The Kyoto Economic Review 78(1): 19-37 (June 2009)

Samuelson’ s Implicit Criticism against Sraffa
and the Sraffians and Two Other Questions†

Yoshinori Shiozawa†

Graduate School of Management, Kyoto University; Department of Commerce, Chuo University

Samuelson’s “A Ricardo-Sraffa Paradigm” (2001) is an implicit criticism against
neo-Ricardians who could not provide any theory beyond the small open economy.
The necessity of constructing a trade theory with traded intermediate products wasevident and was challenged by both strands: neo-Ricardians and mainstream
economists. The attempt failed and the theme has been abandoned since long. My
recent paper shows that a Ricardo-Sraffa type trade theory, with tradedintermediate goods, is possible. In view of this theory, it is shown that theSamuelson’s Conjecture of Limited Substitution is false. The relevance of a newtheory is illustrated by a recent debate on the gains from trade.
Keywords: Ricardo-Sraffa trade theory, intermediate goods, Sraffian bonus
JEL Classification Numbers: F10, F51, B51

1

Introduction

In 2001, Paul Samuelson contributed a paper titled “A Ricardo-Sraffa Paradigm ― Comparing Gains from Trade in Inputs and Finished Goods” to the
Journal of Economic Literature. He chose a rather old topic ― Ricardian trade
theory ― but from a new viewpoint. He compared two scenarios in which we
gain from trade. The first one is Ricardo’s, and the second is supposed to be that of1)
Piero Sraffa . Ricardo viewed trade as an exchange of finished goods such as
wine and linen. Sraffa initiated a new viewpoint that gains from trade are a resultof the trade of input goods. In order to convince readers, Samuelson cites anappropriate example (See Figure 1) where he shows that the gain from input tradeis three times higher than that from the trade of finished goods. Samuelsonreferred to this gain from input trade as the Sraffian bonus.
Apparently, this is in acknowledgement of Sraffa’s 1960 book Production of1)

I have used the word “supposed” because, for the best of my knowledge, Sraffa himself had
never explicitly argued this question.

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Y. Shiozawa

Figure 1: The Sraffian Bonus

Samuelson, “A Ricardo-Sraffa Paradigm,” JEL, 2001

Commodities by Means of Commodities. Samuelson cited Böhm-Bawerk, Irving
Fisher, Piero Sraffa, and Karl Marx as authors who can understand “how labor
working with capital might enjoy higher productivities and higher competitive real
wage rates.” However, he attributed the concept of the input trade to Sraffa. Thisis somewhat unusual because Sraffa did not mention international trade in his 1960
slim book. The entire Sraffa story is Samuelson’s invention. However, it is not
surprising that Samuelson attributed the concept of gains from input trade to
Sraffa.
In his book, Sraffa (1960) rightly shed some light on the importance of inputs.
On the other hand, he was also the editor of the famous Essays andCorrespondence of David Ricardo and presented a new interpretation of Ricardo’s
works. As a result of this and his own book, Piero Sraffa became the patron saintof a new research program, which led to the formation of a school termed byeconomists as neo-Ricardians or Sraffians. However, there was a peculiar lacunain the 1960 book in that the author did not mention the Ricardian idea ofcomparative advantage. This lacuna was not filled by neo-Ricardians, althoughsome of them worked mainly in the field of international trade (See Steedman
1979a and 1979b). They could have studied small open economies. Thus, thetrue enigma lies in (1) why neo-Ricardians could not recognize the importance ofgains from input trade and (2) why they could not present a theory of international

Samuelson’s Implicit Criticism against Sraffa and the Sraffians and Two Other Questions

