Evol. Inst. Econ. Rev. 3(2): 141–187 (2007)

ARTICLEA New Construction of Ricardian Trade Theory—A
Many-country, Many-commodity Case with
Intermediate Goods and Choice of Production
Techniques—
Yoshinori SHIOZAWA
Osaka City University, 3–3–138 Sugimoto, Sumiyoshi-ku, Osaka 558–8585, Japan.
E-mail: y@shiozawa.net
Abstract
Ricardian trade theory is one of the most famous theories of economics but appears to havebeen little developed. Many attempts were made to extend the theory to multi-country,
multi-commodity cases, but none succeeded to construct a general theory that included
intermediate goods. A need to include intermediate goods within the theory was evident, but
hurdles to introduce intermediate inputs were high. Intermediate goods change the entire
structure of analysis; when they are traded, the price of a good is dependent of the prices ofimported inputs. Consequently, prices should be determined simultaneously for prices of all
countries. The present paper has succeeded in overcoming these difficulties and describes
how wages, prices and productions are related. It analyzes the M-country, N-commodity
case with choice of techniques and trade of intermediate goods in general terms, thus
presenting a new basis for international trade theory. New light was shed on topics like gains
and losses from trade, international wage rate discrepancies, and price and quantity
adjustments. On a theoretical plane, the new construction eliminates a traditional weaknessof the Ricardian theory. The traditional Ricardian theory acknowledged labor as the only
input and excluded capital in any form. The new theory, presented here, analyzes capital
goods as traded intermediate inputs.

1.

Introduction

Ricardian trade theory is one of the most famous theories of classical economics. It iscited in many introductory course textbooks on international trade. The logic ofcomparative advantage is even now impressive and profound. A famous nuclear
physicist Stanislav Ulam once asked Paul Samuelson if there is any non-trivial theoremin economics. Many years later Samuelson found a good answer: the theory ofcomparative advantage is one.
Nearly two centuries have passed after the publication of Ricardo’s Principles (1817).

JEL: B12, C65, F10.

Y. SHIOZAWA

In spite of its evident importance and the claims of wide applicability1), the theory
appears to have been little developed. In the 19th century J. S. Mill argued about mutual
demand. In the first half of the 20th century F. D. Graham examined a more general casethan Ricardo’s 2-country, 2-commodity case, but the analysis was always based on
numerical examples. Observations were generalized to the 2-country, many-commodity
case by Harberler (1930). Dornbusch, Fischer and Samuelson (1977) extended theanalysis to the case of continuum of commodities. The many-country, 2-commodity casewas examined by Vainer (1937). In 1950’s and in 1960’s some general analysis for Mcountry N-commodity case appeared. McKenzie (1954b), for example showed theexistence of equilibrium for Graham’s model of international trade. Jones (1961)
discovered a formula that could tell the possible pattern of specialization. After that
period no substantial improvement appeared in this field2).
This does not mean that all the important problems were solved. On the contrary, themodels so far analyzed had two crucial defects. (1) Inputs were restricted to labor as aunique factor and no material inputs were admitted. This implied that intermediate goods
were excluded from any theoretical analysis of international trade. (2) Choice oftechniques was not admitted. This is what is necessary when one wants to analyze
technical change and development.
Intermediate goods change the entire structure of analysis. If intermediate goods were
not traded, the prices of any products of a country can be determined inside that country.
Thus all prices move proportionally to the wage rate of the country. When intermediates
are traded, price of a good is dependent of the prices of inputs and those inputs may be
imported from foreign countries. Consequently, prices should be determined
simultaneously for all countries. The price of a commodity depends not only on the wage
rate of the producing country but may well be dependent on other countries’ wage rates.
The price effects of wage rate changes become very complicated and autarchy prices ofeach country have little connections with international prices. The choice of techniques
adds further complication to this situation.
Hurdles to introduce intermediate inputs were high. The very notion of comparativeadvantage was not clear. As Deardorff (2005a, 2005b) reported, many competing
definitions were proposed without any conclusive arguments. Some examinations based1)
See for example Suranovic (1997–2004) Section 40-1 “The Theory of Comparative AdvantageOverview.”
2)
A unique exception might be Takamasu, 1991, pp. 34–41, in which Takamau examined the caseswith intermediate goods.

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on numerical examples were made but only sporadically (See Amano 1966, Higashida2005, and others). More than twenty years ago, I myself tried unsuccessfully to extend
the Ricardian model to a general case. My paper of 1985 dealt only with the case of two
countries. I occasionally returned to this problem, and recently found a way to overcome
the difficulties that occur with the introduction of intermediate goods. This paper deals
with the many-country, many-commodity case, with traded intermediate goods, andchoice of techniques in general terms. The results were first written in Japanese, but thisarticle is the first publication in English3).
The importance of intermediate goods is apparent. They compose an important part ofworld trade. Recent discussions on fragmentations, outsourcings and intra-industry tradeare all related to intermediate goods. It is relatively unusual to discuss detailed models
without any general theory of intermediate goods. The standard Heckscher-Ohlin theory
is not well fitted to analyze intra-industry trade. Krugman and others presented
explanations from increasing returns to scale, but most of the theory was limited to a
special setting such as two-country and three- or four-commodity cases. The lack of ageneral theory is apparent. Those topics are not explicitly examined in this paper but thetheory presented in this paper may well serve as such a general theory.
The gains from trade, when intermediates are traded, have a different dimension fromthose when final products alone are traded. An elegant example was recently provided by
Paul A. Samuelson (2001). Figure 1 is an illustration depicting to what extent the tradeof intermediate goods enlarges the world production possibility set.
The main themes of this paper are set out as follows. Section 2 is preparatory;
necessary concepts and assumptions are explained. Section 3 is the most complex part ofthe present paper. Subsection 3-1 argues for the existence of a world minimal production
price vector when the wage rate for each country is given. This is only an application ofminimal price theorem for a closed economy but it presents the starting point of all
further discussions of this paper. In subsection 3-2, competitiveness of a technique andother related definitions are presented and the existence theorem of a shared pattern ofspecialization is specified. In subsection 3-3, the notion of modal decompositions isgiven for the wage rate delta. In subsection 3-4, a proof of a more difficult existence
theorem is presented. Finally in subsection 3-5, a result of numerical experimentation is
given.
3)

See Shiozawa (2007). Although this paper and that of Shiozawa (2007) have many common
discussions, they are written as independent papers and in several places the logical constructions aredifferent.

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Y. SHIOZAWA

Fig. 1.

Two Kinds of Gains from Trade

Section 4 discusses the gains from trade. Typical gains from trade are explained insubsection 4.1, but in subsection 4.2 the possibility of unemployment and other losses
from trade is also discussed.
Section 5 discusses the forms of the production possibility set. After an introduction tosome theorems on production possibility points and on maximal points, subsection 5.3
presents an important dual correspondence theorem between two modal decompositions.
Subsection 5-4 presents some conclusions that are easily derived from the dual
correspondence theorem.
Section 6 discusses the famous controversy between Ricardo and Graham on one sideand Mill and others on the other. The conventional focal point was what determines theprices: are supply-side conditions sufficient to determine prices, or do demand conditions
intervene in the determination? A new light is shed on this old debate. Real conflict was
not between supply and demand but between price adjustment and quantity adjustment.
According to the view of dual correspondence theorem, I argue that quantity adjustment
is predominant to price adjustment.
Finally, in section 7, the questions of international wage differentials are approached.
It is obvious that there are large discrepancies in wages between countries. International
trade theory should explain why these discrepancies continue in the presence of (nearlyfree) trade. This is done here, because price determination also implies wage rate
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determination. Thus, policy implications are clear. It is necessary for poor countries tochange their set of techniques4). This may be the most important message of this paper.
The message of the Hechsher-Ohlin model is distorted. If the factor price equation
theorem is applicable, then the wage rates should be equalized for all countries. This istotally contradictory to what is actually observed.
For a considerable time the Heckscher-Ohlin model was regarded as the standard
model of modern trade theory. Discordances in this model with empirical evidence have,on many occasions, been reported. In addition, it has also some serious theoretical
defects. Capital is treated as a factor and it is presumed that factors are not traded.
However, capital goods such as raw materials, parts of machines and durable capital
goods are commonly traded internationally. Not only they are traded, as mentioned
above, they actually represent an important part of world trade. Moreover, the concept ofcapital as a mass, independent of prices, cannot be defended. It may be time to replace
Hechsher-Ohlin theory with extended Ricardian theory.
In the following discussion, the main mathematical tool is real linear algebra or thetheory of linear inequalities. Elementary knowledge of convex polytopes and topology isuseful. The mathematics employed here is not particularly common in recent economics
but all necessary knowledge is classical and obtainable either in Koopmans (1951), in
Gale (1960) or in Nikaido (1961).
2.

Main Scheme of the Theory

The present paper presents a new method for a construction of Ricardian trade theory. By
this method, we can include in the analysis all of the following situation:(a)M-country N-commodity case.

(b)

The choice of techniques.

(c)Intermediate goods (produced and traded).

A system of techniques is a set of techniques, in which each product has a techniquethat produces it. A production belongs to a system of techniques, when it is a sum of
productions and each production belongs to a technique of the system. A system oftechniques is by definition productive when a net product of the production belonging to
the system is positive for all goods. We assume three conditions for production
techniques of each country:(1)Linear production techniques.

4)

Among conditions that determine this set of techniques social technologies, including social
institutions, are as important as physical or material technologies– 145 –

Y. SHIOZAWA(2)

Simple production-type techniques.

(3)

Existence of a productive system.

