (and
similarly for every point on country I's transformation function). The
total of these offer curves can be enclosed in an envelope function which
gives the maximum attainable X for any given Y.
COun+ru I5
Figure 22 - The envelope
Notice that the slope of the offer curve at point U equals the slope
of the transformation function implying no trade at that point.
Furthermore, the envelope has the property that the slope of the
offer curve at its tangency with the envelope will always equal the
slope of the transformation curve. This theorem makes the diagram
quite useful to factor price analysis.
By superimposing country I's indifference curves on the diagram,
we can find its maximum satisfaction point (point Q) at the tangency of
the envelope and indifference curve I-, • The offer curve tangent to Q
goes through point 0 on the transformation curve which is the production
optimizing point for country 1. The monopolist will set the price
44
equal to the slope of the line going from Q to 0, thereby inducing
country 2 to supply OZ of X in exchahge for OZ of good Y. Thus, the
monopolist exports Y and imports X. The marginal revenue for the
monopolist will be the slope of the offer curve at Q, while its mar
ginal cost is the slope of transformation curve at point 0. Bnt
Baldwin (p. 754) proved that these slopes were equal so that
mr » mc for the monopolist. (The international price for Y/X he set
at QZ/OZ, which exceeds these values.) In country 2, the price
QZ/OZ equals marginal cost so that although commodity prices are equal,
factor prices are not: the price maker disrupts Professor Samuelson*s
conclusions.
The final portion of the assumption postulates that the factors
of production will not move between countries. Samuelson (1948, p. 176)
and Laursen (1952, p. 547) show that factor mobility will perform the
function of equalizing factor prices by eliminating the factor endow
ment differentials between countries.
In the symmetrical case depicted in Figure 23, points H and I
depict the autarkic positions in countries 1 and 2. Since the price
of labor is highest in country 2 and the price of land is highest in
country 1, BC of labor will move from country 2 to country 1
(BC - 1/2AC and EF - 1/2DF).
*** coonWt^ X 45
courrhrcji
Figure 23 - Factor mobility
Factor prices are equalized and trade sCpps at point M. Actually this
factor mobility assumption merely allows us to prove the much stronger
theorem that movements of goods suffice to equalize factor prices.
8. Something is being produced in both countries of both commodities and with both factors of production. Each country may have moved in the direction of specializing 6n the commodity for which it has comparative advantage, but it has not moved so far as to be specializing completely on one commodity.
This specialization assumption is very important and has been
touched upon in most of the preceding discussion (See the discussion of
increasing returns). Since the marginal products of the factors depends
only on the factor ratios used in production, then it is obvious that
the endowment ratios of each country place a limit on the possible
factor price adjustment.
In general we can say that non-specialization implies that no
country produce less goods than the total number of factors of production.
As Laing (1961) has pointed out, greater divergences are permitted in the
factor endowment ratios as the production coefficients in the two goods
case show greater disparity. For example, in Figure 24 good Y is land
intensive while good X is labor intensive. The line 0 C shows the most y
extreme T/L ratio which can be used in the production of Y without the
country specializing.
^HRr^'' ^HRr^''
46
<:ourTtry
ttP'pardW AbOxP
Figure 24 - The Laing diagram
0 E is parallel to 0 G. If the 0 origin of country 2 is at H, then
trade will occur and factor prices will be equalized at P in country 1
and P* in country 2. It is clear that the area bounded by 0 G, the
0 0 contract curve in country 1 and 0 E is the only permissible range
for the H points. If a country*s factor endowment ratio places its
0 axis at H', then country 2 will specialize in Y and factor prices
will not be equalized. Clearly, As production functions become
similar, the contract curve approaches the diagonal 0 0 and the y X
region of eligible points for country 2*s factor endowments consistent
with non-specialization collapses into a line. This analysis is
roughly comparable to Travis's theorem of corresponding points and
the eligible possible division of factor endowments between two
countries. The Travis analysis is more readily amenable to stochastic
analysis since we have two areas for consideration: a factor price
47
equalization region without specialization and the rest of the
rectangle (this "area visualization" is useful probabilistically if
factor endowments are distributed at random). The case will not
be dealt with geometrically, however, In any case, we can say that
specialization will not permit the price of the abundant factor in
each country to increase enough to equalize the factor price ratio.
CHAPTER V
CONCLUSION
The eight formal and one implicit hypotheses which Professor
Samuelson used in his proof have been analyzed in their logical
context. We are now prepared to classify them as either inessential,
empirically restrictive, or necessary.
Assxjmptions 1, 2, and 3* were shown to be inessential in the
general proof. The Samuelson proof can easily be extended to K
countries merely by adding equations and unknowns. The probability
that specialization will occur increases but this is related to
peculiarities of the production function and variations in endowment
ratios rather than country numbers. Similarly, the only general
restrictions on the N commodities and M factors case is that N^M.
Though improbable, equalization can occur even when N-' M, as shown
by Vanek. The implicit assvraiption that factor supplies were per
fectly inelastic to factor prices can easily be modified by merely
adding appropriate functional relationships.
Assumptions 6 and 7 can be classified as empirically
restrictive. The hypothesis that land and labor are "qualitatively
identical" is a difficult proposition to substantiate in any given
situation and approaches a pseudo question. The transport cost
assumption can be relaxed without disrupting the theorem's result
but it narrows the geographical relevance from countries to areas
within countries.
Assumption 3", 4, 5, and 8 are considered necessary for the
proof of the theorem. Linearly homogeneous production functions
occupy a central position in the proof. Any other degree of
homogeneity (p) would lead to under- or over-exhaustion of total
48
49
product when factors were paid their marginal return. Euler's theorem
holds where p •• 1. The law of diminishing marginal productivity is
used only to prevent country specialization. It would be possible to
have factor price equalization with specialization but the probability
approaches zero. A reversal 6£cfactor intensities at different factor
price levels is prohibited if the proof is to hold.
This paper has attempted to provide a framework within which
some very eclectic writing in the field of international trade theory
could be unified. Second, it is suggested that transport costs can be
incorporated into the factor price equalization theorem, although at
some loss of generality.
(TEXAS TECHNOLOGICAL COLLEwL
LUBBOCK, TEXAS i-IBRARY
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50
51
Gerakis, Andreas S.: "A Geometrical Note on the Box Diagram," Economica N. W. 28 (Aug., 1961), pp. 310-313.
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