21

trade. In this new theory, the entire economy should be studied wherecommodities are produced by means of commodities, which are produced either inthe country of production or in a foreign country.
In light of this unusual lacuna, Samuelson’ s acknowledgement must be
considered as an implicit criticism of neo-Ricardians. Neo-Ricardians discussed
the Heckscher-Ohlin-Samuelson theory (hereafter, the HOS theory) and strongly
criticized it based on the fact that it relies on the concept of homogeneous capital.
This was, in fact, a part of the famous capital controversy. In this controversy,
neo-Ricardians together with English Cantabridgians were victorious, andSamuelson and his company was defeated (Samuelson 1966). However, neoRicardians could not formulate any constructive theory to replace the HOS theory.
Although Samuelson did not mention the subject of neo-Ricardians, this silence
itself was a form of astute satire.
Logically, this paper consists of four distinct parts. In Section 2, I show how
the lacuna was conceived by neo-Ricardians and how several attempts were made.
Despite their efforts, however, they could not go beyond the small open country
context. In Section 3, I trace how the same topic was conceived by mainstream
economics. In Section 4, based on my previous paper (Shiozawa 2007), I showthat the neo-Ricardian lacuna was now filled by a new trade theory and that
Samuelson’s implicit criticism had been invalid since 2007.
In Section 5, I examine what Samuelson named the Conjecture of Limited
Substitution. Samuelson’s conjecture in a two-country two-good case is false, andI present a counterexample. Next, I show how to apply my general theory to these
simple cases. I also discuss the conjecture in a three-country case, first provided
by Ronald Jones. Although Samuelson’s conjecture was false, a different form of“Limited Substitution Theorem” was obtained as part of the general theory ofinternational trade with intermediate products, as I have indicated at the end of
Section 4.
In Section 6, by way of a conclusion, I cite a recent controversy on the natureof gains from trade. In defending his position, Gregory Mankiw made it clear that
he relies on the Ricardian trade theory, and argued that the Ricardian trade theory
is more appropriate than the HOS theory in the world of the global capital
movement, for the Ricardian trade theory assumes labor force as a uniqueimmobile factor of production. However, there clearly exists a logical flaw in his
contention unless a Ricardian trade theory that can treat input trade is provided.
This small episode clearly shows that the topic discussed here is not only ofhistorical interest but also has actual significance for present-day policy
examinations.

2

A lacuna in Sraffa’s book and its conception

Sraffa’ s 1960 book started that the circular nature of a production system
necessitates abandoning the concept of capital as the primary resource. This very
fact became the starting point of a famous controversy, which is sometimes called22

Y. Shiozawa

the Cambridge-Cambridge Controversy or Capital Measurement Controversy.
Neo-Ricardians criticized the HOS theory. It was an application of the English
Cantabrigians’ criticism of neoclassical orthodoxy. The standard HOS theory
assumes two factors of production: labor and capital. Ian Steedman was the main
figure who led the battle in international trade theory. He presented a numericalexample that showed the re-switching of techniques, and by consequence, capital
reverse. He convincingly showed the logical flaw to the HOS theory. However,this criticism could not change the future flow of the international trade theory andthe HOS theory remained the mainstream theory. The reason behind this state is
obvious. Neo-Recardians could not present a new theory that could replace the
HOS theory.
It is important to note here that Steedman and other neo-Ricardians who
discussed international trade supposed a “small open economy” (Steedman, 1979,
Introduction). In other words, they treated the economy of country A trading withthe rest of the world. The term “small” implied that the influence of country A is
negligible to the world economy. Other people (for example, Gontijo 2000)
claimed that they succeeded in treating the entire world economy, in which two or
more countries engaged in international trade. However, most of them adopted the“rest of the world” concept, in order to make their theory tractable. Few of themtook the initiative to explore the case of many countries in which inputs are traded.
A simple criterion to determine if they have succeeded in formulating an effective
theory would be to examine whether they have succeeded in determining thepossible patterns of trade and specializations. Despite this, in most cases, authors
assigned a particular pattern of trade, but did not clarify why and how that pattern2)
was possible .
If Sraffa’s main contention was the mutual dependence of costs and prices
because of the circular nature of production, then an international trade theory of
the Ricardian type should be constructed on the same principle. However, those
Sraffians who dwell in a small open economy did not (and logically could not)succeed in constructing such a theory. It is clear that there is a lacuna in the
Sraffian research programs. Steedman himself acknowledged this lacuna andwrote about the question twice (Steedman 1999 and Steedman 2002). Maneschi(2004) acknowledged the necessity of extending the “principles of the neoRicardian (or Sraffian) theory of international trade” and treated the case with input
trade but his study is still limited to a small open economy analysis.
Samuelson’s 2001 paper apparently acknowledged Sraffa’s 1960 book, but at
the same time, if read in depth, it was a satire on the inability of neo-Ricardians torecognize and fill the lacuna that might be the most important (and constructive)
challenge for them. No neo-Ricardians seem to have reacted to Samuelson’ s
implicit criticism. One of the few exceptions, are Baldone, Sdogati and Tajoli
2)Sometimes, they claim that they are not interested in determining the pattern of trade andspecializations. However, without solving this problem, a complete theory of international trade
is not possible.