In the case of single country this assumption is called the Leontief-Sraffa model. Thisis merely an international version of the Leontief-Sraffa model, which is also a faithfulgeneralization of the Ricardian trade model. Some additional explanations are needed foreach condition.
(1)

The production technique is linear or maintains constant returns to scale. Each

production of a technique is expressed as a (a, b), where a is a positive real number, a is
an input vector and b an output vector5). All productions are proportional as far as
productions belong to the same technique.
(2)

The production techniques are simple in the sense that net product consists of

only a single kind of goods as far as production belongs to a technique6). With thisassumption any production technique has two kinds of membership: a country and an
industry. A technique is said to belong to industry j when the net product consists ofgoods j. Functions C( ) and G( ) are defined in such a way that we have equationsC(t )h and G(t )j, when the technique t belongs to the country h and belongs toindustry j.
(3)Each country has at least one productive technique system. In some cases, it issufficient to assume that the world as a whole has at least one productive techniquesystem.
When a system of techniques g is given, the matrix A(g ) is usually arranged so as tohave a technique that produces j-th commodity in j-th line. The elements of matrix A(g )are positive for all diagonal entries and non-positive (i.e. zero or negative) for all offdiagonal entries. When the system is productive, A(g ) is invertible and A(g )1 is nonnegative.
The techniques of each country are assumed to be different, reflecting the differencesin each country’s technological knowledge, climate and geography, and resource
allocations. The set of all techniques is denoted by X. Labor power is presumed to be the5)

The present paper deals only with zero profit case. Examination of the positive profit cases is also
possible if one takes as output vector the discounted output vector (1/(1 r))b. In this case, production
possibility set should be interpreted as the set of net surplus product in the growing economy withgrowth rate 1r.
6)
Joint production of the type of electric salt decomposition that produces sodium and chloride at thesame time is excluded. Durable capital goods can be treated as long as they retain constant efficiency
within a prefixed lifetime. For the simplicity of the explanations we assume that production techniques
are all simple in the sense defined above. See Shiozawa (1975).

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unique primary factor, which is fixed to each country and does not move internationally.
Although there is constant immigration between countries, cultural differences areusually large enough to limit the proportion of immigrants to a small part of the
population. Most countries have also a quota system or other measures to limit immigration
into their countries. So the immobile work force assumption remains relevant for thecontemporary world. The labor power of a country i is denoted qi. We set q(qi). The setof techniques X and distribution of labor powers q are two fundamental conditions that
determine all possible states of the economy and are assumed to be fixed7).
The choice of techniques is an essential part of international trade theory. Underassumption (3), each country has a technique that produces each good. Then if there are
M countries in the world, there are M techniques that produce the same good. One should
determine which technique of which country is competitive and which are not. This is
the question of specialization. The determination of pattern of specialization is nothingother than the choice of techniques in worldwide competition. The existence of plural
techniques in a single country for each good makes only a small difference for the
analysis.
3.

Main Process of Analysis of Wage Rates and Prices

3.1

Minimal price vector that corresponds to the wage-rate vector

The key point of the new construction is to start with a column vector w(w h). Here, w his the wage rate of country h, given in terms of a common international currency. We callw a wage rate vector. When there are M countries in the world, the vector is ofdimension M.
The starting with wage rates is inevitable, because any wage-rate vector has acorresponding minimum production-price vector p(p j ), whereas it is exceptional for a
price vector to have a wage-rate vector that generates the price vector p as the minimum
price vector. The linear product of a line vector x and a column vector v of the same
dimension is denoted by x, v.
The main theorem used here is the minimal price theorem.
Theorem 3.1 (Minimal price theorem)
Suppose a country where techniques are linear and simple, and which has a productive
system of techniques. Then there exists a system of techniques g * that gives a minimal
7)

The effects of changing X and q are important targets of analysis but they are beyond the reach ofthis paper.

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Y. SHIOZAWA

price vector p(p j ). This means that for any normalized vector of production
coefficients, the profit inequalitywat , pholds for any technique τ and wage rate w. In particular, if the technique belongs tosystem τ *, the above condition holds with equality. If we pick up techniques that belong
to τ * and arrange the vectors in a matrix form and write it A(τ *), the price vector canbe expressed aspwA(g )11 ,where 1 is a column vector all composed of 1.
This theorem is sometimes called non-substitution theorem. A different expression of
the theorem isp(g *)p(g )for any productive g , when we write p(g ) the minimal-price vector for the system oftechniques g .
This theorem can easily be extended to the case of international trade if the positive
rate-wage vector w(w1, w2, . . . , wM) is given. When the wage-rate vector and a pricevector are given, the excess profit net of capital cost for a technique t is
at , pwC(t ) ,where C(τ ) is the country index of the technique.
Theorem 3.2 (Minimal price theorem for the world)
Suppose that all the assumptions of Section 2 are satisfied and that a positive wage-rate
vector w(w1, w2, . . . , wM) is given. Then there is a system of techniques that gives theminimal production price p(p1, p2, . . . , pN) for the world (the minimal price theorem).
This means that wage vector w and price vector p satisfy the following two propositions:(I)

For any technique, the excess profit net of capital cost is 0 or negative.

(II)

For any industry, there is at least one technique whose excess profit net of

capital cost is equal to 0.
In the following discussion, we simply say “excess profit” instead of saying “excess
profit net of capital cost”, because excess profit is always regarded as net of capital cost.
The above condition (I) can be expressed in a matrix form

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ApIw ,

(1)

if A and I are defined as follows. A is an L-line N-column matrix composed of i-th line atwhere the technique t is numbered i. The i-th line of I is a vector whose elements are 0
for all columns except h, and 1 for the h-th column when the technique t belongs to theh-th country.
Condition (II) cannot be expressed in such a simple way. However, we will later findan equivalent condition so that conditions (I) and (II) hold at the same time.
When wage-rate vector and price vector satisfy inequality (1), they correspond to thepairing of wage-rate and price vectors and are briefly called w-p vector in this paper.
Excess profit of a technique, when it is numbered i, is the i-th line of the expression
IwAp.
3.2

Competitive techniques for a wage-rate vector

For any wage-rate vector w, one can calculate the minimum production-price vector p.
Take a corresponding pairing (w, p) or a w-p vector and fix them for a while. The excess
profit net of capital cost is calculated for each technique. Then we can classify
competitive techniques and non-competitive ones.
Definition 3.1 (Competitive techniques)
Let a w-p vector (w, p) be given. A technique is called competitive when the excess profitof the technique is zero. Otherwise a technique is non-competitive. Two or more
techniques of the same industry, of the same country or of different countries, can becompetitive. For any wage vector w a set of competitive techniques CT(w) is assigned.
Definition 3.2 (Pattern of specialization)
A pattern of specialization is a subset of the set of all techniques. The set of patterns ofspecialization is the power set (X) (or the set of all subsets) of the set of techniques X.
Each wage-rate vector w induces a pattern of specialization which is no other than theset of competitive techniques CT(w).
Note that a set of competitive techniques does not necessarily contain techniques of all
countries. When a pattern of specialization contains for each country at least one
technique of that country, the pattern is called a “shared” pattern of specialization. Whena wage-rate vector w generates a shared pattern of specialization, the vector is said to be“sharing”. The same adjective is applicable for a w-p vector (w, p), when it generates ashared pattern of specialization.
Consider a situation where each country produces something without making a loss at– 149 –

Y. SHIOZAWA

a given w-p vector. “Without making loss” means here that the production of eachcountry is a sum of productions belonging to competitive techniques. Then each country
should have at least one competitive technique. At that situation the set of competitivetechniques must be shared.
Sometimes this sharing wage-rate vector is called the “equilibrium rate”, but thepresent paper does not use this term, because the concept is defined independently fromthe state of supply and demand and remains useful even out with the typical equilibrium
point.
The first problem arises whether there is a wage-rate vector that induces a sharedpattern of specialization. Fortunately, this existence is assured for any economy ofSection 2. This actually leads to the next theorem.
Theorem 3.3 (Existence of shared pattern of specialization)
Consider the situation given at the beginning of Section 2. Let X be a set of techniquesthat satisfies assumptions (1), (2) and (3) of that section. Then there exists a positive
wage-rate vector that induces a shared pattern of specialization.
(Proof of Theorem 3.3)
There are various methods to prove theorem 3.3. Let us cite three of them.
Method 1.

(Combinatorial geometry)

The theorem is equivalent to F. E. Su’s rental harmony theorem. See Su (1999). It is easy
to show these conditions of the theorem starting from the assumptions (1), (2) and (3).
Method 2.

(Modal decomposition, weak version)

With a slight revision of the proof of strong existence theorem, which will be given laterin this section, one obtains a proof of the existence of wage-rate vectors that induce theshared pattern of specialization. In fact, instead of taking sCC(P) for each polytope P of
the cell decomposition, take CT(P) and let wCC(P) be the collection of country numbersthat appear in CT(P). Since the technique of a country is at least competitive for a given
w, wCC(P) is not an empty set. Therefore, if there is no participating wage-rate vector,
one can construct a similar continuous function from D to D/G(D), which leads to acontradiction.
Method 3.

(Geometry of the production possibility set)

Suppose each country, is given a positive quantity of work force. Let q(qi) be a line
vector, where qi is the work force of country i. The proposition 5.4 attests that there– 150 –A New Construction of Ricardian Trade Theory

exists a maximal point y of the production possibility set P in any positive direction.
Therefore, theorem 5.2 provides a w-p vector (w, p) when it satisfies the conditions Ap
Iw and y, pq, w. Proposition 5.3 asserts the existence of an activity vector s which
satisfies equations ysA and qsI. The latter equation means that some technique isactive for each country and one can deduce that each country has at least one competitivetechnique. 
3.3Modal decomposition of wage rate D

Strong interest has been paid for a particular pattern of specialization. That situation isusually called complete specialization. However, this notion is sometimes ambiguous. So
in the present paper different terminology is used.
Definition 3.3 (Strongly competitive country)
Let a list of competitive techniques be given. If for an industry j any country other thancountry i has no competitive techniques of the industry j, then i is called strongly
competitive country for the industry j.
Definition 3.4 (Strongly-shared pattern of specialization)
For any wage-rate vector w, take an paring (w, p) which satisfies the inequality (1).
When the pattern of specialization induced by the vector (w, p) satisfies the following
two conditions:(1) Every country has at least one competitive technique that belongs to thatcountry.
(2) Any competitive technique of the same industry belongs to the same country,
the wage-rate vector w (or the paring (w, p)) is said to induce strongly-shared pattern ofspecialization.
Note that in the definition of the concept strongly-shared pattern of specialization,
nothing is stipulated if there is another competitive technique of country i and of industry
j. Country i may or may not have two or more competitive techniques in industry j. The
next theorem gives a significantly stronger result than theorem 3.3, but does not hold forall situations.
Theorem 3.4 (Existence of strongly-shared patterns of specialization)
Let the economy be that of section 2. Let the number of commodities N be at least equal
to the number of countries M. Techniques are assumed to satisfy conditions (1), (2) and(3) of that section. Suppose, in addition, that for any non-negative wage-rate vector there
is at least one industry for which one country is strongly competitive. Then, there exists– 151 –