Samuelson’s Implicit Criticism against Sraffa and the Sraffians and Two Other Questions

23(2007). The authors claim that their example “builds on a Ricardo-Sraffa approachsimilar to the one developed by Samuelson (2001 and 2004).”
Pasinetti (2006) compared Samuelson and Sraffa and insisted that
Samuelsonian and Sraffian economics run on two different tracks with different
paradigms. Pasinetti admits that Sraffa’s theory is “far from being a complete
theory.” However, he does not mention the possible construction of the RicardoSraffa trade theory, although he cites Samuelson’s 2001 paper.
Neo-Ricardians might have given up constructing a theory that answered
Samuelson’s criticism. However, it was a slightly hasty decision. As I show in
Section 4, the task of constructing a Ricardo-Sraffa type general trade theory was
already achieved by Shiozawa (2007).

3

A topic disappeared or abandoned?

Before presenting a short summary of my results, we shall see how the problem
was conceived in the mainstream trade theory literature.
The necessity of formulating a new trade theory where inputs are traded wasrecognized in as early as the 1960s. Jones (1961) discovered a famous inequality
formula by comparing the permutated products of labor-input coefficients. He
tried to develop his results to “the case where intermediate products are freely
traded.” He only succeeded in treating cases where “the intermediate productstructures are assumed (to be) the same in all countries.” Another case that he wasable to treat was that of an economy where “no products could be traded if they
were to be used as intermediate products in some other country.”
Jones’s paper appeared to be independent of Sraffa’s 1960 book; Jones was
inspired by McKenzie (1954 and 1956). In these papers, McKenzie examined
what he called the Graham Model, which is but a Ricardian model of a manycountry, many-commodity case where labor is the unique input. Using the resultsof the activity analysis developed at that time, McKenzie succeeded in establishingsome fundamental theorems such as the existence of equilibriums and the logicalequivalence of production efficiency and profitability conditions. Furthermore,
McKenzie added some observations to the case where intermediate products weretraded. He observed that “it is not necessary to assume ... that the activities are
actually integrated.” He added that “it is sufficient that intermediate products do
not appear in international trade” (McKenzie 1954, p. 166). If there exist several
methods to produce a commodity, he hints that one can apply Samuelson’s nonsubstitution theorem, given that joint production is absent. However, he was very
well aware that the “simplicity [of the Graham model] is bought at the expense ofprohibiting all trade in intermediate goods” (ibid. p. 180) and “as soon as trade inintermediate products is allowed, the problem loses its special simplicity” (ibidem.
p. 179). McKenzie (1956) even stated in his introduction that “the basic flaw inthe classical treatment of specialization is... the neglect of trade in intermediate
products” (p. 56).
A few years after Jones’ s paper, Chipman (1965) wrote a comprehensive

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Y. Shiozawa

survey of the theory of international trade and published it in Econometrica. He
paid special attention to the mathematical structure of the theory. After reviewing
Jones’s contribution, he wrote an independent section on Trade in Intermediate
Products (§1.8). Chipman emphasized that McKenzie (1954) had stumbled upon
the questions of intermediate products and discovered that “the introduction of
trade in intermediate products necessitates a fundamental alteration in the classicalanalysis.” Chipman added: “It is strange that this had not been noticed before,
despite the fact ... that classical economists used intermediates products such as‘cloth’ and ‘linen’ ”.
Reiter (1953) and McKenzie (1954) remarked that “trade in intermediate
products would enlarge the world production-possibility set” (Chipman, 1965, p.
509). Thus, the “Sraffian bonus” was discovered before Sraffa’s Production ofCommodities by Means of Commodities. However, as Chipman concludes in his
section, “the most interesting consequences of the introduction of trade inintermediate products is ... the possibility of reversal in the pattern ofspecialization” . This is that which McKenzie (1954, pp. 177-179) and Jones(1961, pp. 166-168) have emphasized. McKenzie’s famous remark is that “there
is nothing shocking to common sense in these results. A moment’s reflection willconvince that Lancashire would be unlikely to produce cotton cloth if the cotton
has to be grown in England” (1954, p. 179).
Curiously enough, the theory of traded intermediate goods was not developed
much after 1965. Sporadic works like Chipman (1971), Sanyal and Jones (1982),
and Jones (1994, 1996) appeared; however, it appears that no Ricardo-Sraffa typegeneral theory was presented by mainstream trade theory workers.
In 2000, Ronald Jones published a book under the title Globalization and theTheory of Input Trade, based on the Ohlin Memorial Lectures given in 1997 in
Stockholm. Jones did not intend to overview the trade theory. He treated new
problems that became important through increased mobility of inputs. In fact, thevolume of input trade has increased and various forms of processing trade(including fragmentation and outsourcing) take place more widely than ever.
However, Jones never mentioned any results with regard to the theory ofinternational trade where the intermediate goods or inputs are traded. This factshows the state-of-the-art theory of international trade. The general equilibrium
framework is too general to be used as a basis for determining the pattern ofspecializations.
The interest in trade theory in intermediate products appears to havedisappeared. At the turn of the century, international trade theory, whether it is
neo-Ricardian or mainstream, lacked a theoretical basis when input trade isconsidered.

4

How can a new theory be constructed?

Although it is not an easy exercise, a Ricardian or Sraffian theory ofinternational trade with traded intermediate goods is possible (Shiozawa 2007). It

Samuelson’s Implicit Criticism against Sraffa and the Sraffians and Two Other Questions

25

only requires a bit of unfamiliar mathematics: the theory of linear inequalities andthe theory of convex cones. However, the economic principle is simple. In fact,
the next theorem is obtained as follows:
Theorem 4.1 (Existence of the shared system of prices)
Let e be an economy with M-country and N-commodities and a technology,
which satisfies the following assumptions.
(1) Each country is equipped with a technology that comprises linear methods ofproduction.
(2) Trade is conducted free of transport cost and without any legal restrictions.
(3) Labor is a unique primary factor and does not move across nationalboundaries.
(4) Capital goods are produced and traded across countries and can be used as
inputs of production.
We further assume that(5) each method of production is free of joint production (or the technology is a3)
single product type) ,(6) each country has a productive technique or a productive set of methods, and(7) for each country the maximal rate of profit is higher than a fixed R.
Then, for any profit rate r less than or equal to R, there exists a set of wages
w/pw i and a set of world prices p/pp j, and they satisfy the following
conditions:(8) Each country has at least one competitive method of production in the sensethat the profit rate of the method of production is equal to r, and(9) for all methods of production, the profit rate is less than r.
A few definitions may be useful, for there are different ways for expressing thesame object.
The basic concept in this paper is “method of production” which is called“technique” in Shiozawa (2007). The collection of all the methods of production issometimes called technology of the economy. Each method represents a knowhow, by which one can obtain, after a certain lapse of time, an output b/pb i inplace of consuming inputs a/pa i. The important assumption here is that thetransformation of input into output is linear in the sense that when thetransformation of a into b is possible, then the transformation of ha into hb isalso possible for any non-negative number h. In this case, we say that thesetransformations are productions which belong to the method of production. To
assume linearity of transformation is to assume constant returns to scale. When we
have a set S of methods of production, the transformation of a into b is a3)

If the positive net product of a method of production includes more than two kinds of goods,
the method is said to contain joint production. If the set of methods of production contains no
method with joint production, the technology of the economy is said to be a single product typeor a simple production type.

26

Y. Shiozawa

production belonging to the set S, if pa, b is the sum of productions that belong toa method of production, or an element of the set S . The net product of the
production pa, b is b,a.
Another important assumption is assumption (5). A method is free of joint
production, when the net output comprises a single type of good. The technologyof the economy is a single product type, when any method of production is free of
joint production. Shiozawa (2007) uses the term simple production economy. By
this assumption, one can speak of an industry that produces a specific good. A
method that produces goods j is called a method of industry j. There may be morethan one method that produces the same types of goods or that belongs to the same
industry. Thus, the choice of method of production is usually necessary.
However, this assumption is substantially restrictive, since it excludes the
productions that employ durable capital goods such as machines and equipments.
Theoretically, it is necessary to treat these capital goods as by-products, and thus,
one cannot exclude joint production even when one works in an industry that
produces one specific product. This assumption may then be a grave restriction
while analyzing capital in a logical way. In reality, this restriction is practically
lifted when durable capital goods keep their efficiency constant within thepredetermined limit of durations. The examples of these calculations are provided
in Sraffa (1960, section 82).
A bit of explanation for the difference in terminology will be useful. In
Shiozawa (2007), the methods of production are called techniques. In this article,
the word “technique” is used as a complete set of methods, that is a selection of themethods of production such that one method is chosen in each industry. In
Shiozawa (2007), rather than technique, a word “system of techniques” is used, andthe collection of all methods of production or technology is called “set oftechniques”. Otherwise, there are no major differences in terminology.
A technology is called productive when a net product that belongs to thetechnology is positive. A technique, or a set of methods, is productive when a net
product belonging to the technique is positive. For each technique, a regularmatrix A is associated. If a technique is productive, the associated matrix A is
non-negatively invertible. This implies that A -1 exists, and all these entries are
positive or zeros.
The theorem states that there are a system of wages w and a system of prices pby which each country has at least one worldly competitive method of production,and by consequence, a worldly competitive industry. Note that wages are given bya system of wages. Each country has a uniform level of wages, but they can bedifferent for different countries. The essence of the theorem lies in the fact that
one can find a good system of wages such that the minimum price theorem gives
the requested systems of wages and prices. In addition, note that this theorem does
not refer to the quantities of production and trade. These will be examined andanalyzed when another theorem is obtained.
The above supposed situation is highly general. As the technology does notSamuelson’s Implicit Criticism against Sraffa and the Sraffians and Two Other Questions