Y. SHIOZAWA

an M-dimensional convex cone, in the interior of which the wage-rate vector induces astrongly-shared pattern of specialization8).
Theorem 3.4 is proved with the aid of modal decomposition. Modal decomposition is
also a necessary tool for the understanding of the dual structure which will be explained
in section 5.
Let us first explain the notion of modal decomposition of wage-rate vectors. This canbe defined both on the standard simplex D of wage-rate space or on a non-negative
octant. The definition on the simplex is more visible and easier to imagine9). The
standard simplex D of dimension M1 is defined by the formulaD{(w1, . . . , wM) | w1· · · wM1, w1 0, . . . , wM 0} .
For any vector w on the simplex D, the set of competitive techniques CT(w) isdefined. This is a map from D to the set of all possible patterns of specialization (X).
This map can be in a sense invertible.
Take any pattern of specialization Ptn. For any pattern of specialization Ptn, a subset
R(Ptn) of the simplex D is defined by the following formula:
R(Ptn){w | Ptn  CT(w)} .
This is equivalent to say that R(Ptn) is the set of vectors w(w1, . . . , wM) which satisfy
the conditions:ApIw andA(Ptn)pI(Ptn)w .

(2)Here, A(Ptn) and I(Ptn) are the extracted matrices of A and I for lines belonging to Ptn.
By consequence, R(Ptn) is a convex polytope as the set of solutions of a system of
linear inequalities. Note that inequalities in (2) are all accompanied with equalities. Thisimplies that R(Ptn) is closed. The closed convex polytope R(Ptn) is called modal cellcorresponding to the pattern of specialization Ptn. It may be the empty set, when thesystem of inequalities (2) has no solutions.
8)

In the case where no trade is admitted for intermediates, Jones (1961) presented a necessary andsufficient condition for the existence of strongly-shared pattern of specialization, although in thatpublication he only argued about necessity and no proof of the sufficiency was given. As Higashida(2005) found it, when intermediate goods are admitted in the trade, there exist cases where there are
two or more strongly-shared patterns of specialization.
9)
Modal decomposition can be defined one the set of non-negative price vectors. But that modal
decomposition is much more skewed one. It is only defined on some part of the price simplex. See
Corollary 5.8.

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The collection D of all modal cells which correspond to a pattern of specialization has
the following properties:
1.

All elements of D are convex polytopes.

2.

An empty set is an element of D.

3.

When a polytope P is an element of D, any face of the polytope P is also anelement of D.
4.

If two polytopes P and Q are elements of D, the common set of P and Q is also anelement of D.
5.

Every point of D belongs to an element of D.

A collection D of subsets of a set S, which satisfies five conditions above (when D is
replaced by S for condition 5), is called cell decomposition of the set S. Thus, thecollection D of all modal cells is a cell decomposition of the simplex D. This celldecomposition is called modal decomposition of the wage rate D.
A similar decomposition is possible for a non-negative octant (of high dimension) of
wage-rate space, if one replaces polytopes by convex cones when it is necessary todesignate that the modal decomposition is related to technology.
If a modal cell P corresponds to a pattern of specialization Ptn, then any point of themodal cell induce a pattern of specialization which is a superset of Ptn. This does not
imply that the points of a modal cell have the same pattern of specialization. However, if
the cells are restricted to their relative interior, the correspondence is much more simple.
The relative interior of a polytope P is the set of interior points of P as points of thesmallest affine space that includes P10). The relative interior of the set composed of only
a point is the point itself, or more exactly the set itself. As for modal cells, the relativeinterior of the modal cell R(Ptn) can be expressed as the set of solutions of the following
system of inequalities:
A(X/Ptn)pI(X/Ptn)w andA(Ptn)pI(Ptn)w ,where X/Ptn is the complementary set of Ptn in X. A(S) and I(S) are as before theextracted matrices of A and I for lines belonging to the set S. Any two points in therelative interior of a modal cell have the same pattern of specialization. There is a oneto-one (but not onto) correspondence between the modal decomposition and the set of all
patterns of specialization.
The relative interior of a polytope P is denoted by rel.int(P). With this notation,
10)

An affine space is a subset of a vector space that is obtained by displacing a subspace by a paralleltransportation.

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Y. SHIOZAWA

Fig. 2.

Modal Decomposition of Wage Delta.

relations between patterns of specialization and modal cells take a simple expression. In
fact, for any possible pattern of specialization, one gets the equality
CT 1(Ptn)rel.int(R(Ptn)) .
If one starts from a modal cell P, the same relations can be expressed as follows:Ptn CT(rel.int(P))

and

R(Ptn)P .

When the number of countries is 3, D is a triangle in a plane and can be expressed on asheet of paper. One example of such modal decompositions is given in Figure 2.
The proof of theorem 3.4 requires a lengthy preparation. Before attacking this task, let
us examine how general is the supposition additionally imposed in the theorem. Notethat all w vectors, which are proportional with each other, induce the same set ofcompetitive techniques. It is then sufficient to consider all things on a normalized vector.
Suppose that there exists a point w that does not satisfy the assumed condition. Take
the minimal price vector pp(w) associated to the wage-rate vector w. Then there arefor every industry j at least two competitive techniques belonging to different countries.
For each of these techniques t 1 and t 2, equationswC(t k)at k, p(w)
hold for k1 and 2. At any point w, p(w) is expressed by a system of techniques g insuch a way that– 154 –A New Construction of Ricardian Trade Theoryp(w)A(g )1I(g )w .
Note that the number of combinations that give a system of techniques is finite. If j 1, j 2
are two country numbers of the techniques, this means
w j1w j2at 1at 2, C(g )w ,where C(g )A(g )1 I(g ) is a (M, N) matrix of coefficients. The vector w which satisfies
this equation, lies, in general, in a hyperplane of co-dimension 1 in the standard simplex
D. The N of such hyperplanes do not meet, in general, at a point in common whenNM, for D is of dimension M1. These are requirements that cannot hold, in general,
for an arbitrary given set of techniques of an economy. The expression “in general”
means that by changing one or more coefficient of techniques the proposition is satisfied.
Changing the combination g does not change the situation in terms of generality. One
can always choose points that satisfy a finite number of requirements in theneighborhood of the present coefficients11).
In subsection 3.5 we will estimate explicitly the probability for an economy to satisfy
the existence theorem.
3.4

Existence of strongly-shared patterns of specialization(Proof of theorem)
In this subsection a proof of theorem 3.4 is presented. It proceeds in six steps.
Step 1The set of techniques satisfies conditions (1), (2) and (3) of section 2. Number ofcommodities N is supposed to be equal to or greater than number of countries M.
Suppose, in addition, that for any wage-rate vector there is at least one industry forwhich a country is strongly competitive. For any element P of the modal decompositionD, choose a point w in the relative interior of the polytope P and take the set ofcompetitive techniques. As we have seen above, this set of competitive techniques is
uniquely determined. Then the set of strongly competitive countries for some industry is
uniquely defined and denoted by sCC(P). From the assumption of the theorem, sCC(P)is non-empty.

11)

The expression “in general” is usually ambiguous and often therefore criticized. In the present
argument, generality means that exceptional cases lie only in a set of measure 0 if one introduces anappropriate measure in the space of coefficients. This interpretation is affirmed by the result ofsubsection 3.5.

– 155 –

Y. SHIOZAWA

Step 2For any convex polytope P, the barycenter of P is denoted as G(P). When a list of
country-valued function sCC(P) is given, a continuous function f from D to D it isdefined by the following procedure.
For any barycenter of an element P of modal decomposition D, f B (G(P)) is defined by
the formulafB (G( P)) 1∑ e( j ) .

# ( sCC( P)) j ∈sCC ( P )Here #(S) for a set S stands for the number of elements of S. The vector e( j) is a vertexof D, which is expressed as (d (i, j)) i1, 2, . . . , M for Kronecker’s delta d (i, j)12). The
devisor #(sCC(P)) is necessary only to keep fB(G(P)) on D. The function fB is welldefined over the set of all barycenters. The following steps extend this functionfB

inductively:
1.

Take elements of D of dimension 0 or 0-faces. These are polytopes whose base set

consists of a single point. On the base set of 0-faces, the function f is defined byf(X) fB(X) .
2.

Suppose that function f is defined on the base set of all faces of D whose

dimension is less than d. Take a polytope P among the d-dimensional elements of D. If Q
is a facet of P, the barycenter G(P) is outside Q. Consider a conic polytope R with vertex
G(P) and with base Q. Any point X of R is on a line segment that connects G(P) and apoint Y of facet Q. When X is expressed as XaG(P)(1a)Y, then function f is definedon X byf(X)af(G(P))(1a) f (Y)As function f is defined at G(P) and at any point Y of Q, this is well defined.
3.

Do the same extensions of function f for any facet Q of polytope P. When two

facets Q and Q have a common set P function f is already defined on P and two
definitions coincide on the points of the line that connects G(P) and a point in P. Thus
the definition can be extended on all points of polytope P of dimension d.
4.

A function f is defined over all points of D when the definitions arrives atdimension M1. The obtained function is peace-wise linear and continuous.

12)

Kronecker’s d is defined as d (i, j)1 for ij and d (i, j)0 otherwise.