27

admit joint productions, fixed capitals are logically excluded. However, as I havementioned above, they can be incorporated in the theory as long as we assume thatthe efficiency of the durable capital goods remains constant during the depreciation
period. Each country may have different technologies. As a result, differences inclimate and underground resources can be reflected in the differences of methodsof production. For example, if a country enjoys a good climate for wheat
production, the country has a method of wheat production with lesser coefficients
than those of countries with less fortunate climate.
The above theorem is equivalent to Theorem 3.3 in Shiozawa (2007). The
demonstration is simple when one uses F. E. Su’s rental harmony theorem (Su,1999). Two other versions of demonstrations are possible but they are exempt
here. The existence of the so-called perfect specialization can also be proved when
a generic condition is satisfied (Shiozawa 2007 Theorem 3.4). The vectors w andp , which satisfy the above conditions (8) and (9), are called shared. Sharedvectors form a closed cone of the positive orthant of dimension M+N, where M
is the number of countries and N is the number of goods. Furthermore, we
observe that this shared cone is endowed a mathematical structure called fan. If
one analyzes this structure, one may determine all the possible patterns ofspecializations.
Another fundamental theorem is the next one. Here, I only give the theorem
when the uniform growth rate g of the economy is 0. When the growth rate is
positive, it is necessary to modify the theorem in such a way that the net product ofa production should be redefined as b,p1+g a. The duality between maximal
frontier of productions and the shared cone of wage-price systems is valid only
when the uniform profit rate is equal to the uniform growth rate. As this leads toan unnecessary complication of the formulae, I only state when the growth rate gand the profit rate r are 0. This does not imply that the new trade theory isconstructed only when r and g are 0. This convention is made only for thesimplicity of explanation.
In order to state the theorem, a few new symbols are necessary. Let be an
economy that satisfies the same assumptions as Theorem 4.1. Note that M is thenumber of countries and N is the number of commodities. We introduce a new
symbol T, which is the total number of methods of production of all countries. (Byconsequence, T is at least equal to or greater than N times M ). Let A be thecoefficient matrix with T lines and N columns. By this definition A -1 has no
meaning, since A is no more regular in general. Each line vector of matrix A is
the net product of the production that belongs to each method of production andthat uses one unit of labor as input. Let I be a matrix with T lines and M
columns. The i-th entry of a line of I is 1 only when the line represents a methodof production that belongs to the technology of country i. Finally, let q be thevector representing the amount of labor force of each country.

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Y. Shiozawa

Theorem 4.2 (Maximal point of the production possibility set)
Let be an economy that satisfies the same assumptions as Theorem 4.1. If y
is a maximal point of the production possibility set, then there exists a positive Mcolumn vector w and positive N-column vector p such that
ApAIw and Xy, pY/Xq, wY.
Conversely, if a couple of positive vectors w, p together with a net production
vector y satisfy the above condition, then the net production vector y is one of the
maximal points of the production possibility set.
The shared cone, which may not be convex, is closely related to the production
possibility set. For any pair of vectors w and p in the shared cone, we obtain aproduction possibility set for each country as follows:
If the set of all competitive techniques of a country m is given by j p1, j p2,... , j pk, then the set of competitively produced net products for the country m isgiven by