– 156 –A New Construction of Ricardian Trade Theory

Step 3
The theorem can be demonstrated by contradiction. Suppose there are no elements ofmodal decomposition D that induce strongly-shared pattern of specialization. This means
that, for any polytope P of D, sCC(P) cannot be equal to the set of all country numbers.
Then, the above constructed function f : D→D never takes the value G(D).
In fact, if there is no element of D that induces strongly-shared pattern ofspecialization, there is a country m that is not a member of sCC(P) for any element of D.
Then, for any polytope P of D, f (P) is included in a facet of D. More explicitly, if m isthe number not included in sCC(P), f (P)  D(m) where D(m) is the facet that does notinclude e(m). This proposition can be proved by induction on dimension d for theelements of D.
For d0, the proposition means that f (G(P)) is included in a facet of D(m). This isverified easily by examining the inductive definition of the step 2. Suppose then that the
proposition is verified for all elements of D whose dimension is less than d. For anelement P of D of dimension d let m be a number that is not a member of sCC(P). Then,the base set of P is included in the facet D(m). G(P) and any facet Q of P, are subsets of
P and in this title sCC(G(P)) and sCC(Q) do not contain m as their member. From thisassumption of the induction f (G(P)) and f (Q) are included in D(m). The inductive
definition (2 of the step 2) suggests that f (P) is itself a subset of D(m). So the propositionis asserted and f (D) does not contain G(D).
Step 4
From step 3, if there is no element of D that induces strongly-shared pattern ofspecialization, then one gets a continuous function f : D→D that does not take the valueat G(D). The function f | ∂D, which is the restricted function of f on the boundary ∂D,is homotopic to the identity function from SM2 to SM2. However, the existence of acontinuous function f : D→D/G(D) means that the identity function over SM2 ishomotopic to a constant function. This contradiction follows because we have assumed
that there is no element that induces strongly-shared pattern of specialization. Therefore,
it is assured that there exists at least one element that induces strongly-shared pattern ofspecialization.
Step 5
Let a polytope P of the modal decomposition D induce a strongly-shared pattern ofspecialization. At a point w of P, the wage-rate vector w generates a minimal pricevector p. Given a w-p vector (w, p), any country j has at least one technique, t , that is– 157 –

Y. SHIOZAWA

competitive and no other country has a competitive technique of the same industry. Thisis an open condition in the sense that if w moves in a neighborhood of the original w, theconditions remain valid. So if P induces a strongly-shared pattern of specialization, P
must include the open subset. This means on D that P is of dimension M1.
Step 6
All propositions on D and its convex subset can be translated into propositions on the
non-negative octant and convex cones with vertex 0. The result obtained in step 5 assertsthat there exists an M-dimensional convex cone, in the interior of which the wage-rate
vector induces a strongly-shared pattern of specialization. This completes the proof of
the theorem. 
In the case of an economy that does not admit trade of intermediate goods, there is
only one polytope that induces a strongly-shared pattern of specialization and such apattern of specialization is unique. In the general case, however, where intermediate
goods are traded, an economy might have two or more polytopes that induce stronglyshared pattern of specialization and the patterns of strongly-shared pattern ofspecialization may not be unique.
3.5

Numerical experiments on the occurrence of non-existent cases

Theorem 3.3 holds for all cases given in section 2, whereas theorem 3.4 holds only for an
economy that satisfies the additional condition. As examined above, this condition must
hold in general. This means that if coefficients are randomly chosen from a range of
certain intervals the probability of the cases in which the condition is not satisfied shouldbe zero. With the aid of computers, we can estimate this probability. The only hurdle tosurpass is to code the necessary program.
Table 1 shows some results of numerical experiments. The result given on the thirdline confirms the prediction that the non-existent case probability is zero. In cases ofintegral coefficients the probability may not be zero (the first line). However, if themoving range of coefficients increases, the probability to find a non-existent casebecomes more difficult and approaches the real coefficient setting.
4.

Gains from Trade and Origins of Trade Conflicts

What type of gains do trading countries obtain? What type of sufferings should they
endure, if any harm to trade could exist at all? These are the subjects with which trade
theories have been preoccupied since the very beginning of economics. In most cases,
theorists assured the gains from trade but were very reluctant to admit any harm to trade.

– 158 –A New Construction of Ricardian Trade Theory

Table 1. A result of numerical experiments

Dimensions& fields

Range ofdiagonal
elements

Range ofnon-diagonal
elements

Number ofcases
examined

Non-existent
cases

Probability

3 Integer
3 Integer
3 Real4 Integer[1, 9][1, 18][1, 9][1, 9][0, 5][0, 10][0, 5][0, 5]

100,000
100,000
100,000
100,000

487
0
0*
77

0.0049000.0008(Experiments by Mathematica)* In the course of this examination, it is necessary to check if the inverse of a matrix is nonnegative. In the real coefficients case, the computation is approximate. The matrices that containnegative components whose absolute value is less than 106 were classified as non-negative.

However, in the real world, trade often raised conflicts between countries. It is not rarethat one hears claims that industries of one country are suffering due to the rapid increasein exports from other countries. Some people express these situations as an export ofunemployment.
The majority of economists customarily rebuff these claims by professing that there is
no real trade conflict. In total, trade is beneficial to both countries and complaints are
based on misunderstandings, they suggest. They have no malicious or politicalintentions. They believe what the theory tells to them. They only adopt a standard tool of
economic analysis and apply them to international trade and conclude that there are gains
from trade for both trading countries.
The standard framework of economic analysis is equilibrium analysis and economists
do not regard that the very adoption of this method has a strong influence on the
problem. In equilibrium, everything goes well. No harm occurs. However, this is only aconsequence of the equilibrium framework13). In order to analyze trade conflict it isnecessary to analyze what happens outside the equilibrium. The method of the present
article enables this analysis and thus enlarges the scope of trade theories beyond that ofneoclassical tradition, which is strongly dependent on equilibrium analysis.
In the analysis of gains and losses, it is necessary to look at the situation from thestandpoint of each interest group. In subsection 4.1 the gains for the country as a wholeare examined. In subsection 4.2 the same situation will be examined from the viewpoint
13)

Even a very strong analyst such as Krugman (1996) has some tendency to think and reason in thisway.

– 159 –

Y. SHIOZAWA

of each interest group.
4.1

Gains from trade for the nation

To see what kind of gains we get from trade, suppose for each country an autarky E( j):country j is producing net product y( j), using labor power q( j). If s( j) is the activity
vector of country j theny( j)s( j)A( j) andq( j)s( j)I( j)where A( j) is the net production coefficient matrix for techniques of country j and I( j) is
a column vector of L( j) lines all composed of 1.
Suppose now trade begins. The next theorem follows.
Theorem 4.1 (Trade with constant demand)
Suppose an economy of M-country N-goods with a choice of techniques and in whichintermediates goods are used in production and are tradable internationally. Supposethat a wage-rate vector w(wi) and its associated price vector p induce a shared patternof specialization, then there exists a worldwide production that satisfies the following 5
conditions:1.

The worldwide net production is equal to the sum of the net productions of eachcountry’s autarky.
2.

Each country consumes the same amount of commodities as the autarky.

3.

All goods that are traded internationally are produced by competitive techniqueswith regard to w-p vectors (w, p).
4.

In terms of total values, each country exports as much as it imports.

5.

The total labor time of each country, is less than or equal to the labor time of eachcountry’s autarky case. If activities in one country, include an activity that is notcompetitive with regard to vectors (w, p), then the total sum of labor time of the world isless than the sum of labor time of each country’s autarky.
Note that in the new state, internationally traded commodities are exchanged at thesame price vector p. Prices of other commodities are not specified. They may be tradedwith prices between old autarky prices and international prices. So the situation in thetheorem assumes a double structure: international prices for international trade anddomestic prices for domestic trade and productions.
In the proof of this theorem, we use the next lemma.
Lemma 4.2
Let B be an LN matrix with any line vector bh that has only one positive element and– 160 –A New Construction of Ricardian Trade Theory

all other elements are either zero or negative. Suppose in addition each column has atleast one positive element. If there exists a positive column vector p which satisfies
Bp0or dh, p0 for all line vectors dh of D, then there is a non-zero non-negative vectoru(uh) such that∑ u b 0 .
hhh(Proof of lemma 4.2)
When B is a square matrix, this is a known lemma for a simple production type matrix.
Let A be the matrix of all off-diagonal components of B with signs inverted, in which thenotion of decomposability and indecomposability of A is well known. We say B isdecomposable or indecomposable if A is decomposable or indecomposable, respectively.
When the matrix B is indecomposable, the lemma holds with positive vector u. When B
is decomposable, there is a positive u for the fundamental indecomposable part, and u0
for the remaining part. The proof of this part is omitted for the brevity of the paper. If
matrix B is not square, take a square submatrix of B with the same properties. The
lemma gives a non-negative non-zero vector u and the augmented vector s with zeros foromitted lines satisfies the lemma for the original matrix. (Proof of theorem 4.1)
In this proof, i, j and h denote country number, commodity number and technique
number, respectively. Let y(i)∑h s (i)hah be the country i’s production vectors of theautarky. We adopt the same notations A and I as above14). Let vectors (w, p) induce ashared pattern of specializations. Then an inequality Ap Iw holds. Each technique that
holds with equality is competitive and each technique that holds with strict inequality is
non-competitive.
To prove this theorem, it is necessary and sufficient to show the existence of
production vectors z(i)∑h t(i)hah that satisfy the following conditions:(1) t(i)h 0 for all i.
(2) ∑i z(i)∑i y(i).
(3)

If t(i)h s (i)h, then h is competitive.

(4) z(i)y(i), p0 for all i.
14)

See subsection 3.1, equation (1).

– 161 –

Y. SHIOZAWA(5) ∑h t(i)h  ∑h s (i)h.
(6)

For some country i (5) holds with strict inequality.