 6 s a  6 s Aq .
k

h=1

h

j ph

j ph

k

h=1

j ph

m

Here, a is the net production coefficient vector, and q m is the amount of labor
force for country m, whereas s h indicates the production scale for the technique h.
Worldwide production is given by the Minkowski sum of the production sets for
all countries. Any point thus obtained is proved to be maximal (or efficient), andthey form a cell of the maximal production frontier. A beautiful dual
correspondence theorem is obtained (Shiozawa 2007 Theorem 5.7), and we knowthat a facet (complex of dimension N,1) of the maximal production possibility
set corresponds to a composing straight line of the shared cone. Those lines are ofcourse finite, and the number of facets of the maximal frontier is finite.
An important fact here is that a facet covers a wide range of final demand
compositions; however, the corresponding set of w and p are unique up to scalarmultiplication. This can be called an extended non-substitution theorem for aninternational trade situation. As long as the global demand remains in the same
facet, the relative wages and prices remain constant and quantity adjustment isnecessary, in order that supplies are nearly equal to demands. Although theSamuelson’s Limited Substitution Conjecture is false, as it will be shown in thenext section, the fact shown above can be called “Limited Substitution Theorem,”because wages and prices remain constant, and the set of competitive techniques
remains the same (limited substitution) whenever the final demand changes within
the same facet of the production frontier.
The Ricardo-Sraffa theory does not stop here. Only a special part of the entire
theory is examined. The maximal production frontier provides a set of possible
equilibrium states in which labor is fully employed in all countries and allindustries work competitively. However, the main merits of the Ricardo-Sraffa
theory remain on a different side of analysis. In contrast to the HOS theory, the
new trade theory makes it possible to analyze out-of-equilibrium states.

Samuelson’s Implicit Criticism against Sraffa and the Sraffians and Two Other Questions

29

As an example, consider net production y, which is not in the set of maximal
production frontier. We shall consider a positive system of wages w and apositive system of prices p that satisfy the next condition:
ApAIw.
If y is a production that belongs to the technology of economy , then there exists
a production activity vector s/ps i such thaty/sA and sIAq.
Therefore, we obtain
Xy, pY/XsA, pY/Xs, ApYAXs, IwY/XsI, wYAXq, wY.
Two inequalities cannot be both an equality. In fact, if two equalities are obtained,by virtue of Theorem 4.2, y must lie in the maximal production frontier.
Then, what happens in economy ? One of two inequalities
Xs, ApY?Xs, IwY or XsI, wY?Xq, wY
holds. If the first inequality holds, s t>0 for at least one method of production t
with
Xa t, pY>w c,
where c is the country index of method t. This implies that in some industries, amethod of production is operated even if it is not competitive. The state may
produce a dual structure, or more generally, a differentiated structure inside theeconomy of a country.
If the second inequality holds, then for some countries c,
psI c?q c.
This implies that there is unemployment in country c.
In sum, if net production y is not in the set of maximal production frontier, oneof the next two conditions is violated:(1) Labor is fully employed in all countries.
(2) All production is operated by means of competitive methods of production.
This state may often be observed, since there are many obstructions in order toarrive at a maximal point if one starts from any non-maximal point. One such stateis autarchy. Obstructions may be the difficulties to find out an equilibrium state ofworld trade to finance the change of the production levels. The transition from thepresent state to an equilibrium state may require time to train workers
appropriately. It is normal that one of these obstructions holds. Then, economy
must stay for a long time in a state where either (1) or (2) is violated. In this way,
the out-of-equilibrium state of production and trade is analyzable in the RicardoSraffa framework. However, the purpose of the present paper is not in this task. I
stop here to return to the main flow of the topic.

5

Samuelson’s conjecture is false.

In footnote 5 of the 2001 paper, Samuelson presented what he named
Conjecture of Limited Substitution. It was stated as follows:

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Y. Shiozawa

No matter how many capital-using techniques occur in a 2-good, 2-country
scenario involving fixed localized totals of population, generically (singularcases and joint-products aside), the maximal global production-possibility
frontier will be concave and two-faceted.
The first part of the conjecture, which states that the frontier is concave, is true.
The word “concave” is the short expression of “concave from the origin” andstands for the state that the production possibility set (composed of points maximal
or not) is convex. However, the second part of the conjecture is not true as thefollowing counter-example shows.
Each method of production is represented by a row vector. The first entry islabor input. The second and third entries indicate the net production of goods 1
and 2 for a unit scale of production. The scales are normalized so that the labor
input is unit for a unit production. Let w A and w B be wage rates (measured by aninternational currency). The relevant ratio is relative wage rate w A/w B. We shallassume that w A/1. Wage rate w B takes any positive value.
A Counter- example to Samuelson’s Conjecture
Countries

OutputSymbols of production method

Country A

good 1

A11

(−1

11

A12

(−1

14

0)
−1)

A13

(−1

Country B

Coefficients

good 2A2

20(−1 −7−4)
5)

good 1B1(−1good 2

B2

−1)
4)