If we use expressions s(i)(s(i)h) and the likes, vectors y(i) and z(i) are expressed ass(i)A and t(i)A. Let q(qi) where qi is the maximum labor time available in country i.
Without loss of generality, we suppose∑ s(i )  q .
hihThe production possibility set of the country i is denoted by P(i, qi). This means that∑ t ( i ) a ∈ P ( i, q ) ⇔ ∑ t ( i )  qz(i ) 

h

h

i

h

i

h

with the assumption that t(i)h0 for all h that do not belong to i. Vectors z(i) will befound in the following procedure.
Vectors w and p satisfy the inequality Ap Iw for all lines. Equality holds for linesthat are related to competitive techniques and strict inequality for lines that are related to
non-competitive techniques. Extracting each country’s part of matrix A and I, one gets aninequality
A(i)pwiI(i) .
Here I(i) is a column vector with all entries 1. Set
A(i)pwiJ(i) ,then elements of J(i) are equal to 1 for competitive techniques and smaller than 1 fornon-competitive techniques.
Consider a set E(i) of production activities s(i) belonging to country i which satisfys( j ) J (i ) ∑ s(i ) J (i )  q .
h h

C ( h )i

Then for any activities r(i) in E(i), the net products∑ r ( i )a ( i )x (i ) hhhsatisfy the equality
x(i), py(i), p ,– 162 –i

A New Construction of Ricardian Trade Theory

resulting in two chains of equalities:∑ r a (i), p  w r(i) J  w q ,x(i ), p 

h

i

i

h

i

h

∑ s a (i), p  w s(h) J  w q .

y(i ), p 

h

i

i

h

i

h

Let S{S1, S2, . . . , SM} be a collection of sets whose components are numbers ofcompetitive techniques of the country i and for any element h of set Si letb(h)qi ahy(i) .
Note that y(i) is the net production vector in autarky, and so y(i) is non-negative. The
vector ah has only one positive component and all other components are zeros ornegative. Therefore, b(h) is a simple production type. Moreover,b(h), pqiah, py(i), pqiwi qiwi0 .
Then, the set of vectors b(h) forms a matrix B that satisfies the conditions of lemma 4.2
and there exists a line vector u(uh) such that uB0.
Take a maximal real number h which satisfies for all i the conditionsy( i )  η∑ u b(h) ∈ P(i, q ) .
h

i

(3)

Here h in the summation runs for all competitive h with C(h)i. Let
x (i )  η∑ u b(h) .
hwith the same summation range. Then, the production switches from y(i) toz(i)x(i)y(i) satisfy the conditions of theorem 4.1, when the consumption of country i
remains y(i) and the difference x(i) of net productions and consumptions are tradedinternationally.
Then in fact,(1) ∑i y(i)∑i z(i),(2) y(i)x(i)z(i) for all i.
(3)positive components of x(i) all belong to competitive techniques.

(4) x(i), p0 for all i.
(5)

By definition z(i)y(i)x(i) is included in P(i, qi) and inclusion (3) means that

total labor input of z(i) is equal to or smaller than qi. Moreover, if positive uh exists for– 163 –

Y. SHIOZAWA

Fig. 3.

The same world production with reduced labor input.

country i, inclusion (3) and the relation J(h)I(h) for non-competitive h implies the total
input of labor is smaller than qi. See Figure 3 for clarification. 
Theorem 4.1 asserts that countries can generally obtain gains from trade: the same
consumption with less labor. However, the interests are different with different socialgroups. We will examine this in the next subsection.
4.2

Views from different interest groups

Suppose countries start to trade after autarky. The situation assumed in theorem 4.1 iseasily attainable, if sharing wage rates are found. People then transfer activities from
non-competitive productions to competitive productions since demand levels are alreadyknown. Thus, they can easily find the necessary scales of production according toexisting demand.
The gains from trade are visible for workers who continue to be employed. They getwages at the same rate as before and in a short time prices begin to approachinternational prices. If international prices prevail in a domestic economy, it is evident
that
1
w

i

p

1
wi

p(i )

where p is the international price vector and p(i) is the price of country i when it runs in
autarky. In terms of commodities, real wage increases with increased trade if the strict
inequality phph(i) holds. Even if the prices do not move entirely to p, employed– 164 –A New Construction of Ricardian Trade Theory

workers will get some portion of the benefit.
One problem is that the new situation requires a smaller amount of labor. If eachworker continues to work the same hours per week as before, then some of the workers
should lose work. Thus, in this model, unemployment occurs. People who lose theirwork cannot enjoy cheaper prices, if it is a consequence of trade. By the start of trade, it
is possible that the first effects of trade are the emergence of unemployment. This is oneof the losses due to trade.
For the industrialists also, the effects are different. If they work in a competitive
industry with competitive techniques, they have a chance to expand activities and toacquire more profits. If they work in a non-competitive industry, using non-competitivetechniques, they are obliged to decrease levels of their activity in this industry. Price are
reduced and salable volume decreases and they loose some of their profits.
Of course, as shall be described in the following section, a full employment state does
exist. In the long run, the economy will probably succeed to approach to such a situationand everybody will be happier in this state than in the case of autarky. The question is
how rapid economy can get to this happy situation. If it is a long process that requires
trial and error, people suffer from trade at least for a while.
5.

Production Possibility Set

It is useful to know the shape of the production possibility set. Analysis of the
production possibility set is in one sense simpler than that of wages and prices. The
assumptions imposed on investigated economies are the same as before: there are M
countries, N commodities, and L techniques of simple production type belonging to a
country. Moreover, the following two conditions are assumed: (1) All the production
techniques are simple and each country has for any commodity one technique that
produces the commodity. (2) Each country has a set of techniques that are productive.
Assume each country has a positive amount of labor-power qi. Labor is a unique
primary factor and all other goods can be produced by a combination of a certain
quantity of labor and a set of appropriate amount of various goods. Outputs are tradedand consumed either as consumer consumption or as input to a production in the next
period. In an economy with the set of techniques X{X(i)}, a net production is possible
when it is a non-negative combination of techniques ah, i.e ∑h shah for non-negative si
and labor input for each country is equal to or less than qi. The last condition can be
written as– 165 –

Y. SHIOZAWA∑ s  q for any i .
hih∈Ξ ( i )Let an LN matrix A be the collection of all techniques ah and a LM matrix I, thecollection of vectors of the unit type, meaning that the entries at (h, i) are 1 if they
represent techniques of number h of the country i and 0 for all other entries. Then, apossible net production y and the labor input y0 of the production are given by theformulay sA andy0 sIfor some non-negative L-vector s. In the following discussion production is considered
only as those possible productions and the adjective possible is omitted.
The production possibility set of the economy is denoted P(q). It can be expressed
P(q){yRN | y sA, sIq, s0, sRL}
where q is labor-power vector composed of each country’s labor-power qi. As the set ofpoints satisfying a system of linear inequalities, the production possibility set is a convex
polytope.
5.1

Basic characteristics of production possibility set

The next theorem gives a necessary and sufficient condition that a point of RN is anelement of the production possibility set.
Theorem 5.1 (Production possibility set)
For a point y of RN to be a point of the production possibility set P(q), it is necessary andsafficient that any pairing of column vectors w and p of dimension M and N which
satisfies:
ApIw andw0satisfies the inequality
y, pq, w .
(Proof of 5.1)
If y is a point of P(q), there exists by definition a non-negative activity vector s such thaty sA andsIq .

The last conditions are equivalent to the existence of non-negative t such that– 166 –A New Construction of Ricardian Trade Theoryy sA ands Itq .

Take a (LM)(NM) matrix A I , O E

where O is an MN matrix of elements zeros, and E is an identity matrix of dimension
M. Then, the existence of the above s and t is the existence of a solution A I (s, t )   ( y,q) O E

Minkowsky-Farkas theorem implies that the existence of non-negative solutions (s, t) for
the above system of equations is equivalent to say that any w, p vectors that satisfy thefollowing inequality A I   p 0 O E w (4)satisfies the inequalityp( y,q),    y, p  q, w  0w A maximal point of a production possibility set is a point of the set that has no otherpoints in the production possibility set which is greater by the partial order . A
maximal point is sometimes called efficient production point. In this paper we use
expression “maximal point”. The set of maximal points W is called the maximal frontierof production possibility set or simply the production frontier.
Theorem 5.2 (Maximal point of the production possibility set)
Let y be a maximal point of the production possibility set P(q). Then, there exist a
positive M-column vector w and a positive N-column vector p such that
ApIw andy, pq, w .

(5)

Conversely, if a couple of positive vectors w, p satisfy the above conditions, then a– 167 –

Y. SHIOZAWA

vector y of the P(q) is a maximal point of P(q)15).
Before giving the proof, it is convenient to know a necessary condition for a point be
the maximal point.
Proposition 5.3 (A necessary condition for the maximal point)
If a point y of the production possibility set P(q) is maximal, an activity vector s, which
satisfies the equation ysA also satisfies equation qsI.
(Proof of proposition 5.3)
The proposition can be proved by contradiction. Suppose for a country m, (sI)m qm. As
country m’s system of techniques g is productive, there is a non-negative activity level
vector such that t A(g )z, where z is a positive vector of R N and t Iqm (sI)m. Then, if
we set x (st)A, then (st)Iq. Then, x is a point of P(q), which is strictly superior
to y with respect to order . This is a contradiction. (Proof of theorem 5.2)
Let y be a maximal point of the set P(q). By the definition of maximal point, theinequality A I (s, r ) 
  0 and ≠ 0
y q has no non-negative solution (s, r), In order to prove this, suppose first that r 0. Without
loss of generality, we can suppose that r1. Then,
sAy

and

qsI

sA y

or sI qandholds. This means that production possibility point sA is superior to y. In view of
proposition (5.3), this is a contradiction to the fact that y is a maximal point of P(q).
Suppose now r0. Then from the second column s0. It follows that s0, for s
is non-negative. It is impossible then that sA 0. Therefore, it is impossible for any
non-negative (s, r) to satisfy the inequalities.
Then, by Stiemke-Tucker’s theorem16), a positive solution (p, w) to the inequalities
exists15)
16)

An equivalent theorem has been presented by Takamasu, 1992, Chap.1 Theorem 2.
To be precise, we use a variation given by Nikaido (1961, p 157) lemma 1.
– 168 –A New Construction of Ricardian Trade Theory A I   p 
y q  w   0 exists, where p is an N-column vector and w is an M-column vector. The last statement isequivalent to the conditions
ApIw0andq, wy, p.

On the other hand, as vector y is a point of the production possibility set P(q), fromtheorem 5.1,
y, pq, w.
Combining both inequalities, we obtain
y, pq, w.
Conversely, suppose that positive vectors p and w exist that satisfy those conditions(5). Take any vector z satisfying yz and y z, thenq, wy, pz, p,since p is positive. From theorem 5.1, vector z cannot be an element of P(q). This meansthat the point y that satisfies (5) is a maximal point of P(q). 
Another simple proposition is useful to clarify how maximal points are located.
Proposition 5.4 (Maximal point of a non-negative direction)
Suppose that the labor-power vector q is positive. For any non-negative N-line vector x
there exists a maximal point y of P(q) that is proportional to x.
Another useful proposition is on the uniqueness of p which appears in theorem 5.2.
Proposition 5.5 (Uniqueness of price vector)
Let y be the maximal point of the production possibility set P(q). If, for a pairing ofvectors w and p with w 0, conditions (5) hold, then the vector p is unique for each w.
Proofs of these propositions are simple exercises and are omitted from this paper.
5.2

Production possibility set as the Minkowski sum

In the analysis of the production possibility set P(q), the concept of the Minkowski sum
is useful. First this notion should be defined and how the production frontier is shaped
will be explained.

– 169 –

Y. SHIOZAWA

Definition 5.1 (Minkowski sum)
Let S1, S2 , . . . , SK be sets of a vector space V. The Minkowski sum of these sets is the set
given by the formula
x(i ), x(i ) ∈ Si  .
x ∈ V x 


i

∑

This set is denoted S1S2···SK or ∑i Si.
It is easy to see that the production possibility set P(q) is the Minkowski sum of
production possibility sets P(i, qi) of each country i. It is also easy to see that the
maximal point of P(q) is the sum of the maximal points of P(i, qi). It is important to notethat the sum of the maximal points is not necessarily maximal, even if each one of themis selected from each maximal frontier of P(i, qi).
As we have assumed that each country has a productive system of techniques, themaximal frontier of P(i, qi) is rather simple. P(i, qi) is a closed set that has a non-empty
interior. Thus P(i, qi) is an N-dimensional polytope. The maximal frontier of P(i, qi) is asingle facet of P(i, qi) that has a positive normal vector p(i). This is another expression of
the minimal price theorem, i.e. theorem 3.1. Vector p(i) is proportional to the minimal
price vector of country i. If there is only one technique for each industry belonging tocountry i and if qi1, then the maximal frontier is a (N1)-simplex spanned by Nvectors a1, a2, . . . , aN, if we denote the techniques of industry j by normalized a j. When
there is a choice of techniques, the shape of the maximal facet may have more than N
vertices.
More detailed information can be obtained from theorem 5.2. If a point y of P(q) is
maximal, there exists a positive w and p which satisfy these conditions
ApIw andy, pq, w.

If s is an activity level vector of y, y sA and sIq. Then, if for a technique h an
inequality (Ap)h (Iw)h or ah, pwC(h) holds, sh cannot be positive, forsh(IwAp)h s, IwApsI, wsA, pq, wy, p0 .
This means that for a point y to be maximal there is a pairing of positive vectors, w and p
and y is a convex combination of techniques that are competitive with regard to w and p.
Then let P be the set of all possible shared patterns of specialization. Each element
CM of P has a positive wage-rate vector w and associated price vector p such that the– 170 –A New Construction of Ricardian Trade Theory

set of competitive techniques with regard to w, p is CM. A production set P(CM, q) witha competitive mode CM is defined bysh  qi  .
x x  sA, s  0, sh 0 for h ∉ CM andh∈CM and C ( h )i



∑

The production set P(CM, q) is a face of the production possibility set P(q). In fact, if
vectors w, p induce competitive mode CM, any point y of P(CM, q) satisfies y, pq,w and any element x of P(q) satisfies x, pq, w0. The face P(CM, q) is called
modal cell with competitive mode CM. Thus, the maximal frontier W of a production
possibility set P(q) can be decomposed into a collection of modal cells that have a
different pattern of specialization.
Modal decomposition of the production frontier W is a true cell decomposition thatsatisfies the definition given in section 3.3. In fact, each modal cell is a polytope. Any
two modal cells, if they have common point, intersect at a polytope that is another modal
cell. Lastly, any point of the production frontier W belongs to a modal cell.
In a particular case where countries have no choice of techniques, each modal cell is a
Minkowski sum of simplexes. For example, in the case of three commodities modal cells
are either triangles or parallelograms.
5.3

Dual correspondence between two modal decompositions

As shown in section 5.2 the production frontier W of a production possibility set can be
decomposed into modal cells. Then, we have two cell decompositions: modal
decomposition of the wage rate D and modal decomposition of the maximal frontier Wof the production possibility set. A natural question is how they are related. As we willsee, astonishingly beautiful relations or correspondences exist between them.
There are three levels of correspondences: point-to-point, point-to-set, and set-to-set.
In all these correspondences any two corresponding parts have the same competitivemode of techniques when the sets are restricted to their relative interiors.
Let us start from point-to-point correspondence. For any positive w, the minimal pricevector pp(w) satisfies the inequality ApIw. Then, for any positive wage-rate vector
w a net production y is said to be admissible with regard to w when y is generated only
by a competitive technique with regard to w, p. For a wage rate w there may exist many
admissible productions. So the correspondence from wage rate to admissible productions
is not a function from D to W but a correspondence between D and W. If y is anadmissible production with regard to w, there exists by definition s such that y  s A,

– 171 –

Y. SHIOZAWA

Fig. 4. Modal decomposition of a production possibility set, a projected graph.

qsI and if h is not competitive, then sh 0. Thus the equation y, pq, w holds.
Conversely, if equation y, p(w)q, w holds for any y in W and w in D, then there
exists an activity level vector such that y  s A, qsI and sh 0 for any non-competitiveh. Thus, a necessary and sufficient condition for the admissible correspondence between
y in W and w in D is that they satisfy the equation y, p(w)q, w.
Point-to-set correspondence is simpler than point-to-point correspondence. In fact, if
we set Pc(w){y P(q) | y, pq, w} , Pc(w) is the set of all admissible points of apositive wage-rate vector w. It may well happen that this Pc(w) is empty. For example, if
w does not induce a shared pattern of specialization, some countries do not produce
anything and such a production point cannot be maximal. Indeed, it has been assumed
that q 0 and proposition 5.3 asserts that labor inputs of any country i must be equal to
qi. Therefore, it is necessary for a positive wage rate w to be sharing in order that set
Pc(w) be non-empty.
The above correspondence can be shifted up to the set-to-set level. Let w(1) and w(2)be two wage rates of the same modal decomposition. They each have corresponding
minimal price vectors p(1) and p(2), and their sets of associated competitive techniques
are identical, i.e. CT(w(1), p(1))CT(w(2), p(2)). It is easy to see that Pc(w(1))
Pc(w(2)). Thus, one gets a face-to-face correspondence between elements of modal– 172 –A New Construction of Ricardian Trade Theory

decomposition of wage rate D and faces of the maximal frontier W of the production
possibility set P(q).
Let us now examine the correspondence from the opposite side. Let y be a point of the
production frontier. Then, theorem 5.2 suggests that there are positive vectors w and pwith the conditions
w 0, ApIw andy, pq, w.

(6)

Proposition 5.5 assures that p is unique vis-à-vis w. Thus, p is the minimal price vectorof w. Dual correspondence can be defined as that between the maximal point y and a
wage-rate vector w with pp(w).
Couples w, p with specific conditions (6) may not be unique for a given y. LetCT(w, p) be the set of all competitive techniques for a couple w, p. CT(w, p) varies withdifferent w, p. However, if w(1), p(1) and w(2), p(2) are two couples that satisfy these
conditions (6), there is a third pairing of vectors w(0) and p(0) that satisfy these
conditions (6) such that
CT(w(0), p(0))  CT(w(1), p(1))  CT(w(2), p(2)) .
Indeed it is sufficient to take(w(0), p(0)) a (w(1), p(1))(1  a )(w(2), p(2)) .
for a positive a and smaller than 1. This means that there is a minimal set of competitivetechniques for any couple w, p that satisfies these conditions (6). Thus, a unique set ofcompetitive techniques is defined for a maximal point. This will be named thecompetitive mode of a maximal point.
Definition 5.2 (Competitive mode of a maximal point)
The common set of all competitive technique sets with regard to w, p that satisfies
conditions (6) is called the competitive mode of y. The competitive mode of the maximal
point y is denoted by CM(y).
Once a competitive mode of a maximal point y is defined, the corresponding set ofwage-rate vectors is given by the set{wD | CT(w, p(w))CM(y)} .
This correspondence can be shifted to face-to-face correspondence between modal
decomposition of wage rate D and modal decomposition of the production frontier W. In– 173 –

Y. SHIOZAWA

this face-to-face level, the correspondence is one-to-one and when the modal
decomposition of D is restricted to the sharing wage rate vectors the correspondence
becomes one-to-one onto map.
Note that a set of competitive techniques CT(w) is uniquely defined for any positive
wage rate w. From definition 5.2, a competitive mode CM(y) is given for any maximal
point y. If F is a face of the modal decomposition of the production frontier W, any pointin the relative interior of F has the same competitive mode. This unique competitivemode is defined as the competitive mode of face F.
The same kind of convention is possible for faces of the modal decomposition of wage
rate D. Any point in the relative interior of a face G has the same set of competitivetechniques. We define this unique set as the competitive mode of face G. When thecompetitive mode of face G is shared, it can also be expressed that G is shared.
By this extension of definitions we can finally make a formal definition of dual
correspondence.
Definition 5.3 (Dual correspondence)
Let V be either a point of production frontier W, or a face of modal decomposition W.
Let H be either a point of wage rate D or a face of modal decomposition D. When V andH have the same competitive mode, one states that V and H are connected by the dual
correspondence. It can also be said that V is dual to H and H is dual to V.
To analyze this correspondence in more detail and with rigor, it is inconvenient to seeelements of decompositions directly as sets of price space and of production space. It isalways necessary to question if the corresponding w exists or not. A good method toescape from this inconvenience is to take a higher dimensional double space RMN and
RMN. This is to see a pairing of vectors q and y together on the one hand and a pairingof vectors w and p together on the other hand.
Let X be the set of all techniques of the world. We set two definitions as follows:E O
Cv(X )  (q, y) (q, y)  (t, s)  for some non - negative s and t  .
  I A
E O w Ch(X )  ( w, p)     0 .