6
(−1 −1The switching points of patterns of specializations are given at w B/67/55 ,w /1 , w B/7/12 , and w B/23/72 . At these switching points, three methods
become competitive. Between these switching points, only two methods arecompetitive. The list of competitive methods is given in Table 1.
The pattern of specializations is shared if and only if 67/55@w B@23/72 .
Figure 2 gives an idea of how the methods are related. Country A’s methods areplotted by their second and third coordinates. Country B’ s methods will be
represented by their second and third coordinates multiplied by 1/w B . Thus,
Country B’ s methods move along a straight line when w B changes. The line
connecting the points represents the production possibility set, supported by thethree methods. Key multipliers at the switching point are as follows: 55/67/0.82, 1/1/1, 12/7/1.71, 72/23/3.13.
The production possibility set is determined only when the amounts of laborfor each country are given. We shall simply assume that q A/1 and q B/1. Then,B

Samuelson’s Implicit Criticism against Sraffa and the Sraffians and Two Other Questions

31

Table 1: List of Competitive Methods
B

A

The range of w when w ＝1

The set of competitive methods

B

A11 and A2

B

A11, A2, and B2

w ＞67/55
w ＝67/55
B

67/55＞w ＞1

A11 and B2

B

w ＝1

A11, A12, and B2

B

A12 and B2

1＞w ＞7/12
B

w ＝7/12B

7/12＞w ＞5/14
B

w ＝23/72
B
w ＜23/72

A12, A13, and B2
A13 and B2
A13, B1, and B2
B1 and B2

the facets of the maximal frontier are given at each switching point.
When w B/67/55, methods A 11, A 2, and B 2 are competitive. Country B uses
method B 2 and produces p,1, 4. Country A uses methods A 11 and A 2 and shares
the labor such that the total labor input equals 1. When A 11 uses all the labor forceof the country, the production scale is equal to 1 and the net production is p11, 0.
When A 2 uses all the labor force, the net production is p,7, 5 . Thus, theproduction given by this point w A/1 , w B/67/55 is the line segment thatconnects two points p,7, 5+p,1, 4/p,8, 9 and p11, 0+p,1, 4/p10, 4.
In a similar way, when w A/1 , w B/1 , methods A 11 , A 12 , and B 2 arecompetitive. Thus, the production frontier facet is the line segment, whichconnects p11, 0+p,1, 4/p10, 4 and p14, ,1+p,1, 4/p13, 3 . Whenw B/7/12, methods A 12, A 13, and B 2 are competitive. The frontier facet is the line
segment that connects the points p14, ,1+p,1, 4/p13, 3 andp20, ,4+p,1, 4/p19, 0. When w B/23/72 , methods A 13 , B 1 , and B 2 arecompetitive. Country A produces using only the method A 13. The net product is
p20, ,4. Country B shares labor force in two productions using methods B 1 andB 2. Thus, the line segment is given by the two points p20, ,4+p,1, 4/p19, 0
and p20, ,4+p6, ,1/p26, ,5. All of these results give the total maximal
frontier as it is given in Figure 3.
The above example satisfies the assumptions of Samuelson’s Conjecture. In
fact, it is a two-country two-good case, and the example is generic in the sense thateven if the coefficients are changed within a sufficiently small amount, the numberof facets remains four. Even if we confine frontier only in the positive quadrant,
the number of facets is more than three. This is clearly a counter-example for his
Conjecture.
Samuelson adds a few words for the situation when the number of countries isincreased. He introduces Ronald Jones’ s remark: In a generic three-country(probably two-goods) case, the plausible conjecture will be that the frontier isconcave three-faceted. The above example provides a counter-example to Jones’s

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Y. Shiozawa

Figure 2: A counter-example for the Samuelson’s conjecture

NB. Scales of coordinates are different. The scale of the y- coordinate is 2 times the scale of the xcoordinate.

conjecture about what happens when a third country is added. It is sufficient totake the set of methods for country C p,1, 1, 0 and p,1, 0, 1. It is evident thatfor any pair w A and w B, there exists at least one competitive method for country C.
The profit for a unit production for each method is p 1,w C and p 2,w C. When p 2
is bigger than p 1, the method that produces good 2 is competitive. When p 1 is lessthan p 2, the method p,1, 0, 1 is competitive, whereas method p,1, 1, 0 gives anegative profit. On the other hand, when 23/72?w B?67/55, p 2 is always bigger
than p 1. Thus, it is always method p,1, 0, 1 that remains competitive. Whenthe labor force of country C is 1, its production is always p0, 1. The maximal
frontier is thus shifted by a unit upward, and the form of the maximal frontier
remains the same. However, in this case, the number of facets is four, even if wecount those facets that lie in the positive quadrant. Jones’s conjecture is also false.
In Appendix 2 of his paper, Samuelson reviews the “trade generalization of1960 Sraffa.” He explains the two-country two-good case, and at the end of
Appendix 2 he remarks: “My described techniques can be modified to handle ...
cases of more than two goods and more than two countries.”
This remark is questionable, since the many-good many-country case requires a
new formulation essentially different from the two-good case or the two-country