I
A
p


 



The set Cv(X) is a -cone as a non-negative combination of line vectors of– 174 –A New Construction of Ricardian Trade Theory

E O .
  I AThe set Ch(X) is an -cone as a set of common points of closed half spaces defined by
the column vector of . The main theorem of the polyhedral cone asserts that the twoconcepts, -cone and -cone, are the same. In the sequel, only polyhedral cones are
considered.
Two cones Cv(X) and Ch(X) belongs to spaces of the same dimension, but they areincluded in different spaces that are dual to each other. One is a set of line vectors andthe other is a set of column vectors, although we often write them (w, p) instead ofwriting them in a column form. Points of two spaces are connected by a bilinear form
(q, y), (w, p)y, pq, w .
Writing down matrix multiplication, we see that a point (q, y) of Cv(X) has a nonnegative activity level vector s that satisfies the conditions y sA and sIq. A point (w,p) of Ch(X) satisfies the inequalities
w0

and

ApIw .

It is easy to demonstrate the following two propositions, using the Minkowski-Farkas
lemma in the case of (2):(1)

If a point (q, y) out of origin 0 is in the boundary of Cv(X), then there exists a

point (w(0), p(0)) on the boundary of Ch(X) such that a supporting hyperplane Hw isgiven byy, p(0)q, w(0)0 .
In addition, as Hw passes (q, y), equation y, p(0)q, w(0) holds. If a vector (w(1),p(1)) defines another supporting hyperplane and passes (q, y), then (w(0), p(0))
belongs to the same face of Cv(X) that includes (w(0), p(0)).
(2)

If a point (w, p) out of origin 0 is on the boundary of Ch(X), then there exists a

point (q(0), y(0)) on the boundary of Cv(X) such that a supporting hyperplane Hq isgiven byy(0), pq(0), w0 .
In addition, as Hq passes (w, p), equation y(0), pq(0), w holds. If a vector (q(1),y(1)) defines another supporting hyperplane and passes (w, p), then (q(1), y(1)) belongs– 175 –

Y. SHIOZAWA

to the same face of Ch(X) which includes (q(0), y(0)).
The above correspondence is called polar relations in Ziegler (1995), but in thepresent paper we use the more familiar term “dual relations”, for there is no fear ofconfusion with duality of oriented matroids. Dual correspondence can be shifted up to aset level correspondence between faces of Cv(X) and Ch(X). A face of a cone is acommon set of the cone and a supporting half space. By definition, any face includes theorigin 0 as its element. A face of a cone is itself a cone. The dimension of a cone C isdefined as the maximal number of independent vectors included in cone C and is denotedby dim(C). The relative interior of a cone is the set of vectors that is expressed as the
positive combination of the maximal number of linearly independent vectors of the cone.
Note that the relative interior of a half line as a cone is the half line itself.
From this definition, we obtain the next proposition. A faces Fv of Cv(X) and Fh of
Ch(X) correspond to each other if and only if there are a relative interior point (q, y) of
Fv and a relative interior point (w, p) of Fh such that y, pq, w. When Fv and Fh arein this situation, we also say that face Fv is polar to Fh and vice versa.
Let Fv be polar to Fh, then any vectors selected from Fv are perpendicular to vectors
selected from Fh. Conversely, if Fh is polar to Fv, then any vectors selected from Fh areperpendicular to vectors selected from Fv. In this sense Fv is normal to face Fh andconversely Fh is normal to Fv. Moreover, Fv and Fh generate full space in the sense that
there is no point x out of Fv such that x, u0 for all points u of Fh, and vice versa.
From this, an important formula is obtained for faces that are connected by a dual
correspondence.
Theorem 5.6 (Dual correspondence between cone faces)
For any face Fv of Cv(X) there exists a unique face Fh of Ch(X) such that Fv and Fh areconnected by the dual correspondence. Conversely, for any face Fh of Ch(X) there exists
a unique face Fv of Cv(X) such that Fh and Fv are connected by the dual
correspondence. If Fv and Fh are connected by dual correspondence, thendim(Fv)dim(Fh)MN .
Moreover, if a point (q, y) is in Fv and a point (w, p) is in Fh, and Fv and Fh areconnected by the dual correspondence, theny, pq, w0 .
From this theorem we can easily deduce the next theorem 5.7.

– 176 –A New Construction of Ricardian Trade Theory

Theorem 5.7 (Dual correspondence between modal decompositions)
For any shared face F of modal decomposition of wage rate D there exists a unique face
G of production frontier W such that F and G are connected by the dual correspondence.
For any face G of production frontier W, there exists a unique face F of modal
decomposition of D such that G and F are connected by the dual correspondence. If F
and G are connected by dual correspondence and N is the number of commodities, thendim(F)dim(G)N1 .
Moreover, if w is an element of F and y is an element of G and if F and G are connectedby the dual correspondence, theny, pq, w.
(Proof of theorem 5.7)
When Cv(X) is expressed as a set of vectors (r, x), the production possibility set P(q) is
the restriction of Cv(X) to the subspace rq. A point y of P(q) and therefore a point of
production frontier W is interpreted as a point (q, y) of Cv(X). Suppose a point (q, y)of W is given.
Take a point y in W and a face F that includes y as its relative interior. As a point ofCv(X), (q, y) is included in the relative interior of a face Fv. Then, from theorem 5.6,
there exists a face Fh of Ch(X) which is dual to Fv. Take an element (w, p) of Fh. Since(w, p) belongs to Ch(X), vectors w and p satisfy the conditions w 0, ApIw. From
theorem 5.6, equation y, pq, w holds. From proposition 5.5, p is the minimal pricevector associated with w.
Let us examine the dimensions of F and G. First, if a pair w and p lies in a face Fh,they can be expressed for an appropriately chosen system of techniques g by theequationpA(g )1 Iw .
This g can be fixed as far as the pair w and p remains in the relative interior of the face
Fh. The dimension of Fh is the maximal number of linearly independent vectors in Fh.
Any point of Fh can be expressed as (w, p(w)) where p(w)A(g (w))1 Iw. Then a set ofvectors of Fh has a form{(w(1), p(w(1)), (w(2), p(w(2)), . . . , (w(K), p(w(K))} .

– 177 –

Y. SHIOZAWA

They are linearly independent if and only if w(1), w(2), . . . , w(K) are linearly
independent. Thus dim(Fh)dim(G)1, taking in account that wi are normalized in G
by the condition w1 w2· · · wM 1.
Elements of face F have the form of (r, x). The cone face Fv is a minimal cone in
RMN that includes F. As (q, y) is a relative interior of F in a production space RN, one
can take K linearly-independent vectors d1, d2 , . . . , dK when dim(F)K. One can also
take M linearly-independent vectors e1, e2 , . . . , eM in RM. Then, we can take all points
d1, d2 , . . . , dK, e1, e2 , . . . , eM with origin at (q, y) in Fv. Thus, dim(Fv) dim(F) M.
When all these results are combined, we haveMNdim(Fv)dim(Fh)dim(F)Mdim(G)1 .
As a consequence,dim(F)dim(G)N1 .
When we start from a sharing w we obtain similar results. Thus, the theorem is proved.
The price frontier is the set of price vectors that are an image of function p( ), whichmaps a wage-rate vector w to the minimal price vector associated with w. Function p( )is linear on each face of the modal decomposition of D. Two couples of w and p
belonging to the relative interior of different faces induce different sets of competitivetechniques. Thus the dual correspondence can be defined over modal decompositions ofprice frontiers. Based on what has been observed in the process of the above proof, we
obtain the next corollary.
Corollary 5.8 (Price frontier decomposition)
A face H of the price frontier has the same dimension as the corresponding face G of themodal decomposition of wage rate D. Let H be a face of the price frontier and F a faceof the production frontier W . Suppose F and H are connected by the dual
correspondence, thendim(F)dim(H)N1.
Let x and y be two different points of F, and p and v two different points of H, thenxy, p0

andy, pv0,which implies that faces F and H are perpendicular or orthogonal in N-dimensional– 178 –A New Construction of Ricardian Trade Theory

space17).
5.4

Some simple consequences of the dual correspondence theorem

Dual correspondence theorem is powerful and informative. Let us examine some simple
cases.
Example 5.1 (3-country, 3-commodity case)This is the only case, among many-country, many-commodity cases, where two modal
decompositions can both be shown in plane graphics. Here is the correspondence table offaces.
Modal decompositions
production frontier

wage rate and prices

vertex ↔ facet
line segment ↔ line segment
facet ↔ vertex
Example 5.2 (1-country, many-commodity case)
If there is only a one-country, wage rate D is reduced to a point. Then, there is only one
non-empty face that has dimension 0. If the dimension of the commodity space is N, thendual correspondence theorem 5.7 implies that the production frontier is composed ofonly one facet that has dimension N1. The price vector dual to this facet is unique up
to scalar multiplication. This is the minimal price theorem.
We started from the minimal price theorem (Theorem 3.1). Example 2 shows thattheorem 5.7 is an extension of this minimal price theorem.
Example 5.3 (4-country, 3-commodity case)
The wages and prices of this case were analyzed by Graham and others using numericalexamples. In this case price space has dimension 3. The wage-rate vector moves in athree-dimensioned simplex. Then, from theorem 5.7 we know that the sharing set of D
forms a set of 2 dimensions.
Example 5.4 (M-country, N-commodity case with MN)
This theorem is applicable even for large M and N. If a face F of a production frontier
has dimension d the corresponding faces of the wage space and price space must havedimension N d 1. Of course this dimension must be less than or equal to M1. Then,17)

In fact, F and H lie in different dual different spaces of the same dimension and are situatedperpendicular to each other.

– 179 –

Y. SHIOZAWAd N M. When N is much bigger than M, the faces of the production frontier musthave dimensions no less than NM. This is clearly a singular cell decomposition. On
the other hand, the set of sharing prices has dimensions no greater than M1 in an Ndimensional space. Thus, the prices lie in a very “thin” set. The maximal degrees of
freedom are far smaller than the number of commodities. The possibility of priceadjustment is limited.
6.