Samuelson’s Implicit Criticism against Sraffa and the Sraffians and Two Other Questions

33

Figure 3: Maximal frontier has 4 facets.

case. For example, in the two-country case, one can appeal to the middle-value
theorem, that is, one can find a shared vector by moving wages of a country
continuously from a sufficiently small to a sufficiently large number. The manycountry case requires the existence theorem for a shared pattern of specializations(See Method 1 in the proof of Theorem 3.3 of my paper).
Another point of difference lies in the necessity to analyze the structure of theshared cone, which is closely related to the patterns of specializations. The dual
correspondence between two modal decompositions (one of the wage simplex andanother of the production frontier) is essential to understand that many-good manycountry case, as explained in Shiozawa (2007 Section 5). Samuelson gives no
indication of these ideas. Even for the two-country two-good case, if we know thedual correspondence between modal decompositions, it is easy to guess the generalform of the maximal frontier, and consequently, the maximal number of facets.
Samuelson disregards all these important points of analysis, and it is difficult tobelieve that he has really constructed a Sraffa-type general trade theory for the
many-country many-good case.

6

An episode as a guise of conclusion

As Samuelson’ s example shows, the Sraffian bonus is considerably large.
Samuelson’s implicit criticism against Sraffa and the Sraffians was correct. By its

34

Y. Shiozawa

theoretical and actual importance, a theory of input trade is necessary.
Theoretically, as Sraffa’ s 1960 book reveals, the circular nature of industrial
production is essential. Its actual importance is apparent because the absolutevolume and relative weight of input trade are increasing. Furthermore, processing
4)
trade is becoming more and more important . Processing trade normally takes theform of triangle trade. China imports key devices from Japan, assembles them,and exports the products to the United States. Here, the usually assumed rest-ofthe-world story is no longer valid. We must construct a general theory of input
trade for a many-country many-good case. The first step toward this general
theory has been taken by Shiozawa (2007). This is a reply to Samuelson’ s
criticism, which is now invalid.
The extensions of the Ricardian trade theory to the input trade case have an
unexpected effect on policy discussions, as witnessed by the recent debate between
5)famous economists like Greg Mankiw and Dani Rodrik . The debate started withthe post of Daniel Drezner. Drezner objected to Rodrik’s saying that Rodrik elides
the biggest gain from trade―lower prices. Rodrik responded that (1) trade may
decrease the prices of imported goods but will increase those for export goods and(2) gains from trade for the consumers depend on the relative weight of imported
and exported goods. Mankiw defended the idea that trade can lower prices. By
pointing out that people treat nominal wages as the numeraire, he claimed that all
prices are lowered after trade when they are compared to closed state prices, andthe real wage rate will be increased. Rodrik did not admit Mankiw’s contention
and claimed that the Stolper-Samuelson theorem denies that the real wages ofworkers rise as a result of free trade. Finally, Mankiw posted a short response withthe title “Ricardo vs. Heckscher-Ohlin.” Mankiw argues in this post that he relieson the Ricardian trade theory instead of the more common HOS theory andexplains how this is derived. Further, he contended that the Ricardian trade theory
is more realistic than the HOS theory, as the latter assumes that capital cannot
move from country to country. However, he continues, in today’s global economy,
capital is highly mobile across national borders and concludes that in light of thisfact, in his opinion, the Ricardian theory of trade is more useful than the HOS
theory.
As a student of Ricardian trade theory, I am of the same opinion as Mankiw.
However, it is noteworthy that Mankiw’s contention contains a logical slip. The
traditional Ricardian theory does not assume capital in any explicit form, and thetrade of capital is not taken into consideration. Mankiw’s contention is right only
when a new trade theory of the Ricardo-Sraffa type is constructed. The
construction of a Ricardo-type general theory of input trade is thus relevant totoday’s highly academic discussions, both in theory and policy.

4)

Fragmentation is but a special form of processing trade.
The entire series of interventions is summarized by Mark Thoma in the post of April 28, 2007
in his blog The Economist’s View.
5)

Samuelson’s Implicit Criticism against Sraffa and the Sraffians and Two Other Questions

35

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