Price Adjustment vs. Quantity Adjustment

In the following two sections two major implications of this new construction are
examined.
Since Ricardo first published his work until the present, various interpretations ofRicardian trade theory have been presented. J. S. Mill argued for the influences ofdemand (“reciprocal national demand” theory). F. D. Graham in the second quarter of the20th century pointed out cases where the changes of demand compositions do notnecessarily induce price change and emphasized the importance of the supply side
conditions. This controversy continues to the present. For example, Ikema (1993) and
Minabe (1995) exchanged opinions concerning the stability of international prices. A
long history of analysis can be written concerning this topic. However, my understandingis that the controversy on which of the demand and supply conditions is dominant ismisplaced. The true conflict lies in the role of price adjustment and quantity adjustment
in international trade situations.
Let us limit our observations on the maximal points of the production possibility set
and the wage rates and prices polar to them. My starting point is that, on the production
frontier, almost all points lie in the interior of one of the facets18) (of N1 dimensions )of the modal decomposition. The expression “almost all” is used in a very mathematicalsense. Note that the production frontier is composed of polytopes of dimensions less thanor equal to N1. Introduce any Lebesgue measure of the ordinary type in which any
hypercube of dimension N1 has a positive measure, and then any collections of facesof the modal decomposition are measurable. The set of all faces of dimensions less thanN1 has the measure 0. Therefore, there is a significant chance to choose a point in thisset that is 0 if we measure probability after introducing the Lebesgue measure.
This leads onto a further discussion. Assume a situation where the net production y isequal to the demand of the economy. Almost all cases (of probability 1) y is in the18)
A facet of a polytope is a face of the highest dimension among faces of the boundary. All otherfaces of lower dimensions are simply called faces.

– 180 –A New Construction of Ricardian Trade Theory

interior of a facet. As the interior set is open, any small change in demand keeps the net
production in the same facet. The prices and wage rates that are polar to y remain
constant except for scalar multiplications. This means that in almost all cases priceadjustment has nothing to do with the change in demand. In many analyses various
explanations have been added concerning production adjustments through pricefluctuations. However, when prices return to the original state the only effect of thosefluctuations was to produce a change in the activity levels of production to meet thechanges in demand. Similar effects can be obtained if prices and wage rates remain
constant and quantities are adjusted based on demand. This process is usually called
quantity adjustment.
Traditional discussions focused on which the demand and supply conditions was
dominant in price determination and in determinations of terms of trade and of patternsof trade. The Ricardian theory of trade suggests, however, that there is a predominanceof quantity adjustment over price adjustment.
It appears that this sort of understanding has been accepted since the 1950’s. However,the lack of a general theory restrained the scope of analysts and led to much confusion
and misunderstanding19). People worked mainly either in the price plane, or in the
production plane. They could not truly understand how two entities are connected. For
example, Minabe (1995, Chap. 8) suggested an overview on Graham’s results after
Koopmans (1951) publication on the geometry of polyhedral cones. He came very close
to the duality theorem. He examined mainly two cases: one case where the state is in (theinterior of) a facet and the other where the state is on a 0-dimensional face (i.e. a vertex
or a cross-point of ridges). The former was the case observed by Graham: Prices remain
constant and quantities are adjusted. The latter was suggested by Minabe and was thecase defended by J.S. Mill. In this case, prices may change in a polytope of N1
dimension but the net production or demand must remain at a fixed point. Minabe
alleged that this is the case where the “ ‘reciprocal national demand’ theory iscompletely applicable.” However, this symmetric argument of two cases cannot be
defended.
There is some misunderstanding concerning the significance of the (strongly) shared
19)

Another example: Ikema (1993) contributed to the advancement of Ricardian analysis in the 3country, 3-commodity case, developing a method of graphical analysis. He argued about the necessity
to include imperfect specialization cases in the analysis (“shared pattern of specialization” in thispaper) and yet he could not grasp that his “Graham point” was in fact a general case despite his claimthat it is “very special point”.

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Y. SHIOZAWA

pattern of specialization. As examined in section 3.4, there is in general a wage-rate
vector that leads to a strongly-shared pattern of specialization. If such a vector exists,
vectors of an open domain around it have the same property. It is conspicuous in themodal decomposition of the wage rate D. In fact one or more such domains occupy the
major part of the set of sharing vectors. All other faces have dimensions smaller thanM1. In the wage rate D, the total measure of those faces is 0 if measured by an
ordinary Lebesgue measure. However, this is the story of the wage rate and price side. If
we look at the production and demand side, as theorem 5.7 suggests, the corresponding
faces on the production frontier have only the smallest dimensions, and in the N M casethey shrink to only one or more points. The question is which is more important: Larger
sets in evaluation space or larger sets in real commodity space? Once technology andlabor powers are given, the most important factors that influence the state of economy
are the demands. Of course, demand and prices are connected. One influences the other.
However, formal duality stops here. The demand for a commodity is influenced by manyfactors. The composition of consumer goods is a result of a long history. When one is
obliged to choose a state of an economy, demand comes first and prices follow. Reality
has greater gravitas than price. Thus, the most ordinary situation is that demand andproduction drop in the interior of a facet of the production frontier. As examined above,it is the world where prices play a limited role in any adjustment and the major part ofadjustment is performed by the quantity changes.
7.

Differentials in National Wage Rates

Gains from trade, patterns of specialization and terms of trade are favorite topics ofinternational trade theory. Questions of national wage rate differentials are anotherimportant subject of international trade theory, although little enthusiasm has beenreported for this topic.
This lack of interest can be traced to the founder of the theory of comparativeadvantage. When Ricardo discussed his numerical example, he did not mention wage
differentials, which should have existed at that time between Portugal and Great Britain.
Trade on the basis of comparative advantage is possible only when there are suitablewage differentials. The Heckscher-Ohlin model explains wage differentials as a result of
the different ratios of two production factors. When factor price equalization theorem isvalid, wage rates must be equalized between countries. In this way, modern trade theory
is no more interested in explaining wage differentials between countries. However, thisstate of interest is quite surprising, because in the 20th century a very large discrepancy– 182 –A New Construction of Ricardian Trade Theory

has been observed between developed and developing nations. For example, Japanese
workers once enjoyed a wage rate 70 times that of Vietnamese workers.
An exception to this was the theory of unequal exchange. Arighi Emanuel (1969)
argued that unequal exchange of goods keeps a poor country poor. This theory waswelcomed by dependency theory people and became a part of their theoretical reasoning.
Despite their good will, they were doubly wrong. Their value theory was based on
ethics rather than on economics. They have misidentified the true reasons ofunderdevelopment. As shown in section 4, trade improves the real wage rate as long as
workers remain employed. It is not trade itself that should be accused. Countries should
enhance gains from trade while preventing, for a while, loss from trade. To achieve this,
it is necessary to control the exchange rate of their currency and prepare their industry
for stiffer competition. Despite these misunderstandings, their interest in wage rate
differentiation is justifiable. Wage rate differentiations or the disparity in national wage
rates is one of the most important topics of trade theory. The new Ricardian theory of thepresent paper sheds light on this question.
For simplicity of discussion let the demand of every nation be proportional to apredetermined vector. This is equivalent to assume a constant composition of demand foreverybody. Then, a maximal production of the world can be determined. In general it canbe assumed that this production y is in the relative interior of a facet of the production
frontier. There is only one wage-rate vector w and price vector p that are compatible
with this production. Here the compatibility means that w, p and y statisfy the equation
y, pq, w. That the wage-rate vector w(w1, w2, . . . , wM) is determined means eachnation’s wage rate wi can be determined. Thus, the new theory includes explanations
why national wage rate differentials occur.
There are many factors that influence wage rate differentials. J.S. Mill argued thatprices are dependent on demands. This we cannot deny. When the demand changes andworld production moves from one facet to another, wage-rate vector w changes
accordingly. Change of work power of each country has a similar effect like change ofdemands. Production frontier changes and corresponding facet may change even when
demand composition remains constant. But most important factor in the determination ofeach nation’s wage rate is the technology.
The technology of a country is represented by the set of techniques of that country.
Other factors like natural resources and climate influence the set of techniques but more
and more in recent years the technology as knowledge, tacit and explicit, determines thepossible coefficients of techniques. The level of technology of a country is roughly– 183 –

Y. SHIOZAWA

estimated by the production possibility set of each country. The country that has thebiggest production possibility set for a given amount of labor power has the mostadvanced technology. The story becomes complicated when the ranking differs with thecommodities concerned. However, in all cases, we can say this: when labor productivityof a country is increased uniformly by m and all other conditions are kept constant, thewage rate of the country will be raised to the level m -times the past rate. Thus,
technology matters in the determination of the wage rate of a country. The best way to
improve the wage rate is to raise productivity. The wage represents a substantial part of
the national income. As the division between wage role and profits shows stability for along time, to raise productivity is also a good way to increase national income.
In addition, the Heckscher-Ohlin model, at least in its standard model, assumes that
two countries have the same techniques and explains that disparity is due to thedifference in the factor endowment ratio. In a capital-rich country workers get a betterwage rate whereas in a labor-rich country workers obtain less than their counterparts.
Factor price equalization theorem, although one cannot question its logical consistency,
is also disputable, for wage rate differentiation continues for a long time after trade
opens. This might be another valid reason to return to Ricardian theory in place ofmodern trade theories.
8.

Concluding Remarks

The present paper only shows the possibility of a general theory in the Ricardian
traditions. As it has been shown that the newly constructed theory covers most generalcases: multi-country, multi-commodity, choice of techniques, and trade of intermediate
goods. Starting from a given wage-rate vector one finally analyzes how the wage rates
are determined. Wage rate vectors and prices are divided into an exclusive set ofdifferent competitive modes. The production frontier was also divided into an exclusive
set of different competitive modes. Entities with the same competitive mode are
mutually related with clear economic meanings. The new construction, presented in thispaper, with the help of dual correspondence theorem could clarify the old dispute over
the roles of supply and demand conditions. It may also provide an explanation regarding
how and why the wage rate differentials are formed and preserved. From suchobservations an important policy implication can be deducted: the importance oftechnology and technology development. However, all these are only the starting pointof the new theory. Many problems remain to be solved; differences between small andbig countries, the effects of tariffs, the effects of transportation cost decrease, the effects– 184 –A New Construction of Ricardian Trade Theoryof change (not choice) of techniques, wage rate improvement and so on. These are vast
subjects and yet I believe a firm foundation has been laid for these new investigations.
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